Theoretical Aspects of Chemical Reactivity

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19~ T H E O R E T I CA L A N D C O M P U TAT I O NA L C H E M I S T RY

Theoretical Aspects of Chemical Reactivity

T H E O R ET I CA L A N D C O M P UTAT I O NA L C H E M I ST RY

S E R I E S E D IT O R S

Professor P. Politzer Department of Chemistry University of New Orleans New Orleans, LA 70148, U.S.A. VOLUME 1 Quantitative Treatments of Solute/Solvent Interactions P. Politzer and J.S. Murray (Editors) VOLUME 2 Modern Density Functional Theory: A Tool for Chemistry J.M. Seminario and P. Politzer (Editors) VOLUME 3 Molecular Electrostatic Potentials: Concepts and Applications J.S. Murray and K. Sen (Editors) VOLUME 4 Recent Developments and Applications of Modern Density Functional Theory J.M. Seminario (Editor) VOLUME 5 Theoretical Organic Chemistry C. Párkányi (Editor) VOLUME 6 Pauling’s Legacy: Modern Modelling of the Chemical Bond Z.B. Maksic andW.J. Orville-Thomas (Editors)

VOLUME 7 Molecular Dynamics: From Classical to Quantum Methods P.B. Balbuena and J.M. Seminario (Editors) VOLUME 8 Computational Molecular Biology J. Leszczynski (Editor) VOLUME 9 Theoretical Biochemistry: Processes and Properties of Biological Systems L.A. Eriksson (Editor) VOLUME 10 Valence Bond Theory D.L. Cooper (Editor)

Professor Z.B. Maksić Rudjer Bošković Institute P.O. Box 1016, 10001 Zagreb, Croatia VOLUME 11 Relativistic Electronic Structure Theory, Part 1. Fundamentals P. Schwerdtfeger (Editor) VOLUME 12 Energetic Materials, Part 1. Decomposition, Crystal and Molecular Properties P. Politzer and J.S. Murray (Editors) VOLUME 13 Energetic Materials, Part 2. Detonation, Combustion P. Politzer and J.S. Murray (Editors) VOLUME 14 Relativistic Electronic Structure Theory, Part 2. Applications P. Schwerdtfeger (Editor) VOLUME 15 Computational Materials Science J. Leszczynski (Editor) VOLUME 16 Computational Photochemistry M. Olivucci (Editor) VOLUME 17 Molecular and Nano Electronics: Analysis, Design and Simulation J.M. Seminario (Editor) VOLUME 18 Nanomaterials: Design and Simulation P.B. Balbuena and J.M. Seminario (Editors) VOLUME 19 Theoretical Aspects of Chemical Reactivity A. Toro-Labbé (Editor)

19~ T H E O R ET I CA L A N D C O M P UTAT I O NA L C H E M I ST RY

Theoretical Aspects of Chemical Reactivity

Edited by Alejandro Toro-Labbé Laboratorio de Química Teórica Computacional (QTC) Facultad de Química Pontificia Universidad Católica de Chile Santiago, Chile

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-52719-6 ISBN-10: 0-444-52719-2 ISSN: 1380-7323

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Contents

Preface

vii

1 Chemical reactivity and the shape function P. Geerlings, F. De Proft, and P. W. Ayers 2 Density functional theory models of reactivity based on an energetic criterion Andrés Cedillo

1

19

3 The breakdown of the maximum hardness and minimum polarizability principles for nontotally symmetric vibrations Miquel Torrent-Sucarrat, Lluís Blancafort, Miquel Duran, Josep M. Luis, and Miquel Solà

31

4 Classification of control space parameters for topological studies of reactivity and chemical reactions Bernard Silvi, Isabelle Fourré, and Mohammad Esmail Alikhani

47

5 Understanding and using the electron localization function Patricio Fuentealba, E. Chamorro, and Juan C. Santos 6 Electronic structure and reactivity in double Rydberg anions: characterization of a novel kind of electron pair Junia Melin, Gustavo Seabra, and J. V. Ortiz

57

87

7 Using the reactivity–selectivity descriptor f(r) in organic chemistry Christophe Morell, André Grand, Soledad Gutiérrez-Oliva, and Alejandro Toro-Labbé

101

8 The average local ionization energy: concepts and applications Peter Politzer and Jane S. Murray

119

9 The electrophilicity index in organic chemistry Patricia Pérez, Luis R. Domingo, Arie Aizman, and R. Contreras

139

10 Electronic structure and reactivity of aromatic metal clusters Paulina González, Jordi Poater, Gabriel Merino, Thomas Heine, Miquel Solà, and Juvencio Robles v

203

vi

Contents

11 Small gold clusters form nonconventional hydrogen bonds X-H· · ·Au: gold–water clusters as example E. S. Kryachko and F. Remacle

219

12 Theoretical design of electronically stabilized molecules containing planar tetracoordinate carbons Alberto Vela, Miguel A. Méndez-Rojas, and Gabriel Merino

251

13 Chemical reactivity dynamics in ground and excited electronic states P. K. Chattaraj and U. Sarkar 14 Quantum chemical topology and reactivity: A comparative static and dynamic study on a SN 2 reaction Laurent Joubert, Ilaria Ciofini, and Carlo Adamo 15 A quantitative structure–activity relationship of 1,4-dihydropyridine calcium channel blockers with electronic descriptors produced by quantum chemical topology U. A. Chaudry, N. Singh, and P. L. A. Popelier Index

269

287

301

319

Preface

We provide in this book a broad overview of recent theoretical and computational developments in the field of chemical reactivity. This volume contains contributions written by eminent specialists, which deal with various aspects of the subject, going from theoretical developments to applications in interesting molecular systems and clusters. These chapters provide an authoritative overview of research and progress in the field of chemical reactivity. As the use of reactivity descriptors is growing up, its fundamental theoretical aspects challenged theorists; this aspect is reviewed in various chapters where traditional concepts are revisited and explored from new viewpoints and new reactivity descriptors are proposed. Applications in the frontiers of reactivity principles, introducing the dynamic and the statistical viewpoints to chemical reactivity or challenging traditional concepts such as aromaticity make this book an essential source of inspiration and reference material for researchers and graduate students. The contents of this book cover theoretical developments and applications. In Chapter 1 (Geerlings, De Proft, Ayers), a review of the shape function, a whole theory of chemical reactivity, is presented. In Chapter 2 (Cedillo), the energy stabilization is shown to play a key role in identifying reactivity parameters. Chapter 3 (Torrent-Sucarrat, Blancafort, Duran, Luis, Solà) presents a critical review of the validity of the principles of maximum hardness and minimum polarizability. In Chapter 4 (Silvi, Fourré, Alikhani), catastrophe theory is introduced as a mathematical tool in the study of the energetic and mechanism of chemical reactions. Chapter 5 (Fuentealba, Chamorro, Santos), brings back the localization concept in chemistry through sharp definitions and applications of the electron localization function (ELF). In Chapter 6 (Melin, Seabra, Ortiz), the ELF is revisited from the perspective of the electron propagator to characterize the reactivity of anions. A new reactivity–selectivity descriptor is introduced in Chapter 7 (Morell, Grand, Gutiérrez-Oliva, Toro-Labbé) and used to explain the mechanism of classical reactions in organic chemistry. In Chapter 8 (Politzer, Murray), the local ionization energy concept is introduced and used to characterize molecular reactivity toward electrophiles. The reactivity and selectivity of molecular structures involved in an important number of reactions in organic chemistry are reviewed from the perspective of the electrophilicity power in Chapter 9 (Pérez, Domingo, Aizman, Contreras). Chapter 10 (González, Poater, Merino, Heine, Solà, Robles) analyzes reactivity descriptors to explain and assess aromaticity in metal clusters. In Chapter 11 (Kryachko, Remacle), nonconventional hydrogen bonds between gold clusters and water molecules are analyzed in light of through-bond interactions that induce the interesting proton-acceptor ability of gold atoms. A fascinating quest to find new molecular structures containing planar tetracoordinate carbons is presented in Chapter 12 (Vela, Méndez-Rojas, Merino). Chapter 13 (Chattaraj, Sarkar) presents the chemical reactivity descriptors for ground and excited states in a dynamical context. Molecular dynamic simulations are used in Chapter 14 (Joubert, Ciofini, Adamo) to analyze the covalent and electrostatic interactions that characterizes the mechanism of SN 2 reactions. The book is closed with Chapter 15 (Chaudry, vii

viii

Preface

Singh, Popelier) where quantitative structure–activity relationships based on different kind of electronic descriptors are applied to challenging systems of calcium channels blockers. I wish to thank all the forty-five authors who contributed to this volume for their great scientific work and willingness to be part of this project. Special thanks goes to Dr. Soledad Gutiérrez-Oliva (QTC@PUC) for her valuable assistance in putting together this volume. Alejandro Toro-Labbé QTC, Pontificia Universidad Católica de Chile

Theoretical Aspects of Chemical Reactivity A. Toro-Labbé (Editor) © 2007 Published by Elsevier B.V.

Chapter 1

Chemical reactivity and the shape function a

P. Geerlings, a F. De Proft, and b P. W. Ayers

a

Eenheid Algemene Chemie (ALGC), Faculty of Sciences, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium and b Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S4M1

Abstract Conceptual density functional theory (DFT) offers an elegant way to predict and interpret the outcome of a chemical reaction in terms of the properties of the reactants. The properties of interest are usually response functions with respect to perturbations, with the perturbations of greatest interest being changes in the external potential, in the number of electrons, capturing the essential chemical characteristics of the relevant reactive agent. Within DFT, the electron density function r has always been considered as the (simplest) carrier of information on the system. In recent years, however, the shape function, r (defined as r/N , i.e. the density per particle; N being the number of electrons), has been invoked as an even simpler function characterizing a given system. In this contribution, the properties of this function are reviewed. Its importance for understanding molecular structure and reactivity is highlighted, linking variations in the shape function to more conventional reactivity descriptors in DFT and providing explicit methods for describing chemical reactivity using the shape function only. Applications in the field of atomic and molecular similarity analysis are given, and a shape function-based analogue of Mezey’s holographic electron density theorem is discussed.

1. Introduction It is now widely recognized that density functional theory (DFT) [1] revolutionized quantum chemistry [2]. The recognition, in the Hohenberg and Kohn (HK) theorems [1], that the electron density function r could replace the wave function xN for an 1

2

Chemical reactivity and the shape function

accurate description of an N -electron system paved the way for a computationally attractive scheme that undoubtedly increased the overall influence of quantum chemistry for the practicing organic, inorganic and biochemist, by allowing relatively accurate computations on systems of increasing size in reasonable computing times. These developments in what has been termed [3] computational DFT [4] have been accompanied by a series of remarkable developments in the field of conceptual DFT [5–7]. Following Parr’s identification of the electronic chemical potential entering the DFT analogue of the Schrödinger equation as the negative of the derivative of the energy with respect to the electron number, and hence as the electronegativity [8], a variety of reactivity descriptors [5–7] have been proposed in this context (hardness, softness and electrophilicity) to be used as such (in their global and local forms) or in the context of a principle (Sanderson’s electronegativity equalization principle [9], the hard and soft acids and bases principle [10,11] and the maximum hardness principle) [12]. In 1983, Parr and Bartolotti focused on the density per particle function r, defined as r/N and normalized to 1, and termed it the shape factor in the context of laying out rules for computing changes in functionals of when they are re-expressed as functionals of N and . This quantity, which could intuitively be associated with the redistribution of the total number of electrons N among the different parts of a system giving rise to the shape of the distribution, received little attention up to the mid-1990s when two papers, one by Baekelandt, Cedillo and Parr [13] and another by Geerlings and De Proft [14], highlighted the role of the shape function in hardness and electronegativity, respectively. In 2000, Ayers [15] proved that for a finite Coulomb system, the density per particle determines the value of any observable quantity and hence can be placed on equal footing with the density itself. This contribution paved the way for more intensive search on the properties of r and the advantages of using r instead of r; it also put previous findings, for example in the field of similarity analysis, in a broader perspective. The present contribution aims to review the properties of the shape function and to illustrate with recent examples its fundamental importance and to put older examples in a broader context.

2. Basics: The shape function as a carrier of information 2.1. Definition Following Parr and Bartolotti [16], we define the shape function r as r =

r N

(1)

with obvious properties that the function is everywhere non-negative and normalized to 1 r ≥ 0∀r

rdr = 1

(2)

P. Geerlings et al.

3

2.2. For a finite coulombic system (r) determines both the external potential v(r) and the number of electrons N This important property has been proven by Ayers in two steps. The fact that r determines vr can most easily be seen by referring to the Wilson argument for DFT [17] which is, in turn, based on Steiner’s corollary [18] to Kato’s theorem [19] stating that the positions of cusps in the density r locate the nuclear positions, R , and the atomic numbers of the nuclei, Z , can be determined from the cusp conditions Z = −

1 1 r ¯ 2 r r − R r = R

(3)

where r ¯ denotes the spherical average of the electron density around the point R . Because N is a constant, r shows cusps at the same positions as r. Inserting (1) into (3), one obtains Z = −

r ¯ 1 1 2 r r − R r = R

(4)

showing that the cusps of determine also the nuclear charges. A knowledge of all R

and Z determines the external (Coulombic) potential. vr = −

Z

r − R

(5)

The proof that r determines N is more intricate and will not be reproduced in all details. It combines the convexity postulate on the E = EN curve and the properties of the long-range behaviour of the electron density function. All experimental or theoretical results on atomic and molecular systems reveal a particular property of the behaviour of the energy of a system as a function of its number of electrons (cf. Figure 1). The decrease in a system’s ground-state energy due to the addition of an electron successively diminishes as its number of electrons increases until the system becomes unbound. In terms of the notation of Figure 1, EN0 −2 − EN0 −1 > EN0 −1 − EN0 > EN0 − EN0 +1

(6)

and N0 is a reference value. In terms of (vertical) ionization energies, one then obtains IN0 −1 N0 −2 > IN0 N0 −1 > IN0 +1 N0

(7)

The ionization energy, Ia a−1 , determines the long-range behaviour of the system with N = a electrons [20]. In terms of the logarithmic derivative for large r, one gets lim

r→

ln r r

√ = − 8I

(8)

4

Chemical reactivity and the shape function E E1

•

E2

•

E3 E4

• • N0–2

N0–1

N0

N0+1

N

Figure 1 Variation of the energy, E, of an atom or molecule with the number of electrons, N , at constant external potential

with r denoting the spherical coordinate that measures the distance to the average particle position. Combining (1) and (8), it is easily seen that √ ln r lim (9) = − 8I r→ r The results of (7) and (9) can now be combined as follows. Suppose that a single r, generating a unique vr, is associated with two N values, N1 and N2 N1 = N2 . By virtue of (9) these two systems with the same shape function would then have the same ionization energies, yet have different number of electrons for the same external potential. It is easily seen, for example in an atom where the external potential is governed by the nuclear charge, that this situation is impossible: for a given nuclear charge two atoms with a different number of electrons (say A A+ ) will necessarily have different ionization energies. Combining all this it is seen that, in analogy with HK’s first theorem, r determines both vr and N and hence the Hamiltonian. Consequently, r determines all observable properties. This remarkable result can be placed in an experimental context looking at the setup of X-ray experimental measurements, where it is commonplace to think that with present day techniques, accurate electron density functions r can be obtained (which in its turn would be sufficient to identify the molecule via HK). In fact, the densities finally obtained in such an experiment are the result of a complex transformation of refraction data whose intensities are measured and manipulated relative to each other [21,22]. X-ray experiments thus initially yield the shape function.

2.3. One step further: A holographic density theorem for the shape function The role of r as a carrier of information can be taken one step further by involving the so-called holographic density theorem. This theorem goes back to Riess and Münch [23]

P. Geerlings et al.

5

and Bader and Becker [24] and was recently refined by Mezey [25,26]. In its most refined form, the holographic density theory states that if the electron density, r, is known in any subdomain of an unconfined Coulomb system (e.g. an atom, molecule or cluster), then the electron density in all other subdomains, r, can be determined from the electron density in the specified region, . (Figure 2b is a pictorial representation of the holographic density theorem.) The only restriction on the subdomain is that cannot have zero volume. In other words, r determines r and as, via HK, r determines the Hamiltonian (Figure 2a), r determines all properties of the system among others the energy: the HK theorem functional relationship E = E

(10)

(a) Hohenberg–Kohn

Nuclei Compatibility

•

•

• ••

•

•

•

ρ(r) for a ground

• With a single external potential v (r) v (r),N

→E = Eν[ρ]

•

state

(b) Mezey: Holographic density theorem Ω (nonzero volume subdomain)

• • • • •• • • ρ(r)

•• ρΩ (r)

•

•

→E = Eν[ρ Ω]

(c) Ayers: Shape function

• •

• •

•

•

•

•

σ(r)

→E = Eν[σ]

•

v (r),N

(d) Combining (b) and (c)→ Holographic shape theorem •

••

• •

σΩ (r )

• ••

• •

σ(r )

•

•

→E = Eν[σΩ ]

v (r),N

Figure 2 From Hohenberg–Kohn to a holographic shape theorem

6


can be converted into E = E

(11)

This remarkable property has been called the holographic density theorem on the basis it indicates that the entire electron density can be reconstructed from the electron density in some small fragment of the system. To state it very simply, information on r can be gained by considering r, the density in a finite, otherwise arbitrary subdomain . From Section 2.2., knowledge of r is sufficient for knowing the Hamiltonian, or stated in terms of the energy functional E = E r

(12)

As Mezey’s arguments for the holographic property for r can easily be transferred to r, a combination of (11) and (12) then leads to the relation E = E r

(13)

yielding a holographic theorem for the shape function [27]. The shape function in any subdomain of a boundaryless Coulomb system with nonzero volume completely determines the Hamiltonian and therefrom the wavefunction, energy and all other observable properties (Figure 2d). All this points to the importance to look for similarity in shape when considering similarity of atoms, molecules, etc.

2.4. A variational theorem for and its use in computational DFT Recall that the equation FHK + r =

(14)

can be considered as the DFT counterpart of the time-independent Schrödinger equation H = E

(15)

Here, FHK is the HK functional. Both (14) and (15) are derived from the variational principles for the energy (in terms of the electron density and wave function, respectively), and both and E enter the equations as Lagrange multipliers associated with normalization constraints. The equal status of r and r as information carriers suggests that, just as for r, we can derive a variational equation for r. Starting from equation (12), minimizing the energy of the system with respect to the shape function under the constraint that the shape function should at all times integrate to unity (1) yields

E− rdr − 1 = 0

(16)

P. Geerlings et al.

7

where is a Lagrangian multiplier corresponding to in (14) upon variation of r. One easily obtains T Vee + + Nvr =

(17)

with T = T the kinetic energy functional and Vee the electron–electron interaction energy. Equation (17) is the shape function analogue of (15) and calls for a r turning the 1.h.s. into a constant. In (15) can be written as E/r. Similarly, in (17), =

E r

which by virtue of the chain rule can be written as N: E x E = = x = N r x r

(18)

(19)

as x x =N = N x − r r r

(20)

Equation (19) provides the quantitative link between the variational principle for the wave. Recently the possibility of using the shape function to set up a variational theorem has also been stressed by Bultinck and Carbo [28]. Equation (17) calls for the development of functionals for the energy and other quantities interns of the shape function. Work in this direction has been undertaken by one of the authors, showing that atomic properties (specifically the kinetic and exchange energies) have simple approximate expressions in terms of the shape function [29] and supporting the proposition that computational shape-functional theory represents an alternative to DFT for atomic and molecular systems.

3. The shape function as a primer for reactivity descriptors and its use in a conceptual shape function theory 3.1. Some earlier work: The role of the shape function in electronegativity and hardness. Starting from the E = EN v and E = EN (we note that itself is dependent on N ) relationships, one can express the change in energy from one ground state to another as: E E dE = dN + rr (21) N r N or

dE =

E N

dN +

E rr r N

(22)

8


As now can be considered to be a functional of N and = N

(23)

r dN + r r

r N

(24)

one gets dr =

r N

Combining (22) and (24) and equating expressions in dN one gets E r E E = dN + r N N r N N As now r = N N

r N

=

1 1 1 fr − 2 r = fr − r N N N

(25)

(26)

equation (24) becomes E 1 fr − rdr = − N r N

(27)

The electronegativity at constant external potential v and constant shape factor is related through a function involving the deviation of Fukui function from the average electron density per electron [14]. The quantity E/r has been identified as the Lagrangian multiplier being equal to N (cf. equation (19) [30]. This result bears complete analogy with the second derivative case, the hardness, treated by Baekelandt, Cedillo and Parr [13] and yielding: (28) = − hrfr − rdr with 1 hr = N

r

(29) N

containing the functional derivative with respect to the shape function of the chemical potential , which is itself the first derivative of the energy with respect to N . The quantity hr has been advocated to be a useful definition of the local hardness, whose ambiguity had been raised before [31–34], as (30) = hrfrdr = hrrdr (31) Note that De Proft, Liu and Parr [35] provided an alternative definition for the local hardness in the isomorphic ensemble as − gr with gr = vr/N .

P. Geerlings et al.

9

E r equal to r dr (see Section 3.2) has been r N r N shown to be roughly proportional to the global hardness of the system (or inversely proportional to its polarizablity) [36]: the more polarizable a system, the smaller the energy change associated with a change in shape as intuitively expected. Under the influence of an external perturbation (e.g. due to the external electric field), a polarizable system will change its shape more easily. (For a more detailed account see [37].) In the next paragraph, variations of the energy with shape functions are placed in a broader context, dropping the constraint of constant N and looking for analogues of the E/r functional derivatives to other ensembles. Its analogue

3.2. Variation of state functionals with the shape function The results about E/rN and /rN mentioned in the previous paragraph can be put in a more general context by considering the fundamental relationship E = E v, where now N and v are considered to be functionals of , giving rise to a composite functional E = EN

(32)

which can be seen as the state functional for the canonical ensemble. Other ensembles however exist focusing on [38] other pairs of variables which can be used to describe the passing from one ground state of a system to another as can happen, for example, in a chemical reaction. Through the appropriate Legendre transformation one obtains • the grand canonical ensemble state function

= E − N

(33)

• the isomorphic ensemble FN = E − N

rrdr

(34)

• the grand isomorphic ensemble R = F − N

(35)

where the corresponding couples of variables v N , and can be written as functionals of (with trivial results in the case of the variable itself). Considering the canonical ensemble one then obtains dEr =

E rdr r

(36)

10


and

E r

N E r

+ dr

r r r N r

= + r dr

r r N

=

E N

(37)

Note that the chemical potential (at constant ) is appearing together with a second term (taken at constant N ) which was seen in Section 3.1. to be related to the hardness of the system. Following the same ansatz for the canonical ensemble one obtains: d r = rdr (38) r and

r

=

r N r − Nfr dr − N r N r

(39)

where use has been made of the relation /r = fr. Analogous relationships can be obtained for F and R [30] yielding, however, no new descriptors in the r.h.s. of the analogue of equations (37) and (39). These relationships link variations in the shape function to the reactivity descriptors commonly used in DFT and provide explicit methods for describing chemical reactivity in terms of the shape function yielding ‘conceptual shape function theory’. Thus, it is clearly seen that the shape function not only determines all the physical properties of an isolated molecule but, since reactivity descriptors can be explicitly constructed from the shape function, also its chemical properties. As the resulting equations are not always that simple to apply at first sight, we pass in next paragraph to some pragmatic procedures for extracting descriptors from the shape function.

3.3. From the Fukui function and hardness to the complete set of shape function reactivity descriptors On the basis of the foregoing the idea can be used if indeed, knowing , all properties of a system are determined, including its reactivity descriptors such as hardness, softness, Fukui function and, by combining the last two descriptors, local softness. In this paragraph, an overview will be given on pragmatic steps which can be taken to do so. It is well known that the electron density r at large r can be written, for both an atomic and molecular system, as [20]: r = ANe−

√ 8Ir

(40)

where the pre-exponential factor AN is assumed to be a function of the number of electrons of the system.

P. Geerlings et al.

11

The Fukui function at large r values then becomes: √ −8r I fr → ANe− 8Ir √ 2 8I N

(41)

or, taking the logarithmic derivative, 1 √ ln fr → − 8Ir r r

(42)

The asymptotic behaviour of the Fukui function is then √ ln fr → − 8Ir r→ r lim

(43)

Dividing the expression for the long-range behaviour for the Fukui function by the electron density yields: fr 1 AN 8r I → − √ (44) r AN N N 2 8I which upon differentiation with respect to r gives: fr 2 I → −√ r r 2I N

(45)

The l.h.s. is now independent on r and is clearly a measure for the hardness of the system. It is now easy to rewrite the derivative in the l.h.s. of (45) ⎞ ⎞ ⎛ ⎛ N +N ⎜ ⎟ ⎜ ⎜ ⎜ 1 N N ⎟ ⎟ ⎟ (46) ⎠ = r ⎝ ⎠ = r N r ⎝ N N Showing that knowledge of leads to I/Nv directly related to a measure of the chemical hardness and so to the total softness S [30]. On the other hand, the Fukui function can easily be obtained from the shape function as: fr =

r r = rN = r + N N N N

(47)

and the local softness can be obtained as: sr = Sfr

(48)

Note that the chemical hardness as such is not obtained but that the derivative I/Nv may be written as I 1 + = + = (49) N 2 N 2

12


3 E where the third-order energy derivative = may be expected to be small [39]. N 3 Use was made of the finite difference approximations for and yielding: I +A = (49a) 2 I −A = (49b) 2

where I and A are the corresponding vertical ionization energy and electron affinity, respectively (for a detailed account see [5]). In the same spirit, knowledge of I via the r analogue of r r = BNe−

√ 8Ir

(50)

gives information on I via the long-range behaviour of r and so, as A is typically smaller than I, or . Once known, and can be combined into the electrophilicity [40] =

2 2

(51)

which can also be turned in its local analogue [41] by multiplication with the Fukui function: r = fr

(52)

yielding the complete spectrum of global and local reactivity descriptors in use until now [5].

4. Applications 4.1. Quantitative similarity as shape similarity The search for the degree of similarity of molecules is a topic not only of academic, but also of industrial interest. For example, in pharmacology, one of the basic ideas is that similar molecules (in case drugs) (re)act in the similar ways. Since the early 1960s, intense research has been focussed on using quantum chemical techniques to quantify molecular similarity, leading to various proposals for quantum molecular similarity indices (QMSI). The most general formalism has been proposed by Carbo and coworkers starting from the idea that if two molecules A and B are similar, their electron density functions A r and B r should be similar. To put it in a mathematical context, the integral ZAB =

A r − B r2 dr

(53)

P. Geerlings et al.

13

should be minimized, yielding in a natural way the demand for a maximal value of ZAB defined as (54) = A rB rdr ZAB which upon normalization yields the Carbo similarity index [42]: r rdr ZAB = A B 2 A rdr 2B rdr

(55)

As now (cf. (1)) A r = NA A r B r = NB B r

(56)

with NA and NB the number of electrons of molecules A and B, respectively, one gets: ZAB

=

A rB rdr = ZAB B2 rdr

A2 rdr

(57)

i.e. the Carbo index is identical with a shape function-based similarity [43-44]. This result is not completely unexpected in view of the information content in r discussed above. It means that the vast majority of studies on molecular similarity [45], which all use Carbo-type indices can be considered as studies on shape function similarity. It means that similarity in shape is a fundamental issue to look at, both locally and globally.

4.2. Quantum similarity of atoms Although the literature shows abundant example of studies of quantum similarity of molecules [45], analogous investigations on atoms are nearly nonexistent. Below we briefly describe how a shape function-based approach may lead to results which are more appealing than those on the density giving a first example of the advantage of using r as an alternative to r. Using the Carbo index, identical for density and shape, the similarity index for a given atom in its ground state invariably shows a nearest neighbour effect as shown in Figure 1 for the noble gases [46]. The more the atomic number of the atom differs from the reference value the lower the similarity, which obscures any periodic trends. This basic feature of the periodic table could however be regained by invoking aspects from information theory [47,48]. We therefore turned to the evaluation of the information discrimination S, defined below for a continuous probability function Pk x S

Pk P0

=

Pk x ln

Pk x dx P0 x

(58)

14


where Pk x and P0 x are normalized probability distributions, the reference distribution or prior being P0 x. The choice for Pk x as A r was at first obvious, the Pk x function being chosen as the density function of the noble gas preceding the atom A under consideration but renormalized so as to give the same number of electrons upon integration as A r. Pk x →

NA r N0 0

(59)

Evaluation of SA =

A r ln

A r dr NA 0 r N0

(60)

then yields the results in Figure 3, wherein periodicity is regained. Reformulating (60) directly in terms of the shape function gives: SA =

A r ln

A r dr 0 r

(61)

ID

yielding (cf. Figure 4) a more pronounced periodicity, with the distance between successive atoms in a given period gradually decreasing from first through the fourth rows with decreasing slope when passing from first to fourth row. One thereby regains one of the basic characteristics of the periodic table namely that the evolution in (many) properties through a given period slows down when going down in the table. These results are reminiscent of the study on information theory for atoms in molecules by Parr and Nalewajski [49] in which the most favourable matching of two electron densities i and 0 (the reference density) normalized to different total number of electrons Ni and N0 is achieved when the corresponding shape functions i r and 0 r are equal.

0

10

20

30 Z–1

40

50

60

Figure 3 Information discrimination (ID) vs. Z for atomic densities with the noble gas of the previous row as reference

ID

P. Geerlings et al.

15

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 50

0 Z

Figure 4 Information discrimination (ID) vs. Z for atomic shape function with the noble gas of the previous row as reference

All in all, r is not just r deprived from its N dependency! contains information on N and (1) can therefore be written as: r = N rN

(62)

It may be hypothesized that in this way the use of the shape function may be more useful for properties where the massive direct N dependency of r may obscure the essence, but where the finer details on the N dependency are retained in the shape function N r.

5. Conclusions The density per particle of shape function r is seen to be a simpler alternative for the density as the fundamental carrier of information in the study of the electronic structure of atoms and molecules, up to the limit that the shape function holographic theorems can be formulated. Just as in DFT, a variational theorem can be written down and an alternative set of reactivity descriptors can be conceived. The long-range behaviour of the function yields information which reconciles density and shape function theory. The example of similarity analysis shows that in many, if not all, cases looking for molecules or atomic similarity nearly coincides with looking for similarity in shape. A final question remains: studies gathered in this contribution clearly showed that the shape function is an alternative to the electron density function both from computational and from conceptual point of view – would there now be any reason to prefer the shape function to the electron density in practice? A first example has been given in Section 4.2. in the similarity study on atoms. This domain needs to be addressed much more broadly in the future: recent results by these authors [50] in a comparative study of atomic ionization potential functionals of the electron density and the shape function show that the shape function may be more powerful in predicting those periodic properties, where the strong dependence of the electron density on the number of electrons may obscure the essential characteristics.

16


Acknowledgements PG and FDP are grateful to the VUB and to the Fund for Scientific Research-Flanders (Belgium) (FWO) for continuous support to the ALGC Research Group. PWA thanks the NSERC and the Canada Research Chairs for funding and PG and FDP for enabling a 2-month research stay at the VUB in the autumn of 2004.

References 1. Hohenberg, P.; Kohn, W. Phys. Rev. B 1964, 136, 864. 2. Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. 3. Parr, R. G.; Yang, W. Annu. Rev. Phys. Chem. 1995, 46, 701–728. 4. Koch, W.; Holthausen, M. C. A Chemist’s Guide to Density Functional Theory, 2nd Edition, Wiley-VCH, Weinheim, 2001. 5. Geerlings, P.; De Proft, F.; Langenaeker, W. Chem. Rev. 2003, 103, 1793–1873. 6. Geerlings, P.; De Proft, F.; Langenaeker, W. Adv. Quant. Chem. 1999, 33, 303–328. 7. Chermette, H. J. Comput. Chem. 1999, 20, 129–154. 8. Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801–3807. 9. Sanderson, R. T. Science 1951, 114, 670–672. 10. Pearson, R. G. J. Am. Chem. Soc. 1963, 85, 3533–3539. 11. Pearson, R. G. Chemical Hardness, Wiley-VCH, Weinheim, 1997. 12. Pearson, R. G. J. Chem. Educ. 1968, 45, 981. 13. Baekelandt, B. G.; Cedillo, A.; Parr, R. G. J. Chem. Phys. 1995, 103, 8548–8556. 14. DeProft, F.; Geerlings, P. J. Phys. Chem. A 1997, 101, 5344–5346. 15. Ayers, P. W. Proc. Natl. Acad. Sci. U. S. A. 2000, 97, 1959–1964. 16. Parr, R. G.; Bartolotti, L. J. J. Phys. Chem. 1983, 87, 2810–2815. 17. Wilson, E., B. quoted by P. O. Löwdin, Int. J. Quant. Chem. 1986, 519, 19. 18. Steiner, E. J. Chem. Phys. 1963, 39, 2365. 19. Kato, J. Commun. Pure. Appl. Math. 1957, 10, 151. 20. Handy, N. C. European Summer School in Quantum Chemistry 2000, Lund University, Chapter 10. 21. Stout, G. M.; Jensen, L. M. X-Ray Structure Determination: A Practical Approach, 2nd Edition, John Wiley, 1989. 22. Stevens, E. D.; Coppens, P. Acta Cryst. 1975, A31, 512. 23. Riess, J.; Münch, W. Theor. Chim. Acta 1981, 58, 295. 24. Bader, R. F. W.; Becker, P. Chem. Phys. Lett. 1988, 148, 452–458. 25. Mezey, P. G. Mol. Phys. 1999, 96, 169–178. 26. Mezey, P. G. J. Chem. Inf. Comp. Sci. 1999, 39, 224–230. 27. Geerlings, P.; Boon, G.; Van Alsenoy, C.; De Proft, F. Int. J. Quant. Chem. 2005, 101, 722–732. 28. Bultinck, P.; Carbo, R. J. Math. Chem. 2004, 36, 191. 29. Ayers, P.; Phys. Rev. A 2005, 71, 062506. 30. De Proft, F.; Ayers, P. W.; Sen, K. D.; Geerlings, P. J. Chem. Phys. 2004, 120, 9969–9973. 31. Ghosh, S. K.; Berkowitz, M. J. Chem. Phys. 1985, 83, 864. 32. Ghosh, S. K. Chem. Phys. Lett. 1990, 172, 77–82. 33. Harbola, M. K.; Chattaraj, P. K.; Parr, R. G. Isr. J. Chem. 1991, 31, 395–402. 34. Langenaeker, W.; De Proft, F.; Geerlings, P. J. Phys. Chem. 1995, 99, 6424–6431. 35. De Proft, F.; Liu, S.; Parr, R. G. J. Chem. Phys. 1997, 107, 3000–3006.

P. Geerlings et al. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

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Vela, A.; Gazquez, J. L. J. Am. Chem. Soc. 1990, 112, 1490–1492. Fuentealba, P. J. Phys. Chem. A 1998, 102, 4747–4748. Nalewajski, R. F.; Parr, R. G. J. Chem. Phys. 1982, 77, 399. Fuentealba, P.; Parr, R. G. J. Chem. Phys. 1991, 94, 5559–5564. Parr, R. G.; Szentpaly, L. v.; Liu, S. J. Am. Chem. Soc. 1999, 121, 1922–1924. Domingo, L. R.; Aurell, M. J.; Perez, P.; Contreras, R. Tetrahedron 2002, 58, 4417–4423. Carbo, R.; Leyda, L.; Arnau, M. Int. J. Quant. Chem. 1980, 17, 1185–1189. Boon, G.; De Proft, F.; Langenaeker, W.; Geerlings, P. Chem. Phys. Lett. 1998, 295, 122–128. Boon, G.; Langenaeker, W.; De Proft, F.; De Winter, H.; Tollenaere, J. P.; Geerlings, P. J. Phys. Chem. A 2001, 105, 8805–8814. Carbo, R.; Besalu, E.; Girones, X. Adv. Quant. Chem. 2000, 38, 1. Borgoo, A.; Godefroid, M.; Sen, K. D.; De Proft, F.; Geerlings, P. Chem. Phys. Lett. 2004, 399, 363–367. Kulback, S.; Leibler, R., A. ANN. Math. Stat. 1951, 22, 79. Kulback, R., A. Information Theory and Statistics, Willey, New York, 1959. Nalewajski, R. F.; Parr, R. G. Proc. Natl. Acad. Sci. U. S. A. 2000, 97, 8879–8882. Ayers, P. W.; De Proft, F.; Geerlings, P. submitted.

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Chapter 2

Density functional theory models of reactivity based on an energetic criterion Andrés Cedillo Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa San Rafael Atlixco 186, Iztapalapa DF 09340, México

Abstract Density functional theory is a very fruitful theoretical tool to understand and develop reactivity concepts. Several quantities have been defined to construct a theoretical framework of the intrinsic response of chemical species, using either intuitive or formal procedures. However, this framework is far from a final and complete form, and no systematic procedure is known to generate the appropriate quantities to describe a specific problem. This work shows that some of the most successful reactivity parameters appear from a procedure where energy stabilization takes a relevant place.

1. Introduction Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn–Sham method;1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.78 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9 Usually reactivity parameters have been associated with the response of electronic properties of the system to the changes in the independent variables. Then reactivity parameters are identified with response functions, and they are represented by derivatives of electronic properties. 19

20

DFT models of reactivity

Within the Born–Oppenheimer approximation, the number of electrons, N , and the external potential (from nonelectronic forces), r, completely determine the Hamiltonian operator, and, as a consequence, they also determine all the properties of the system. Specifically, the ground-state energy, E, is a concave function of the number of electrons and a convex functional of the external potential, E = EN The former dependence comes from experimental evidences, consecutive ionization potentials always increase, the latter was demonstrated by Lieb several years ago.10 Variations in the independent variables lead to a change in the energy which is approximated by a differential form, E

E N + rdr = N + r rdr E ≈

N r N The derivative with respect to the number of electrons,

E ≡

N is called the electronic chemical potential, or shortly the chemical potential.3 For non degenerate ground states, the functional derivative with respect to the external potential corresponds to the electron density, E r = r N Then and r represent the response of the energy to the changes on N and r, respectively. Following the same procedure, changes of other electronic properties can be analyzed. Variations of the chemical potential and electron density possess special interest,

≈ N + rdr = N + fr rdr

N r N r

r r ≈ N + r dr = fr N + r r r dr

N r N The response of the chemical potential to changes in the number of electrons is called the chemical hardness,4 2

E

= > 0 ≡

N

N 2 The positive sign of chemical hardness is a consequence of the concavity of the energy with respect to the number of electrons. The Fukui function, fr, appears in both equations since it represents the sensitivity of the chemical potential to the changes in the external potential and that of the density with respect to the number of electrons,6 E

r

fr ≡ = = r N

N

N r N

Andrés Cedillo

21

The response of the density to the changes in the external potential, r 2 E r r ≡ = r N rr N is the density response kernel. Since and r are the first derivatives of the energy, their corresponding variations only involve the three second derivatives of the energy, namely the density response kernel, Fukui function and hardness. In addition to the canonical representation, where the independent variables are N and r, the convexity of the functional EN allows the existence of other representations.8 The grand potential,

E ≡ EN − N = EN − N

N provides an appropriate representation for open systems, the grand canonical representation. Previous equation corresponds to a Legendre transformation, which generates an alternative representation for ground states. In this case, we have the following expressions,

≈ + rdr = −N + r rdr

r

N N N ≈ + rdr = S − sr rdr

r r

r r dr = sr − r r r dr + r ≈ r

For this representation, −N and r represent the response of the grand potential to the changes of and r, respectively,

= −N = r

r As a consequence, for an open system, the number of electrons and the density depend on and r. The sensitivities of N and r correspond to the second derivatives of the grand potential, and they are the chemical softness,7 2

N

S≡ =−

2 the local softness,7 sr ≡

r

N =− r

=

r

and the softness kernel,8 r r ≡ −

r r

=−

2 rr

22


Derivatives from two representations are not independent; they can be related by a reduction of derivatives scheme, analogous with the thermodynamical procedure.9 For the previous representations, one finds that the second derivatives of the grand canonical representation can be written in the following form,

N 1 1 S= = =

N

r

N

r sr ≡ = = Sfr

N −1

r

N r

r r − r r ≡ − =

r N r = Sfrfr − r r From these equations, one can easily find that sr = r r dr where we used the following properties, r r dr = 0

S=

srdr

frdr = 1

The same procedure can be applied for derivatives from other representations. Electronic properties obtained by differentiation are usually classified by its dependence on the position. Global properties have the same value everywhere, such as the chemical potential, hardness and softness. Electron density, Fukui function and local softness change throughout the molecule, and they are called local properties. Finally, kernel properties depend on two or more position vectors, like the density response and softness kernels. Global parameters describe molecular reactivity, local properties provide information on site selectivity, while kernels can be used to understand site activation. The chemical behaviour of a given species strongly depends on the nature of the other molecules involved in the interaction. For a specific type of reaction, an appropriate model is needed to simulate the chemical environment of the species of interest. In the present work, the interest is focused on the initial response of the molecule to a particular type of chemical situation, independent of the value of those parameters that characterize one specific reaction. In other words, the intrinsic capabilities of the chemical species are studied and modelled as derivatives of the electronic properties with respect to an appropriate independent variable. For example, in those processes where charge transfer is involved (such as Lewis acidity and basicity, electrophile– nucleophile interactions and coordination compounds), the number of electrons must be an independent variable; when a small molecule interacts with a very large counterpart (such as enzyme–substrate interaction and adsorption on solid surfaces), the chemical potential of the large partner will be imposed on the small molecule, and its number of electrons will not be independent. In this work, energetic criteria will be applied to support the use of derivatives, coming from some ground-state representation, to describe the intrinsic reactivity of molecules in specific types of reactions.

Andrés Cedillo

23

2. Electronegativity and related quantities When two different atoms or fragments form a covalent bond, each one attracts the shared electron pair with different strength, giving rise to a polar bond. Pauling11 originally proposed a model to quantify this property, named electronegativity. Over the years, other electronegativity scales have been defined, using different ways to compute it.12 Electronegativity differences drive many chemical processes. Lewis acid and base model characterize acids as electron-deficient species, while bases are electron donors. Since Lewis acids show a stronger attraction for the electrons than Lewis bases, the electronegativity of the acids must be larger than that of the bases. Coordination compounds are formed by one or more Lewis acids, usually metal cations, and some ligands, Lewis bases. In organic reactions, an electrophile represents an electron-deficient species, or Lewis acid, while a nucleophile corresponds to an electron donor, or Lewis base. One molecule may have several acid and basic sites. In order to characterize the nature and strength off those sites, local properties are needed. Identification of electrophilic and nucleophilic regions of the molecule is very useful in the prediction of the initial steps of chemical reactions.

2.1. Global analysis Consider a process where a set of chemical species will receive a small portion of electron transfer, dN > 0. When the external potential remains constant for all the species, the changes of the energy are given by dEi = i dN In a species with the ability to accept charge, the energy must decrease after the electron-transfer process, and its chemical potential must be negative. The corresponding stabilization for the set of molecules becomes −dEi = − i dN > 0 Then, the negative of the chemical potential is proportional to the stabilization of a chemical species when it receives electrons. Likewise, when the same species donate electrons, dN < 0, their energies increase, dEi = − i −dN > 0 The negative of the chemical potential also measures the resistance of the chemical species to deliver electrons. From this behaviour, the negative of the chemical potential is identified with the electronegativity,3

E 1 = − = − ≈ I + A

N 2 The last approximation corresponds to the finite differences approach to the first derivative, and I and A represent the ionization potential and electron affinity of the

24


species, respectively. This approximation corresponds to Mulliken’s electronegativity approach.13 In the interaction of two molecules with different chemical potentials (or electronegativities), neglecting the effects from the changes in the external potential, electron transfer from the species with higher chemical potential (lower electronegativity) to the species with lower chemical potential (higher electronegativity) is energetically preferred. Along this electron-transfer process, the chemical potential of both species also changes, d i = i dN where is the chemical hardness, which is a positive quantity, and it can be evaluated by a finite differences scheme, ≈ I − A. The chemical potential of the acceptor increases, while that of the donor decreases, until both become equal (chemical potential equalization).14 At this point, electron transfer ends. Electronegativity equalization methods have been very useful to predict atomic or fragment charges for complex systems.15 In these methods, it is shown that the maximum stabilization is obtained when the electronegativity of each fragment becomes equal. Chemical potential of a species measures the initial affinity for the electrons; however, during the electron transfer, this affinity is modified. Chemical hardness, , modulates this variation. Ignoring the external potential effects, the change in the energy, up to second order, is given by 1 E ≈ N + N2 2 Two electron acceptors (Lewis acids), with the same chemical potential (electronegativity), will show different behaviour in the charge transfer process. The species with larger hardness (harder species) easily satisfies its affinity for electrons, while the species with lower hardness (softer species) keeps its affinity for larger amounts of charge transfer, see Figure 1. Donating behaviour for bases shows that for the same change in the energy, the soft base is able to transfer a bigger amount of charge than the hard one. Then hard species are associated with small amounts of charge transfer, while for soft

Figure 1 Change in the energy for two species with the same chemical potential 1 > 2

Andrés Cedillo

25

species with large. This accordance is consistent with Pearson’s hard and soft acids and bases principle.16 An electronegative species increases its chemical potential when it receives charge up to the time that its affinity for electrons is satisfied, that is when the chemical potential vanishes. This situation corresponds to a minimum in Figure 1. Using the second-order truncated Taylor expansion, one finds that the maximum amount of charge that a Lewis acid can accept is Nmax = − S = S This expression represents the charge capacity of the species. It is important to note that a very hard species, even when its electronegativity is large, can have a small value of Nmax , on account of the small value of its softness, see Figure 1. This situation is found on the chemical behaviour of the halogens, fluorine atom is more electronegative and harder than chlorine atom, however chloride ion is a weaker base that fluoride ion. When both atoms accept one electron, chloride ion’s chemical potential is still negative, while fluoride’s becomes positive; furthermore, stabilization of chlorine is larger than that of fluorine. The maximum stabilization of a species in a charge transfer process can be also evaluated, 1 1 ≡ − Emin = − E Nmax = 2 S = Nmax 2 2 This quantity has been identified as an electrophilicity index.5 Note that it is constructed from two important features of an electrophile, electronegativity and charge capacity, which act cooperatively. One can observe that global reactivity parameters represent the response of an electronic property to the changes of the independent variables. The physical meaning of these quantities can be rationalized by proposing specific situations where a few quantities are involved. Also specific combinations of these parameters are useful to describe important chemical trends.

2.2. Local analysis A single molecule can exhibit both acidic and basic conducts on different sites. Global properties are unable to provide this kind of description, in this case, local properties are useful. Fukui function represents, either the response of the chemical potential to variation in the external potential or the response of the density to changes in the number of electrons.6 The latter is important in charge transfer descriptions. Fukui function minimizes a hardness functional,17 Hg ≡ grr ˜ r gr dr dr where r ˜ r is the hardness kernel,8 r ˜ r ≡

2 F rr

26


F is the Hohenberg–Kohn universal functional and gr represents any function that integrates to unity,

grdr = 1

By the way, the extreme value of the hardness functional is equal to the chemical hardness,17 Hf = frr ˜ r fr dr dr = Then energy and density come from the variational principle of the energy, while Fukui function and hardness are obtained from the minimization of the hardness functional. Recently, it has been shown that the previous variational principle for the Fukui function can be derived from a minimization of the energy.18 When an electronic system changes its number of electrons in dN , the variation of the density that minimizes the energy follows the Fukui function, r = frdN That is, the most favourable way to distribute (or extract) dN electrons in a chemical species is guided by the Fukui function. From Janak’s extension for noninteger occupation numbers,19 the Fukui function can be written as the sum of a frontier orbital density and relaxation terms, ⎧ N

i r − ⎪ ⎪N r + dN < 0 ⎪ ⎨

N i=1 fr = ⎪ N

i r + ⎪ ⎪ ⎩N +1 r + dN > 0

N i=1 where i is the density associated with the i-th Kohn–Sham spin orbital. In consequence, Fukui function is considered as a generalization of the frontier orbital theory,20 since it includes terms coming from the redistribution of the electronic density induced by the addition or removal of electrons. Note that discontinuity on the density, as a function of the number of electrons, leads to a couple of expressions. When the number of electrons increases, a nucleophilic attack or acidic behaviour, nucleophilic Fukui function may be related to the LUMO density, +

f r ≡

r

N

+ ≈ LUMO r

If the number of electrons decreases, an electrophilic attack or basic behaviour, electrophilic Fukui function may be related to the HOMO density,

r − − f r ≡ ≈ HOMO r

N

Andrés Cedillo

27

Fukui function is usually computed by a finite differences procedure, f − r ≈ N r − N −1 r f + r ≈ N +1 r − N r where N represents the density of a system with N electrons. Donor or basic sites of chemical species are characterized by large positive values of the Fukui function f − , while acceptor or acidic sites correspond to large values of f + . Since Fukui function integrates to unity, it cannot be used to compare sites from different molecules. In many cases, for large systems, it is found that Fukui function distributes along the molecule. Local softness also satisfies a variational principle,18 the variation of the density that minimizes the grand potential, when the chemical potential changes in d , is given by r = srd From the relation between Fukui function and local softness, electrophilic and nucleophilic local softnesses can be computed. Donor and acceptor sites can also be identified by large values of both types of local softnesses; in addition, it can be used to compare sites of different molecules and to identify which one is softer or harder. The electrophilicity index can also be extended to a local context,21 and a comparison of the electrophilicity of sites in different species can be made.

3. Interactions with nuclei Changes in the nuclei are associated with variations in the external potential. For example, nuclear displacements are related to Hellman–Feynman forces; protonation reactions (addition of a nucleus) are used to study Lewis acidity and basicity in gas phase and solution. In a protonation reaction, a hydrogen nucleus is added to a chemical species. The proton can be placed in several positions, and the molecular energy determines stable isomers and the most stable protonated form. The addition of one proton, at the position R0 , is represented by a change in the external potential given by r = −

1 r − R0

while the number of electrons remains constant. The change in the molecular energy, Emol , includes two terms, one for the electronic part, E, and the other for the nuclei, Enn , Emol = EN + − EN + Enn The nuclear contribution is an electrostatic term, Enn =

M

Z =1 R − R0

28


where M is the original number of nuclei in the species. For the electronic part, a Taylor series expansion is used,

1 r r r r dr dr + · · · 2 r 1 r r =− dr dr + · · · dr + r − R0 2 r − R0 r − R0

EN + − EN =

r rdr +

Then, the change in the molecular energy, which depends on R0 , can be approximated, up to second order, by 1 Emol R0 ≈ R0 + R0 2 where is the molecular electrostatic potential, R =

r Z − dr r − R =1 R − R M

and the last term is an integral of the density response kernel, R =

r r dr dr < 0 r − Rr − R

The sign of the last integral is a consequence of the convexity of the energy as a functional of the external potential. Stable protonated isomers are associated with minima of Emol . A first-order approximation predicts that protonation occurs at places where the molecular electrostatic potential is a minimum.22 Since, in neutral species, negative values of the electrostatic potential are usually associated with lone pairs or electron-rich regions,23 this approximation is very rational. At this level, the change in the energy, Emol R0 ≈ R0 , represents the electrostatic interaction of the proton with the nuclei and the unperturbed electron density. Relaxation of the density, induced by the presence of the proton, is taken into account by higher-order terms. The second-order term, related to the density response kernel, contributes with an additional stabilization from the initial response of the density to the new positive charge. At the present, there is no simple procedure to compute the response kernels, neither its contributions to energy, and fundamental studies on this direction are desired. Nucleophiles or Lewis bases are involved in many chemical processes, including protonation, then the characterization of the nucleophilicity should take into account the response functions associated with variations of the external potential.

4. Concluding remarks The use of an energy stabilization procedure allows the identification of some response functions as reactivity parameters. In other cases, energetic criteria were used to find insights into the physical meaning. Since the intrinsic reactivity framework is far from

Andrés Cedillo

29

being complete, the use of this kind of procedure can be useful to develop the appropriate parameters to describe specific chemical situations. At the present, the finite differences approximation used to evaluate global and local properties seems to be reasonable. However, theoretical developments to find useful computing schemes for kernels are needed, as well as models to understand and apply kernels to specific reactivity problems, such as the site activation case. Many chemical reactivity aspects are beyond the present framework. Since this methodology studies the intrinsic or initial response, properties associated with advanced steps of the reaction, such as those related with the transition state or the reaction product, may be inaccurately described. The same can be said about situations where the Born–Oppenheimer approximation fails, like tunnelling contributions.

Acknowledgements This work received financial support from CONACYT grant 39622-F.

References 1. See for example: Parr, R. G. and Yang, W., Density Functional Theory of Atoms and Molecules, Oxford, New York (1989); Gross, E. K. U. and Dreizler, R., Density Functional Theory, Springer, Berlin (1993). 2. Sen, K. D. and Jørgensen, C. K. (Eds.), Electronegativity, Structure and Bonding 66, Springer, Berlin (1987); Sen, K. D. (Ed.), Chemical Hardness, Structure and Bonding 80, Springer, Berlin (1993); Pearson, R. G., Chemical Hardness: Applications from Molecules to Solids, Wiley-VCH, Weinheim (1997). 3. Iczkowski, R. P. and Margrave, J. L., J. Am. Chem. Soc. 83 3547 (1961); Parr, R. G., Donnelly, R. A., Levy, M. and Palke, W. E., J. Chem. Phys. 69 4431 (1978). 4. Parr, R. G. and Pearson, R. G., J. Am. Chem. Soc. 105 7512 (1983). 5. Parr, R. G. von Szentpály, L. and Liu, S., J. Am. Chem. Soc. 121 1922 (1999). 6. Parr, R. G. and Yang, W., J. Am. Chem. Soc. 106 4049 (1984). 7. Yang, W. and Parr, R. G., Proc. Natl. Acad. Sci. U.S.A 82 6723 (1985). 8. Berkowitz, M. and Parr, R. G., J. Chem. Phys. 88 2554 (1988). 9. Nalewajski, R. F. and Parr, R. G., J. Chem. Phys. 77 399 (1982); Nalewajski, R. F., J. Chem. Phys. 78 6112 (1983); Cedillo, A., Int. J. Quant. Chem. Suppl. 28 231 (1994). 10. Lieb, E. H., Int. J. Quant. Chem. 24 243 (1983). 11. Pauling, L., The Nature of the Chemical Bond, 3rd ed., Cornell, Ithaca (1960). 12. See for example: Sanderson, R. T., J. Chem. Educ. 65 112 (1988); Sanderson, R. T., J. Chem. Educ. 65 227 (1988); Huheey, J. E., Inorganic Chemistry: Principles of Structure and Reactivity, 2nd ed., pp. 159–173, Harper & Row, New York (1978). 13. Mulliken, R. S., J. Chem. Phys. 2 782 (1934). 14. Sanderson, R. T., Science 121 207 (1955). 15. See for example: Mortier, W. J., Ghosh, S. K. and Shankar, S., J. Am. Chem. Soc. 108 4315 (1986); Van Genechten, K., Mortier, W. J. and Geerlings, P., J. Chem. Phys. 86 5063 (1987); Baekelandt, B. G., Mortier, W. J., and Schoonheydt, R. A. Structure and Bonding 80 187 (1993); Nalewajski, R. F., Korchowiec, J. and Zhou, Z. Int. J. Quant. Chem. Suppl. 22 349 (1988). 16. Pearson, R. G., J. Am. Chem. Soc. 85 3533 (1963); Pearson, R. G., Hard and Soft Acids and Bases, Dowden, Hutchinson and Ross, Stroudsburg, PA (1973).

30 17. 18. 19. 20.


Chattaraj, P. K., Cedillo, A. and Parr, R. G., J. Chem. Phys. 103 7645 (1995). Ayers, P. W. and Parr, R. G., J. Am. Chem. Soc. 122 2010 (2000). Janak, J. F., Phys. Rev. B 18 7165 (1978). Fukui, K., Theory of Orientation and Stereoselection, Springer, Berlin (1973); Fukui, K., Science 218 747 (1982). 21. Fuentealba, P. and Contreras, R., in Reviews of Modern Quantum Chemistry, Sen, K. D. (Ed.), pp. 1013–1052, World Scientific, Singapore (2002); Chattaraj, P. K., Maiti, B. and Sarkar, U., J. Phys. Chem. A 107 4973 (2003); Cedillo, A. and Contreras, R. (to be published). 22. Politzer, P. and Daiker, K. C., J. Chem. Phys. 68 5289 (1978). 23. See for example: Murray, J.S. and Sen, K.D. (Eds.), Molecular Electrostatic Potentials: Concepts and Applications, Elsevier, Amsterdam (1996).


Chapter 3

The breakdown of the maximum hardness and minimum polarizability principles for nontotally symmetric vibrations Miquel Torrent-Sucarrat, Lluís Blancafort, Miquel Duran, Josep M. Luis, and Miquel Solà Institut de Química Computacional and Departament de Química, Universitat de Girona, E-17071 Girona, Catalonia, Spain

Abstract In this chapter, we review our latest results on the validity of the maximum hardness and minimum polarizability principles in nontotally symmetric vibrations. These nuclear displacements are particularly interesting because they keep the chemical and external potentials approximately constant, thus closely following the two conditions of Parr and Chattaraj required for the strict compliance with the maximum hardness principle. We show that, although these principles are obeyed by most nontotally symmetric vibrations, there are some nontotally symmetric displacements that refuse to comply with them. The underlying physical reasons for the failure of these two principles in these particular nuclear motions are analyzed. Finally, the application of this breakdown to detect the most aromatic center in polycyclic aromatic hydrocarbons is discussed.

1. Introduction During the past two decades, the density functional theory (DFT) [1,2] has revolutionized the theoretical studies of chemical reactivity in many areas of chemistry, varying from inorganic to organic chemistry and from material science to biochemistry [3,4]. Based on the famous Hohenberg and Kohn theorems [5], DFT methods have become an attractive alternative to the traditional ab initio correlated calculations, due to the correctness and swiftness of their results. This crucial role of the DFT in chemistry was recognized in 1998, with the Nobel Prize awarded to the founding father of DFT, Professor Walter Kohn [6]. 31

32

Breakdown of the MHP and MPP for nontotally symmetric vibrations

On the basis of the idea that the electron density is the fundamental quantity for describing atomic and molecular ground states, Parr and co-workers were able to give mathematical definitions for chemical concepts, which were already known and had been in use for many years in various branches of chemistry (e.g. electronegativity and hardness), thus affording their calculation and quantitative use. This important branch of the DFT, called conceptual DFT by its main protagonist, R. G. Parr, has been developed since the late 1970s and early 1980s [1]. The central quantities of conceptual DFT [7,8] are the response functions. These functions are the response of the chemical system to perturbations in its number of electrons, N , and/or the external potential, vr , which is the potential acting on an electron at position r due to the nuclear attraction plus other external forces that may be present. Assuming differentiability of the electronic energy, E, with respect to N and vr , a series of response functions appear, probably the most important being the electronic chemical potential, , and the hardness, , defined as [9] E and (1) = N vr 2 E = (2) = N 2 vr N vr A finite difference approximation leads to the following working definitions of these quantities 1 = − I + A and 2 = I − A

(3) (4)

where I and A are the first vertical ionization potential and the electron affinity of the neutral molecule, respectively. Equations (3) and (4) can be approximated by the energy of the frontier HOMO H and LUMO L molecular orbitals using the Koopmans’ theorem I ≈ − H and A ≈ − L [10] 1 = L + H and 2 = L − H

(5) (6)

Associated with these properties, important chemical reactivity principles have been rationalized within the framework of conceptual DFT: the hard and soft acids and bases principle (HSAB) [9], the Sanderson electronegativity equalization principle (EEP) [11], the maximum hardness principle (MHP) [9,12,13], and the minimum polarizability principle (MPP) [14]. The aim of this chapter is to revise the validity of the last two principles in nontotally symmetric vibrations. We start with a short section on the fundamental aspects of the MHP and MPP (section 2). Section 3 focuses on the breakdown of these principles for nontotally symmetric vibrations, while section 4 analyses the relationship between the failure of the MHP and the pseudo-Jahn–Teller (PJT) effect. A mathematical procedure that helps to determine the nontotally symmetric distortions of a given molecule that produce the maximum failures of the MPP or the

M. Solà et al.

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MHP is derived in section 5. From the results obtained with this procedure, we develop a set of simple rules that allow predicting a priori without calculations the existence of vibrational modes that break the MPP (section 6). Finally, we show in section 7 how the nuclear displacements that break the MPP can be a useful indicator of local aromaticity in polycyclic aromatic hydrocarbons.

2. The maximum hardness and the minimum polarizability principles In 1987, Pearson stated for the first time the MHP under the form that “there seems to be a rule of nature that molecules arrange themselves to be as hard as possible” [13]. The MPP was formulated on the basis of the MHP and a linear [15,16] (or cubic according to other authors [14,17–19]) relationship between the softness (the inverse of hardness) and the polarizability. The MPP affirms that the natural evolution of any system is toward a state of minimum polarizability. The polarizability is usually taken as one-third of the trace of the polarizability tensor, =

1

xx + yy + zz 3

(7)

Both principles have been applied successfully to the study of molecular vibrations [20–23], internal rotations [24–29], excited states [30,31], aromaticity [32], and different types of chemical reactions [33–42]. A formal proof of the MHP based on statistical mechanics and the fluctuation–dissipation theorem was given by Parr and Chattaraj [43] under the constraints that and vr must remain constant upon distortion of molecular structure. There is no single chemical process that satisfies these two severe constraints. However, relaxation of these restrictions seems to be permissible, and it has been found that in most cases the MHP still holds; even though, the chemical and external potentials vary during the molecular vibration [20–23], internal rotation [24–29], or along the reaction coordinate [33–42]. Hereafter, we will refer to the generalized MHP (GMPH) or MPP (GMPP) as the maximum hardness or minimum polarizability principles that do not require the constancy of and vr during molecular changes. It is worth nothing that some failures of the GMHP and GMPP in some chemical reactions and excited states have been reported [44–50]. In most of these cases, it has been argued that the and vr change perceptibly during the process. Thus, these situations do not break the strict MHP, because this principle is rigorously valid only under constant and vr [43]. Recently, an alternative and simpler proof of the MHP has been given by Ayers and Parr [51]. The molecular motion along a nontotally symmetric vibration is an interesting case in point to analyze from the viewpoint of the MHP and MPP [20]. Let us start with a molecule in its equilibrium geometry and make a displacement from equilibrium along a nontotally symmetric normal mode (see Figure 1). In this kind of distortion, positive and negative deviations from the equilibrium structure along nontotally symmetric vibrational modes yield molecular configurations that have the same , and average potential of the nuclei acting on the electrons ven . Then, if Q represents a nontotally symmetric normal mode coordinate, it follows that /Q = 0 /Q = 0 /Q = 0, and ven /Q = 0 at the equilibrium geometry. After a

34

Breakdown of the MHP and MPP for nontotally symmetric vibrations O

O H

H Q0

α η H

O H

η

η

Energy Energy

α

α

Figure 1 Schematic representation of the behavior of the energy, hardness, and polarizability in the antisymmetric and symmetric stretching vibrational modes of water

small displacement, Q, from the equilibrium geometry, the average external potential may be written as: en 0 Q + · · · (8)

en = ven + Q Qe and, therefore, since en /QQe = 0 it is found that for small nontotally symmetric displacements, en is approximately constant. The same applies to the chemical potential. Hence, for small distortions along nontotally symmetric normal modes, and en are roughly constant, becoming the nuclear motion that most closely follows the two conditions of Parr and Chattaraj [20,43]. According to the GMHP and GMPP, the equilibrium hardness must be a maximum, and the polarizability a minimum, for any nontotally symmetric distortion (see Figure 1). To our knowledge, all numerical calculations of hardness and polarizability along the nontotally symmetric vibration modes carried out before our studies confirmed this point [20–23]. In contrast, for totally symmetric distortions, near the equilibrium geometry, the hardness keeps increasing steadily as the nuclei approach each other. Indeed, neither nor or en shows any sign of maximum or minimum at the equilibrium geometry (see Figure 1). This is not a violation of the strict MHP since neither the chemical nor the external potentials are kept constant during this kind of distortion.

3. Breakdown of the MHP and MPP for nontotally symmetric vibrations: some examples In February 2000, after a seminar given by Jaque in our institute entitled “On the validity of the minimum polarizability principle in molecular vibrations and internal rotations,” some of us started a discussion about the validity of the MPP in bond length

M. Solà et al.

35

alternation (BLA) modes of molecules such as benzene or pyridine. The argument was that BLA distortions in aromatic rings reduce the delocalization of the -electrons and, consequently, the polarizability of the molecule. As a result, nontotally symmetric BLA vibrational modes should disobey the GMPP and, likely, the GMHP. So, we decided to study the behavior of some of these BLA vibrational modes, finding for the first time the existence of a nontotally symmetric vibrational mode (the b2 normal mode of pyridine at 13044 cm−1 ) that breaks the GMHP and GMPP [52]. In contrast to the examples of breakdown of the GMHP and GMPP reported to that point, this was an example of a failure of these principles for the most favorable case in which and vr keep almost constant, indicating that the strict version of the MHP cannot be straightforwardly generalized. In Figure 2a, it is clearly seen that this b2 vibration mode has the characteristic atomic displacements of BLA modes. The breakdown of the GMHP implies an increase of the hardness along the b2 normal mode of pyridine. This augment in the hardness value can be attributed to a larger destabilization of the LUMO than of the HOMO. The destabilization of HOMO is expected in nontotally symmetric displacements and leads usually to a decrease of the HOMO–LUMO gap and hardness. The destabilization of the LUMO is somewhat unexpected and can be understood by observing the shape of this orbital (Figure 2c) and the BLA distortion of the b2 vibrational mode (Figure 2a). The increase in the LUMO energy is basically produced by the interaction between the lobe on N and the lobes on the C atoms adjacent to the N atom as the molecule vibrates. When the N atom and an adjacent C atom approach, the increase in the antibonding interaction is more important than the reduction of the same antibonding interaction as the other adjacent C atom and N move away. The same is true for the C atom located in para position with respect the N atom. As a result, the LUMO is destabilized during the BLA vibration, and the hardness increases in contradiction with the GMHP. As expected from the inverse relationship between hardness and polarizability, it is found that the polarizability value diminishes during the b2 displacement from the equilibrium structure, in disagreement with the GMPP. At first view, one may attribute the breakdown of the GMHP and GMPP in this normal mode to the fact that and r change significantly during this

N

H

C

C

C

C H

H

C

H

H

(a)

(b)

(c)

Figure 2 (a) Schematic representation of the displacement vector corresponding to the b2 normal vibration mode of pyridine at 13044 cm−1 . Representation of the B3LYP/6-31++G(d,p) isosurfaces 0.1 (gray) and −01 (black) a.u. of the (b) HOMO and (c) LUMO

36


vibration. However, this is not the case, and we have found that variations in and

r are similar among all the nontotally symmetric modes of pyridine. It is worth noting that the existence of BLA modes that do not comply with the GMHP and GMPP is not a particular feature of pyridine. We have found that several -conjugated organic molecules such as benzene, naphthalene, anthracene, or phenanthrene possess BLA modes that violate the GMHP and GMPP, irrespective of the method of calculation used [52,53]. However, less expected is the generalization of this breakdown to non--conjugated or even non--bonded organic and inorganic systems. Actually, we have found that systems like the hydrogen fluoride tetramer, diborane, and anionic aluminum tetramer also present nontotally symmetric vibrations, which break the GMHP and GMPP [54]. Although they are not -conjugated organic species, all these systems possess a certain degree of cyclic electron delocalization, even in the case of hydrogen-bonded species such as the hydrogen fluoride tetramer. However, we have also identified some molecules with an almost completely localized electronic structure (cyclobutane and chair cyclohexane) that do not follow the GMHP [54]. These results are relevant since it is important to know the applicability limits of the GMHP and GMPP. We have also found that the molecular distortions that disobey the GMPP are not necessarily the same as those that break the GMHP. To get some insight into the origin of this discrepancy, we have started our analysis from the exact expression of the isotropic static electronic polarizability derived from perturbation theory: 1 o ˆ n 2

= n 3 n=0

(9)

In this equation, o and n are the wavefunctions of the ground and nth excited states, respectively, and n = En − Eo /, where Eo and En are the energy of the ground and nth excited states. Considering now that the only non-negligible excited state in (9) corresponds to the HOMO to LUMO excitation, and making the assumption that the energy difference between the ground and this excited state equals the HOMO– LUMO gap, one finds that the static electronic polarizability is proportional to the softness, i.e.

∝S=

1 L − H

(10)

Since we have used (6) for the calculation of the hardness, if (10) was correct, then we should find the same anti-GMPP and anti-GMHP modes. However, for several systems (e.g. fluoride tetramer), this is not the case. Less crude approximations are given by the following proportionality relationships:

∝

H L+1

1 − i i=H−1 j=L j

(11)

M. Solà et al.

37

and

∝

N H

1 i=1 j=L j − i

(12)

In (11), one makes the hypothesis that all excited states that do not include HOMO−1 and HOMO to LUMO and LUMO + 1 orbital excitations can be neglected and that ˆ n 2 term is identical for all selected excitations. The latter assumption is the o the only approximation considered in (12). The results show that neither the softness nor the approximation of (11) yields the correct sign to detect the GMPP/anti-GMPP character of the nontotally symmetric vibrations analyzed. However, (12) provides the correct sign for most nontotally symmetric distortions. These results cast serious doubts about the general validity of taking for granted the proportionality between polarizability and softness, which is the main assumption made in the formulation of the MPP [54].

4. Breakdown of the MHP and the PJT In 1992, Pearson and Palke already suggested a possible relationship between the MHP and the PJT effect [20]. Thus, after finding that the GMHP may be disobeyed in some nontotally symmetric vibrations, we were interested to study whether the PJT coupling is the reason behind the violation of the GMHP along the nontotally symmetric vibrations. The PJT effect [55,56] is usually formulated using second-order perturbation theory, where the energy, EQ, is expanded about the point of minimum energy Q0 2 2

U

U

Q EQ = EQ0 + Q 0

0 + 0

2

0 Q 2 Q 2

U

Q 0

k Q + +··· (13) E0 − Ek k where Q = Q−Q0 represents a nuclear displacement coordinate and U is the energy of the potential energy surface (PES). In the calculation of the derivatives of U with respect to Q in (13), it is here assumed that the kinetic and the electronic–electronic energy keep constant along Q and, therefore, that only the nuclear–electronic and nuclear–nuclear parts of the PES change along Q. If the original wave function, 0 , is nondegenerate, the linear term is zero at the minimum geometry. Writing (13) in a more compact form, we obtain (14) EQ = EQ0 + f00 Q2 + f0k Q2 + · · · k

where the sum of the two second-order terms, f00 and f0k , gives the force constant of the vibration, Q. At the minimum, the first second-order term, f00 , is positive [55]. The last term, f0k , corresponds to the PJT coupling, and it implies an energy stabilization of the ground state due to an electronic coupling between nondegenerated, but close in energy, states (normally between the ground state and the first excited state). This

38


term only appears for nontotally symmetric vibrations with the same symmetry as the product 0 k . The f0k value is negative for the ground state of the molecule, because E0 − Ek < 0, and therefore it lowers the force constant of the ground state, producing smoother PES. In contrast, there will be an inverse effect on the PES of the first excited state, E1 Q. If we assume that the coupling between the ground state and the first excited state is bigger than that between the first excited state and the remaining excited states, the f1k term will be positive, E1 − E0 > 0. Thus, the force constant increases, and the PES of the first excited state becomes steeper. In conclusion, there will be a widening of the energy gap due to PJT effects along the nontotally symmetric vibration, as shown in Figure 3. For the analogy between the PJT effect and the breakdown of the GMHP to be valid, using Koopmans’ approximation to evaluate hardness, several other conditions have to be fulfilled: first, we assume that the energy gap between the ground and excited state equals to the HOMO–LUMO gap; second, the excited state coupled to the ground state has to be dominated by the HOMO–LUMO excitation; and finally, the minimum geometry and “uncoupled” (without PJT) force constants of the two coupled states have to be similar. Because these assumptions are only approximate, only a substantial PJT effect will induce the exception to the GMHP in nontotally symmetric vibrations. The PJT contribution was calculated subtracting the CASSCF coupled (with PJT) frequencies to the CASSCF “uncoupled” (without PJT) frequencies, using symmetryrestricted CASSCF wave functions [56]. This procedure was applied to some nitrogen heterocycles (pyridine, pyrazine, and pyrimidine), showing that the PJT coupling is the reason behind the violation of the GMHP along their BLA modes. For instance, the b2 distortion that breaks the GMHP of the pyridine shows a PTJ effect of 7775 cm−1 . On the other hand, the b2 symmetric BLA mode of the pyridazine obeys the GMHP and shows a substantially smaller PJT effect 1251 cm−1 [57].

E1 Uncoupled (no PJT)

η1 > η0

E0

Coupled (includes PJT)

η0 ≈ εL – εH ≈ E1 – E0

Q0

Q

Figure 3 Pseudo-Jahn–Teller (PJT) effect between 0 and 1 along a nontotally symmetric mode. Dashed-line parabolas: “uncoupled” potential energy surfaces (without PJT). Full-line parabolas: coupled potential surfaces (with PJT)

M. Solà et al.

39

5. Diagonalization of the polarizability and the hardness Hessian matrix with respect to the vibrational normal coordinates To find the nuclear displacements that have a more marked breakdown of the GMPP and GMHP character, we diagonalize the Hessian matrix of the hardness [54] and polarizability [53] with respect to the nontotally symmetric normal coordinates, whose elements are obtained as 2 kl = and (15) Qk Ql 2

kl = (16) Qk Ql with k and l running over the nontotally symmetric modes. The negative/positive sign of the diagonal terms of matrix indicates us whether a nontotally symmetric vibrational mode has GMHP/anti-GMHP character. The eigenvectors obtained in the diagonalization of are linear combinations of vibrational modes (for a given eigenvector all implicated vibrational modes belong to the same symmetry species) giving the distortions that produce the largest hardness changes. In this sense, the sign of the eigenvalue characterizes the GMHP or anti-GMHP character of the distortion along a given eigenvector. If an eigenvector has a positive eigenvalue, the equilibrium structure will represent a minimum of hardness along this distortion, which consequently breaks the GMHP. Likewise, a positive/negative sign of the diagonal terms of matrix tells us whether a certain nontotally symmetric vibrational mode has GMPP/anti-GMPP character. Finally, a negative eigenvalue in the diagonalization of implies a non-fulfillment of the GMPP along this distortion. Figure 4 displays the two nontotally symmetric vibrational modes of benzene that do not fulfill the GMPP at the HF/6-31G level (1b2u and 2b2u ) and the two corresponding postdiagonalization distortions (1b2u and 2b2u ) [53]. The 1b2u displacement 0762 · 1b2u +0648 · 2b2u presents a negative eigenvalue. In this distortion, only the nuclear displacements of the carbon atoms are significant, showing a clear BLA character. This fact corroborates the importance of the BLA distortions of the heavier atoms on the breakdown of the GMPP. In contrast, the 2b2u displacement −0648 · 1b2u + 0762 · 2b2u has now a positive eigenvalue, and in this case, the displacements of the carbon atoms have been reduced and those of the hydrogen atoms have been increased. The diagonalization of and with respect to the nontotally symmetric normal coordinates has been applied to 44 and 11 molecules, respectively, including aromatic, non--conjugated, and non--bonded organic and inorganic molecules [53,54,58]. The obtained results show that, while the diagonal elements of and matrices are strongly dependent on the methodology used to compute them, the eigenvalues of the diagonalization of and are almost totally independent of the method and basis set used. The diagonalization of and increases the absolute value of the most relevant eigenvalues. This concentration of information facilitates the analysis of the results and will be essential in the following two sections. The diagonalization of and is also basic to detect molecular distortions that do no follow the GMPP in some species (e.g. borazine and trimer HF at the HF/cc-pVTZ level) and the GMHP in some other systems (e.g. cyclobutane and byciclo[1.1.0]butane at the HF/cc-pVTZ level) [54].

40

Breakdown of the MHP and MPP for nontotally symmetric vibrations H

H

C

H

H

C

C

C

C

H

H

C

C

H

C

C

C

H

H

1b2U′ (0.762·1b2u + 0.648·2b2u)

1b2U

H

H

H

C

H

C

C

C

C C

H

C

H

H

H

C

C

C

C C

H

H

H

2b2U

H

C

H

H

H

C

2b2U′ (–0.648·1b2u + 0.762·2b2u)

Figure 4 Schematic representation of the displacement vectors corresponding to the two b2u studied normal vibration modes of benzene (1b2u and 2b2u ) and the two postdiagonalization nuclear distortions of benzene (1b2u and 2b2u ). The displacement vectors have been calculated at the HF/6-31G level

6. Simple rules to predict without calculations the existence of vibrational modes that break the MPP With the gathered information of the previous section, we have derived a set of simple rules that, for a given -conjugated molecule, allow to a priori predict the existence of nontotally symmetric anti-GMPP distortions without the requirement to perform calculations [53]. Rule A: The molecule should have a BLA movement. Rule B: Draw all possible BLA movements: (a) If all BLA movements transform as the totally symmetric representation, the GMPP is obeyed by all nontotally symmetric modes of the studied system. (b) If one or more BLA movements do not transform as the totally symmetric representation, the GMPP is disobeyed by some of the nontotally symmetric vibrational modes of the studied molecule.

M. Solà et al. Pyrazine D2h

41 N

N

Pyrimidine C2ν

Pyridazine C2ν

N

b2u

N

b2

N N

a1

Benzocyclobutadiene C2ν

a1

a1

Naphthalene D2h

ag

b2u

Figure 5 Different schematic BLA modes that can be drawn for pyrazine, pyrimidine, pyridazine, benzocyclobutadiene, and naphthalene

These rules have successfully been applied to 33 -conjugated molecules[53,58]. For instance, Figure 5 displays the schematic BLA distortion that is possible to draw by just looking at the geometry of the pyrazine, pyrimidine, pyradizine, benzocyclobutadiene, and naphthalene. The first rule is fulfilled by all these molecules, because it is possible to draw a BLA movement in all systems. In accordance with rule B, we can affirm that pyrazine, pyrimidine, and naphthalene possess nontotally symmetric movements that disobey the GMPP because these systems at least present one BLA movement that do not transform as the totally symmetric representation. In contrast, the BLA movements of the pyradizine and benzocyclobutadiene belong to the totally symmetric representation; therefore, the GMPP is obeyed by all their nontotally symmetric vibrational modes. The GMPP is directly not applicable in the case of totally symmetric vibrations, simply because the potentials cannot be considered along these and external constant chemical modes /Qsym Q = 0 /Qsym Q = 0, and ven /Qsym Q = 0 , disobeying the e e e two conditions of Parr and Chattaraj.

7. Breakdown of the GMPP as an indicator of the most aromatic center In the previous sections, we have seen that the BLA vibrational motions that disobey the MPP in -conjugated molecules are distortions of the equilibrium geometry that produce a reduction of the polarizability due to the localization of -electrons [53]. In aromatic systems, this concentration of -electrons is expected to be especially

42


important in the region with a more delocalized electron cloud. Thus, the nuclear displacements that break the GMPP should have larger components in those rings that possess the highest local aromaticity. To investigate this hypothesis, we have applied the diagonalization of to a series of 20 polycyclic aromatic hydrocarbons with welldefined local aromaticities [58]. The results obtained indicate that the predictions of the most aromatic center given by this new method and those expected from NICS calculations fully coincide. As an example, the Figure 6 depicts the postdiagonalization nuclear distortions with the most marked anti-GMPP character for phenanthrene and pentacene (the numbers inside the molecular rings being their NICS values). As can be seen in Figure 6, the phenanthrene and pentacene present two and three different types of aromatic rings, respectively. The postdiagonalization distortion of phenanthrene shows a negative eigenvalue with a clear BLA movement in the most aromatic center. In addition, the method of diagonalization of in the pentacene molecule allows by looking at the b2u distortion of Figure 6 to determine the relative aromaticity order of the three rings. While ring A contains fixed atoms, rings B and C show significant displacements of the carbon atoms, and although only the most aromatic ring C displays a BLA distortion.

–5.9

–9.9

Phenanthrene C2ν

b2

–12.5 C

Pentacene D2h

–10.0 B

–3.8 A

b2u

Figure 6 Schematic representation of the displacement vectors corresponding to the nontotally symmetric postdiagonalization nuclear distortions that show the most marked anti-GMPP character of phenanthrene and pentacene obtained at the HF/6-31G level. The depicted displacement vectors of the non-hydrogen atoms have been multiplied by two in the representation for seek of clarity. The numbers inside the molecular rings are the NICS values of these rings calculated at the HF/6-31+G(d) level with the HF/6-31G optimized geometry

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43

–2.7 21.0 a1 Benzocyclobutadiene C2ν

Figure 7 Schematic representation of the displacement vectors corresponding to the totally symmetric postdiagonalization nuclear distortions that show the most marked anti-GMPP character of benzocyclobutadiene obtained at the HF/6-31G level. The depicted displacement vectors of the non-hydrogen atoms have been multiplied by two in the representation for seek of clarity. The numbers inside the molecular rings are the NICS values of these rings calculated at the HF/6-31+G(d) level with the HF/6-31G optimized geometry

This result is consistent with the NICS values, indicating that ring C is more aromatic than B, and this, in turn, more aromatic than A. At the end of the previous section, we have concluded that BLA movements of the benzocyclobutadiene belong to the totally symmetric representation, indicating that the GMPP is obeyed by all nontotally symmetric modes. Thus, with the present methodology, it is not possible to ascertain the most aromatic center in this molecule. As an alternative way, we have investigated whether the diagonalization of with respect to the totally symmetric normal coordinates can provide information that cannot be obtained in this case from the nontotally symmetric vibrations [58]. The totally symmetric distortions at the equilibrium geometry are neither a maximum nor minimum of properties such as , or en . Notwithstanding, their eigenvalues apprise the curvature of the polarizability along these symmetric displacements and indicate whether the equilibrium geometry is near or far from a polarizability maximum or minimum. As can be seen in the Figure 7, the application of this method to the benzocyclobutadiene helps to determine the aromatic character of the different rings. Certainly, it is possible to apply the diagonalization of the polarizability Hessian along totally symmetric modes to the rest of the molecules, although the results obtained do not provide much more information than those derived from the diagonalization along nontotally symmetric modes.

8. Conclusions In this chapter, we have presented an overview of our research on the breakdown of the maximum hardness and minimum polarizability principles in nontotally symmetric vibrations. Although these nuclear displacements hold the most favorable conditions for the fulfillment of these two principles, it has been shown that there are a number of aromatic, -conjugated, non--conjugated, or even non--bonded organic and inorganic

44


molecules that possess nontotally symmetric molecular distortions that do not follow these two principles. For aromatic and -conjugated molecules, we have devised a set of rules to predict without calculations the existence of vibrational modes that disobey the minimum polarizability principle. We have also proved that the diagonalization of the polarizability and hardness Hessian matrices facilitates the analysis of the breakdown of these two principles in nontotally symmetric nuclear motions. In addition, it has been found that the polarizability and the softness (the inverse of the hardness) are not always proportional, and, consequently, there are vibrational modes that follow or disobey only one of the two principles. We have also discussed the relationship between the breakdown of the maximum hardness principle and the PJT effect, showing that vibrational modes that suffer a large pseudo-Jahn–Teller effect are good candidates for disobeying this principle. Finally, we have provided exhaustive evidence about the breakdown of the minimum polarizability principle as a tool to predict the most aromatic ring in a given polycyclic aromatic molecule.

Acknowledgments Support for this work from the Spanish Ministerio de Ciencia y Tecnología (projects CTQ2005-08797-C02-01 and BQU2002-03334) and from the DURSI (Generalitat de Catalunya) (project 2005SGR-00238) and the use of the computational facilities of the Catalonia Supercomputer Center (CESCA) are gratefully acknowledged. M.S. thanks the DURSI for financial support through the Distinguished University Research Promotion, 2001.

References 1. RG Parr, W Yang: Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. 2. W Koch, MC Holthausen: Chemist’s Guide to Density Functional Theory, Wiley-VCH, Weinheim, 2000. 3. T Ziegler, Chem. Rev. 91 (1991) 651. 4. M Torrent, M Solà, G Frenking, Chem. Rev. 100 (2000) 439. 5. P Hohenberg, W Kohn, Phys. Rev. B 136 (1964) 864. 6. W Kohn, Rev. Mod. Phys. 71 (1999) 1253. 7. H Chermette, J. Comput. Chem. 20 (1999) 129. 8. P Geerlings, F De Proft, W Langenaeker, Chem. Rev. 103 (2003) 1793. 9. RG Pearson: Chemical Hardness: Applications from Molecules to Solids, Wiley-VCH, Oxford, 1997. 10. T Koopmans, Physica (Utrecht) 1 (1934) 104. 11. RT Sanderson, Science 121 (1955) 207. 12. RG Pearson, J. Chem. Educ. 76 (1999) 267. 13. RG Pearson, J. Chem. Educ. 64 (1987) 561. 14. PK Chattaraj, S Sengupta, J. Phys. Chem. 100 (1996) 16126. 15. P Politzer, J. Chem. Phys. 86 (1987) 1072. 16. A Vela, JL Gázquez, J. Am. Chem. Soc. 112 (1990) 1490. 17. TK Ghanty, SK Ghosh, J. Phys. Chem. 97 (1993) 4951. 18. P Fuentealba, O Reyes, J. Mol. Struct.: THEOCHEM 101 (1993) 65.

M. Solà et al. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58.

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P Fuentealba, Y Simón-Manso, J. Phys. Chem. A 101 (1997) 4231. RG Pearson, WE Palke, J. Phys. Chem. 96 (1992) 3283. G Makov, J. Phys. Chem. 99 (1995) 9337. S Pal, N Vaval, R Roy, J. Phys. Chem. 97 (1993) 4404. PK Chattaraj, P Fuentealba, P Jaque, A Toro-Labbé, J. Phys. Chem. A 103 (1999) 9307. S Gutiérrez-Oliva, JR Letelier, A Toro-Labbé, Mol. Phys. 96 (1999) 61. T Uchimaru, AK Chandra, S Kawahara, K Matsumura, S Tsuzuki, M Mikami, J. Phys. Chem. A 105 (2001) 1343. R Parthasarathi, J Padmanabhan, V Subramanian, B Maiti, PK Chattaraj, J. Phys. Chem. A 107 (2003) 10346. J Cadet, A Grand, C Morell, JR Letelier, JL Moncada, A Toro-Labbé, J. Phys. Chem. A 107 (2003) 5334. PK Chattaraj, S Gutiérrez-Oliva, P Jaque, A Toro-Labbé, Mol. Phys. 101 (2003) 2841. S Gutiérrez-Oliva, A Toro-Labbé, Chem. Phys. Lett. 383 (2004) 435. PK Chattaraj, A Poddar, J. Phys. Chem. A 103 (1999) 1274. P Fuentealba, Y Simón-Manso, PK Chattaraj, J. Phys. Chem. A 104 (2000) 3185. ZX Zhou, RG Parr, J. Am. Chem. Soc. 111 (1989) 7371. P Jaque, A Toro-Labbé, J. Phys. Chem. A 104 (2000) 995. D Datta, J. Phys. Chem. 96 (1992) 2409. T Kar, S Scheiner, J. Phys. Chem. 99 (1995) 8121. PK Chattaraj, A Cedillo, RG Parr, EM Arnett, J. Org. Chem. 60 (1995) 4707. TK Ghanty, SK Ghosh, J. Phys. Chem. 100 (1996) 12295. A Toro-Labbé, J. Phys. Chem. A 103 (1999) 4398. PK Chattaraj, P Fuentealba, B Gómez, R Contreras, J. Am. Chem. Soc. 122 (2000) 348. U Hohm, J. Phys. Chem. A 104 (2000) 8418. P Jaque, A Toro-Labbé, J. Chem. Phys. 117 (2002) 3208. TK Ghanty, SK Ghosh, J. Phys. Chem. A 106 (2002) 4200. RG Parr, PK Chattaraj, J. Am. Chem. Soc. 113 (1991) 1854. E Sicilia, N Russo, T Mineva, J. Phys. Chem. A 105 (2001) 442. T Kar, S Scheiner, AB Sannigrahi, J. Mol. Struct.:THEOCHEM 427 (1998) 79. M Solà, A Toro-Labbé, J. Phys. Chem. A 103 (1999) 8847. LT Nguyen, TN Le, F De Proft, AK Chandra, W Langenaeker, MT Nguyen, P Geerlings, J. Am. Chem. Soc. 121 (1999) 5992. B Gómez, PK Chattaraj, E Chamorro, R Contreras, P Fuentealba, J. Phys. Chem. A 106 (2002) 11227. B Gómez, P Fuentealba, R Contreras, Theor. Chem. Acc. 110 (2003) 421. LT Nguyen, F De Proft, MT Nguyen, P Geerlings, J. Chem. Soc., Perkin Trans. 2 (2001) 898. PW Ayers, RG Parr, J. Am. Chem. Soc. 122 (2000) 2010. M Torrent-Sucarrat, JM Luis, M Duran, M Solà, J. Am. Chem. Soc. 123 (2001) 7951. M Torrent-Sucarrat, JM Luis, M Duran, M Solà, J. Chem. Phys. 117 (2002) 10561. M Torrent-Sucarrat, M Duran, JM Luis, M Solà, J. Phys. Chem. A 109 (2005) 615. RG Pearson, J. Am. Chem. Soc. 91 (1969) 4947. MJ Bearpark, L Blancafort, MA Robb, Mol. Phys. 100 (2002) 1735. L Blancafort, M Torrent-Sucarrat, JM Luis, M Duran, M Solà, J. Phys. Chem. A 38 (2003) 7337. M Torrent-Sucarrat, JM Luis, M Solà, Chem. Eur. J. 11 (2005), 6024.



Chapter 4

Classification of control space parameters for topological studies of reactivity and chemical reactions a

Bernard Silvi, a Isabelle Fourré, and b Mohammad Esmail Alikhani

a

Laboratoire de Chimie Théorique (UMR-CNRS 7616) and Laboratoire de Dynamique, Interactions et Réactivité (UMR-CNRS7075), Université Pierre et Marie Curie, 4 Place Jussieu 75252-Paris cedex, France

b

Abstract The gradient dynamical system and the catastrophe theories are two very useful and complementary mathematical tools for the study of the energetic and mechanisms of chemical reactions. We propose a classification of the potential functions and of the control space parameters. It emerges that the structural stability is a central concept for the understanding of chemical reactions and of chemical reactivity.

1. Introduction Most of recent ideas on the analysis of potential energy hypersurfaces have their origin in 1970 Fukui’s paper [1] on the formulation of the reaction coordinate. In this article, Fukui introduced the concept of intrinsic reaction coordinate defined as “a curve passing through the initial and transition points and orthogonal to energy equipotential contour surface.” Accordingly, he emphasized the gradient of the Born–Oppenheimer potential energy and the related mathematical tools enabling its analysis. During the same decade, the use of orthogonal trajectories to describe the electron density distribution was pioneered by Richard Bader [2]. In 1977, Collard and Hall [3] explicitly identified the dynamical system theory as the relevant mathematical framework for such studies. They defined the intrinsic reaction coordinate as “the union of two orthogonal trajectory segments from the saddle point.” They also pointed out the possibility of catastrophe situations to occur on reaction surfaces. The growing importance of differential geometry 47

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Classification of control space parameters

for the study of potential energy surfaces is testified by a series of papers published by Tachibana and Fukui [4–6] and by Mezey [7–11]. More recently, Margalef–Roig et al. [12] have investigated the profiles of global properties such as energy, hardness, and chemical potential with respect to the reduced reaction coordinate [13–16] and have shown that there is a local bifurcation condition in the parameter space which corresponds to a bifurcation catastrophe. It is then possible to identify topological regions within which chemical processes follow or not the Hammond postulate [17] or the principle of maximum hardness [18]. Another interesting paper discussing the evolution of energy hypersurfaces in terms of catastrophe theory concepts has been published by Wales [19]. This article shows that the Hammond postulate can be considered as a consequence of the fold catastrophe properties. The characterization of reaction mechanisms is another fruitful application of Thom’s catastrophe theory [20] initiated by Richard Bader and his co-workers. The Atoms in Molecules (AIM) analysis enables a rigorous approach of molecular structures, thanks to the definition of the molecular graph [21], i.e. the set of the bond paths linking the nuclei of the considered system. The evolution of the molecular graph along the reaction coordinate is achieved by bifurcation catastrophes which account for the changes in the bonding. The fold catastrophe is illustrated by the C2 approach of O1 D to H2 to form a water molecule and corresponds to the breaking of the H–H bond followed by the formation of two O−H bonds, whereas the elliptic umbilic appears to be the signature of a ring formation [21]. This technique has been successfully applied to ring formation and to isomerization [22] but fails to describe some elementary chemical processes such as a bond dissociation. Malcolm and Popelier [23] considered the full topology of the Lr function defined as minus the Laplacian of the electron density r to investigate the inversion of ammonia evidencing three successive catastrophes, an elliptic umbilic, a triple fold, and a triple dual cusp. The topological analysis of the Becke and Edgecombe localization function ELF [24] provides an alternative approach more directly connected with the chemical concepts of bond and lone pairs [25,26] and which therefore nicely accounts for the evolution of the bonding along a reaction coordinate [27]. This method has been applied to a large sample of chemical reactions such as bond formation/dissociation [27], isomerizations [28], electron harpooning [29], proton transfers [30,31], two state reactions [32], the Diels–Alder cycloaddition [33], and the 1-3 dipolar cycloaddition [34]. In the context of the catastrophe theory, the evolution of a system depends upon a set of parameters called control space parameters. The present chapter presents a classification of the control space parameters according to their physical nature.

2. An overview of the mathematical background 2.1. Topological definitions Dynamical system. A dynamical system is a vector field of class C1 bound onto a manifold M. Such a vector field has no discontinuities. To any point m belonging to the M manifold corresponds one vector Xm and only one. M is a subset of q and Xm has q components x1 x2 xq . Moreover, the vector field depends upon control parameters, c , which belong to the set W referred to as the control space of dimension k. In the case of applications related to physics or chemistry, such parameters may be

B. Silvi et al.

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related to internal characteristics of the system or to external constraints (i.e. electric or magnetic field, external pressure, and temperature). The solutions of the system of equations dm/dt = Xm are locally unique, and therefore there is only one trajectory passing through m. The trajectories are determined by integrating dm/dt = Xm with respect to the fictitious time variable t. The limit sets of M of mt for t → ± are called the and limit set. Gradient dynamical system. The vector field of a gradient dynamical system is the gradient of a function called potential function, i.e. Xm = Vxi c , where xi implicitly denotes the set of the q variables of q defining the point m of the manifold M and where c stands for the control space parameters. Critical points. The critical points (or limit points) of a dynamical system are the points of M for which Xmc = 0. A critical point is either an or an limit of a trajectory. The subset of points of M by which are built trajectories having mc as limit is called the stable manifold of mc ; the unstable manifold of mc is the set for which mc is an limit. The dimension of the unstable manifold is the index of the critical point. The set of the critical points of a dynamical system satisfies the Poincaré–Hopf formula: −1IP = M P

where IP is the index of the critical point P and M the Euler characteristic of the manifold. A critical point of index zero is an attractor of the dynamical system. The stable manifold of an attractor is called the basin of the attractor. The stable manifold of a critical point of index greater than zero is a separatrix, it is the border of two or more basins. The index of a critical point mc of a gradient dynamical system is the number of positive eigenvalues of the matrix of the second derivatives of the potential function at mc . In this case, a critical point is said to be hyperbolic if none of the eigenvalues is zero.

2.2. Elementary catastrophe theory Up to now, the quantities c have been implicitly considered as constants. Elementary Catastrophe Theory studies how a gradient dynamical system changes as the control parameters c changes in the special case where k = dimW ≤ 5. In this context, the evolution of the equilibria can be studied by considering the behavior of the Hessian matrix Hij = Vxk c /xi xj of Vxi c . If det Hij c x=xs = 0 then it is said that the critical point is hyperbolic; in the other case it is called non-hyperbolic. The configuration of the control parameters c∗ for which detHij c∗ x=x s = 0 is called bifurcation point. The set of c for which the Hessian matrix of a given critical point is nonzero defines the domain of stability of the critical point. A small perturbation of s Vxi c∗ brings the system from a domain of stability to another. If none of the critical points of the system changes then is located in a domain of structural stability. Thom’s theorem states that in the neighborhood of xs c∗ after a smooth change of the variables, the potential can be written as: n V xi c = u x1 x1 c + i c xi2 (1) i=l+1

The symbol = means equal after a smooth change of variables.

50

Classification of control space parameters Table 1 R. Thom’s nomenclature of elementary catastrophes Name

Fold Cusp Swallow tail Hyperbolic umbilic Elliptic umbilic Butterfly Parabolic umbilic

Codimension

Corank

1 2 3 3 3 4 4

1 1 1 2 2 1 2

Universal unfolding x3 + ux x4 + ux2 + vx x5 + ux3 + vx2 + wx x3 + y3 + uxy + vx + wy x3 − xy2 + ux2 + y2 + vx + wy x6 + ux4 + vx3 + wx2 + tx x2 y + y4 + ux2 + vy2 + wx + ty

In this equation, ux1 x1 c is the universal unfolding of the singularity, it is a polynomial function of degree higher than 2 of a “canonical” form depending upon the l variables with zero eigenvalues, l is called the corank, the i ’s are the n − l nonzero eigenvalues. The unfolding contains all the information about how Vxi c may change as the control parameters change. Thom has classified these universal unfoldings according to their corank and to the dimension of the control space W called the codimension. Thom’s classification is reported in Table 1.

3. Applications to chemical reactions and reactivity At a microscopic level, two kinds of functions can be derived from quantum mechanics. On the one hand are those corresponding to global properties such as the energy, expectation values of operators, chemical potential, or global hardness as defined within the conceptual DFT context [35] and on the other hand are local properties such as electron density distributions, the electron localization function ELF [24], local hardness [36,37], and the Fukui functions [38,39].

3.1. Global and local functions A global property function is usually expressed as the expectation value of an operator or as the derivative of such an expectation value with respect to an internal or external parameter of the system. In the Born–Oppenheimer approximation, the electronic wave function depends parametrically upon the coordinates of the n nuclei, and therefore a set of the 3n–6 linearly independent nuclear coordinates constitutes the natural variables for such a choice of the potential function. However, the manifold M on which the gradient vector field is bound can be defined on a subset of q provided q ≤ 3n–6, for example the intrinsic reaction coordinate (unstable manifold of a saddle point of index 1 of the Born–Oppenheimer energy hypersurface) or the reduced reaction coordinate.

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The local functions depend upon at most 3N coordinates of the electron configuration space. This number can be reduced to 3k by integration over the positions of N –k electrons. For example, in the case of spinless reduced density distribution functions [40]: N! k r1 rk = · · · x1 xk xk+1 xN N − k! ∗ x1 xk xk+1 xN d1 dk dxk+1 dxN In this equation, xi denotes the space ri and spin i coordinates of the electron labeled by i. As other examples of local functions, we can mention the local softness [41], the local hardness [36], the Fukui function [38], the Becke and Edgecombe electron localization function ELF [24], the spin pair composition [42], or the electron localizability indicator of Kohout et al. [43].

3.2. Classification of control space parameters Global and local functions depend upon a very large and generally indefinite number of parameters which may be quantifiable or not and which may account in principle for anything. The study of the evolution of the dynamical system upon variations of its control space parameters intends to answer two questions: • Is the dynamical system structurally stable with respect to a given perturbation? Or in other words, do two different values of control space parameters belong to the same domain of structural stability? • Which type of catastrophe might be expected and which parameters are responsible for it? An explicit continuous variation of the control space parameters, which implies that these latter may be expressed in terms of real numbers, is only required for the determination of the boundaries of the structural stability domains. In such cases, some of the parameters may take nonphysical values when varied, and the actual calculations of the potential function have to be achieved by “adiabatic connection”-like techniques. The classification proposed here is based on the interpretation of the results rather than on the mathematical properties of the parameters, if any. It consists of three main classes: the “internal,” the “external,” and the “methodological” classes of parameters. The “internal” parameters are those which describe the system itself, isolated or not, i.e. • its chemical composition given by the set of the atomic numbers ZA , or less precisely by the nature of substituents, • the electronic state determined by the number of electron N and the spin operators quantum numbers S and MS , • the set of the nuclear coordinates RA in the case of local functions. The “external” parameters describe the interaction of the system with the surrounding such as the solvent effect which can be accounted for by reaction field methods, the pressure, or the temperature. Finally, the “methodological” class of parameters address the reliability of a given level of calculation or of a given model.

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3.3. Examples of applications The following examples consider both global and local functions as well as both quantifiable and nonquantifiable parameters and are organized according to the classification described above. The first case we want to mention is the structural stability of the ELF gradient field with respect to the chemical composition in the second period 10 electrons hydrides AHn CH4 NH3 H2 O and HF. Figure 1 displays the localization domains of these molecules. The three former molecules have one core basin C(A) and four valence basins which are either monosynatic V(A) or protonated disynaptic V(A, H), hydrogen fluoride has two valence basins instead of four. The ELF gradient field is therefore structurally stable within a rather large stability domain which encompasses CH4 NH3 , and H2 O; it becomes unstable when the system passes to a continuous group symmetry because the ELF function is totally symmetric. The structural stability of the ELF gradient field of N -electron systems with respect to the composition supports the “least topological change” rule [44] which enables to predict the favored protonation site of a base.

Figure 1 Localization domains of CH4 NH3 H2 O, and HF

The rotational barrier of glyoxal and oxalyl halides has been investigated by Margalef– Roig et al. [12] who have shown that a change of substituent yields a bifurcation.

X

O X

O

O C C

C C O

X

X

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Energy (kJ)

12.00

8.00

4.00

0.00 0

π/2

π Torsional angle

3π/2

2π

Figure 2 Energy profiles along the torsional angle of oxalyl fluoride (- -), oxalyl chloride − and oxalyl bromide · · ·

Figure 2 displays the energy profiles of the transcis rotational isomerization of oxalyl fluoride, oxalyl chloride, and oxalyl bromide. For all molecules the cis conformation is the absolute minimum, whereas the trans conformation is either a secondary minimum (oxalyl fluoride) or a maximum (oxalyl chloride and oxalyl bromide). Oxalyl bromide presents a shallow minimum corresponding to a gauche conformer. Substituting F by Cl yields a cusp catastrophe which changes the two maxima at ±/2 and the minimum at 0 into a maximum at 0. The substitution of Cl by Br is responsible for a dual-fold catastrophe in which two wandering points near ±2/3 give rise to a new minimum (gauche conformation) and a new maximum. The characterization of chemical reaction mechanisms by catastrophe theory in the AIM and ELF frameworks has a common methodological background. Technically, the AIM or ELF analysis is performed for a series of structures calculated along the reaction path by the IRC method, the turning points between structural stability domains are then located and the catastrophe identified when the two successive domains belong to the same Born–Oppenheimer energy surface. Only three elementary catastrophes have been recognized so far in the chemical reactions: the fold, cusp, and elliptic umbilic catastrophe. The fold catastrophe transforms a wandering point (i.e. a point which is not a critical one) into two critical points of different parity. Its unfolding is x3 + ux x is the direction of the eigenvector corresponding to the eigenvalue of the Hessian matrix which changes sign and u is the control space parameters which governs the discontinuity. For u > 0, the first derivative is positive for all x, the catastrophe takes place at u = 0 for which both first and second derivatives are zero, and for u < 0 there are two critical points at x = ± u3 . The proton-transfer reaction AH + B A− + BH+ [30,31,45] involves two successive fold catastrophes: the first corresponds to the A–H bond breaking and

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the second to the formation of the B–H bond. The cusp catastrophe transforms a critical point of a given parity into two critical points of the same parity and one of the opposite parity. It occurs in the dissociation of a covalent bond into neutral moieties. Finally, the elliptic umbilic catastrophe changes the index of one critical point by 2. In the ELF framework, elliptic umbilic catastrophes occur when the number of basin remaining constant the synaptic order of a valence basin changes as it is the case for dative bond dissociations [27]. The changes of potential energy curves with respect to surrounding effects have been investigated by Tapia et al. in the 1970s [46], who presented the proton potential curves of the water dimer for different solute–solvent coupling parameters. Below a critical value of this parameter, there is only one minimum corresponding to the hydrogen bonded complex, above this value a second minimum appears which accounts for the formation of an ionic pair. More recently, Gómez and Pacios have reported similar findings for the 4-methyl-imidazole-aspartate complex. Figure 3 displays the proton potential energy profiles of the HCl H2 O complex calculated by the Onsager method for different values of the dielectric constant. A fold catastrophe takes for ∼ 50 in which a wandering point yields a minimum (close to rCl–H = 22 Å) and a maximum at rCl–H = 18 Å. Another example is provided by pressure-induced phase transitions in which a cusp catastrophe transforms a double-well of the potential energy curve into a single-well and vice versa. In a wide pressure range, between 10 and 100 GPa, stishovite is the stable modification of silica. At about 100 GPa, a phase transition from P42 /mnm (rutile) to PmnmCaCl2 occurs which corresponds to the twinning of the initial tetragonal cell

–537.10

+

−537.13 +

−537.15 +

−537.18 Energy (a. u.)

+

+

−537.20

+

+

= 0.0

+

+

+

+

−537.23

+

+ +

+

+ +

−537.25

+ +

= 5.0

+

+

+

+

+

−537.28

+

+ + +

+

+

+

+

+

+ +

+ + + + +

+

+ + +

+

+

+ +

+

+

+

+ + +

+

−537.30

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= 50.0

+ + +

+

+ +

+ + +

+

+ + + +

+

+ + +

−537.33 1.20

+

1.60

2.00

2.40

r (Cl−H)

Figure 3 Energy profiles for proton transfer in the ClH H2 O complex. The effect of environment is accounted for by the Onsager model with different values of the solvent dielectric constant

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into two equivalent orthorhombic cells, one with a > b and the other with b > a. This phase transition has been studied at the periodic Hartree–Fock level by Jolly et al. [47], who have shown that above 100 GPa the enthalpy curve becomes double-welled. Alternatively, the increase of pressure may lower the barrier of a double-well potential as it is the case for ice VIII [48].

4. Conclusion In this chapter, we have tried to convince the reader of the usefulness of the dynamical system theory for chemical reactivity studies. Indeed, it is possible to predict which changes may be achieved when internal, external, or methodological parameters are varied from the shape of energy surface or from the topologies of local functions. The structural stability of the gradient vector fields of global and local functions describing chemical systems appears to be an important concept which has to be considered to understand the reactivity. Moreover, the application of the catastrophe theory to chemical reactions enables the description of the mechanisms [27–34,49–52].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Fukui, K.; J. Phys. Chem. 1970; 74, 4161. Bader, R. F. W.; Acc. Chem. Res. 1975; 8, 34. Collard, K.; Hall, G. G.; Int. J. Quant. Chem. 1977; 12, 623. Tachibana, A.; Fukui, K.; Theor. Chim. Acta (Berlin) 1978; 49, 321. Tachibana, A.; Fukui, K.; Theor. Chim. Acta (Berlin) 1979; 51, 189. Tachibana, A.; Fukui, K.; Theor. Chim. Acta (Berlin) 1979; 51, 275. Mezey, P. G.; Theor. Chim. Acta (Berlin) 1981; 58, 309. Mezey, P. G.; Theor. Chim. Acta (Berlin) 1981; 60, 97. Mezey, P. G.; Theor. Chim. Acta (Berlin) 1982; 62, 133. Mezey, P. G.; Theor. Chim. Acta (Berlin) 1983; 63, 9. Mezey, P. G.; Potential Energy Hypersurfaces; Elsevier, Amsterdam 1987. Margalef-Roig, J.; Miret-Artés, S.; Toro-Labbé, A.; J. Phys. Chem. A 2000; 104, 11589. Cardenas-Jiron, G. I.; Lahsen, J.; Toro-Labbé, A.; J. Phys. Chem. 1995; 99, 5325. Cardenas-Jiron, G. I.; Toro-Labbé, A.; J. Phys. Chem. 1995; 99, 12730. Cardenas-Jiron, G. I.; Gutierrez-Oliva, S.; Melin, J.; Toro-Labbé, A.; J. Phys. Chem. A 1997; 101, 4621. Toro-Labbé, A.; J. Phys. Chem. A 1999; 103, 4398. Hammond, G. S.; J. Am. Chem. Soc. 1953; 77, 334. Pearson, R. G.; Chemical Hardness Applications from Molecules to Solids; Wiley-VCH, Weinheim 1997. Wales, D. J.; Science 2001; 293, 2067. Thom, R.; Stabilité Structurelle et Morphogénèse; Intereditions, Paris 1972. Bader, R. F. W.; Nguyen-Dang, T. T.; Tal, Y.; J. Chem. Phys. 1979; 70, 4316. Tal, Y.; Bader, R. F. W.; Nguyen-Dang, T. T.; Ojha, M.; Anderson, S. G.; J. Chem. Phys. 1981; 74, 5162. Malcolm, N. O. J.; Popelier, P. L. A.; J. Phys. Chem. A 2001; 105, 7638. Becke, A. D.; Edgecombe, K. E.; J. Chem. Phys. 1990; 92, 5397. Silvi, B.; Savin, A.; Nature 1994; 371, 683.

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26. 27. 28. 29. 30. 31. 32.

Häussermann, U.; Wengert, S.; Nesper, R.; Angew. Chem. Int. Ed. Engl. 1994; 33, 2069. Krokidis, X.; Noury, S.; Silvi, B.; J. Phys. Chem. A 1997; 101, 7277. Krokidis, X.; Silvi, B.; Alikhani, M. E.; Chem. Phys. Lett. 1998; 292, 35. Krokidis, X.; Silvi, B.; Dezarnaud-Dandine, C.; Sevin, A.; New J. Chem. 1998; 22, 1341. Krokidis, X.; Goncalves, V.; Savin, A.; Silvi, B.; J. Phys. Chem. A 1998; 102, 5065. Krokidis, X.; Vuilleumier, R.; Borgis, D.; Silvi, B.; Mol. Phys. 1999; 96, 265. Michelini, M. C.; Sicilia, E.; Russo, N.; Alikhani, M. E.; Silvi, B.; J. Phys. Chem. A 2003; 107, 4862. Berski, S.; Andrés, J.; Silvi, B.; Domingo, L.; J. Phys. Chem. A 2003; 107, 6014. Polo, V.; Andres, J.; Castillo, R.; Berski, S.; Silvi, B.; Chem. Eur. J. 2004; 10, 5165. Geerlings, P.; De Proft, F.; Langenaeker, W.; Chem. Rev. 2003; 103, 1793. Berkowitz, M.; Ghosh, S. K.; Parr, R. G.; J. Am. Chem. Soc. 1985; 107, 6811. Berkowitz, M.; Parr, R. G.; J. Chem. Phys. 1988; 88, 2554. Parr, R. G.; Yang, W.; J. Am. Chem. Soc. 1984; 106, 4049. Ayers, P. W.; Levy, M.; Theor. Chem. Acc. 2000; 103, 353. MacWeeny, R.; Methods of Molecular Quantum Mechanics; Academic Press, London; 2nd edn. 1989. Yang, W.; Parr, R. G.; Proc. Natl. Acad. Sci. U.S.A. 1985; 82, 6723. Silvi, B.; J. Phys. Chem. A 2003; 107, 3081. Kohout, M.; Pernal, K.; Wagner, F. R.; Grin, Y.; Theor. Chem. Acc. 2004; 112, 453. Fuster, F.; Silvi, B.; Chem. Phys. 2000; 252, 279. Alikhani, M. E.; Silvi, B.; J. Mol. Struct. 2004; 706, 3. Taoia, O.; Poulain, E.; Sussman, F.; Chem. Phys. Lett. 1975; 33, 65. Jolly, L.-H.; Silvi, B.; D’Arco, P.; J. Chim. Phys. 1993; 90, 1887. Besson, J. M.; Pruzan, P.; Klotz, S.; Hamel, G.; Silvi, B.; Nelmes, R. J.; Loveday, J. S.; Wilson, R. M.; Hull, S.; Phys. Rev. 1994; B49, 12540. Krokidis, X.; Moriarty, N. W.; Lester, Jr., W. A.; Frenklach, M.; Chem. Phys. Lett. 1999; 314, 534. Michelini, M. C.; Russo, N.; Alikhani, M. E.; Silvi, B.; J. Comp. Chem. 2004; 25, 1647. Santos, J. C.; Polo, V.; Andrés, J.; Chem. Phys. Lett. 2005; 406, 393. Santos, J. C.; Andrés, J.; Aizman, A.; Fuentealba, P.; Polo, V.; J. Phys. Chem. A 2005; 109, 3687.

33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.


Chapter 5

Understanding and using the electron localization function a

Patricio Fuentealba, b E. Chamorro, and b Juan C. Santos

a

Departamento de Física, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Casilla 653 and b Departamento de Ciencias Químicas, Facultad de Ecología y Recursos Naturales, Universidad Andrés Bello, Av. República 275, Santiago, Chile

1. Introduction The applications of quantum mechanics to chemistry have primordially two goals. First, to provide the numerical value of observables which can be confronted with experimental measurements and, second, to help in understanding many empirical concepts widely used in chemistry. The electron localization function, ELF, enters in the second goal. It helps in understanding the empirical concept of electron localization, specially the pair electron localization in the spirit of Lewis structures. This paper is not an attempt to review all of the applications of the ELF, rather the aims are to explain in an easy way with as less as possible of mathematical formalism the significances of the ELF and, more important, how to use it. Hence, from the very beginning, we give answer to a common question of chemists in front of theoretical paper: Why should I bother trying to understand this function, when I do not have any chance to apply it and when I do not have any software to calculate it? Well, in this case, everybody can calculate the ELF using the TOPMOD software developed by Silvi and co-workers1 which is free. It uses the output of popular programs for electronic structure calculation, i.e. the commercial package of programs GAUSSIAN2 and the free software GAMMES3 to obtain the ELF and elaborate the necessary topological analysis. The ELF is also implemented in some other packages designed for periodic systems4 or can also be calculated using the interfacing program DGRID5 for GAUSSIAN,2 MOLPRO6 and ADF.7 A new algorithm to fast calculation of the 3D and 2D arrays needed for the topological analysis of ELF is also available.8 The ELF was introduced by Becke and Edgecombe9 and first applied to a great range of systems, from atoms to molecules and solids, by Savin et al.10 Some years later, Silvi 57

58

Understanding and using the ELF

and Savin11 proposed a topological classification and rationalization of the ELF which helps in giving a quantification of the chemical concepts associated with the function. After that, and with the launch of the TOPMOD1 free software, there have been many applications of the ELF to molecules, clusters and solids. One of the key concepts to understand the significance of the electron localization is the Fermi hole, which is a direct consequence of the Pauli exclusion principle.12 It permits to answer the difficult question pointed out by Lewis13 when he introduced in chemistry the concept of the electron pair. His question is as follows: If the only law of nature acting on the electron’s movement is the Coulomb’s law which is repulsive between a pair of electrons, then how is it possible to find a pair of electrons moving together in a certain region of space? In fact, Lewis postulated the failure of the Coulomb’s law for small distances. Remember that at this time the quantum mechanics was not fully understood, and later, Lewis changed his position. In fact, it is important to notice that, in a rigorous sense, it does not make any sense to talk about a localized electron because it goes against the Heisenberg’s uncertainty principle. A nice explanation to all of this is given by Bader,14 and we follow it. The existence of a localized electron pair implies that there exists a high probability of finding two electrons of opposite spin in a given region of the space and for which there is a small probability of exchange with other electrons that are outside of this region. To understand this, it is necessary to think that as an electron moves through space, it moves also with its Fermi hole. The Fermi hole related to a given electron in a given position in the space is a distribution function which measures the decrease due to the Pauli principle in the probability of finding another electron of the same spin at some position in the space. Hence, it depends on the position of the reference electron, say r1 , and on the position of the other electron, say r2 . Therefore, for a given spin, the diagonal of the Fermi hole, r1 = r2 , should be equal to the total density for the given spin at this point. This is the only way to be sure that the probability of finding another electron of the same spin at this position is zero, condition necessary to respect the Pauli principle. For the same reason, for a fixed r1 , the position of the reference electron, the integration of the Fermi hole over r2 corresponds to the removal of one electron of identical spin. Hence, the Fermi hole describes the spatial delocalization of the charge of the reference electron. As written by Bader, ‘an electron can go only where its hole goes and, if the Fermi hole is localized, then so is the electron’. One example: an electron of a given spin moving near a nucleus. There is a big attractive potential acting on the electron, and the potential well to get outside this region is also big. Its Fermi hole is strongly localized around this region of the space. Suppose now that the Fermi hole is so localized on this region that everywhere on this region it equals its maximum value, the total density of the given spin, then, all other electrons of this spin are completely excluded from this region. The same result would be obtained for an electron of the other spin if it happens to be in the same region of the space. Hence, this pair of electrons of different spin is confined to move on this particular region of the space in the vicinity of a nucleus where all the other electrons irrespective of its spin are excluded. Since repulsions between the electrons act in opposition to this effect, the localization can never be perfect. It is important to notice that the Fermi hole does not produce an attraction between a pair of electrons of different spin, it rather produces an extra repulsion between the pair of electrons and all the rest. In general, this argumentation is correct for the inner 1s2 pair of electrons, and may be for the core electrons of any system, but it is clearly difficult to apply to the

P. Fuentealba et al.

59

valence electrons which are weakly bonded. Therefore, it is important to have in mind that the image of localized electrons in a bond or as a lone pair are only good models in order to understand the chemistry, but they do not have any physical realization. In the next section, the principal ingredients involved in the ELF will be explained, and their relation with chemical concepts will be clarified. Then, a brief comparison of the ELF with other theoretical related tools, like the atoms in molecules model of Bader, will be done. Next, some elementary concepts from the mathematical theory of topological analysis will be in a rather crude way presented. After that, some applications, extensions and results will be discussed, focusing in particular on applications developed at our group.

2. ELF developments 2.1. Becke’s proposal and interpretation In studying the correlation among electrons, it was very early realized that because of the Pauli principle the movement of electrons of the same spin is strongly correlated than the one between electrons of different spin. Therefore, it seems convenient to study the electron pair density for electrons of the same spin and for electrons of different spin, separately. The electron pair density, r1 r2 , gives us the probability of finding an 2 electron of spin at point r1 when a second electron of spin is located at point r2 . Because the electron–electron interaction depends only on the distance between the electrons and not on the angular orientation, it is convenient to change the coordinate system to the ones defined by r = ½r1 + r2 and s = r1 − r2 . The advantage of the new coordinate system is that now the electron interaction does not depend on six variables (r1 and r2 ) but only on four (r and s). Hence, only the spherical average electron pair density is necessary. This is defined as

r1 s = 2

1 2 r1 sds 4

(1)

where the integration is over the angles of the vector s. Now, Becke and Edgecombe2 preferred to work with the conditional pair density for electrons of the same spin which is a measure of the conditional probability of finding an electron at position r2 when with certainty there is an electron of the same spin at position r1 . It is given by P r s =

r s 2av r

(2)

where r is the electron density of electrons with spin which in the Kohn–Sham approximation is given by r =

i r 2

(3)

i

with the Kohn–Sham orbitals given by the set i , and the sum is over all the occupied orbitals with spin .

60


Then, they proposed to examine the Taylor series expansion of the spherical average conditional pair density in the vicinity of the point s = 0 where one is measuring the short-range behaviour of the electron at point r2 approaching the reference point r1 . The leading term of the Taylor series expansion is given by 1 2 2 1 s + · · · · · P r s = − 3 4

(4)

where is the positive definite kinetic energy density defined by =

i 2

(5)

i

After Becke and Edgecombe, the Taylor expansion contains all the electron localization information. The smaller the probability of finding the second electron near the point r the more localized is the reference electron. Hence, electron localization is directly related to the bracket enclosed in the right hand side of (4) D = −

1 2 4

(6)

which is a non-negative quantity.15 One can also easily demonstrate that D vanishes for the hydrogen atom and also for the helium atom in the Hartree–Fock (HF) approximation, and it is also expected to be negligible in the regions near the nuclei, where one finds the most localized electrons. Therefore, it is reasonable to expect that the quantity D will be small in the regions of the space where the probability of finding a localized electron or a localized pair of electrons is high. However, the function D can have very high values in other places, and one does not know how near to zero should be to consider an electron to be localized. Hence, Becke and Edgecombe proposed two additional scaling rules. The homogeneous electron gas, the kinetic energy density of which is given by D0 = cf 5/3 , is used as a reference, and for numerical convenience the function is mapped to one which is defined between zero and one. Hence, they proposed the following ELF: ELF = 1 + 2 −1

(7)

= D /D0

(8)

where

In this way, the following inequality is obeyed 0 ≤ ELF ≤ 1

(9)

and a value of the ELF close to one corresponds to a region of the space where there is a high probability of finding electron localization, whereas an ELF value close to one-half corresponds to a region of electron gas-like behaviour. The so defined function is independent of any unitary transformation of the orbitals, and, in principle, it is


61

derivable from the electron density. In fact, it is only necessary to have ones of the currently used procedures to obtain the Kohn–Sham potential from the density to get from the potential, via the Kohn–Sham equations, the necessary orbitals to calculate the function as it was done by Kohout and Savin.16 Moreover, the function can be, in the same indirect way, computed from the experimentally obtained density. There have been approximate determinations of the ELF using electron densities derived from X-ray diffraction data,17 and also slight modifications of the ELF to bypass the use of orbitals. They have also the interesting property of relating the ELF to other physical concepts. The first one18 proposed to use the sum rule for the exchange hole to model directly the pair conditional probability. Other used an approximate kinetic energy functional to relate the ELF to the electrostatic potential19 and, finally, the ELF has also been related to the concept of electronic temperature.20 This connection between local temperature and electron localization has been recently reviewed.21 We should also mention the recently introduced electron localizability indicator (ELI),22 based on a functional of the same-spin pair density which is related to the ELF within an HF approximation, and it differs from the ELF in the case of correlated wave functions.

2.2. Savin’s interpretation The so presented ELF is mainly based on an interpretation of the conditional pair probability density for electrons of the same spin. A conceptually different interpretation was put forward by Savin et al.23 who realized that the term D could be generalized for any density independent of the spin as D=

1 1 2 i 2 − 2 i 8

(10)

One can easily verify that for a closed-shell system, both expressions are the same. The main point now is to recognize the second term of the right hand side of the last equation as the von Weizsacker kinetic energy density,24 which is exact for the hydrogen atom, exact for the helium atom in the HF approximation and exact for a system of bosons. Remember that boson particles can occupy all of them the same quantum state being in this case the perfect localization. The first term of the right hand side represents the kinetic energy density of the molecule under study. Hence, in the region of the space where there is a high probability of finding a localized electron pair, the von Weizsacker kinetic energy density will be a good model, and the function D will have a value near zero. Within this interpretation, the ELF is formally a measure of the excess of kinetic energy density due to the Pauli exclusion principle. Now, it should be clear that the concept of electron pair localization is nothing else but a manifestation of the Pauli exclusion principle.

2.3. Numerical stability One important characteristic of the ELF is its numerical stability with respect to the theoretical level at which the electron density and the molecular orbitals have been

62


C(O)

V1(O)

V(H1, O)

V(H2, O)

V2(O)

B3LYP

HF

STO-3G

AM1

Figure 1 H2 O isosurfaces calculated at different levels of theory

calculated. Contrary to most of the population analysis that depend seriously on the basis set used or the method of calculation (HF, BLYP, B3LYP, etc.), the ELF is rather independent of these changes. As an example, one can see in Figure 1 an isosurface of the ELF for the water molecule calculated at different levels, from AM1 semiempirical method to some of the currently used functionals with a relative big basis set 6-311 + +G∗∗ . The qualitative view of the picture is in all cases the same. Hence, the chemical interpretation of the ELF is also the same. Indeed, there are a series of works where the ELF has been calculated, specially for solids, using en extended Hückel method.2526 This good property of the ELF can be at first glance explained looking at the qualitative behaviour of the total electron density, which is also very independent of the level of calculation. Of course the numerical values can be different, so are the expectation values based on the density. But the maxima and minima are all almost at the same positions. More precise, all the critical points are almost at the same positions. Next, it will be shown that mathematically this means that the function is topologically invariant to the level of calculation. Burdett and McCormick gave a more precise explanation to this invariance.27 They concluded that the ELF is primordially based on the nodal properties of the occupied orbitals of the system. Hence, what matters is the symmetry of the orbitals which is independent of the level of calculation. According to this view, electrons are localized in regions of the space where there are significant electron density and few nodes from all of the occupied orbitals. The number of nodes is also important in order to understand the ELF for transition metals, as it will be shown later.

2.4. Analogies and differences with the atoms in molecules (AIM) model The atoms in molecules (AIM) model of a molecule proposed by Bader14 is completely based on the properties of the electron density. The maxima and minima of the density are used to define a volume in the space, which can be associated with a particular atom in a molecule. Notice that such atoms in a molecule are clearly not spherical, and they extend over all the space with sharp boundaries between two atoms. The information about the zones of depletion or accumulation of charge is extracted from the Laplacian of the density, 2 r . The Laplacian of a function determines where the function will locally augment its value, 2 r < 0, and where it will decrease its value, 2 r > 0. This cannot be made using just the density because the density is a monotonically decreasing function. Hence, the regions of the space where 2 r < 0


63

represent the regions of charge concentration. Bader preferred to work with the negative of the Laplacian, − 2 r , so that a maximum of this function denotes a position at which the electron density is maximally concentrated. It has been found that the local maxima of the function correspond very well to the regions of bonding according to the VSEPR model of Gillespie.28 Hence, the regions of charge concentration defined by the Laplacian of the electron density correspond to the regions of space dominated by the presence of a single pair of electrons. In this sense, one can think that both functions, the ELF and the negative of the Laplacian of the electron density, have the same information. Bader et al.29 have done an extensive study of the topological characteristics of both functions. They found that, in general, both functions are homeomorphic, it means that in all the studied cases both functions present the same number of maxima and minima and they are located almost on the same regions. However, there are systematic differences. In all cases, the radial distance from a nucleus at which the maxima of the ELF is found is always greater than in the negative of the Laplacian of the electron density, and, more important, for a single covalent bond the ELF presents a clear region of the space between both atoms with the only exception of bonds to hydrogen, whereas the negative of the Laplacian of the electron density yields to separate regions associated with each of the participating atoms, and it does not give a clear visualization of the simple covalent bond. In a later work, Bader and Heard30 concluded that the ELF has no direct relationship to the conditional pair density for same-spin electrons. Hence, it seems reasonable to use both functions as complementary tools. As an example, Llusar et al.31 used both functions to study the metal–metal bond in a series of dimers and complexes, and Chesnut and Bartolotti32 used also both functions to study the aromaticity in some substituted cyclopentadienyl systems. More recent examples include the study of unusual bonding in some Bismuth-bridged binuclear molybdocene complexes,33 the nature of bonding in the Leflunomide and some of its biological active metabolites,34 the reactivity of hydroxyperoxy radical,35 the bonding in the singlet and triplet gas-phase − − 36 the absorption of ion/molecule reactions of NbO− 3 NbO5 , and NbO2 OH2 with O2 , Pd on MgO(0 0 1), -Al2 O3 0 0 0 1 and SiO2 surfaces,37 the bonding interactions metal-carbonyl,38 the gas-phase acidity of some oxyacids39 and the nature of bonding of the three-centre-four-electron bond.40

2.5. Results in atoms As a first example of how the ELF works, some results in atoms are presented. Although in the original paper, Becke and Edgecombe9 presented results for some atoms, it was the work of Kohout and Savin41 which showed all the potential of the ELF in explaining the atomic electron structure. The electrons occupying orbitals with the same principal quantum number define each atomic shell, and the concept is primordial to explain the periodic properties. However, it is not so clear from the theoretical point of view. The electron density alone does not show any shell structure, it decays exponentially. It is necessary to use the radial distribution 4r 2 r to see the atomic shell structure.4243 However, it fails for the heavier atoms. On the other side, the ELF is not only able to show the shell structure but is also able to give the radius of each shell r s and the amount of electrons q s on each shell.20 The maxima of the ELF indicate the shells. They are separated by the minima of the ELF which are the radius of each shell. The number of

64

Understanding and using the ELF Table 1 The ELF shell radii and electron numbers

Li 2 S F 2 P Na 2 S Cl 2 P Cr 7 S Cu 2 S Br 2 P

qk

rk

qL

rL

qM

20 21 22 22 22 22 22

153 034 026 015 010 008 007

10 69 79 79 80 83 85

214 082 047 036 028

10 69 123 172 172

rM

237 217 110

electrons in each shell is calculated by the numerical integration of the electron density between the boundaries of the shell. In Table 1, some representative results are shown. One can see that always the innermost K-shell has around two electrons, the following shell, the L-shell, has almost always around eight electrons with systematic deviations for the heavy atoms. Kohout and Savin41 concluded that the ELF is able to resolve the atomic shell structure for all atoms from Li to Sr, and it also gives for each shell electron numbers close to those given by the periodic table of the elements. Small deviations were also explained due to the influence of core-valence separation, especially when d electrons are present.44 In Figure 2, the ELF curves for some selected atoms are depicted. Here, one can use the spherical symmetry to plot it as a function of only one variable, the distance to the nuclei. It is interesting to note that for the alkaline metal atoms the ELF does not decay to zero when the distance goes to infinity. The same fact occurs for the alkaline-earthmetal atoms and for any spherical system with an outer shell formed only by electrons on s-orbitals. In fact, for the hydrogen atom, the ELF is equal to one everywhere. On the other side, one can perfectly see the shell structure even for rubidium atom. The same is true for the noble gas atoms and, as shown by Kohout and Savin,22 for all the atoms to Sr.

2.6. Topological tools The ELF is a scalar function of three variables, and in order to obtain more information from it, it is necessary to use a mathematical approach called differential topology analysis. This was first done by Silvi and Savin,11 and later on extended by them and co-workers.4546 Unfortunately, one cannot visualize in a global way a three-dimensional function. Usually, one resorts to isosurfaces like the ones in Figure 1, or to contour maps. A three-dimensional function has a richer structure than a one-dimensional function, and their mathematical characterization introduces some new words which are necessary to understand in order to go further. It is the purpose of this section to explain this new terminology in a manner as simpler as possible. Let us begin with a one-dimensional (1D) example, a function fx like the one in Figure 3. The function has three maxima and two minima characterized by the sign of the second derivative. In three dimensions (3D) there are more possibilities, for there are nine second derivatives. Hence, one does not talk about maxima but about attractors. In 1D, the attractors are points, in

65

1.00

1.00

0.75

0.75

ELF

ELF


0.50

Li

Na 0.50

0.25

0.25

0.00

0.00 0

1

2

3

4

5

0

6

2

1

3

1.00

0.75

0.75

ELF

ELF

1.00

K

0.50

5

6

4

5

6

4

5

6

Rb 0.50

0.25

0.25

0.00

0.00 1

0

2

4

3

5

1

0

6

3

2

r [au]

r [au] 1.00

1.00

0.75

0.75

Ar

ELF

Ne 0.50

0.25

0.50

0.25

0.00 0

1

2

3

4

5

0.00

6

0

r [au]

1

2

3

r [au] 1.00

0.75

Kr ELF

ELF

4

r [au]

r [au]

0.50

0.25

0.00 0

1

2

3

4

5

6

r [au]

Figure 2 The atomic shell structure determined by ELF

66

Understanding and using the ELF f (x ) 1

.

Atractor

.

.

f-domain

x0 x x

α

0 A

Basin

B

Figure 3 Localization domains of fx

3D they can be a point, a surface or a volume, so are also the minima. In 1D, every attractor is surrounded by two minima (suppose the limit values at zero and infinity are minima), which encloses a line (see Figure 3); in 3D they are surrounded by a surface that encloses a volume. In 1D, mathematically one characterizes the basins saying that it is the line formed by all the points around the attractor such that the point plus the first derivative is closer to the attractor. In 3D, one follows the gradient of the function saying that the basin is formed by all the points enclosed in the volume formed by the gradient lines ending up in the attractor. One can try to visualize it: look at the 1D function in Figure 3, rotate around the y - axes in 360 and you will have a 2D function and the basin indicates in the Figure 3 will be a surface. One more rotation will produce a 3D function and the basin will be a volume. The concept of isosurface has also its correspondence in 1D. Look in Figure 3 at the points where the function has the value fx = x0 . The lines joining the points are the equivalent to the volume enclosed by an isosurface. They are called the f-domain. In Figure 3 there are three f-domains at fx = x0 . However, if one goes to a lower value of the function, close to fx = 0, one will have only one f-domain. There is an important difference in both cases. In the first case, at fx = x0 , the f-domains enclose only one attractor each, but in the second case, close to fx = 0, the only f-domain encloses the three attractors. The f-domains are said to be reducible when they contain more than one attractor and irreducible when they contain one attractor. The basins formed by the irreducible f-domains have a clear chemical meaning and mathematical characterization. There are basically two types of basins.45 If the basin contains a nucleus (except a proton) it is a core basin (C). Otherwise it is a valence basin (V). Always a valence basin will be connected with at least one core basin. A basin representing a lone electron pair will be connected with only one core basin and its attractor will be called monosynaptic. A basin representing a covalent bond will be connected with two core basins, and its attractor will be called disynaptic. There are also higher synaptic orders. Hydrogen is a particular case because it does not have core electrons. Therefore, it is an exception. The bond of any atom to hydrogen appears as a basin that contains the proton and, in general they have a great volume. Silvi has recently discussed the usefulness of the synaptic order concepts in the context of multicentre bonding analysis.47 For the


67

graphical visualization, it is convenient to have a colour code associated with each type of basin. For example, in the water molecule (Figure 1) there is one core basin for the oxygen K-shell denoted C(O) of violet colour. Two basins containing the protons and associated with the O–H bond are denoted VH1 O and VH2 O and they are blue in colour, and there are two basins corresponding to the lone pairs V1 O and V2 O yellow in colour. In the last example, it is clear that a graphical representation of the ELF is able to give a qualitative picture of the type of bonds in a molecule and the regions of the space where it is possible to find electron pairs. However, this is only qualitative information. To get a more quantitative knowledge about the localization one has to go to the integrated density and other properties.45 Remember that in the atomic case the integration of the electron density between two minima of the ELF yields the number of electrons on each shell. Using the new terminology, we have integrated the electron density over each basin. Hence, in molecules, the integration of the electron density over a basin yields the average number of electrons on this basin. In particular, for a basin labelled A , its average number of electrons or electron population is given by N˜ A = r dr (11) A

Therefore, one expects that this number closely correlates with the empirical chemical knowledge about the number of electrons participating in a bond. For example, for the water molecule, the basin associated with the oxygen core has an average electron population of 2.08 electrons. The basins associated with the oxygen–hydrogen bonds have an average electron population of 1.78 and, finally, the basins associated with the electron lone pairs have an average electron population of 2.18. It is important to remember that this is only an average which has a quantum uncertainty. The uncertainty is given by the variance or fluctuation of the population. It is defined as 2 N˜ A =< N 2 >A − < N >2A

(12)

where N represents the electron number operator and the brackets < >A mean an integration over the volume of the basin. Since N 2 is a bielectronic operator one cannot evaluate the average using only the density. It is necessary to use the second-order density matrix. The quantity 2 was investigated by Bader in the context of the atoms in molecules model48 and by Savin et al. in the context of the ELF.45 The fluctuation 2 is an extensive quantity, and, therefore, it depends on the number of atoms of the system making difficult the comparison among molecules with different number of basins. Hence, it is convenient to introduce the relative fluctuation48 =

2 N˜ N˜

(13)

which is a positive quantity and lower than one. An analysis of the electron delocalization based on the definition of a covariance operator has been recently discussed by Silvi, both in the AIM and in the ELF frameworks.49 Another useful concept of topological theory, which can help in quantifying the information carried on the basins, is the concept of bifurcation. At very low values of the ELF, you have only one f-domain which is reducible (it contains more than one

68

Understanding and using the ELF C(A) xo V(A,B) f (x ) ˜ 0

C(B)

Figure 4 Bifurcation diagram of fx

attractor), and you will look at just one surface containing all the system. Increasing the value of the ELF, at the point of the minima, the basins separate and more than one f-domain appears. This is called a bifurcation, and it is clearly visualized by a bifurcation diagram like the one in Figure 4. This is the bifurcation, diagram of the function depicted in Figure 3. The first bifurcation corresponds to the separation of the disynaptic basin and the core basin of point B and occurs at the global minimum of the function. The second bifurcation appears later on and corresponds to the separation of the central disynaptic basin and the core basin associated with the point A. Hence, the points of bifurcation correspond to the minima of the function. In the case of the ELF, the lower the bifurcation point the more localized are the corresponding basins. This kind of analysis based on bifurcations is connected to the concept of synaptic order previously defined,4749 and it has been applied to the study of electron localization in some simple chemical systems45 as it will be noted below. A recent application showing the usefulness of this type of analysis has been reported by Silvi concerning the bonding nature of the VOx and VOx + x = 1 − 4 molecular systems.50

2.7. Other developments Topological analysis of the ELF constructed from density components has also been evaluated. Separations of the – spin4151 and the – 52 electron contributions to density have been recently reported. Although the total ELF is not recovered by the addition of the individual components, the separation is a useful tool to evaluate some important electronic aspects of different classes of chemical systems as radicals or aromatic species. The separation of spin components has been used to evaluate the radical characteristics of aromatic chemical systems.51 The topological analysis was made over a separated density constructed from the and components. In this way, it was possible to visualize the degree of localization of the unpaired electron mainly in open-shell systems (see Figure 5). Hence, the and spin separation of the ELF provides more insights into chemical bonding structure. In particular, the ELF provides a qualitative description of the space region where is most probable to find an unpaired electron.51 As an example, Figure 5 shows the ELF and ELF for the para-benzine radical in its singlet and triplet states which are known to be very near in energy. The para-benzine system has a particularly selective bioactivity and has been widely exploited for anticancer drug design.53 Further, from a purely theoretical point of view the para-benzine system is a challenge to current calculation methods. The system undergoes a spatial symmetry


69 ELFα

ELFβ

Singlet

Triplet

Figure 5 ELF isosurfaces of the para-benzine in singlet and triplet states

breaking, which means that the wave function obtained from standard single-determinant methods fails to transform as an irreducible representation of the molecular point group. Crawford et al. published a thorough study on this problem.54 From Figure 5 one can see that the ELF and ELF are different. One electron with spin is located on the carbon atom at the bottom, whereas the other electron with spin appears to be located on the carbon atom at the top of the figure. Of course, this is an artifact of the calculation because the probability of having an electron with and spin is the same. Kraka et al.55 demonstrated that the on-top density discussed by Perdew et al.56 is able to reproduce the correct symmetry of the Hamiltonian. In the same way as the -separation has been performed, one can proceed to a -separation.52 This separation has been used to evaluate the aromaticity of organic molecules and clusters. An index of aromaticity was proposed using a scale based on the bifurcation analysis of the ELF constructed from the separated densities. In principle, the total ELF has no information about and bonds, it depends only on the total density. Hence, the ELF does not show clear differences between both kinds of bonds. However, the topological analysis over separated densities, ones formed by the -orbitals and the other ones formed by the -orbitals, yields the necessary information.52 Of course, this is possible only for the molecules which present the symmetries, i.e. planar molecules. The bifurcation analysis of the news ELF and ELF can be interpreted as a measure of the interaction among the different basins and chemically, as a measure of electron delocalization.45 In this way, the and aromaticity for the set of planar molecules described in the Scheme 1 has been characterized.52 The aromatic rings present the highest and the antiaromatic systems the lowest bifurcation values of ELF . The bifurcation of the ELF occurs close to 0.75, except in system where sigma delocalization exists. In this way, the aromaticity of polycyclic aromatic hydrocarbons was well predicted, and also the aromaticity of new molecules important contributions was corroborated. In the all-metal aromatic compound Al2− 4 to stability from the two aromatic electrons and the system in the plane of the molecule were observed. The isosurfaces of the total ELF, ELF and ELF functions are showed in the Figure 6. By this way, one can predict the existence of or aromaticity, but it is difficult to predict the net aromaticity of species that presents aromaticity and antiaromaticity at the same time as it occurs in systems like Al4− 4 . In this case, the average value of the

70


0.91; 0.76

0.78; 0.76

0.70; 0.77

0.75; 0.76

0.64; 0.75

CO OC OC

B B

B B

CO

Al

B B

CO

Al

N

0.85; 0.68

0.11; 0.79

Scheme 1

N

N

Al

CO

0.82; 0.75

N

N

Al

2-

0.78; 0.81

0.99; 0.88

0.15; 0.78

0.35; 0.73

0.57; 0.77

Bifurcation values of ELF and ELF functions for some aromatic and antiaromatic

molecules Total ELF

ELFπ

ELFσ

Figure 6 ELF isosurfaces of Al2− 4 cluster

ELF and ELF bifurcations was used to construct a general scale to measure the global aromatic character of a molecular system.57 This general scale predicted the electron delocalization characteristic of known organic and metallic aromatic and antiaromatic systems. On a different development, recently an extension of the ELF to the time-dependent density functional formalism has been presented.58 With the advent of attosecond laser pulses, the information of the time scale and temporal order of the different bond breaking or bond formation processes will be important, and this is the kind of information one can extract from the time-dependent version of the ELF.

3. Some applications 3.1. Molecular geometry and bond types One of the first and most direct applications of the ELF is to the explanation and confirmation of the VSEPR model of Gillespie.28 This was first done by Savin et al.10 In molecular systems rich in electrons, the molecular distribution of the pairs of electrons


71

ab2

BeCl2

ab3

BCl3

ab4

CH4

ab5

PCI5

ab6

SCI6

ab3e

NH3

ab2e3

XeCl2

ab3e2

ClF3

ab4e

SF4

ab5e

BrF5

Figure 7 ELF isosurfaces for some molecules with typical geometries predicted by VSEPR model.

(lone and bonding) predicted by the VSEPR method can be visualized by means of the topological description of the ELF (see Figure 7). The classical representation of systems without lone pairs in the central atom abx x = 2–6 is very clear. In the systems with lone pairs on the central atom, it is important to appreciate the antiposition of the ELF attractors in molecules of type ab3 e2 , which is more distinctive than the VSEPR model. In molecules with three lone electron pairs at the central atom ab2 e3 , the attractors are cylindrical in shape. There are also other studies of special molecules like the hexafluoride of xenon and related molecules.59 The relevance of the octet rule in hypervalent molecules in the context of the ELF has been also studied.60 Indeed, the molecules that present a non-VSEPR geometry have been also rationalized by means of the ELF.61 The typical kind of bonding in organic molecules can be easily seen in Figure 8. The C−H simple bond is represented by a disynaptic basin V(C,H) in blue colour. The C−C simple bonds are described by a circular region (green colour) between two core regions (violet colour). The C−C double bond is shown as a double region perpendicular to the line joining the atoms, and the attractors corresponding to C−C triple bond are distributed in a cylindrical shape. The two lone pairs of electrons in the oxygen atoms are clearly in antiposition as was commented before for systems abx e2 .

72


Methane

Ethane

Propane

Ethene

Propene

Allene

Ethyne

Propyne

Dimethyl-ether

Propenaldehyde

Formaldehyde

Figure 8 ELF isosurfaces of some classical organic molecules

The ELF is not only used to explain or to visualize the type of bond in the wellknown molecules. It is also useful to clarify the type of bond in new molecules. For instance, it has been found that the new NaS N2 material has all the characteristics of a Zintl phase,62 that the phase transition in iodine under compression is due to the presence of the lone pairs,63 that the magnetic cluster Fe4 is characterized by a large delocalization of the electron density64 and that the contribution to bonding from the d electrons in the iridium atom in the new material IrGa2 is important.65 The localization nature of bonding at the superconducting stannide SrSn4 has been also explained, just like the homonuclear multiple bond between main group elements other than carbon, in particular the possible existence of a Ga–Ga triple bond in some new complexes.66

3.2. The nature of pericyclic and pseudopericyclic bonding at concerted transition states ELF analysis has been probed to be a powerful scheme to elucidate between the pericyclic and pseudopericyclic character of bonding at concerted transition states. In this context, ELF methodology has been recently applied to get new insights concerning the nature of bonding at the transition states of the thermal electrocyclization =CH2 ) and its heterosubstituted analogues, (2Z)-2,4,5of (Z)-1,2,4,6-heptatetraene (X= =O) and (2Z)-2,4,5-hexatrien-1-imine (X= =NH) (see Scheme 2).67 hexatrienal (X= It is known that a pseudopericylic pathway is characterized by the lack of cyclic orbital overlap, that is, there exists one or more disconnections in the bonding along the cyclic array of interacting centres.6869 Other characteristics which make interesting these type of reactions are that they have nearly planar and no anti-aromatic transition states, providing low activation barriers. On the other hand, pericyclic reactions involve no disconnection in the cyclic array of overlapping orbitals, non-planar allowed or forbidden transition states (depending on the number of electrons) with characteristic activation


C4

73

C4

C5

C3

C6 C2

C3

X7

C1

Pericyclic TS

C5 C6

C2 C1

C4

C5

C3

X7

C6 C2

X7

C1

Pseudopericyclic TS

Scheme 2 Transition structures for the thermal electrocyclization of (Z)-1,2,4,6-heptatetraene (X= CH2 ) and its heterosubstituted analogues, (2Z)-2,4,5-hexatrienal (X= O) and (2Z)-2,4,5hexatrien-1-imine (X= NH) energy barriers.69 In the present case,67 ELF results provide further evidence in support =NH, X= =O and X= =CH2 cases. of a single disrotatory pericyclic interaction for the X= This conclusion is based on the fluctuation analysis of electron density (i.e. covariance contributions interpreted in terms of delocalization) at the cyclic reaction centre upon the formation of the new C2−X7 bond. The fluctuation pattern is found to be very similar =NH and in the three cases. Furthermore, it is found that lone pair populations when X= = = X O are lower than 5%, and it seems to play, as suggested first by Rodriguez-Otero and Cabaleiro-Lago,70 only a stabilizing role in the global electrocyclization process. These findings based on ELF analysis contribute to the relevant controversy concerning the pericyclic or pseudopericyclic intimate nature of bonding at these heterocyclic transition structures (TSs).7172 It has also been previously shown73 that the fluctuation pattern of electron density in the ELF basins provides a consistent description of pseudopericyclic and pericyclic bonding in concerted processes such as thermal chelotropic decarbonilation reactions.74 Experimental support for planar pseudopericyclic transition states in chelotropic decarbonilations has been recently reported.75 ELF picture of bonding reveals that for the eight transition states analysed (see Scheme 3), the departing CO can be visualized in terms of a carbon monoxide structure with a ‘lone pair’ region on the carbon atom. Based upon the average bonding contributions (in per cent) to the ‘lone pair’ region centred at the carbon atom of the CO-leaving group and bifurcation diagrams, a clear distinction between the pseudopericyclic and pericyclic topologies can be achieved. Henceforth, this type of covariance analysis based on the ELF topology could constitute a complementary value to the traditional Woodward–Hoffmann symmetry-orbital rules.76

3.3. The ELF and the chemical bonding along reaction processes Some of the concepts of the catastrophe theory77 have been recently invoked in connection with ELF topological analysis to study the changes of bonding characteristics along chemical processes. In this context, Krokidis and co-workers have investigated the ammonia inversion, the breaking of the ethane C−C bond, and the breaking of the dative bond in NH3 BH3 ,78 the proton transfer in malonaldehyde,79 the proton transfer in the protonated water dimmer,80 the isomerization mechanisms in XNO (X=H Cl),81

74

Understanding and using the ELF Transition states for some thermal decarbonilations 17.3%

O

30.9%

O

O

26.7% 25.1%

O

O

O

30.9%

9.8% N 7.5% H Pseudopericyclic: two orbital disconnections O

17.3% Pericyclic: zero orbital disconnections

O

53.7%

0.2% O

22.4 %

19.0 %

O

O

O

38.0%

O

7.0 %

5.9%

7.1%

O

Pseudopericyclic: one orbital disconnection

Scheme 3 Analysis of the pericyclic and pseudopericyclic nature of bonding for some thermal decarbonilations and more recently the nature of the electron transfer and three-electron bonding in the reaction of Li with Cl2 .82 The nature of the change in chemical bonding along the reaction coordinate of some simple pericyclic reactions has been also explored.83 For instance, ELF analysis has been found useful for describing the bond breaking/bond forming at concerted transition states. In particular, a concerted antarafacial bonding nature for the [1s,3a]hydrogen, a biradical interaction for the [1a,3s]methyl and an ionic interaction for the [1a,3s]fluorine sigmatropic rearrangements in the allyl system (see Scheme 4) have been fully characterized through the examination of integrated densities over the ELF basins and their related variance properties. On the one hand, the fluctuation of electron density forms a cyclic and characteristic antarafacial pattern of bonding in the first case, while on the other, monosynaptic X H H

H H

C3

C2

C3

C2

H

H

X

H C1 H

C1 H

X = CH3, F

Scheme 4

H

X=H

[1s,3a]hydrogen, [1a,3s]methyl and [1a,3s]fluorine sigmatropic rearrangements in the allyl system. ELF analysis reveals a concerted interaction in the first case, a biradical interaction in the second one and a ion-pair interaction in the last one


75

basins, localizing approximately one electron, have been found on the methyl and allyl fragments. In the case of the fluorine migration, the ELF basin structure allows us to determine a charge separation of 0.6e between negative-charged fluorine and a positive allyl fragment. A deeper understanding of these molecular reaction mechanisms can thus be achieved in terms of the rearrangement of electron density among the ELF basins. The [1a,3s] sigmatropic shift of the fluorine atom in the 3-fluorpropene system has been also previously discussed in detail. The transition state has been thoroughly characterized in terms of a ion-pair structure with a charge separation of 0.6e, and the changes in the bonding characteristics along the intrinsic reaction coordinate reaction (IRC) path (see Schemes 5a and 5b) were described in terms of the ELF basin properties, i.e. electron populations, variance and delocalization indexes.84 We would also like to mention here that ELF analysis has been also found valuable into the detailed description of intramolecular proton-transfer reactions in some thiooxalic =O-CS-XH, (1) X= =O Y= =O; (2) X= =O, Y= =S; (3) X= =S, acid derivatives HY-C= =S, depicted in Scheme 6.85 Y= In all these cases, ion-pair transition structures have been characterized, and the intimate nature of bonding has been discussed using the electron properties arising from ELF analysis. In particular, charge separations of 0.48e, 0.42e and 0.18e can be deduced from the basin structure and populations for the oxygen to oxygen, sulfur to oxygen and sulfur to sulfur proton-transfer transition states, respectively.

3.4. Reactions yielding aromatic products 3.4.1. Radical reactions: Bergman reaction The reaction mechanism of the Bergman cyclization of the (Z)-hexa-1,5-diyne-3-ene to yield p-benzyne (Scheme 7 and Figure 9) has been studied recently in the framework

–217.00

Energy, au.

–217.02

–217.04

–217.06

–217.08

–217.10 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

IRC reaction coordinate

Scheme 5a

Total energy along the IRC for the [1a,3s]F sigmatropic shift

76


RC

Structure

ELF isosurface

–5.0

–3.0

0 (TS)

+3.0

+5.0

Scheme 5b

H1

O

X

Y

S

ELF isosurfaces of some selected points of IRC H2

Scheme 6

H1

O

X

Y

S

H2

Proton transfer of thiooxalic acid derivatives H C

H

C H

C

H C

C

C H

C

H

C C

C C

H

C H

(Z)-hexa-1,5-diyn-3-ene

Scheme 7

1,4-dehydrobenzene (p-benzyne)

Bergman reaction of (Z)-hexa-1,5-diyne-3-ene

of the ELF.86 In this study, the evolution of the bonding along the reaction path is modelled by the changes in the number and synaptic order of the ELF valence basins, and each topological configuration comprised a structural step or stability domain.


77

TS

–230.83

Energy, (au)

–230.84

–230.85

III –230.86

I

–230.87

II

IV

V

–230.88 –8

–7

–6

–5

–4

R x,

–3

amu1/2

–2

–1

0

1

2

bohr

Figure 9 IRC profile of the Bergman cyclization of (Z)-hexa-1,5-diyn-3-ene

Five domains of structural stability of the ELF along the intrinsic reaction path were determined. The first is the most costly in terms of energy, and it presents strongly geometry changes with small variations in the population of ELF basins. In the second step appears two monosynaptic basins V(C2) and V(C5) in the backside of C2 and C5 , enhancing the biradicaloid character of the moiety. The TS or third step shows a strong electron rearrangement with participation for the first time of the double bond C3 –C4 . In step IV, two monosynaptic basins are created on C1 and C6 atoms by electron transfer from the initial triple bonds basins as a prelude to cyclization. In this step, characteristics of biradical system are appreciated. The last step corresponds to the closure of the ring by the formation of a disynaptic basin VC1 C6 to reach the electronic characteristics of the p-benzyne. The ELF basin isosurfaces of the steps described before are depicted in Figure 10. In addition, the separation of the ELF into in-plane ELF and out-ofplane ELF contributions allowed discussing the aromaticity profiles (see Figure 11). aromaticity appears in the vicinities of the TS, and it is provided exclusively by the delocalization of the in-plane electronic system, while aromaticity takes place in the final stage of the reaction path, once the ring has been formed, being maximum in the p-benzyne. A related study but concerning the 1,3-dipolar cycloaddition of fulminic acid and ethyne was previously reported.87 3.4.2. Trimerization of acetylene The reaction mechanism and the development of the aromaticity along the trimerization of acetylene to yield benzene (Scheme 8, Figure 12) have been analysed by the ELF in the same framework of structural stability domains described before.88 The electronic rearrangements associated with bond breaking/forming processes determined five steps along the intrinsic reaction path, the ELF isosurfaces that characterized

78


(a) V(H9,C4)

C(C5)

(c)

(b)

C(C6) V(H10,C6)

C(C4)

V(C5)

V(C5)

V(C5,C6) V1,2(C3,C4)

V(C4,C5)

V(C1,C2)

C(C3) V(H8,C3)

C(C2)

V(C3,C4)

V(C2,C3)

V(H7,C1)

C(C1)

V(C2)

V(C2)

(f)

(e)

(d)

V(C5) V(C4,C5)

V(C6) V(C1)

V(C5,C6)

V(C1,C6)

V(C1,C6)

V(C2,C3) V(C2)

V(C1,C2)

Figure 10 ELF isosurfaces for: (a) (Z)-hexa-1,5-diyn-3-ene, (b) step II, (c) step III (TS), (d) step IV, (e) step V and (f) p-benzyne

1.0

III (TS) Bifurcation value

0.8

ELFσ

0.6

Average σ−π 0.4

ELFπ 0.2

I

II

IV

V

0.0

R x, amu1/2 bohr

Figure 11 Evolution of the ELF ELF and average ELF –ELF along the IRC path of Bergman cyclization

each step are shown in the Figure 13. In the first step, there is an approximation between acetylene moieties without rearrangement of electron density between ELF basins. In step II, which includes the TS, each valence basin corresponding to a triple bond is transformed into two degenerate basins out of plane of the molecule. This


79 H9

C3

H8

C4 H10

C2

H11

C1

C5 C6

H7

H12

Scheme 8

Trimerization of acetylene

–231.95 –232.00

Energy (au)

–232.05 –232.10

III

–232.15

II

–232.20

I

–232.25

IV

–232.30

V

–232.35 –12

–10

–8

–6

–4

–2

0

2

4

6

8

R x (amu1/2bohr)

Figure 12 IRC profile of the trimerization of acetylene

step is characterized by the deformation of the acetylene units in order to reduce the closed-shell repulsive interaction. Step III corresponds to the simultaneous formation of a monosynaptic basin over each carbon atom preparing for the cyclization of the system. In the fourth step occurs the formation of a six-membered cycle, associated with the formation of three disynaptic basins representing new C–C covalent bonds. In the last step, the three pairs of disynaptic basins are transformed into monosynaptic basins, and the valence basin populations are equalized resembling the transformation of non-aromatic into aromatic benzene. The analysis of in-plane ELF and out-of-plane ELF contributions (Figure 14) shows that the TS has a low -electron delocalization which is assigned to a process of through-space electron delocalization without -aromatic character. At the end of step IV, ELF increases sharply while ELF decreases slowly, according with the beginning of diatropic current as was previously observed by the analysis of magnetic properties.89 The increase of ELF bifurcation values in this step of the reaction does not reveal aromaticity according to the separated ELF scale. The aromaticity only is developed at the final stage of the reaction.

80

Understanding and using the ELF (b)

(a) V(C3,C4)

V(C,H)

V1,2(C3,C4)

V(C,H) C(C3) C(C2) V(C1,C2)

V(C,H)

C(C4)

V1,2(C1,C2)

C(C1)

C(C5) C(C6)

V(C,H) V1,2(C5,C6)

V(C,H) V(C5,C6)

V(C,H)

(d)

(e) V(

) ,C 3 C2 V(

) C3 V( C 2) V(

(c)

) ,C 3 C2

V(C4,C5)

V(C4)

V(C1,C2)

C

C

V(

V(

V(

) ,C 6 C5

1 ,C 6)

V(C4,C5) V(

1 ,C 6)

V(C5) V( C C1 ) 6)

) ,C 4 C3 ( V

Figure 13 ELF isosurfaces for: (a) acetylenes moieties, (b) step II (TS), (c) step III, (d) step IV and (e) step V (benzene) 1.0

ELFσ ELFπ ELFσ −π average

ELF bifurcation

0.8

0.6

0.4

V IV

0.2

III II I

0.0 –12

–10

–8

–6

–4

–2

0

2

4

6

8

R x (amu1/2bohr)

Figure 14 Evolution of the ELF , ELF and average ELF –ELF along the IRC path of trimerization of acetylene

3.5. Applications to atomic clusters The atomic clusters are defined as a conglomerate of atoms from as few as two to hundreds of them. They are usually produced by evaporation of the metal, and most of them are highly reactive. Hence, most of them exist only in gas phase. One of


81

the most known exceptions is the fullerene, which is so stable that crystallize easily. Besides its importance in the nanotechnology, they are from a theoretical point of view interesting due to the variety of bonding they present. For instance, the magnesium dimer, the smallest magnesium cluster, presents a van der Waals type of bonding, the bigger clusters, say Mg10 , present a covalent bond and when the cluster is growing at same point the bond should be metallic in character like the solid. To elucidate the type of bonding in clusters, the ELF is of high utility. It is important to notice that most of them do not follow the Lewis rules of bonding, and they are not described by a simple Lewis structure. For instance, think on the Li4 cluster. On the ground state, it is a rhombus. Hence, each lithium atom is surrounded by two other lithium atoms, and there are four valence electrons to bind all the atoms. It is impossible to draw a Lewis structure. In Figure 15, one can see an ELF isosurface for Li4 . There are two valence basins surrounding three lithium atoms each. The population of each basin is of around 2e. Therefore, the picture is very clear. The Li4 cluster is held together through two two-electron-three-centre bonds. In an exhaustive work, Rousseau and Marx90 used the ELF to understand the variations in the type of bonding of lithium clusters, nanoclusters, bulk metal and surfaces. They concluded that the electrons prefer to localize in the interstitial regions, leading to multicentre bonding for both the clusters and the solids, including their surfaces. For nanoscale clusters Li40 , only the surface presents strong localization and the interior displays localization properties similar to the bulk metallic solid. The ELF not only permits to understand the type of bonding but also to do some predictions about the reactivity of the clusters.91 Taking again Li4 as an example, one can try to predict at which position will a hydrogen atom bind to the cluster, on top of a lithium atom, in a bridge position between two atoms or from above the four atoms?. The hydrogen atom is more electronegative than a lithium atom. Therefore, the hydrogen atom will take charge from the cluster, and the bond will be polar. Hence, it is reasonable to think that the hydrogen atom will approach the cluster on the regions where it is most probable to find the valence electrons. Looking at the ELF isosurface, one can predict that the hydrogen atoms will attach to the cluster at a bridge position between two lithium atoms. In Figure 15 one can see an ELF isosurface for the hydrogenated cluster of Li4 H2 . Each hydrogen basin has a population of 2e and the two-electron-three-centre basin at the top of the figure remains with a population of around 2e. Note that the cluster is divided into two regions, one with delocalized electrons and one with localized electrons, as should be in a model for a metallic–insulator interphase. A special branch of the atomic cluster research is the study of clusters of the metal transition atoms, especially because of their catalytic properties. However, for the transition metal atoms, the application of the ELF deserves to be carefully analysed. One

Li4

Li4H2

Figure 15 ELF isosurface of Li4 and Li4 H2

82


observes two characteristic features in systems formed by transition metal atoms. The ELF presents very low values almost everywhere, and the number of attractors augments considerably. For instances, in a recent study of the ground state of Fe4 the ELF analysis64 reveals 21 attractors, and it presents always values lower than 0.5, the electron gas reference value. These two features obviously complicate the analysis.92−94 Kohout et al. have carefully examined this point.95

4. Conclusions and outlook Some general aspects related to the derivation, and interpretations of ELF analysis, as well as some representative applications have been briefly discussed. The ELF has emerged as a powerful tool to understand in a qualitative way the behaviour of the electrons in a nuclei system. It is possible to explain a great variety of bonding situations ranging from the most standard covalent bond to the metallic bond. The ELF is a well-defined function with a nice pragmatic characteristic. It does not depend neither on the method of calculation nor on the basis set used. Its application to understand new bond phenomenon is already well documented and it can be used safely. Its relationship with the Pauli exclusion principle has been carefully studied, and its consequence to understand the chemical concept of electron pair has also been discussed. A point to be further studied is its application to transition metal atoms with an open d-shell. The role of the nodes of the molecular orbitals and the meaning of ELF values below 0.5 should be clarified.

Acknowledgements Our thanks to Profs. A. Savin and B. Silvi (Paris) for introducing us to the world of the electron localization function. We acknowledge the support received from MIDEPLAN and CONICYT through the Millennium Nucleus for Applied Quantum Mechanics and Computational Chemistry, project P02-004-F. J.C.S. thanks Universidad Técnica Federico Santa Maria for the support through grant UTFSM 130423. E.C. is also grateful to Universidad Andrés Bello for the support received through the project UNAB DI 16-04. We finally thank Fondecyt for financial support through grant nos. 1030173 and 1050294.

References and notes 1. S. Zoury, X. Krokidis, F. Fuster and B. Silvi, Computers & Chemistry (Oxford) (1999) 23 597. See also: http://www.jussieu.fr/silvi 2. The oficial WEB site for this commercial software is http://www.gaussian.com 3. The General Atomic and Molecular Electronic Structure System (GAMESS): www.msg.ameslab.gov/GAMESS/GAMESS.html 4. For instance: (a) The Car-Parrinello Molecular Dynamics code (CPMD, http://www.cpmd.org/ http://www.cpmd.org/) implements an ELF for the valence electrons, both in a spin-polarized average of ELF as well as the separate and orbital contributions as it has been proposed by M. Kohut and A. Savin, Int. J. Quant. Chem. 60 (1996) 875; (b) The VASP package


5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

83

(http://cms.mpi.univie.ac.at/CMSPage/main/) also implements the ELF in a framework of mechanical molecular dynamics (MD) using pseudopotentials and a plane wave basis set. Utility program written by M. Kohout, Max Planck Institute for Chemical Physics of Solids, Dresden, Germany. For contact and to obtain the program, please E-mail to: [email protected] with the Subject: DGrid The official WEB site for this commercial software is: http://www.molpro.net/ The official WEB site for this commercial software is: http://www.scm.com/ P. Soler, F. Fuster and H. Chevreau, J. Comput. Chem. 25 (2004) 1920 A. D. Becke and K. E. Edgecombe, J. Chem. Phys. 92 (1990) 5397 A. Savin, R. Nesper, S. Wengert and T. Fassler, Angew. Chem. Int. Ed. Engl. 36 (1997) 1808 B. Silvi and A. Savin, Nature 371 (1994) 683 W. Pauli, Z. Physik 36 (1926) 336 G. N. Lewis, J. Am. Chem. Soc. 38 (1916) 762 R. F. W. Bader, Atoms in Molecules. A Quantum Theory, Clarendon Press, Oxford, 1990 Y. Tal and R. F. W. Bader, Int. J. Quant. Chem. Quant. Chem. Symp. 12 (1978) 153 M. Kohout and A. Savin, Int. J. Quant. Chem. 18 (1997) 1431 V. Tsirelson and A. Stash, Chem. Phys. Lett. 351 (2002) 142 S. R. Gadre, S. A. Kulkarni and R. K. Pathak, J. Chem. Phys. 98 (1993) 3574 P. Fuentealba, Int. J. Quant. Chem. 69 (1998) 559 P. K. Chattaraj, E. Chamorro and P. Fuentealba, Chem. Phys. Lett. 314 (1999) 114 P. Ayers, R. Parr and A. Nagy, Int. J. Quant. Chem. 90 (2002) 309 M. Kohout, K. Pernal, F. Wagner and Y. Grin, Theor. Chem. Acc. 112 (2004) 453 A. Savin, H. J. Flad, J. Flad, H. Preuss and H. G. von Schnering, Angew. Chem. 104 (1992) 185 C. Von Weizsacker, Z Phys 96 (1935) 431 A. Burckhardt, U, Wedig, H, G. von Schnering and A. Savin, Z. Anorg. Allg. Chem. 619 (1993) 437 F. Zurcher, S. Leoni and R. Nesper, Z. F. Kristalog. 218 (2003) 171 J. K. Burdett and T. A. McCornick, J. Phys. Chem. A102 (1998) 6366 R. J. Gillespie, Molecular Geometry, van Nostrand Reinhold, London, 1972; R. J. Gillespie and R. S. Nyholm, Rev. Chem. Soc. 11 (1957) 239 R. F. W. Bader, S. Johnson, T. H. Tand and P. L. A. Popelier, J. Phys. Chem. A 100 (1996) 15398 R. F. W. Bader and G. L. Heard, J. Chem. Phys. 111 (1999) 8789 R. Llusar, A. Beltran, J. Andres, F. Fuster and B. Silvi, J. Phys. Chem. A105 (2001) 9460 D. B. Chesnut and L. J. Bartolotti, Chem. Phys. 257 (2000) 175 S. Roggan, G. Schnakenburg, C. Limberg, S. Sandhoefner, H. Pritzkow and B. Ziemer, Chem. Eur. J. 11 (2004) 225 J. J. Panek, A. Jezierska, K. Mierzwicki, Z. Latajka and A. Koll, J. Chem. Inf. Comp. Sci. 45 (2005) 39–48. (a) A. Bil and Z. Latajka, Chem. Phys. 305 (2004) 243 (b) A. Bil and Z. Latajka, Chem. Phys. 303 (2004) 43 J. R. Sambrano, L. Gracia, J. Andres, S. Berski and A. Beltran, J. Phys. Chem. A 108 (2004) 10850 J. R. B. Gomes, F. Illas and B. Silvi, Chem. Phys. Lett. 388 (2004) 132 J. Pilme, B. Silvi, M. E. Alikhani, J. Phys. Chem. A 107 (2003) 4506 J. Boily, J. Phys. Chem. A 107 (2003) 4276 J. Molina and J. A. Dobado, Theor. Chem. Acc. 105 (2001) 328 M. Kohout and A. Savin, Int. J. Quant. Chem. 60 (1996) 875. R. J. Boyd, Can. J. Phys. 56 (1978) 780. A. M. Simas, R. P. Sagar, A. C. Ku and V. H. Smith Jr., Can. J. Chem. 66 (1988) 1923.

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44. 45. 46. 47. 48.

M. Kohout and A. Savin, Int. J. Quant. Chem. 18 (1997) 1431. A. Savin, B. Silvi and F. Colonna, Can. J. Chem. 74 (1996) 1088. S. Noury, F. Colonna, A. Savin and B. Silvi, J. Mol. Struct. 450 (1998) 59 B. Silvi, J. Mol. Struct 614 (2002) 3 R. F. W. Bader, in Localization and Delocalization in Quantum Chemistry, vol. 2. Edited by O. Chalvet et al. Reidel, Dordrecht, 1976 B. Silvi, Phys. Chem. Chem. Phys. 6 (2004) 256 M. Calatayud, J. Andres, J. Beltran and B. Silvi Theor. Chem. Acc. 105 (2001) 299 J. Melin, P. Fuentealba, Int. J. Quant. Chem. 92 (2003) 381 J. C. Santos, W. Tiznado, R. Contreras, P. Fuentealba, J. Chem. Phys. 120 (2004) 1670 K. C. Nicolau, A. L. Smith, S. Wendeborn and C. K. Wang J. Am. Chem. Soc. 113 (1991) 3106 T. D. Crawford, E. Kraka, J. F. Stanton and D. Cremer, J. Chem. Phys. 114 (2001) 10638 J. Grafenstein, A. Hjerpe, E. Kraka and D. J. Cremer, J. Phys. Chem. A104 (2000) 1748 J. P. Perdew, A. Savin and K. Burke, Phys. Rev. A51 (1995) 4531 J. C. Santos, J. Andres, A. Aizman and P. Fuentealba, J. Chem. Theory Comput. 1 (2005) 83 T. Burnus, M. A. L. Marquez and E. K. U. Gross, Phys. Rev. A 71 (2005) 10501 Y. Simon-Manso and P. Fuentealba, Theochem 634 (2003) 89 S. Noury, B. Silvi and R. J. Gillespie, Inorg. Chem. 41 (2002) 2164 R. J. Gillespie, S. Noury, J. Pilme and B. Silvi, Inorg. Chem. 43 (2004) 3248 F. Dubois, M. Schreyer and T. F. Fassler, Inorg. Chem. 44 (2005) 477 A. Savin J. Phys. Chem. Solids 65 (2004) 2025 S. Berski, G. Gutsev, M. Mochena and J. Andres, J. Phys Chem. A 108 (2004) 6025 M. Bostrom, Y. Prots and Y Grin, Solid State Sci. 6 (2004) 499 H. Grutzmacher and Th. Faessler, Chem. Eur. J. 6 (2000) 2317. E. E. Chamorro and R. Notario, J. Phys. Chem. A. 108 (2004) 4099 J. A. Ross, R. P. Seiders and D. M. Lemal, J. Am. Chem. Soc. 98 (1976) 4325 See for instance: (a) D. M Birney, Org. Lett. 6 (2004) 85; (b) C. Zhou and D. M Birney, J. Org. Chem. Soc. 69 (2004) 86; (c) C. Zhou and D. M. Birney, J. Am. Chem. Soc. 125 (2003) 15268; (d) C. Zhou and D. M. Birney, J. Am. Chem. Soc. 124 (2002) 5231; (e) W. W. Shumway, N. K. Dalley and D. M. Birney J. Org. Chem., 66 (2001) 5832; (f) D. M. Birney, J. Am. Chem. Soc. 122 (2000) 10917; (g) D. M. Birney, X. L. Xu and S. Ham, Ang. Chem. Inter. Ed. Engl. 38 (1999) 189 J. Rodriguez-Otero and E. M. Cabaleiro-Lago Chem. Eur. J. 9 (2003) 1837 (a) A. R. De Lera, R. Alvarez, B. Lecea, A. Torrado and F. P. Cossio Ang. Chem. Inter Ed. Engl. 40 (2001) 557; (b) J. Rodriguez-Otero and E. M. Cabaleiro-Lago Angew. Chem., Int. Ed. 41 (2002) 1147; (c) A. R. De Lera, F. P. Cossio, Angew. Chem., Int. Ed. 41 (2002) 1150; (d) J. Rodriguez-Otero and E. M. Cabaleiro-Lago Chem. -Eur. J. 9 (2003) 1837 (a) E. Matito, M. Sola, M. Duran and J. Poater, Comment on the “Nature of Bonding in the Thermal Cyclization of (Z)-1,2,4,6-Heptatetraene and its Heterosubstituted Analogues” J. Phys. Chem. A (2005) in press. (b) E. E. Chamorro and R. Notario, J. Phys. Chem. A (“Reply to ‘Comment on the “Nature of Bonding in the Thermal Cyclization of (Z)-1,2,4,6Heptatetraene and its Heterosubstituted Analogues” J. Phys. Chem. B 109 (15) 7594–7595 (April 21, 2005) E. Chamorro, J. Chem. Phys. 118 (2003) 8687 D. M Birney, S. Ham and G. R. Unruh, J. Am. Chem. Soc. 119 (1997) 4509 H. X. Wei, Z. Chun, S. Ham, J. M. White and D.M. Birney, Org. Lett. 6 (2004) 4289 (a) R. B. Woodward and R. Hoffmann J. Am. Chem. Soc. 87 (1965) 395; (b) R. Hoffmann and R. B. Woodward, J. Am. Chem. Soc. 87 (1965) 2046; (c) R. Hoffmann and R. B. Woodward, J. Am. Chem. Soc. 87 (1965) 4389; (d) R. Woodward and R. Hoffmann, Angew. Chem. Int. Ed. 8 (1969) 781

49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69.

70. 71.

72.

73. 74. 75. 76.


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77. (a) R. Thom 1975. Structural Stability and Morphogenesis: An Outline of a General Theory of Models. London: Benjamin; (b) R. Thom 1983. Mathematical Models of Morphogenesis, Chichester: Horwood.; (c) R. Gilmore, Catastrophe Theory for Scientists and Engineers Dover Publications; Reprint edition (July 1, 1993); (d) P. T. Saunders, An Introduction to Catastrophe Theory. Cambridge University Press (June 30, 1980) 78. X. Krokidis, S. Noury and B. Silvi, J. Phys. Chem. A 101 (1997) 7277 79. X. Krokidis, V. Goncalves, A. Savin and B. Silvi, J. Phys. Chem. A 102 (1998) 5065 80. X. Krokidis, R. Vuilleumier, D. Borgis and B. Silvi, Mol. Phys. 96 (1999) 265 81. X. Krokidis, B. Silvi and M. E. Alikhani, Chem. Phys. Lett. 292 (1998) 35 82. X. Krokidis and A. Sevin, Progress in Theoretical Chemistry and Physics 2 (2000) (Quantum Systems in Chemistry and Physics, Vol. 1), 345 83. E. Chamorro, J. C. Santos, B. Gomez, R. Contreras and P. Fuentealba, J. Phys. Chem. A: 106 (2002) 11533. 84. E. Chamorro, J. C. Santos, B. Gomez, R. Contreras and P. Fuentealba, J. Chem. Phys. 114 (2001) 23 85. E. Chamorro, A. Toro-Labbe and P. Fuentealba, J. Phys. Chem. A 106 (2002) 3891 86. J. C. Santos, J. Andres, A. Aizman, P. Fuentealba and V. Polo, J. Phys. Chem. A 109 (16) (2005) 3687–3693 87. V. Polo, J. Andres, R. Castillo, S. Berski and B. Silvi, Chemistry-A European journal 10 (2004) 5165 88. J. C. Santos, V. Polo and J Andres, Chem. Phys. Lett 406 (4–6) (2005) 393–397 89. R. Havenith, P. Fowler, L. Jenneskens and E. Steiner, J. Phys. Chem. A 107 (2003) 1867 90. R. Rousseau and D. Marx, Chem. Eur. J. 6 (2000) 2982 91. P. Fuentealba and A. Savin, J. Phys. Chem. A 105 (2001) 11531 92. R. Llusar, A. Beltran, J. Andres, F. Fuster and B. Silvi, J. Phys. Chem. A105 (2001) 9460 93. Michelini, N. Russo, M. Alikhani and B. Silvi, J. Comput. Chem. 25 (2004) 1647 94. J. Pilme, B. Silvi and M. Alkhani, J. Phys. Chem. 107A (2003) 4506 95. M. Kohout, F. Wagner and Y. Grin, Theor. Chem. Acc. 108 (2002) 150



Chapter 6

Electronic structure and reactivity in double Rydberg anions: characterization of a novel kind of electron pair Junia Melin, Gustavo Seabra, and J. V. Ortiz Department of Chemistry, Kansas State University, Manhattan, KS 66506-3701, USA

Abstract A double Rydberg anion (DRA) consists of a stable cationic core and two electrons in a diffuse Rydberg orbital. These anions correspond to a local minimum on a potential energy surface where more stable isomers may exist. Experimental and theoretical works have contributed to a better understanding of the unusual electronic structure of these molecules. With electron propagator calculations and analysis of the electron localization function, some relationships between electronic structure and reactivity in DRAs are considered.

1. Introduction In a pioneering photoelectron study of the anion–molecule complex H− NH3 , workers at Johns Hopkins University discovered a low-energy peak that could not be assigned to a hot band of this anion–molecule complex.1 The invariance of the latter peak’s position with respect to deuteration, which eliminated hot bands of the anion–molecule complex from consideration, led these workers to propose the presence of another isomer in the mass-selected ion sample. They subsequently proposed that this feature pertained 2 to a tetrahedral NH− 4 anion. Perturbative electron propagator calculations provided an accurate assignment of the photoelectron spectrum, ascribing the two principal peaks to a H− NH3 complex and the low-energy peak to a tetrahedral anion.3 The Dyson orbital corresponding to the latter feature has a1 symmetry and exhibits NH antibonding and 87

88

Electronic structure and reactivity in DRAs

HH bonding relationships between diffuse s functions. Such phase relationships and the predominance of diffuse s functions on hydrogens explained the sharpness of the corresponding photoelectron peak. This description also validated the use of the term double Rydberg anion (DRA),24 for two electrons are found in a Rydberg-like orbital that is distributed on the periphery of a closed-shell cation, NH+ 4. Several theoretical works were published on other simple hydrides and, in addition − 5−13 to tetrahedral NH− for C3v OH− 3 and tetrahedral PH4 . 4 DRAs have been found − − − Geometry optimizations of C3v SH3 and linear structures of FH2 and ClH2 encountered transition states (TS) instead of DRA minima.10 The same study provided harmonic − − vibrational frequencies for NH− 4 OH3 , and PH4 . − − Recent calculations on NH3 R and OH2 R anions, where R = CH3 NH2 OH, and F, have identified other stable anions of this type.14 Here, AHn substituents replace − hydrogens from the parent species, tetrahedral NH− 4 , and C3v OH3 . Dyson orbitals for − electron detachments from stable anions such as NH3 CH3 are delocalized over the periphery of the entire species. This result generalizes previous studies where large molecular cations were found to accommodate a diffuse, Rydberg electron that is spread over the periphery of the entire cationic kernel.15 Covalent and ionic bonding that involves Rydberg-like orbitals has been explored as well.16 Experiments with higher resolution on N2 H7−17 were quickly followed by electron propagator calculations 18 that confirmed the existence of an anion–molecule complex, H− NH3 2 , and two DRAs with vertical electron-detachment energies placed symmetrically about the position of the low-energy peak in the NH− 4 spectrum. One of these N2 H7− species is a complex consisting of the tetrahedral DRA and a coordinated ammonia molecule. The other features a hydrogen bond between two N atoms in a structure that resembles the N2 H7+ NH+ 4 – NH3 complex. The Dyson orbital for anion electron detachment in the latter isomer is localized on the three nonbridging hydrogens attached to the ammonium fragment’s N atom. Vibrational satellites of each of the three vertical peaks also were assigned. Agreement of equally high quality was obtained for vibrational satellites seen in the NH− 4 spectrum. These works established the existence of a novel variety of electron pair in DRAs. Extensions of traditional electron pair concepts are clearly needed for these anions. The electron localization function (ELF) 19 is an interesting and robust descriptor of chemical bonding, which has been successfully applied to a wide variety of molecular systems.20−24 This function, which is based on a topological analysis of a quantum function related to Pauli repulsion, describes the degree of localization (or delocalization) of electron pairs within the molecular space. Section 2 explains the theory behind electron propagator calculations and the ELF. Section 3.1. contains results of ELF analysis for NH3 R− DRAs (with R = H CH3 NH2 OH) and molecular complexes of N2 H7− . After validation of the topological ELF analysis in the characterization of Rydberg electrons, the next step (described in Section 3.2.) is to study the reaction path that connects a DRA with a global minimum in the corresponding potential energy surface. In particular, we studied the reaction − profile between tetrahedral NH− 4 and the H NH3 complex. Our goal is to find the TS and determine the activation energy for this reaction. Through electronic structure calculations at different geometries along the energy profile, we expect to find when the double Rydberg character of NH− 4 is lost in favor of electronic distributions that are characteristic of ion–molecule complexes.

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2. Theory 2.1. Electron propagator theory Electron propagator calculations 25−28 of electron-binding energies (that is, electronattachment and -detachment energies) may be based on one-electron equations which read F + Dyson Dyson = Dyson Dyson (1) i i i i where F , the Fock operator, depends on the first-order density matrix of a reference state, is the energy-dependent, nonlocal correlation operator known as the self-energy, Dyson is a self-consistent eigenvalue and Dyson is the corresponding eigenfunction i i known as the Dyson orbital. The eigenvalues equal electron-binding energies, and the Dyson orbitals are defined by the following equations for electron-detachment energies Dyson x1 = i

√ N N x1 x2 x3 xN ∗ × iN −1 x2 x3 xN dx2 dx3 dxN

(2)

and electron-attachment energies Dyson x1 = i

√

N +1

iN +1 x1 x2 x3 xN +1

× N∗ x2 x3 xN +1 dx2 dx3 dxN +1

(3)

where xj is the space-spin coordinate of electron j, N is an initial state with N electrons and iN ±1 is the i-th final state with N ± 1 electrons. Diagonal approximations neglect off-diagonal matrix elements of the self-energy operator in the canonical, Hartree– Fock (HF) basis. Perturbative arguments underlie the second order, third order, P3 and OVGF diagonal approximations of the self-energy operator. The more advanced Brueckner Doubles T1 (BD-T1) method 25 does not make the diagonal approximation and includes partial, infinite-order contributions to the self- energy operator. The latter corrections are generated by the use of the Brueckner Doubles coupled-cluster wave function to describe the initial state and by other techniques, which consider final-state orbital relaxation and differential correlation effects. The norm of the Dyson orbital, pi , reads 2 pi = Dyson x1 dx1 (4) i and is an index of the qualitative validity of the perturbative arguments that are made in the diagonal approximations. In the latter methods, the Dyson orbital equals the square root of the pole strength times a canonical, HF orbital. In contrast, Dyson orbitals generated with nondiagonal methods are expressed as a linear combination of orbitals pertaining to the reference state, which may be of the HF or Brueckner varieties.

90


2.2. Electron localization function The ELF function is defined according to

Tr

r = 1 + TTF r

2 −1 (5)

and is interpreted as a local measure of Pauli repulsion.19 In this equation, Tr represents the difference between the kinetic energy density of the real system and the exact kinetic energy density of a fictitious bosonic system with the same electronic density. Therefore, r measures the degree of electron pairing with respect to a homogeneous electron gas whose kinetic energy density is given by the Thomas–Fermi model, TTF r. ELF values are close to 1 for localized electron pairs, whereas small values of this function r < 05 correspond to highly delocalized electron density. The topological analysis of the ELF provides a useful partition of molecular space into subsystems, called basins. These nonoverlapping regions are categorized as either core or valence basins, where valence basins are labeled by the number of connections with core basins, or synaptic order. A monosynaptic basin, VX1 , typically describes lone pairs that belong to X1 . A disynaptic basin, VX1 X2 , describes bonds between X1 and X2 atomic centers. Basins associated with more than three atomic cores are called polysynaptic. Of greatest importance in the following discussion are the asynaptic basins, which describe electrons that are not connected with atomic centers.2129 Basins provide not only a useful qualitative picture of the electron pairs in a molecule, but they also have welldefined properties.30 For example, electron population of a given basin is obtained by integrating the electron density over its volume i , N˜ i =

rdr

(6)

i

The relative fluctuation of these populations, , has been suggested as a measure of the degree of electron delocalization.31 It is defined as, N˜ i =

2 N˜ i N˜ i

(7)

where 2 N˜ i is the variance or quantum uncertainty associated with N˜ i .

2.3. Methods of calculation The following strategy for electronic structure calculations on DRAs was employed. Geometry optimization and harmonic frequency analysis for the cations were performed at the HF level with a standard Pople basis set.1418 These structures were used as initial guesses in the optimization of the respective anions, where the 6-311 + +Gdp basis set, which includes diffuse functions, was used. By this stage, optimizations and frequency calculations could be refined using a higher level of theory; therefore, MP2 and QCISD calculations were performed for all the molecular systems. The diffuse

J. V. Ortiz et al.

91 NH4−

NH3CH3−

NH3NH2−

NH3OH −

Figure 1 Optimized structures for NH3 R− systems

nature of the highest occupied molecular orbital in the DRAs requires that the basis set be supplemented with an additional set of diffuse functions. Exponents for diffuse Gaussian functions were obtained by multiplying the smallest exponent with a given angular dependence by 1/3. In this manner, sp functions on nitrogen atoms and additional s functions on hydrogen were added. The ELF analysis was performed on the electron density obtained from single point calculations at the MP2 level with a bigger basis set, 6-311+ + G(2df,2p) augmented with extra diffuse functions, at the anion equilibrium geometries (see Fig. 1). Gaussian03 32 was employed for all these calculations, whereas the ELF analysis was carried out with the TopMod 33 package of programs and Vis5d software 34 for visualization. − The TS for the internal conversion of the tetrahedral NH− 4 DRA into the H NH3 complex was obtained at the MP2 level with the usual basis set treatment, and it was characterized by a unique imaginary frequency 525i cm−1 . The reaction profile was obtained by performing an intrinsic reaction coordinate (IRC) calculation 35 upon the TS structure. Some points on this profile were chosen for a full analysis with electron propagator methods, including several diagonal self-energy approximations and the BD-T1 method.25 Finally, topological analysis of the ELF was performed on various points along the reaction path.

3. Results and discussion 3.1. ELF analysis of DRA systems In this section, ELF analysis for NH3 R− systems (with R = H CH3 NH2 OH) and molecular complexes of N2 H7− is presented. The color convention for all pictures is green for NH bonds, blue for lone pair electrons, and red for Rydberg electrons. Note that in some figures there are small red dots which correspond to core electrons. These features will be ignored in tables and in qualitative descriptions. In general, all the systems analyzed with the ELF show asynaptic basins, which by definition are associated with none of the atomic cores in the molecule. Because Rydberg electrons and asynaptic basins are absent in the parent cations, asynaptic basins can be assigned to Rydberg electrons in the uncharged and anionic species. High fluctuation values found for these basins reflect a high degree of delocalization. ELF results for NH− 4 in Table 1 show four equivalent bisynaptic basins corresponding to NH bonds, which are equivalent in tetrahedral symmetry. The electron population is about 2.0 electrons for each of them, and fluctuation values are in agreement with typical NH single bonds. Four asynaptic valence basins also are found. With a 0.24

92

Electronic structure and reactivity in DRAs Table 1 ELF Analysis of NH3 R− , where R = H, CH3 NH2 and OH Molecule

1

4

2

3

6 4

5

3

1

2

4

5 N2 N1

3

1

2

4

3

2

N

2

V(H1,N) V(H2,N) V(H3,N) V(H4,N) V(Asyn) V(Asyn) V(Asyn) V(Asyn)

202 202 202 202 024 024 024 024

079 079 079 079 021 021 021 021

039 039 039 039 089 089 089 089

V(H1,N) V(H2,N) V(H3,N) V(H4,C) V(H5,C) V(H6,C) V(N,C) V(Asyn) V(Asyn) V(Asyn)

207 207 207 204 204 204 178 046 047 046

082 083 082 064 064 064 097 036 037 036

04 04 04 031 031 031 054 078 078 078

V(H1,N1) V(H2,N1) V(H3,N1) V(H4,N2) V(H5,N2) V(N1,N2) V(N2) V(Asyn)

208 211 208 201 201 154 224 103

079 081 08 079 078 091 096 055

038 038 038 039 039 059 043 054

V(H1,N) V(H2,N) V(H3,N) V(H4,O) V(N,O) V(O) V(O) V(Asyn) V(Asyn)

211 214 211 17 119 238 242 114 026

081 081 08 081 079 107 108 054 023

038 038 038 048 066 045 045 048 088

Basin

1

electron population, these basins have a very high value: 0.89 on a scale of 0 to 1. Figure 2 shows the graphic representations of the ELF analysis for this anion. As has been mentioned above, the ELF approaches unity for localized electron pairs. N core electrons and NH bonds can be identified clearly at r = 08. Rydberg electrons are highly delocalized when r = 03, and they are located preferentially close to hydrogen atoms (Fig. 2b). A spherical distribution of Rydberg basins around the cation core is observed even with a smaller isosurface where r = 020. Figure 2c is in agreement with the analysis based on the Dyson orbital for electron detachment from the anion.18

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93

η(r ) = 0.8

η(r ) = 0.3

η(r ) = 0.2

(a)

(b)

(c)

Figure 2 Topological analysis of ELF function for NH− 4

For the NH3 CH− 3 anion, valence basins describing NH, CH, and NC bonds are found. Three asynaptic basins account for Rydberg electrons. The populations for these basins are slightly higher than their counterparts in the NH− 4 anion, but values are smaller. Figure 3a depicts the ELF topology for this system at different isosurface values. Rydberg basins again are highly delocalized for r = 030 and are located in regions close to H atoms belonging to the NH3 fragment. At lower r values, Rydberg basins are expanded until they completely envelop the NH3 portion of the anion. In NH3 NH− 2 , a lone electron pair is present which reduces the symmetry of the system. ELF analysis shows three bisynaptic basins corresponding to NH bonds in the NH3 fragment. Only two of them are equivalent in electron population, but all three have the same fluctuation value. There are two NH basins that belong to the NH2 fragment and another bisynaptic region describing the NN bond. Rydberg electrons are contained in a unique asynaptic basin which contains only 1.03 electrons. The fluctuation value assigned to this basin is comparable with those of the NN and NC bonds. Figure 3b shows the corresponding pictures for this system. Rydberg electrons can be identified at r = 04. Exploring the lowest values for the ELF isosurface, it is possible to see how Rydberg electron density encloses the anion and interacts even with the H atoms of the NH2 substituent while leaving free the region around the lone pair. The NH3 OH− anion behaves similarly. NH bonds, with about the same electron populations and values as those of the previous cases, are found. Two monosynaptic η(r)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(a)

(b)

(c)

− − Figure 3 Topological analysis of ELF function for (a) NH3 CH− 3 , (b) NH3 NH2 , and (c) NH3 OH

94


basins describing lone pairs on the oxygen concentrate high electron population, but their fluctuations reveal delocalization. Rydberg electrons in this case are split into two basins. In Figure 3c, the Rydberg basin with larger population is located in the opposite direction with respect to the lone pairs and interacts with H1 and H3 of the NH3 fragment. The remaining asynaptic basin, with a very small population and high delocalization, appears for r = 04 − 03. From a general point of view, Figure 3 shows the evolution of the isosurface value for the last three anionic systems. It is clear that the presence of lone pairs has an important effect on the Rydberg electrons’ localization. Repulsions with lone pair electrons distort the Rydberg electrons toward the more positive regions of the anion. Results for N2 H7− complexes are summarized in Table 2 and Figure 4. The most stable arrangement corresponds to two ammonia molecules coordinated to a hydride Table 2 ELF Analysis of N2 H7− Isomers Complex 4

7

N2

N1

3

6 5

2 1

5

1 N2

N1 2

3 6 7

4

1

5

N1

6

4

7

3 2

N2

Basin

N

2

V(H1) V(H2,N1) V(H3,N1) V(H4,N1) V(H5,N2) V(H6,N2) V(H7,N2) V(N1) V(N2)

283 148 190 189 148 190 189 220 220

090 090 076 075 090 075 076 096 096

032 060 040 040 061 040 040 044 044

V(H1,N1) V(H2,N1) V(H3,N1) V(H4,N1) V(H5,N2) V(H6,N2) V(H7,N2) V(N2) V(Asyn) V(Asyn) V(Asyn)

205 203 197 204 201 192 196 209 034 035 032

084 079 078 079 080 078 079 100 029 029 028

041 039 039 039 040 040 040 048 084 084 085

V(H1,N1) V(H2,N1) V(H3,N1) V(H4,N1) V(H5,N2) V(H6,N2) V(H7,N2) V(N2) V(Asyn) V(Asyn) V(Asyn) V(Asyn)

204 203 203 201 195 194 193 214 041 037 024 022

080 080 080 080 077 076 076 095 033 031 021 020

039 040 039 040 039 039 039 045 081 083 089 090

J. V. Ortiz et al. Optimized structures

95

η(r ) = 0.8

η(r ) = 0.35

η(r ) = 0.3

η(r ) = 0.25

η(r ) = 0.2

η(r ) = 0.15

(a)

(b)

(c)

Figure 4 Topological analysis of ELF function for (a) H− NH3 2 , (b) bridge, and (c) NH− 4 NH3

(Fig. 4a). Topological analysis of this ionic structure does not find any asynaptic basin. Therefore, one may conclude that this complex has no Rydberg character. Another minimum that is less stable by 0.5 eV also was found. In this case, the NH− 4 anion is bridged to an ammonia molecule. Three Rydberg basins, with almost the same population and fluctuation, are present. The highly delocalized electrons of these basins can be visualized only when r = 02 or less. Finally, a third structure exhibits coordination between NH− 4 and NH3 fragments. Four asynaptic basins are present in this structure. Two of them, identified as occupying the region between the molecules, are almost equivalent in population and slightly more localized than the other two. The isosurfaces depicted in Figure 4c show how two Rydberg basins appear at r = 03, whereas at r = 02 the NH− 4 molecule is completely covered by Rydberg electron density. From the last picture, one can conclude that this complex corresponds to a NH− 4 DRA coordinated to an ammonia molecule.

3.2. Transformation of a DRA to an ion–molecule complex Figure 5 shows the energy profile for the reaction. A sharp increase of energy is observed in the pathway from the reactant to the TS. Enlargement of the NH bond distance occurs with retention of C3v symmetry. After the TS, energy decays slowly, but with more drastic geometry changes. The detached H4 atom (see Table 3 for numbering) travels around the NH3 fragment, leaving the C3v axis soon after the TS. Table 3 summarizes the geometries for stationary points on this profile. The TS has geometrical parameters that resemble those of the tetrahedral form of NH− 4 , but with an obvious elongation between H4 and the nitrogen atom. The final product is a complex between a hydride anion and an ammonia molecule. The latter species has NH bond lengths that are close to those of

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Electronic structure and reactivity in DRAs –56.91 –56.92

Energy (au)

–56.93

–56.94 –56.95

–56.96 –56.97 –56.98 –4

–2

0

2

4

6

8

10

Reaction coordinate − Figure 5 Reaction profile for internal conversion of NH− 4 DRA to H NH3 complex with respect to an arbitrary reaction coordinate

Table 3 Geometrical parameters of reactant, transition state, and product 4 4

N 1

N 1

N-H2 N-H4 H1 -N-H2 H1 -N-H4 H4 -N-H1 -H2

1.0182 1.0182 109.47 109.47 120.00

3

1

3

N

1.0218 1.7501 108.73 110.20 120.88

3

4

1.0141 2.8340 103.97 99.95 −10318

Distances in Å, angles in degrees.

an isolated NH3 molecule, but with HNH angles that are smaller. The activation energy for this reaction, Eact , has been determined to be 0.59 eV (13.58 kcal/mol), whereas the energy difference between reactant and product, ER , is −068 eV (−1559 kcal/mol). These values include zero-point energy corrections. Bowen’s experimental work assigned an electron-detachment energy of 0.47 eV to the tetrahedral NH− 4 anion, whereas the main peak, corresponding to an electron-binding energy of the H− NH3 complex, was determined to be 1.11 eV. Table 4 lists vertical electron-detachment energies (VEDEs) calculated with different approximations of electron propagator theory for the reactant, the product and ten other molecular structures along the reaction path, including the TS. As has been discussed elsewhere, due to strong electron correlation in the DRA, Koopmans values do not agree closely with the experiment. Nevertheless, these values are included in the table as a comparative reference for an uncorrelated method. Propagator calculations within the diagonal selfenergy approximation, namely, second and third order, OVGF and P3, provide better descriptions than HF orbital energies, but still do not give an accurate account of the

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97

Table 4 Vertical electron-detachment energies (in eV), with different approximations, along the reaction path KT

2nd

3rd

OVGF

P3

BD-T1

React 2 3 4

−0229 −0530 (0.898) −0435 (0.846) −0242 −0553 (0.893) −0436 (0.844) −0282 −0509 (0.848) −0316 (0.798) −0358 0002 (0.759) −0017 (0.714)

−0359 −0356 −0219 −0021

(0.754) −0459 (0.860) −0477 (0.856) (0.759) −0481 (0.855) −0472 (0.859) (0.717) −0408 (0.80) −0285 (0.796) (0.701) 010 (0.736) −0125 (0.686)

TS 6 7 8 9 10 11

−0382 −0410 −0795 −0953 −1170 −1411 −1563

−0023 −005 −0417 −0565 −0777 −1024 −1191

(0.712) (0.727) (0.893) (0.895) (0.890) (0.888) (0.893)

0122 0223 0097 −0184 −0481 −0779 −0943

(0.752) (0.749) (0.805) (0.832) (0.850) (0.862) (0.868)

0002 0001 −0311 −0428 −0634 −0881 −1064

Prod −1629 −0993 (0.870) −1165

(0.718) (0.732) (0.905) (0.903) (0.893) (0.889) (0.894)

0187 0249 −001 −0214 −0454 −0712 −0883

(0.745) (0.759) (0.905) (0.909) (0.902) (0.897) (0.901)

−0138 −0159 −0343 −0393 −0574 −0817 −1001

(0.677) (0.662) (0.747) (0.755) (0.796) (0.813) (0.828)

(0.90) −1265 (0.899) −0968 (0.906) −1102 (0.793)

Pole strength values are in parentheses.

photoelectron spectrum. For instance, second order and P3 underestimate the VEDE of the H− NH3 complex, but third order and OVGF overestimate this transition energy. As for the reactant, second order predicts a higher VEDE, and the remaining methods are below the experimental value. A global inspection of these diagonal methods reveals how the electron-binding energy decreases from the reactant to the TS, reaching a negative value for the latter structure and its neighbors. The OVGF approximation is an exception, for it predicts a small but still positive VEDE at the TS. Pole strengths present interesting trends, for they are mostly around 0.85 in the reactant, decreasing quickly as the reaction evolves to the TS structure. Afterward these values increase again, until they reach 0.9 in the product. The definition of the pole strength parameter in propagator theory implies that when it is close to unity, Koopmans’s approximation is qualitatively valid. Therefore, the small values obtained around the TS reveal that the overlap between Dyson and occupied HF molecular orbitals is poor in this region, and a better electron correlation treatment is required. Note that the smallest pole strength is placed just before the TS and not on that point as might be expected. The reason for this finding is that the TS structure is no longer a DRA. This hypothesis will be supported in the following discussion of ELF results. Electron-detachment energies obtained with the BD reference state merit attention, since this method goes beyond the diagonal self-energy approach and has a more flexible treatment of correlation. For reactant and product, BD-T1 values show an excellent agreement with the experimental spectrum. Binding energies are small around the TS, but are all positive. The lowest ionization energy belongs to the structure previous to the TS, but the minimum value for the pole strength occurs afterward. The pole strength of the reactant is higher than that for the product. Results of ELF topological analysis are depicted in Figure 6. For simplicity, not all of the calculated structures have been included, but the picture illustrates how the basin distribution varies along the reaction path. Because of the high delocalization of Rydberg electrons, the plots correspond to a small ELF value, r = 025, in all cases. For the reactant, NH− 4 , four peripheral Rydberg basins around the cation core are found,

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Electronic structure and reactivity in DRAs –56.91 –56.92

Energy (au)

–56.93 –56.94 –56.95 –56.96 –56.97 –56.98 –4

–2

0

2

4

6

8

10

Reaction coordinate − Figure 6 ELF analysis along the reaction profile for internal conversion of NH− 4 DRA to H NH3 complex

just as has been described in previous section. The following structure shows just two red basins, where the Rydberg electrons are polarized along the C3v axis. As the energy increases, Rydberg electrons are more localized on the leaving H atom and are found finally within a monosynaptic basin that corresponds to a hydride anion. At the TS structure, the asynaptic basins are gone and lone pair electrons on nitrogen appear in blue. The reaction has evolved to the product, where H− and NH3 fragments may be identified clearly.

4. Conclusions The present study confirms the existence of a novel variety of electron pair. In contrast to the bonding pairs of Lewis and Langmuir and to the lone pairs of Moffitt, the diffuse − electron pairs of NH− 4 and N2 H7 are built chiefly of extravalence atomic functions and occupy the periphery of molecular cations. The concept of a Rydberg electron pair may lead to the prediction or observation of similar species or it may eventually yield to more generalized qualitative concepts of electronic structure. The reaction profile that is displayed in the figures demonstrates the presence of a considerable barrier to the rearrangement of NH− 4 from a tetrahedral, double Rydberg structure to an anion–molecule complex. In the TS, C3v symmetry applies as one of the NH distances is markedly longer than the others. An additional reduction of symmetry occurs after the transition state with the formation of the hydride–ammonia complex. Calculation of accurate electron-detachment energies along the entire reaction path requires the use of a highly correlated electron propagator approximation. Pole strengths associated with these transition energies have a minimum value near the transition state and indicate that correlation effects are largest at these geometries. Dyson orbitals for the electron-detachment energy also differ most from HF orbitals at these structures. The delocalized amplitudes of the Dyson orbital associated with the DRA have become more localized on the leaving hydrogen in the transition state.

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Analysis of the electron localization function for geometries of the same reaction path provides an alternative, but compatible, perspective on the evolution of electronic structure. The novel asynaptic basin of the DRA, which represents a pair of electrons that is delocalized over the periphery of the ammonium cation core, is transformed into a conventional, monosynaptic basin that is associated with the departing hydrogen at the geometries near the transition state. After the transition state, the NH− 4 system may be described as a hydride–ammonia complex.

Acknowledgments We acknowledge the support of the National Science Foundation through grant CHE0135823 to Kansas State University.

References 1. J. V. Coe, J. T. Snodgrass, C. B. Freidhoff, K. M. McHugh and K. H. Bowen, J. Chem. Phys. 83, 3169, (1985). 2. J. T. Snodgrass, J. V. Coe, C. B. Freidhoff, K. M. McHugh and K. H. Bowen, Faraday Disc. Chem. Soc., 86, 241, (1988). 3. J. V. Ortiz, J. Chem. Phys. 87, 3557, (1987). 4. J. Simons and M. Gutowski, Chem. Rev., 91, 669, (1991). 5. H. Cardy, C. Larrieu and A. Dargelos, Chem. Phys. Lett., 131, 507, (1986). 6. D. Cremer and E. Kraka, J. Phys. Chem., 90, 33, (1986). 7. M. Gutowski, J. Simons, R. Hernandez and H. L. Taylor, J. Phys. Chem., 92, 6179, (1988). 8. M. Gutowski and J. Simons, J. Chem. Phys., 93, 3874, (1990). 9. J. V. Ortiz, J. Chem. Phys., 91, 7024, (1989). 10. J. V. Ortiz, J. Phys. Chem., 94, 4762, (1990). 11. N. Matsunaga and M. S. Gordon, J. Phys. Chem., Vol. 99, 12773, (1995). 12. J. Moc and K. Morokuma, Inorg. Chem., 33, 551, (1994). 13. G. Trinquier, J. P. Daudey, G. Caruana and Y. Madaule, J. Am. Chem. Soc., 106, 4794, (1984). 14. H. Hopper, M. Lococo, O. Dolgounitcheva, V. G. Zakrzewski and J. V. Ortiz, J. Am. Chem. Soc., 122, 12813, (2000). 15. A. I. Boldyrev and J. Simons, J. Chem. Phys., 97, 6621, (1992). 16. (a) A. I. Boldyrev and J. Simons, J. Phys. Chem., 96, 8840, (1992). (b) J. S. Wright and D. McKay, J. Phys. Chem., 100, 7392, (1996). (c) A. I. Boldyrev and J. Simons, J. Phys. Chem. A, 103, 3575, (1999). 17. S. J. Xu, J. M. Niles, J. H. Hendricks, S. A. Lyapustina and K. H. Bowen, J. Chem. Phys., 117, 5742, (2002). 18. J. V. Ortiz, J. Chem. Phys., 117, 5748, (2002). 19. A. D. Becke and K. E. Edgecombe, J. Chem. Phys., 92, 5397, (1990). 20. A. Savin, A. D. Becke, J. Flad, R. Nesper, H. Preuss and H. Von Schnering, Angew. Chem. Int. Ed. Engl., 30, 409, (1991). 21. A. Savin, O. Jepsen, J. Flad, R. Nesper, O. K. Andersen, H. Preuss and H. G. Von Schnering, Angew. Chem., 31, 187, (1992). 22. E. Chamorro, J. C. Santos, B. Gomez, R. Contreras and P. Fuentealba, J. Chem. Phys., 114, 23, (2001) 23. P. Fuentealba and A. Savin, J. Chem. Phys. A, 105, 11531, (2001).

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24. J. Melin and P. Fuentealba, Int. J. Quant. Chem., 92, 381, (2003). 25. J. V. Ortiz, Adv. Quant. Chem., 33, 35, (1999). 26. A. M. Ferreira, G. Seabra, O. Dolgounitcheva, V. G. Zakrzewski and J. V. Ortiz, in Quantum Mechanic Predictions of Thermochemistry Data” edited by J. Cioslowski (Kluwer, Dordrecht, 2001), p. 131. 27. J. Linderberg and Y. Öhrn, in Propagators in Quantum Chemistry, Second Edition (Wiley, Hoboken, New Jersey, 2004). 28. J. V. Ortiz, in Computational Chemistry: Reviews of Quantum Current Trends, Vol 2, edited by J. Leszczynski (World Scientific, Singapore, 1997), p. 1. 29. A. Savin, B. Silvi and F. Colonna, Can. J. Chem., 30, 1088, (1996). 30. B. Silvi and A. Savin, Nature, 371, 683, (1997). 31. R. F. W. Bader, in Localization and Delocalization in Quantum Chemistry, edited by O. Chavet et al. (Riedel, Dordrecht, 1976) 32. Gaussian 03, Revision B.03, M. J. Frisch, G. W. Trucks, H. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, Jr., J. A. Montgomery, T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P.Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, J. A. Pople, Gaussian, Inc., Pittsburgh PA, 2003. 33. S. Noury, X. Krokisdis, F. Fuster and B. Silvi, Comput. Chem. Oxford., 23, 597, (1999) 34. B. Hibbard, J. Kellum and B. Paul, vis 5d, version 5.2: Visualization Project, University of Wisconsin-Madison Space Science and Engineering Center, 1990. 35. C. Gonzalez and H. B. Schlegel, J. Chem. Phys., 90, 2154, (1989) and J. Chem. Phys., 94, 5523, (1990).


Chapter 7

Using the reactivity–selectivity descriptor f r in organic chemistry ab

Christophe Morell, a André Grand, b Soledad Gutiérrez-Oliva, and b Alejandro Toro-Labbé a

Département de Recherche Fondamentale sur la Matière Condensée, Service de Chimie Inorganique et Biologique, LAN (FRE2600), CEA-Grenoble, 17, rue des Martyrs, 38054 Grenoble Cedex 9, France and b Laboratorio de Química Teórica Computacional (QTC), Facultad de Química, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

1. Introduction Besides the yield, there are many requirements for a chemical reaction to be interesting in organic synthesis, the regio- and stereo- selectivity1 are crucial factors to understand reaction mechanisms. Indeed, one of the most exciting challenges for a chemist is to control the regio- and stereo- selectivity of the chemical species involved at the different steps of the overall synthesis process. A variety of well-known organic reactions such as additions to alkenes,1 Diels Alders (DA) reactions,1−3 electrophilic aromatic substitution,4 cyclo-additions4 and nucleophilic additions1 on ketones or aldehydes present relatively high regio- and/or stereo- selectivity that have been rationalized using different theoretical tools and models. In this context, it can be observed that there is a lack of a theoretical model that allows the rationalization of reaction mechanisms in terms of a universal reactivity–selectivity descriptor. In this contribution, a theoretical model in which regio- and stereo- selectivity are characterized through a unique reactivity– selectivity descriptor defined in terms of the electronic properties of the species involved is proposed and then applied to analyse few classical organic reactions. The mechanisms of chemical reactions and the reactivity properties of the molecules involved started to be elucidated through the analysis of the wave functions defining the quantum state of molecular systems,5−7 for instance the Fukui’s frontier molecular orbital (FMO) theory89 has been very successful in rationalizing organic reactions basically through the analysis of the in- and out-of-phase overlap between the highest occupied molecular orbital (HOMO) of the nucleophile and the lowest unoccupied molecular 101

102

Reactivity–selectivity descriptor fr in organic chemistry

orbital (LUMO) of the electrophile; in this context, quite complicated mechanisms, such as the electrocyclic ring-opening of cyclobutene,10 have been explained. On the other hand, few empirical principles and postulates such as the hard and soft acids and bases (HSAB) principle proposed by Pearson11 or the Hammond postulate12 have been used to rationalize chemical behaviours or to explore transition state structures and energies.13 However most of these principles and postulates remained empirical until a branch of density functional theory (DFT)14−16 called conceptual DFT 17−21 has been developed and applied to chemistry. Keystone developments of the theory of chemical reactivity are due to Robert G. Parr and co-workers who provided the theoretical basis to formal definitions of empirical concepts that were already used in chemistry such as electronegativity , hardness and softness S = 1/.1422 A set of global and local descriptors of the reactivity of molecular systems emerged on the basis of DFT,14−16 and the formal link between DFT and classical chemistry was achieved through the definition of the chemical potential as the negative of the electronegativity = −.23 The chemical potential, originally introduced as a Lagrange multiplier to allow minimization of the energy functional of the electron density E under the condition that the density integrates to the total number of electrons, characterizes the escaping tendency of the electronic cloud.14 Chemical hardness2425 can be understood as the resistance to charge transfer, and the principle of maximum hardness (PMH) proposed by Pearson 26 provides the conceptual framework to interpret this property: it establishes that maximum stability is obtained when the system reaches the maximum hardness. Since minimum energy is related to maximum hardness, reactive systems are expected to be unstable and soft.2426−28 Also, the Pearson’s HSAB principle,11 stating that hard acids prefer to coordinate with hard bases while soft acids prefer to coordinate with soft bases, gives an indication of the coordination preference of molecules in terms of the relative values of hardness and softness of the reaction partners.11242529 In this context, the HSAB principle provides an interpretative basis for important reactivity descriptors derived from DFT.112430 Formally and are defined as the first and second derivatives of the energy functional E ≡ E N r with respect to N , the number of electrons.14 The electron density r and the so-called Fukui function fr are derivatives of the energy functional with respect to the external potential r;31 these are local functions that characterize the selectivity concept that distinguishes the different atoms within the molecule. Reactions involving ionic compounds are often governed by hard–hard interactions32−34 that are basically of electrostatic nature; in such cases the electron density

r and the local charges that can be obtained through condensation of it34 are good descriptors of the specific interactions that drive the whole reaction.34 Reactions involving non-charged species are in general governed by soft–soft interactions due to orbital overlapping; in these cases, the Fukui functions, which are in fact representations of the frontier electronic densities,3233 are appropriate to describe them.35 In summary, global descriptors S define molecular reactivity, while local descriptors f characterize the nucleophilic or electrophilic behaviour of specific regions within the molecule 2830 . Descriptors that combine both global and local properties make the link between reactivity and selectivity and are referred to as reactivity–selectivity descriptors. For instance, local softness sr = frS 22 is a reactivity–selectivity descriptor in which the Fukui function distributes the global softness all over the molecule, thus defining local softness at every point within the molecular topology. In particular, the

A. Toro-Labbé et al.

103

definition of local softness has made possible to propose a local version of the Pearson’s HSAB principle that is frequently used as an interpretative basis for reactivity–selectivity descriptors.36 Recently, a new local index that connects selectivity and reactivity has been defined as the variation of the global hardness due to a change in the external potential.37 More specifically, the index describes the response of the hardness when the molecule is under an external field. In this paper, we provide several examples of the use of this new reactivity–selectivity descriptor by revisiting few textbook’s classical organic chemical reactions. This chapter is organized as follows. In Section 2, the basic definitions of the DFT descriptors are reminded, and the master equations that allow defining the new index are presented. The formal relation between the new index and the PMH is also discussed in Section 2. Section 3 deals with the computational details of the theoretical calculations presented and discussed in Section 4 where some classical organic reactions are studied in the light of the new reactivity–selectivity descriptor. Section 5 contains some concluding remarks.

2. Theoretical background 2.1. General definitions The Hohenberg–Kohn theorems of DFT1516 state that a complete characterization of an N -particle wave function and energy requires the knowledge of the number of electrons N and the external potential r due to electron–nuclei interactions. In this context; the energy, a functional of the electron density, can be represented simultaneously as a function of N and a functional of r E ≡ E N r.14 This is the starting point of conceptual DFT to provide the theoretical framework for rationalizing chemistry since the descriptors of chemical reactivity and selectivity are response functions towards variation of N and r. The response to changes in the number of electrons produces global electronic properties that characterize the reactivity of the system, while the response to changes in the external potential produces local electronic properties that rationalize the selectivity concept. Assuming that the external potential remains constant during a variation of N , the response of the system is measured at first order by the chemical potential and at second order by the molecular hardness:2325 E = (1) N r and

=

2 E N 2

= r

N

(2) r

The variation of the energy with respect to the external potential is measured by the electronic density r 14 at first order and by the Fukui function fr 31 at second order. Both electronic density and Fukui function are local quantities:14 E (3)

r = r N

104


and

fr =

r N

=

r

r

(4) N

The electronic density provides the site reactivity information of ionic systems, while the Fukui function is better suited when dealing with neutral species. Owing to the discontinuity of the derivative of (4), two different Fukui functions can be defined by applying the finite difference approximation:8 + r + + ≈ L r (5) f r = = r N N r − r − − = ≈ H r (6) f r = N r r N At a point r f + r measures the reactivity towards a nucleophilic attack that results in an electron increase in the system;838 f − r measures the reactivity towards electrophilic attacks which results in an electron decrease in the system.838 The reactivity–selectivity descriptor fr is defined as:37 fr = f + r − f − r

(7)

such that when fr > 0 then the point r favours a nucleophilic attack, whereas if fr < 0 then the point r favors an electrophilic attack. Therefore, positive values of fr identify electrophilic regions within the molecular topology, whereas negative values of fr define nucleophilic regions. Since the Fukui functions are positive 0 ≤ fr ≤ 1, the numerical values of fr lie within the interval −1 1 −1 ≤ fr ≤ 1. The following normalization condition applies: frdr = 0 (8) This condition may provide a criterion to check the validity of the numerical values of Fukui functions.37

2.2. The principle of maximum hardness The index fr is related to the absolute hardness through the following equation:1437 fr fr = = (9) r N N r and the integral form of (9) can be written as: = fr rdr

(10)

Equation (10) shows that for a chemical reaction, it is possible to characterize the variation of the hardness of the reacting molecules at any point along the reaction


105

coordinate by monitoring simultaneously fr and r. Moreover, it has been shown that when an anionic/cationic species approaches the positive/negative region of fr, it produces an increase in the hardness of the molecule such that > 0, in agreement with the PMH.26 In contrast to this, when an anionic/cationic species approaches the negative/positive region of fr, it produces a decrease in the hardness.37

3. Computational details The geometries of the molecules under study were fully optimized at the HF/6-311G∗∗ level.39 Since the following analysis is based on the sign of fr, the numerical quality of fr is crucial to obtain the right selectivity trend of the systems. Then the Fukui functions f + r and f − r have been determined by using the spin density of the N + 1 and N − 1 systems, as suggested by Galván et al.40 All the calculations have been performed using the Gaussian03 package.41 The results are displayed in the figures with the following colour code: dark grey for electrophile regions fr > 0 and light grey for the nucleophile regions fr < 0.

4. Applications 4.1. Regio-selective addition to alkenes: The Markovnikov’s rule The addition of an electrophile compound to an asymmetrical alkene is so highly regioselective that it can be quoted as regio-specific.14 In its original version, the so-called Markovnikov’s rule 42 refers to the addition of hydrogen halides on alkenes as when an asymmetrical substituted alkene reacts with a hydrogen halide, the hydrogen is added to the carbon that has the greater number of hydrogen substituents, and the halide goes to the carbon having the fewer hydrogen substituents. This rule has been extended to any double bond within a molecule substituted by one or more electron-donating groups (EDG).143 The explanation of the high regio-selectivity can be given either through the stability of the carbocation formed or through the asymmetry of the HOMO of the alkene, though the interpretation using the FMO theory is sometimes not straightforward. Figure 1 displays the HOMO of propene (methyl ethene), it can be observed that both substituted and unsubstituted carbons are equally probable to be attacked by an electrophile. However, when propene reacts with a hydrogen halide, it is expected from the Markovnikov’s

Figure 1 Map of the HOMO orbital of propene at HF/6-311G** level

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rule that the halide, a nucleophile, goes to the carbon bearing the methyl group and the hydrogen to the terminal carbon. The decomposition of the molecular orbital HOMO expressed in the atomic basis set shows that the contribution of the p orbital localized on the carbon bearing the methyl group is 0.43 and the contribution of the p orbital localized on the unsubstituted carbon is 0.45, a very small difference that do not allow to decide unambiguously the actual site for electrophilic addition. However, on the basis of this small difference, it has been established that the hydrogen, an electrophile, will attack the unsubstituted carbon.14 The use of the reactivity–selectivity descriptor fr lifts the ambiguity. Figure 2 displays fr for propene, it can be seen that the unsubstituted carbon of propene is a nucleophile (fr < 0, yellow) that will react with an electrophile, the hydrogen of the halide. On the other hand, the substituted carbon is an electrophile (fr > 0, red) and will react with the halide, as expected from the Markovnikov’s rule. Discrimination between the nucleophilic and electrophilic power of the two carbons is very clear when using the reactivity–selectivity descriptor fr. The regio-selectivity of alkenes substituted by electron withdrawing groups (EWGs),144 such as cyano, carboxy or nitro groups, is expected to be opposite to that of propene. The nucleophile prefers binding to the carbon away from the EWG.45 Figure 3 shows the fr maps of two different alkenes monosubstituted by EWGs (carboxy and cyano groups). It can be seen that the substituted carbon is now a nucleophile (yellow) and the unsubstitued carbon is an electrophile (red). Thus whatever the substituent is, the selectivity of monosubstituted alkenes is correctly described by fr.

Figure 2 fr map of propene

Figure 3 fr map of alkenes monosubstituted by an EWG (carboxy and cyano groups)


107

4.2. Addition of borane to alkenes: an extension of the Markovnikov’s rule The hydroboration of alkenes is a very important way to convert alkenes in other organic groups.1 For example, the hydroboration of an alkene followed by the oxidation with hydrogen peroxide and sodium hydroxide produces an alcohol1 in a highly stereo- and regio-selective reaction. As can be seen in Figure 4, the hydroboration–oxidation of 1-methylcyclopentene produces only the trans-2-methyl-cyclopentanol with a yield of 86%.45 This result implies that the boron atom and the hydrogen atom are added to the double bond simultaneously on a syn mechanism.46 A concerted mechanism is invoked for these additions – it is shown in Figure 5.1 Boron attacks the carbon bearing the hydrogen atom, while a hydrogen is added to the second carbon involved in the double bond. At first sight, the Markovnikov’s rule do not hold in this case. Figure 6 displays the fr maps of borane and 1-methylcylopentene, boron appears in dark grey as an electrophile (fr > 0), whereas the three hydrogens are nucleophiles (fr < 0); this is in agreement with the observed relative electronegativity of boron and hydrogen: B < H. In 1-methylcylopentene, the substituted carbon is an electrophile (dark grey) and the unsubstituted carbon is a nucleophile (light grey). The matching between electrophilic and nucleophilic regions (dark ↔ light) of the two molecules predicts that boron will prefer binding to the unsubsituted carbon, while hydrogen will prefer binding to the substituted carbon. On the other hand, the overlap between the appropriate regions of fr belonging to both alkene and borane proposes a concerted mechanism in which the boron and the hydrogen atoms attack simultaneously on the same side of the double bond forming the cyclic intermediate as shown in Figure 5. In summary, the regio-selectivity within the frame of the Markovnikov’s rule is correctly described by fr; on the other hand, the stereo-selectivity is also well characterized through the overlap between the appropriate regions of the fr descriptor of the molecular partners.

Me

H

Me H

1) BH3

OH

2) NaOH 86 %

Figure 4 Scheme of hydroboration of methylcyclopentene

H

H Me

B H

Figure 5 Transition structure of the hydroboration of methylcyclopentene

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Figure 6 fr map of borane and 1-methylcylopentene

4.3. Regio- and stereo-selectivity of Diels-Alders (DA) reactions Another kind of reaction involving alkenes in which a concerted mechanism is often invoked are the cyclo-additions.13 DA reactions are one of the most important cycloaddition reactions in organic chemistry.3 Indeed, since cyclation and polymerization are competitive reactions, it is very useful and conceptually important to find a way to produce ring compounds with regio- and stereo-selectivity controlled at will. The regio-selectivity of the DA reaction is well known, according to the electron-donating or -withdrawing character of the functionalizing group on the diene or on the dienophile, the main product of the DA reaction will be an ortho- or para- oriented cyclo-adduct. Four types of products classified in terms of its regioselectivity have been encountered in DA reactions – they are summarized in Figure 7. In the following paragraphs, we illustrate the four types of DA reactions by using as EDG the methoxy group OCH3 and as EWG the cyano group (CN). Since DA reactions involve mainly the carbons C1 C4 of the diene and C1 C2 of the dienophile (Figure 8), the following discussion is focused on those carbons. Figure 9 displays the fr density maps for two reactions – in the left panel the diene and the dienophile are monosubsituted in position 1 by OCH3 and CN and in right panel the diene and the dienophile are monosubsituted in position 1 by CN and OCH3 . In both cases, the carbon C1 of the diene and the C1 of the dienophile have opposite behaviour. When the former is electrophile/nucleophile, the latter is nucleophile/electrophile. Similarly carbons C4 of the diene and carbon C2 of the dienophile present opposite behaviour, then when the nucleophilic part of one molecule reacts with the electrophilic part of the other molecule, it can be observed that the fr descriptor match perfectly to produce in both cases the ortho adduct. If the subsituent groups are in position 2, then the electrophilic and nucleophilic regions are exchanged in both the diene and the dienophile, as shown in Figure 10. The para adduct is produced in both cases because C1 of the dienes which is a nucleophile/electrophile reacts with the C2 of the dienophiles which is an electrophile/nucleophile. These theoretical predictions are in perfect agreement with the experimental results summarized in Figure 7.


109

Type A EDG

EDG EWG

EWG

+

Type B EWG

EDG

EDG

+ EWG

Type C EWG

EWG EDG

EDG

+

Type D EDG

EWG

EWG

+ EDG

Figure 7 Main products of Diels–Alder reactions

More complex cases can also be handled correctly by fr; indeed the regioselectivity of the DA reaction involving a diene with a hetereoatom such as sulphur, can also be predicted.47 The reaction of (E)-4-aminobut-3-ene-2-thione with acroleine produces 4-amino-3,4-dihydro-6methyl-2H-thiopyran-3-carbaldehyde (ADMTC)47 . This is the ortho adduct, as can be seen in Figure 11. The fr maps of both reactants are displayed in Figure 12, and from the matching of nucleophilic and electrophilic regions of the partner molecules, the ortho adduct is correctly predicted. Primary and secondary interactions. Figure 13 displays the fr maps for the interaction of cyclopentadiene and cyclopentene; here only primary interactions are observed between the fr electrophilic and nucleophilic regions of the diene and the dienophile. Therefore, both endo and exo stereoisomers are equally expected. However, in spite of the steric hindrance, in most cases the DA reaction produces the endo stereoisomer.1

110


X

C2

C1

Y

C′1 +

C3

X

C1 +

C′2

Y

C′1

C2 C3

C′2

C4

C4

Figure 8 Numbering of diene and dienophile with X = OCH3 /CN and Y = CN/OCH3

Figure 9 fr map of diene and dienophile monosubstituted in position 1 by EDG and EWG

Figure 10 fr map of diene and dienophile monosubstituted in position 2 by EDG and EWG

NH2

NH2

O

O

+

H3C

S

(E)-4-Aminobut-3-ene-2-thione

H3C Acroleine

S

4-Amino-3,4-dihydro-6-methyl-2H-thiopyran-3-carbaldehyde

Figure 11 Diels–Alder reaction between 4-aminobut-3-ene-2-thione and acroleine


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Figure 12 fr map of 4-aminobut-3ene-2-thione and acroleine

Figure 13 Primary interactions between fr of cyclopentadiene and cyclopentene

Figure 14 Primary and secondary interactions between frontier orbital of two cyclopentadienes

This amazing stereo-selectivity has been rationalized by Woodward and Hoffmann48 using the FMO theory, it involves secondary interactions between nonreacting parts of the two molecules (see Figure 14). An analysis based on the overlap between fr of both reactants may explain the stereo-selectivity in terms of the secondary interactions that take place between nonreacting moieties. The fr map of two cyclopentadiene molecules is displayed in Figure 15. The primary interactions are identified by full lines, while the secondary interactions are displayed by dashed lines. It can be observed that the secondary interactions between electrophilic and nucleophilic regions stabilize the adduct that leads to the production of the endo isomer.

112


Hydrogen linked to sp3 carbon

Figure 15 Primary and secondary interactions between the fr of two cyclopentadienes

4.4. Basicity of cyclopentadiene The dark grey colour on the hydrogen bonded to the sp3 carbon indicates that they are electrophiles (Figure 15), and so nucleophiles or bases can attack them. In general, hydrogen bonded to sp3 carbon atoms should not have a strong reactivity since the electronegativity of carbon and hydrogen atoms is quite close to each other49 . However, comparison of the pKa of cyclopentadiene1 pKa = 16 with the average pKa of hydrocarbons pKa = 40 confirms the intrinsic acidic behaviour of the hydrogen atoms bonded to the sp3 carbon. If a proton is removed from the cyclopentadiene (acid/base reaction), the resulting product is the anion cyclopentadienyl whose stability is explained by its aromaticity.50 Indeed, the electronic pair on the former sp3 carbon is conjugated with the two other double bonds, giving rise to a cyclic compound with six electrons that, according to the Huckel’s rule, constitutes an aromatic system.14

4.5. Regio-selectivity of the electrophilic aromatic substitution (EAS) For a monosubstituted benzene, according to the donor or withdrawal character of the substituent, the EAS will be oriented in ortho, meta or para position with respect to the substituent. The molecules chosen to test the predicitive capabilities of fr in studying electrophilic aromatic substitution are aniline, phenol, benzaldehyde and cyanobenzene, whose reactivity and orientations are well known.14 Since NH2 and OH are EDGs they will orientate the EAS in positions ortho and para of the phenyl moiety. On the other hand, since CHO and CN are EWGs they are meta-orienting groups. Thus, these groups make the aromatic ring very poor in electrons with respect to benzene, and, therefore, they strongly deactivate the ring, i.e. reactions proceed much slower in rings bearing these groups compared to the same reactions in benzene. Figure 16 displays a map of the nucleophilic/electrophilic behaviour of the different sites within the molecule according to the fr descriptor. For both aniline and phenol, the regions with fr < 0 (light grey) where an electrophilic reaction should take place are located in positions ipso, ortho and para. In both molecules, the meta position with fr > 0 is reserved for a nucleophilic reaction. For benzaldehyde and cyanobenzene, the result is opposite, ortho and para positions are dark grey coloured, while the meta position is light grey. In summary, regarding the electrophilic aromatic substitution, the fr index indicates that ipso, ortho and para positions are active sites in both


113

Figure 16 fr map of aniline, phenol, cyanobenzene and benzaldehyde

aniline and phenol, while the meta position is the only reactive site in benzaldehyde and cyanobenzene. These results are in perfect agreement with experimental results.51

4.6. Reactivity of ketones and aldehydes The ketones and aldehydes have basically two different sites where a reaction with a nucleophile can take place: the carbon of the carbonyl group and the hydrogen atom bonded to the carbon.14 If a nucleophile reacts with the carbon of the carbonyl group several products can be formed.14 For instance, if the nucleophile is a hydride anion, a reduction of the double bond occurs.52 If a base reacts with the hydrogen atom bonded to the carbon, the result is an enolate anion.53 Figure 17 summarizes these possibilities. The FMO perspective indicates that the probability of an attack by a nucleophile on the carbon atom is higher than that on the oxygen atom. The fr maps of a ketone and an aldehyde are displayed in Figure 18 where it should be noted that the carbonyl carbon is the most electrophilic atom of the molecules. Although in a lower extent, the hydrogen atoms of the methyl groups are also electrophiles; this explains the fact that these hydrogens react easily with bases to produce enolate anions.154 Finally, the Nu H

R″

C

Nu O

O C

R R′

H

C C

R R'

R″ Base

–

O C C

R″

R R′

Figure 17 Schematic view of the reactivity of ketones and aldehydes

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Figure 18 fr map of propanone and ethanal

behaviour of the carbonyl oxygen is also quite well characterized as nucleophilic. In summary, the three main reactive sites of aldehydes and ketones are correctly described by fr.

4.7. Reactivity of enolate anions Many reactions of ketones and aldehydes as for instance the aldol condensation or the haloform reaction 55 are initiated by enolate anions. In these cases, a proton linked to the carbon is removed, producing an enolate that reacts with electrophiles 14 . It has been found that in gas phase, the oxygen is the most reactive site, whereas the carbon is mostly attacked in condensed phase. =CO-CH3 − ; it Figure 19 displays the map of fr for the enolate anion H2 C= can be observed that fr identifies both the carbonyl oxygen and the carbon as being the most reactive sites for electrophilic attacks. Although both sites present similar characteristics in terms of fr, their reactivities towards electrophiles are correctly characterized when taking into account the medium where the reaction actually takes place. In condensed phase, soft–soft interactions are predominant, and they are correctly described by the Fukui functions themselves and fr;56 the condensed values of the fr descriptor57 (fo = −055 and fc = −095) indicate that the carbon is the most reactive site for an electrophilic attack. In contrast to this, in gas phase, the alkylation is driven by hard–hard interactions which are of electrostatic nature and therefore are well described through the atomic charges.58 The Mulliken charges on the oxygen is qo = −069 and that on the carbon is qc = −050; this indicates that in gas phase the preferred site for an electrophilic attack will be the oxygen. So, even in situations where it might be difficult to characterize local selectivity among sites presenting similar

Figure 19 Map of the fr descriptor of the enolate anion H2 C = CO-CH3 −


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reactive behaviour, the fr descriptor provides a valuable tool for a first identification of the sites. Application of extra chemical criteria should lead to the right result.

5. Concluding remarks In this chapter 7 the reactivity–selectivity descriptor fr that characterizes the reactive behaviour of the different topological regions within a molecule has been used to study the reactivity and selectivity of various classical organic reactions. It has been shown that the intrinsic electrophilic/nucleophilic character of the different molecular regions determined through fr predicts correctly the regio- and stereo-selectivity of these reactions. It is important to mention that in contrast to the FMO theory where the couple nucleophile–electrophile needs to be identified to perform the molecular orbital analysis, the use of the reactivity–selectivity descriptor fr does not need an a priori classification of the reacting molecules since all molecules present topological regions in which both nucleophilic and electrophilic characters are locally allowed. It has been stressed that in the rationalization of the mechanism of a chemical reaction characterized by soft–soft interactions, the only thing that matters is the matching of the electrophilic region of one molecule with nucleophilic region of the other; this complies with the PMH 26 and with the HSAB postulate.11 In the case of hard–hard interactions extra chemical considerations should be added for a correct identification of the reactive sites.

Acknowledgements The authors wish to acknowledge the financial support from FONDECYT through project N 1060590, FONDAP N 11980002 (CIMAT) and Programa Bicentenario en Ciencia y Tecnología (PBCT), Projecto de Inserción Académica N 8. S.G-O acknowledges the financial support from Núcleo Milenio de Mecánica Cuántica Aplicada y Química Computacional, Código P02-004-F.

References 1. (a) Anslyn EV, Dougherty D: Modern Physical Organic Chemistry, California, 2006, University Science Books (b) March J: Advanced Organic Chemistry, Ed 4, New York, 1992, Wiley & Sons, Inc. 2. Carruthers W: Cycloaddition Reactions in Organic Synthesis, Oxford, 1990, Pergamon Press plc. 3. Boger DL, Weinreb SM: Hetero Diels–Alder Methodology in Organic Synthesis, London, 1987, Academic Press. 4. Carey AF, Sundberg RJ: Advanced Organic Chemistry. Part A: Structure and Mechanisms, Ed 4, New York, 2000, Springer. 5. Szabo A, Ostlund NS: Modern Quantum Chemistry, Introduction to Advanced Electronic Structure Theory, Ed 2, New York, 1989, McGraw-Hill. 6. Levine IN: Quantum Chemistry, Ed 5, New York, 2000, Prentice Hall. 7. McQuarrie DA: Quantum Chemistry, Oxford, 1983, Oxford University Press.

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8. (a) Fukui K, Yonezawa T, Shingu H, J Chem Phys 20:722, 1952 (b) Fukui K, Yonezawa T, Nagata C, Shingu H, J Chem Phys 22:1433, 1954 (c) Fukui K, Science 217:747, 1982. 9. Fukui K: Theory of Orientation and Stereoselection, Berlin, 1970, Springer. 10. Woodward RB, Hoffman R, Angew Chem Int Ed Engl 8:781, 1969. 11. Pearson RG: Hard and Soft Acid and Bases Principle in Organic Chemistry, Stroudsberg, PA, 1973, Dowden, Hutchinson & Ross Inc. 12. Hammond GS, J Am Chem Soc 77:334, 1955. 13. (a) Solá M, Toro-Labbé A, J Phys Chem A 103:8847, 1999 (b) Bulat F, Toro-Labbé A, J Phys Chem A 107:3987, 2003. 14. Parr RG, Yang W: Density Functional Theory of Atoms and Molecules, New York, 1989, Oxford University Press. 15. Hohenberg P, Kohn W, Phys Rev B 136:864, 1964. 16. Kohn W, Sham LJ, Phys Rev A 140:1133, 1965. 17. Parr RG, Yang W, Annu Rev Phys Chem 46:701, 1995. 18. Ayers P, Parr RG, J Am Chem Soc 122:2010, 2000. 19. Kohn W, Becke AD, Parr RG, J Phys Chem 100:12974, 1996. 20. De Proft F, Geerling P, Chem Rev 101:1451, 2001. 21. Geerling P, De Proft F, Langenaeker W, Chem Rev 103:1793, 2003. 22. Yang W, Parr RG, Proc Natl Acad Sci USA 82:6723, 1985. 23. Parr RG, Donnelly RA, Levy M, Palke WE, J Chem Phys 68:3801, 1978. 24. Pearson RG: Chemical Hardness: Applications from Molecules to Solids, Weinheim, 1997, Wiley-VCH Verlag GMBH. 25. Parr RG, Pearson RG, J Am Chem Soc 105:7512, 1983. 26. Pearson RG, J Chem Educ 64:561, 1987. 27. Datta D, J Phys Chem 96:2409, 1992. Chattaraj PK, Proc Indian Natl Sci Acad 62:513, and references therein, 1996. 28. (a) Gutiérrez-Oliva S, Letelier JR, Toro-Labbé A, Mol Phys 96(1):61, 1999 (b) Toro-Labbé A, J Phys Chem A 103:4398, 1999. 29. (a) Chattaraj PK, Lee H, Parr RG J Am Chem Soc 113:1855, 1991 (b) Chattaraj PK, Schleyer P v R, J Am Chem Soc 116:1067, 1994. 30. (a) Pérez P, Toro-Labbé, Contreras R, J Phys Chem 103:11246, 1999 (b) Chattaraj PK, Fuentealba P, Jaque P, Toro-Labbé A, J Phys Chem 103:9307, 1999. 31. Parr RG, Yang W, J Am Chem Soc 106:4049, 1984. 32. Klopman G: Chemical Reactivity and Reaction Paths, New York, 1974, Wiley. 33. Klopman G, J Am Chem Soc 90:223, 1968. 34. Melin J, Aparicio F, Subramanian V, Galván M, Chattaraj PK, J Phys Chem A 108:2487, 2004. 35. Gazquez JL, Mendez F, J Phys Chem 98:4591, 1994. 36. Li Y, Evans JNS, J Am Chem Soc 117:7756, 1995. 37. Morell C, Grand A, Toro-Labbé A, J Phys Chem A 109:205, 2005. 38. (a) Chattaraj PK, Pérez P, Zevallos J, Toro-Labbé A, J Phys Chem A 105:4272, 2001 (b) Chattaraj PK, Gutiérrez-Oliva S, Jaque P, Toro-Labbé, Mol Phys 101(18):2841, 2003. 39. (a) Hehre WJ, Ditchfield R, Pople JA, J Chem Phys 56:2257, 1972 (b) Francl MM, Pietro WJ, Hehre W, Binkley JS, Gordon MS, Frees DJ, Pople JA, J Chem Phys 77:3654, 1982 (c) Hariharan PC, Pople JA, Theor Chim Acta 28:213, 1973. 40. Galván M, Vela a, Gázquez JL, J Phys Chem 92:6470, 1988. 41. Frisch MJ, Trucks GW, Schlegel HB et al., Gaussian, Inc, Pittsburgh PA, 2003. 42. Markovnikov W, Justus Liebigs Ann 153:256, 1870. 43. (a) Isenberg N, Grdinic M, J Chem Educ 46:601, 1969 (b) Grdinic M, Isenberg N, Intra-Sci Chem Rep 4:145, 1970. 44. (a) Simpkins NS, Tetrahedron 56:6951, 1990 (b) Fuchs PL, Braish TF, Chem Rev 86:903, 1986 (c) Bernasconi CF, Tetrahedron 45:4017, 1989.


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45. Brown HC, Zweifel J, J Am Chem Soc 83:2544, 1961. 46. (a) Brown HC, Vara-Prasad JVN, Zee SH, J Org Chem 51:439, 1986 (b) Brown HC, Sharp RC, J Am Chem Soc 88:5851, 1966 (c) Pelter A, Smith K, Brown HC: Borane Reagents, New York, 1988, Academic Press. 47. Pradére JP, N’Guessan YT, Quiniou H, Tonnard F, Tetrahedron 31:3059, 1975. 48. (a) Woodward RB, Hoffmann R, J Am Chem Soc 87:4388, 1965 (b) Alston PV, Ottenbrite RN, Cohen T J Org Chem 43:1864, 1978. 49. (a) Hafner K, Kafber KH, Konig C, Kreuder M, Ploss G, Schulz G, Sturm E, Vopel KH, Angew Chem 75:35, 1963 (b) McLean S, Haynes P, Tetrahedron 21:2343, 1965. 50. (a) Webster OW, J Org Chem 32:39, 1967 (b) Bordwell FG, Drucker GE, J Org Chem 46: 632, 1981. 51. Ingold CK: Structure and Mechanism in Organic Chemistry, Ed 2, Ithaca, New York, 1969, Cornell University Press. 52. Seyden-Penne: In Reduction in Organic Chemistry, Chichester, 1984, Ellis Horwood. 53. D’Angelo J, Tetrahedron 32:2979, 1976. 54. Fleming I: Frontier Orbitals and Organic Chemical Reactions, London,1976, John Wiley & Sons Inc. 55. Nielsen AT, Houlihan WJ, Org React 16:1, 1968. 56. (a) Chattaraj PK, González-Rivas N, Matus MH and Galvan M, J Phys Chem A 109:5602, 2005 57. Bulat FA, Chamorro E, Fuentealba P, Toro-Labbé A, J Phys Chem A 108:342, 2004. 58. Hocquet A, Toro-Labbé A, Chermette H, J Mol Struc 686:213, 2004.



Chapter 8

The average local ionization energy: concepts and applications Peter Politzer and Jane S. Murray Department of Chemistry, University of New Orleans, New Orleans, LA 70148, USA

1. Orbital ionization energies The ionization energy I of an N-electron atom or a molecule having energy E is a well-defined property: I = EN − 1 − EN

(1)

I has been measured experimentally at a high level of accuracy for a large number of atoms and molecules [1]. For molecules, it can be important to distinguish between the adiabatic and the vertical ionization energies. The former corresponds to the neutral molecule and the ion, both being in the ground states, which may or may not have similar geometries. The latter refers to the situation in which the positions of the nuclei are the same in the ion, as in the neutral molecule, which may not be the ion’s ground vibrational state. Our interest shall be in vertical ionization energies. Computationally, either I can be obtained via (1) by evaluating the appropriate EN − 1 and EN . However, another approach is used very frequently. In Hartree– Fock (HF) theory, it follows directly from the formalism that the vertical ionization energy Ii of any electron i would equal the negative of its orbital energy i if all of the orbitals of the system were unaffected by the loss of the electron. Koopmans’ theorem assures the stability of the one from which the electron is lost [2,3], and thus the approximation Ii ≈ −i

(2)

is a common one at the HF level. Then, the molecular vertical ionization energy I corresponds to i for the highest occupied orbital. 119

120

Average local ionization energy

In reality, the remaining orbitals do undergo some changes when an electron is removed from one of them. By ignoring this, which stabilizes the positive ion, (2) overestimates Ii . On the other hand, HF theory does not take account of electronic correlation, the inclusion of which would lower the energies of both the neutral system and the ion, but more so the former, since it has a larger number of electrons. Thus, neglecting correlation causes Ii to be underestimated. These two errors accordingly cancel to some extent. For some assessments of the accuracy of (2), see Politzer and Abu-Awwad [4] and Maksic and Vianello [5]. The physical significance, if any, of the orbital energies in Kohn–Sham density functional theory has been the subject of considerable analysis and discussion [6–19]. Perdew et al. concluded that (2) is true for the highest occupied exact Kohn–Sham orbital, but that nothing can be stated rigorously concerning the others [7–11,13]. Kleinman has dissented concerning I = −highestKS [12]; however it was supported by Harbola [15]. Gritsenko et al. [18,19] have argued that (2) applies to all of the Kohn–Sham orbital energies when computed at a high level of accuracy, giving particularly good estimates of Ii for the valence orbitals; they have also presented a spin-density functional analogue of Koopmans’ theorem [20,21]. In practice, it has been found that HF and typical Kohn–Sham procedures (e.g. BP86, B3PW91, etc.) produce valence orbital energies having magnitudes that tend to be larger and smaller, respectively, than the experimental ionization energies of the electrons [4,8,10,14,16]: iKS < Ii < iHF

(3)

The iKS , including that for the highest occupied orbital, are likely to be 2–3 eV less than the Ii , deviating more than the iHF . However, the iKS show the interesting feature that, for a given exchange/correlation functional, the difference Ii − iKS is fairly uniform for all of the valence orbitals in a molecule. This suggests that (a) the error is somewhat systematic and (b) the relative magnitudes of the iKS should be physically meaningful. It could also be inferred that it should be possible to devise schemes for converting the magnitudes of normal Kohn–Sham orbital energies into good approximations of electronic ionization energies. This has indeed been done, for example by Jellinek and Acioli [22]. The same has been achieved more empirically [17] by changing appropriately the weighting parameters in Becke’s hybrid exchange/correlation functional [23].

2. Local ionization energies The preceding discussion has related ionization energies to orbitals, which is what has usually been done. An alternative approach, which has also proven to be very useful, is to associate ionization energies with specific sites within the system, i.e. a local ionization energy [24]. It is this latter property that shall be the focus of this chapter. We introduced the concept of an average local ionization energy Ir within the framework of HF theory [24]. It is defined by, Ir =

i ri i

r

(4)

P. Politzer and J. S. Murray

121

In (4), r is the total electronic density, while i r is that of the ith occupied orbital. On the basis of (2), we interpret Ir as the average energy required to remove an electron at the point r in the space of a system of nuclei and electrons. Gal and Nagy pointed out that (4) can be written in a slightly modified form [25], r =

i ri i

r

(5)

where r is viewed as the average local orbital energy. This makes plausible the observed parallels between Ir and the local temperature Tr [25,26], which is related to the kinetic energy per electron at the point r [27]. Ir and Tr show similar behaviour for atoms [25,26] as well as for molecules [25], except in the bond regions.

3. Ir and atomic shell structure One of the most characteristic and fundamental features of atoms is their shell structure. This is the basis for the periodicity in their properties that has facilitated establishing a significant degree of order in a vast array of laboratory observations. The importance of shell structure has stimulated numerous efforts to relate it in some manner to atomic electronic densities. It has been known for some time that r itself does not serve this purpose, since it decreases monotonically with radial distance from the nucleus [28–32]. An alternative (noting that the atom’s electronic density is spherically averaged, i.e. r = r [33]) is the radial density, Dr = 4r 2 r. This is necessarily zero at the nucleus but then goes through one or more maxima and intervening minima, depending upon the number of electrons in the atom [34,35]. It is tempting to interpret these minima as marking the boundaries between shells, and this has indeed been investigated in detail [34–39]. However, this straightforward approach does not satisfactorily distinguish between successive shells starting with the M, N pair, presumably due to interpenetration [40]. Analogous problems are encountered when 2 r [41,42] and r/r [43] are tested as indicators of shell structure. A generally more successful approach involves the quantity Vr/r, where Vr is the electrostatic potential due to the nucleus and electrons of the atom, Vr =

Z r dr − r r − r

(6)

with Z being the nuclear charge. It was observed some time ago [44] that Vr/r has maxima at those radial distances at which the radial density Dr has minima. This is particularly noteworthy because both Vr and r are monotonically decreasing [28]. Sen et al. have confirmed that Vr/r, which they label the average local electrostatic potential, does separate the outer shells and also predicts approximately their expected occupancies [45–47]. Kohout and Savin [48] and Pacios and Gómez [49] have reported, respectively, that the electron localization function and a gradient term in the Fukui function are effective in demonstrating shell structure. It is interesting to note that both of these depend upon

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the quantities r 2 /r and 2 r, and thus can be viewed as related to the local kinetic energy density [11,25,43,48,50] and hence to the local temperature Tr [25–27]. Since, as already mentioned, this behaves for atoms in a manner similar to the average local ionization energy, it might be anticipated – in retrospect – that both Tr and Ir will reflect shell structure. Indeed they do [26,51], as was in fact anticipated by Ghosh and Balbás’ study of krypton [50]. Without being aware of these links, we found in 1991 that Ir for atoms decreases in a roughly stepwise fashion with radial distance from the nucleus [51]. There are alternating regions of slowly and rapidly decreasing Ir. If the points of inflection are taken to be the boundaries between shells, then the expected number of these is obtained for most atoms, and their integrated occupancies are close to the formal ones. (Interpenetration of shells is always a factor among the transition elements [40].) Sen et al. have pointed out a few atoms in the second transition series for which the N,O separation does not occur [38]. This may be a consequence of the level of the HF wave functions. This study verified that the average local ionization energy is relatively constant within an atomic shell, that this can be used to distinguish the shell and that Ir as defined by (4) is capable of revealing these features. This is relevant to the application of Ir to be discussed in the next section. The average local ionization energy and the average local kinetic energy (or local temperature) are able to resolve atomic shell structure, as is the ratio Vr/r. Does there exist some connection? The answer is yes; Ghosh and Balbás pointed out [50] that this is provided by the Euler equation of density functional theory [11], which relates T r /r to Vr, where T r is the kinetic energy functional. An interesting point is that it is the properties related to energy rather than electronic density that are most effective with respect to shell structure.

4. Ir and electronegativity Pauling introduced his electronegativity scale in 1932 [52,53], although the concept was already mentioned by Berzelius nearly a century earlier (cited by Allen [54]). Pauling’s reasoning was in a molecular framework: The difference between the energy of a heteronuclear bond and the average of the corresponding homonuclear ones is due to the ionicity of the former and can therefore be expressed quantitatively in terms of the difference in the electronegativities of the two atoms. During the past 70 years, there have been numerous efforts to, in effect, reproduce Pauling’s scale, but using only the properties of the individual atoms. Some of these electronegativity formulations are at least in a molecular context; several, for example, seek to estimate the attractive potential or force felt by an electron at the atom’s covalent radius [55–58], thus trying to satisfy Pauling’s description of electronegativity as ‘the power of an atom in a molecule to attract electrons to itself’ [59]. Definitions based on valence-state ionization potentials and electron affinities [60–62] are also viewing the atom in a molecular environment. In contrast, other approaches to electronegativity deal with the atom in its ground state. One of the most prominent of these is due to Parr et al. [11,63]. In the course of their development of density functional theory, they equated the electronegativity


123

to the negative of the chemical potential , and approximated both in terms of the ground-state ionization potential I and electron affinity A: = − ≈

I +A 2

(7)

Another ground-state representation of electronegativity, which has achieved considerable acceptance, was proposed by Allen [54,64–66], who argued that it should be the average ionization energy of the valence electrons of the free atom. This was initially designated as the ‘spectroscopic electronegativity’ [64], but later as the ‘configuration energy’ CE [54,65,66]. For the representative (i.e. non-transition) elements, CE =

n s s + np p ns + n p

(8)

in which ns and np are the numbers of s and p valence electrons and s and p are the differences in the multiplet-averaged total energies of the atom and the appropriate monopositive ion in their ground states. These are to come from spectroscopic measurements. Allen et al. have demonstrated that the CE scale is consistent with various chemical and physical properties [54,64–66] and that it correlates well with the updated Pauling and the Allred-Rochow. Extending (8) to transition elements poses an obvious problem, because it is often not clear how many s and d electrons should be included. Allen et al. addressed this by means of a computational technique that yields fractional subshell occupancies [66], which can then be combined with spectroscopic data. However, this ignores the role of interpenetration involving lower-lying subshells, which can be significant [40,67] for representative elements as well. We have recently pointed out that Ir affords a means of avoiding the need to identify and count valence electrons [68]. If Ir is computed on an outer surface of the atom, then it is fully consistent with Allen’s focus upon the average ionization energy of the valence electrons but without requiring that these be specified. We chose to compute Ir on the surface defined by the 0.001 au (electrons/bohr3 ) contour of r; for molecules, this typically encompasses at least 98% of the electronic charge [69]. The Ir, now labelled I S = 0001, was obtained for H–Kr with Clementi’s extended-basis-set HF wave functions [70], except for hydrogen, for which we took the exact i and i r. It is important to note that I S = 0001 is invariably larger than i for the highest occupied orbital (except for the special case of hydrogen). This is due to the contributions of lower orbitals, which do have some electronic density at the r = 0001 au surface. This shows that I S = 0001 inherently takes interpenetration of subshells into account. Our I S = 0001 correlates very well with Allen’s CE, which are his electronegativities. For the representative elements through Kr, R2 = 0976; when the first transition series is included, R2 = 0959 [68]. The largest discrepancy is for zinc, which we find to be more electronegative – on a relative basis – than do Allen et al. [66]. This can be explained by noting that they take its valence shell to consist of only the 4s electrons, whereas we find significant contributions from lower subshells. The choice of the atomic surface on which to compute Ir is somewhat arbitrary, but the observation that Ir is approximately constant within a shell, discussed in the preceding section, indicates that other outer contours of r, e.g. 0.002 or 0.0015 au,

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would predict the same trends. The radii of the r = 0001 au surfaces [68] confirm that these do fall within the outermost shells of the atoms [51]. Furthermore, we have shown in studies of molecules that different outer contours produce similar trends in predicted properties [71]. The use of I S = 0001 to represent electronegativity is fully within the spirit of Allen’s approach and is simply an alternative means of implementing it. The procedure is purely computational, all atoms are treated in the same straightforward manner, and – most important – there is no need to enumerate valence electrons.

5. Ir and molecular reactivity towards electrophiles In analysing, interpreting and predicting the reactive behaviour of molecules, we have made extensive use of two local properties: the average local ionization energy Ir, defined by (4), and the electrostatic potential Vr that is produced by the nuclei and electrons, Vr =

A

r dr ZA − RA − r r − r

(9)

In (9), ZA is the charge on nucleus A, located at RA . Vr is a physical observable, which can be obtained experimentally, by diffraction methods [72–74], as well as computationally. Its magnitude and sign at any point r depend upon the relative contributions of the nuclear and the electronic potentials. In applying Ir and Vr to molecular reactivity, we have found it expedient to compute them on an outer surface of the molecule, since this is what is encountered by an approaching reactant. Molecular surfaces have often been defined by means of intersecting spheres centred on the nuclei [75–78], having perhaps the respective van der Waals radii. We prefer to again follow Bader et al. [69] and use the 0.001 au contour of r. This has the advantage that it reflects features such as lone pairs and strained bonds, which are specific to the molecular charge distribution. The notations I S r and VS r indicate that the properties have been calculated on the molecular surface. We normally determine VS r at the HF STO-5G∗ level; a minimum basis set has been found, by us and by others [79,80], to be quite satisfactory for this purpose. For I S r, on the other hand, a larger basis set is needed, e.g. 6-31G∗ [4]. The HF procedure has usually been utilized for I S r, but it has been shown that a Kohn–Sham density functional technique, in general, does equally well [81,82]; the magnitudes are different, but the trends (which is what is important for establishing relative site reactivities) tend to be the same. It has been our experience that I S r and VS r play different but complementary roles with respect to molecular reactivity [71,83–85]. VS r is effective for treating noncovalent interactions, which are primarily electrostatic in nature [74,86–89]. For instance, a variety of condensed-phase physical properties – boiling points, critical constants, heats of phase transitions, solubilities and solvation energies, partition coefficients, surface tensions, viscosities, diffusion constants and densities – can be expressed quantitatively in terms of one or more key features of VS r, such as its maximum and minimum, average deviation, positive and negative variances, etc. [80,90–92]. Hydrogen bond donating


125

and accepting tendencies correlate well with the magnitudes of, respectively, the most positive and the most negative values of VS r VSmax and VSmin [93]. The electrostatic potential has also been invoked successfully for ‘molecular recognition’ processes, as between an enzyme and substrate or drug and receptor, in which interaction is promoted by mutually appropriate patterns of positive and negative potentials [73,74,79,80,94–96]. However, for interactions that involve covalent bond formation/disruption and/or charge transfer, VS r is much less reliable; the electrostatic aspect is often not dominant. A good example is electrophilic attack on monosubstituted benzene derivatives, C6 H5 X. This might be expected to take place at the site(s) of the most negative VSmin of each molecule. For X = NH2 , OH, F, Cl and NO2 , the electrostatic potentials on the molecular surfaces are shown by Politzer et al. [85]. In each case, there is a negative region, with a local VSmin , above and below the ring, reflecting the electrons, although it is quite weak for the strongly electron-withdrawing X = NO2 . In no instance does it clearly indicate certain carbons to be favoured for electrophilic substitution. Furthermore, the overall most negative VSmin is invariably associated with the substituent X. (For phenol, for example, the HF/6-31G∗ VSmin of the OH is −304 kcal/mole; those of the ring are −193 kcal/mole [85].) On this basis alone, it would be anticipated that an approaching electrophile would interact with the X group. Yet it is well known that electrophilic substitution generally occurs on the ring [97], at sites that depend upon the nature of X. To explain this, we turn to I S r. Its relationship to molecular reactivity is based on the premise that the lowest values of I S r, the I Smin , reveal the locations of the least tightly bound, most reactive electrons, most readily available for transfer or sharing with an electrophile. For the C6 H5 X molecules, the lowest I Smin are found to be near ring carbons [24,81,85,98], fully consistent with the known ortho-, para- or meta-directing tendencies of X, even the unusual meta, para combination of X = NH+ 3 . Comparison of the I Smin with those of benzene also indicates, correctly, whether the ring has been activated or deactivated towards electrophiles. These may initially be attracted to the vicinity of the substituent X, which has the most negative electrostatic potential; however the electrostatic interaction is not sufficient to compensate for the I Smin of X being considerably higher than those of the ortho, para or meta carbons of the ring. (Looking again at phenol, the HF/6-31G∗ I Smin of the OH is 15.8 eV; those of the ortho and para carbons are 11.6 eV [85].) Protonation is probably the electrophilic process most likely to occur at X, due to the high electron affinity of H+ (13.6 eV [1]); it is especially favoured when X = NH2 [99], for which I Smin is relatively low [81,85]. We have also extended these analyses to heterocyclic aromatic systems, such as azines and N-oxides [98,100–102]. In these, it was the influence of the nitrogen(s) or N+ –O− upon the remainder of the ring that was being examined by looking for the I Smin ; the results were in agreement with what has been observed. Given the success of I S r in predicting the directing and activating/deactivating behaviour of substituents on benzene rings, it was natural to investigate possible relationships between I S r and the Hammett constants m and p as well as Taft’s inductive constant I [103,104]. These are well-established experimentally-based measures of the electron-donating and -withdrawing powers of substituents; the first two focus on aromatic systems, while the third is more general. There are indeed good-to-excellent linear correlations between these constants and the I Smin at appropriate molecular sites [24,81,83,98,105], which have allowed us to predict the former for groups for which they were not known [24,102].

126


It seems reasonable to anticipate that enthalpy of protonation, Hpr , and pKa can be related to I Smin , since H+ can be regarded as the ultimate electrophile. This expectation has been verified (despite the potential complication that pKa involves aqueous solution), but with an interesting qualification. Within each of several different categories of acids (and even for some different types taken together), it was found that both Hpr and pKa could be represented well in terms of the I Smin of the conjugate base (which is what interacts with the H+ ) [71,83,98,101,105–107]. This has permitted the prediction of a number of unknown pKa values [83,100–102,106,107]. However, these studies dealt almost entirely with acids involving only elements in the first row of the periodic table. When we addressed a set of nine hydrides and their conjugate bases, encompassing three rows and three columns of the periodic table, we obtained good correlations for Hpr and pKa with I Smin within each horizontal row separately, but not for all three together [71]. This is because I Smin does not show the correct trends within the vertical columns. This can be seen for the hydride conjugate bases in Table 1. Hpr and pKa decrease in magnitude from left to right horizontally and from top to bottom vertically. I Smin would therefore be expected to increase in both directions – it does so horizontally, but it decreases vertically. On the other hand, the variation of VSmin from top to bottom indicates a weakening electrostatic interaction, fully consistent with the trends in H pr and pK a . Thus, by expressing Hpr and pKa in terms of both ISmin and VSmin , Hpr = 1 I Smin + 1 VSmin + 1

(10)

pKa = 2 I Smin + 2 VSmin + 2

(11)

it was possible to achieve R2 between 0.925 and 0.994 [71]. Table 1 Experimental protonation enthalpies and pKa and computed HF/6-31 + G ∗ I Smin and VSmin for conjugate bases of nine hydrides abc Conjugate base Hpr pKa I Smin VSmin

NH− 2 −4037 34 1.52 −1731

OH− −3908 16 3.39 −1819

F− −3714 3.2 5.64 −1782

Conjugate base Hpr pKa I Smin VSmin

PH− 2 −3709 29 1.46 −1327

SH− −3511 7.0 3.07 −1379

Cl− −3337 −7 4.85 −1423

Conjugate base Hpr pKa I Smin VSmin

AsH− 2 −3621 − 1.43 −1287

SeH− −3427 4 2.91 −1318

Br − −3236 −95 4.48 −1330

a

Calculated data are from [71], which also give sources of experimental values. Hpr and VSmin are in kcal/mol; I Smin is in eV. c The pKa are for the neutral hydrides. b


127

These hydrides illustrate the complementarity of I S r and VS r in the context of molecular reactivity. Within a horizontal row, the electrostatic aspect is not determining; the variation in VSmin is small and not consistent with the trends in Hpr and pKa . However, these can be explained nicely in terms of I Smin , the charge transfer/sharing aspect. Within a vertical column, on the other hand, the situation is exactly reversed, and electrostatics are dominant. Another example of I S r − VS r complementarity is provided by the heterocycles furan (1) and pyrrole (2). The locations of their computed I Smin and VSmin are given in Table 2 [85]. Both have relatively high I Smin near the heteroatoms and lower ones near the carbons of furan and the C − C bond midpoints of pyrrole. The VSmin values are associated with the C − C midpoints and the oxygen of furan; there is none by Table 2 Computed HF/6-31G∗ I Smin and VSmin of furan, pyrrole, guanine and cytosine Molecule

I Smin (eV)

VSmin (kcal/mol)

C 112 O: 16.7

C − Cb −170 O: −246

C − C:b 10.6 N: 16.7

C − C:b −281

C5 : 11.5 N7 C8 :c 12.3 N3 131 O: 13.8

N3 −18 O −63

C5 120 N3 121 O: 13.0

C5 −1 O −69

O

α β 1, furan H N α β 2, pyrrole

H

3

N

H2N 2

N9

4

8

H

N1

5

6

N 7

O

3, guanine N

H2N

3

4

2

5

1N

6

O

H

4, cytosine a

Data are from [85] for 1 and 2, [84] and [85] for 3 and 4. Above bond midpoint. c Approximately midway between N7 and C8 on molecular surface. b

128


the nitrogen of pyrrole, which has a planar configuration and hence no well-defined lone pair. In the gas phase, furan is known to undergo electrophilic substitution preferentially at the positions, pyrrole at the [97,108]. For furan, 1, this can be explained by noting that the electrostatically most favoured approach is to the oxygen but interaction there is inhibited by the high I Smin , in contrast to that at the neighbouring C. In the case of pyrrole, the I Smin is compatible with either or substitution, but the VSmin will guide an electrophile to . The lower carbon I Smin of pyrrole is consistent with its greater observed reactivity [97,108]. Finally, Table 2 also includes the computed I Smin and VSmin of guanine (3) and cytosine (4) [84,85]. Experimentally, it is known that the primary sites for electrophilic attack are N7 of guanine and N3 of cytosine [109–113]; of secondary importance are the oxygens on each molecule and N3 of guanine. It is interesting that none of these positions feature the lowest I Smin in these molecules, which are at C5 in each instance. The reason why there is little or no attack by electrophiles at these carbons is the absence of significant negative electrostatic potentials to initially attract them. It was shown by Murray et al. [84] that the strong negative VS r in guanine and cytosine is associated with the N7 /O portion of the former, and the N3 /O of the latter. Within these regions, the nitrogens have lower I Smin than do the oxygens (Table 2), and thus are more likely to undergo reaction, although some does occur at the oxygens. There is also some at N3 of guanine, which has both a (weakly) negative VSmin and a relatively low I Smin . These examples show the complementary roles of I S r and VS r with respect to reactivity towards electrophiles. When it is a radical that is approaching, however, then I S r would be expected to be the determining factor. Indeed, C5 has been observed to be a favoured site for the interactions of guanine and cytosine with radicals [114–116].

6. Ir and bond, radical characterization In general, I S r can be used quite effectively to distinguish between localized and delocalized electrons. Consider ethylene and benzene as prototypes of the two extremes. Ethylene, in which the electrons are primarily localized between the carbons, has =C double bond [117]; in contrast, benI Smin above and below the midpoint of the C= zene, in which they are completely delocalized, has no I Smin associated with the bonds, only with the carbons [24,81,82]. The polycyclic aromatic hydrocarbons encompass the full range of intermediate situations. It has long been recognized that it is chemically and structurally incorrect to imply that all of the rings are similar in aromatic character [118,119]. Clar argued, for instance, that anthracene, which has 14 electrons, should not be depicted as 5, which would require 18, but rather as 6A − 6C. These structures suggest that the bonds C1−C2 C3−C4 C6−C7 and C8−C9 have more double bond character than do the others. This is confirmed by the experimental geometry of anthracene [1]; those bonds are =C double bonds, while the others 1.356 A in length, close to the 1.32 A average of C= are all at least 1.400 A, similar to the typical aromatic 1.39 A. Those four are also the only bonds in anthracene having an I Smin near the midpoint [82].


129 9

10

1

8

2 3

7 6

5

4

5, incorrect

6A

6B

6C

This was found to be a general pattern within a series of eight polycyclic aromatic hydrocarbons plus benzene [82]. Those bonds that the resonance structures predict to have more localized electrons, i.e. greater double bond character, are experimentally significantly shorter than the others and also are the ones having I Smin near their midpoints. There is in fact an excellent correlation between the computed bond lengths and I Smin R2 = 0972. For several of these molecules, however, there are I Smin associated with certain carbons, as well as with some of the bonds. As in the cases of guanine and cytosine, discussed earlier, such carbons are possible targets for attack by electrophiles (if appropriate channels of negative VS r are present) or by radicals. Thus, we have proposed that anthracene, for example, should be represented by an additional structure, 6D [82], having two radical sites:

6D

(6D was also advocated by Polansky and Derflinger [120].) The importance of 6D is confirmed by the chemical behaviour of anthracene [97,121], such as relatively facile addition across the 5,10 positions. We have cited considerable additional chemical evidence supporting the presence of the predicted radical sites in the other molecules studied [82]. Another category of molecules having I Smin near bond midpoints is strained hydrocarbons with three-membered rings [83,117], e.g. cyclopropane (7), bicycle-[1.1.0]butane (8) and tetrahedrane (9) but not cyclobutane (10) or cubane (11). In triprismane (12), the bonds at the ends of the molecule have midpoint I Smin , but not the three connecting ones.

7

8

9

10

11

12

130


In extensive studies of strained hydrocarbons [79,122–127], we have shown that their C−C bonds have negative potentials (which is unusual for C−C single bonds) and that these are stronger for three-membered rings, which also have a higher degree of strain than four-membered ones. These factors presumably account for the I Smin noted above. In all of these respects, the three-membered rings are similar to ethylene, and indeed both 7 and 8 have some olefin-like reactive properties [128–131]. An interesting case is [1.1.1]propellane (12). There has in the past been some controversy over whether there is or is not a bond between the central carbons [132–135]. If there were, then the molecule would have three three-membered rings, and we would expect them to have bond I Smin . Since none were found [117], we have concluded that there is no central C−C bond. The only I Smin values are to the outsides of these two carbons, which is consistent with our earlier argument that the molecule has biradical character [126], as depicted in 13. There is supporting evidence: [1.1.1]propellane undergoes polymerization at these carbons [136–138], and they react with N2 O4 to yield the dinitro derivative [139].

13

7. Ir and polarizability Polarizability is a second-rank Cartesian tensor, with components ij , which governs the lowest-order response of an atom’s or molecule’s electronic density to an external electric field [140]. For a static field, the tensor is symmetric, i.e. ij = ji . Our concern in this discussion shall be with the scalar, or average, polarizability : =

1 xx + yy + zz 3

(12)

can be linked to several key aspects of chemical reactivity, such as hardness, softness and charge capacity [11,141–144]. There are also ‘principles’ of maximum hardness [145–147] and minimum polarizability [149–152], which state that systems tend naturally to evolve towards those states. It has long been recognized that the polarizabilities of atoms and molecules can be correlated with their volumes [153–158]. For a uniform conducting sphere of radius R and volume V , the relationship is exact [87]: = R3 =

3V 4

(13)

Using a statistical model, Dmitrieva and Plindov derived, for a neutral atom with radius r0 [159], = fZr03

(14)


131

in which fZ is a slowly varying function of the nuclear charge. As an approximation, therefore, ∼ r03

(15)

We have investigated the validity of (15) for ten different measures of atomic radius [160]. R2 ranged from 0.207 to 0.976. This work furthermore supports (14) but suggests that fZ is periodic as well as slowly varying. It is the outermost, least tightly held electrons that are primarily affected by an external electric field [161,162], and hence it is reasonable to think that polarizability may correlate inversely with ionization energy. Thus, it may not be surprising that Dmitrieva and Plindov were able to derive [159], for neutral atoms, = gZ/I 3

(16)

gZ being again a slowly-varying function, so that, ∼ 1/I 3

(17)

Equation (17) was also obtained by Hati and Datta [163]. The existence of inverse − I relationships for atoms, such as (17), has been verified empirically on several occasions [51,159,160,163,164]. It can be argued, however, that (15) and (17) are really not independent of each other, because of the inverse variation of I with atomic size, which can be seen from the asymptotic dependence of r upon radial distance from the nucleus [29,165–167]: r ∼ exp −22I05 r

(18)

For molecules, I shows no consistent pattern with respect to either or V [158]. It is also much more limited in range; within one representative group of 29 molecules, 24 have I between 8 and 11 eV, while increases from 2.26 to 1657 A3 and V from 34.12 to 17375 A3 in the same subset [158]. For these 29 molecules, R2 = 0553 for vs 1/I and 0.523 for vs 1/I 3 . On the other hand, the inclusion of I improves the − V correlation for molecules [158], but even more effective is the introduction of the average of I S r over an outer molecular surface [157,158], such as the r = 0001 au. Using this to determine both I Save and V , R2 increases from 0.960 for vs. V to 0.984 for vs. V/I Save , and the root mean error decreases from 0.76 to 0.48 eV. Furthermore, some significant − V outliers, such as C6 F6 and H5 C2 2 O, fit much better into − V/I Save . Both V and I Save are anomalously large for these molecules, relative to the trend in ; thus using the ratio V/I Save essentially compensates for the anomalies. Even though V is clearly a much more important determinant of than is either I or I Save for molecules, we do feel that on a local level, the tightness of binding is still the key to polarizability. Accordingly we have proposed, already several years ago [51,168], that Ir can be regarded as a measure of local polarizability, low Ir indicating high r and vice versa. This should not be inconsistent with the recognized dependence of on volume, because we are comparing equal volume elements dr. Our local

132


polarizability can be regarded as related to Jameson and Buckingham’s ‘polarizability density’ [169], satisfying, = rdr

(19)

On a qualitative level, an analysis of several molecules has demonstrated that linking r inversely to Ir seems to be quite reasonable [158]. The portions of molecules that are expected to be the more polarizable, e.g. electrons and lone pairs, do have the lowest I S r, while acidic hydrogens, which have lost much of their electronic density, have high I S r. Furthermore, the regions of lowest I S r coincide with the largest components of the experimental polarizabilities. Some quantitative support has come from a study comparing the I S r of the nucleotide bases to the computed interaction energies of each molecule, with a test charge placed at various points around it [84]; the magnitudes of these energies were intended to indicate the response to the polarizing effect of the test charge. An approximate form of (19) in terms of finite increments is, =

i

(20)

i

where the i are the polarizabilities of portions of the molecule, e.g. chemical groups. The idea underlying (20) is not new. Since the polarizabilities of molecules correlate with their volumes [153–157], and the latter can be expressed as sums of atomic and/or group contributions [155,170], it seems natural to try to also develop transferable atomic (valence state), group and/or bond polarizabilities that can be utilized to estimate molecular values [155,156,171,176]. These efforts have had considerable success. We have accordingly examined the feasibility of extending our −V/I Save correlation to atoms and groups that are components of molecules. Initially we looked at CH3 , NH2 , OH and NO2 [177]; later we added F, Cl, Br, CN and SH [176]. In each instance, we took a series of at least 18 representative molecules containing the atom or group in question and computed V/I Save for it in each molecule, again for the r = 0001 au surfaces. The first issue was whether the results for each atom or group are sufficiently constant within its series, and thus transferable. Our criterion for this was the standard deviation of V/I Save for each atom or group. The best transferability was found for NO2 , for which was just 1.8% of the mean value of V/I Save ; the poorest was for Cl, 6.5%. This range can be explained by noting that NO2 is nearly always a strong withdrawer of electronic charge [105], whereas Cl, as well as others such as Br and NH2 , can function as both donors and acceptors and are therefore more influenced by their molecular environments. Overall, the transferability was concluded to be satisfactory. The key point then was to assess how well these mean V/I Save represent the relative polarizabilities of the atoms and groups. For this purpose, we took Miller’s atomic (valence state) and group polarizabilities as the standards [175]. Excluding NO2 , which is not included in his tabulation, R2 for the V/I Save vs Miller correlation is 0.963. This allows us to predict Miller for the NO2 group: 27 A3 . It might be argued that it would be more consistent with (19) to obtain the polarizabilities of parts of molecules by integrating dr/Ir over their volumes, or perhaps by


133

using V/I, where I is the average of Ir, rather than V/I Save . Perhaps one of these would indeed be preferable. On the other hand, since it is the outermost electrons that are most relevant to polarizability [161,162], it could also be claimed that V/I Save is fine except that V should correspond to only the outer portion of the group or atomic volume. These are all options worthy of investigation.

8. The present, the future The average local ionization energy Ir has many interesting and significant aspects and applications. It is related to local temperature and atomic shell structure, it is linked to electronegativity and shows promise as a measure of local polarizability. It permits the characterization of bonds and radical sites, and – in conjunction with volume – the prediction of molecular and group polarizabilities. Finally, it is an effective guide to reactivity towards electrophiles, especially when complemented by the electrostatic potential. All of these areas continue to be studied. There are also new directions to be explored. One of these is the use of Ir to help understand what is occurring during the course of a chemical reaction [179]. This involves monitoring the variation in I Save along the reaction coordinate for each of the participants. This is done in the context of the ‘reaction force’, a very intriguing approach to analysing the course of a process [179–182]. Another new line of enquiry is the possible relationship between I Save /V and hardness, which is suggested by the –V/I Save [157,158] and the –hardness [143] correlations. Thus, there remains much to be investigated!

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

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Chapter 9

The electrophilicity index in organic chemistry a

Patricia Pérez, b Luis R. Domingo, c Arie Aizman, and d R. Contreras

a

Departamento de Física y Departamento de Química, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile b Instituto de Ciencia Molecular, Departamento de Química Orgánica, Universidad de Valencia, Dr. Moliner 50, 46100 Burjassot, Valencia, Spain c Departamento de Química, Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile and d Departamento de Química, Facultad de Ciencias, Universidad de Chile, Casilla 653-Santiago, Chile

Abstract We review in this chapter the applications of theoretical scales of global and local electrophilicity to rationalize the reactivity and selectivity for a significant number of reactions in organic chemistry. The model is based on the global electrophilicity index, formerly introduced by Maynard et al. and further formalized by Parr et al. The global electrophilicity index categorizes, within a unique absolute scale, the propensity of electron acceptors to acquire additional electronic charge from the environment. The local extension of this index provides useful information about the active sites of electrophiles, thereby allowing the characterization of the intramolecular selectivity in these systems. These concepts will be illustrated for a series of chemical reactions in organic chemistry, including polar cycloadditions and electrophilic addition reactions, substrate selectivity in electrophilic aromatic substitution (Friedel–Crafts) reactions, hydrolysis of carbonyl compounds, the reactivity of carbenes and carbenium ions and the superelectrophilicity of dicarbenium, oxonium and dicarboxylic acids. 139

140

Electrophilicity index in organic chemistry

1. Introduction The understanding of the course of organic chemical reactions was stimulated with the development of the valence electronic Lewis’ theory1 and the general acid–base theory of Lowry and Brönsted.2 On the basis of these electronic models, Ingold3 in the 1930s introduced the concepts of electrophile and nucleophile for atoms and molecules. These terms are associated with electron-deficient and electron-rich species, respectively. From that time, there have been several attempts to classify organic molecules within empirical (hopefully absolute) scales of electrophilicity and nucleophilicity. However, this objective is difficult to reach if one considers that a universal electrophilicity/nucleophilicity model should accommodate, within a unique scale, substrates presenting a large diversity in electronic and bonding properties, without mentioning the presence of the medium. Recent works of Mayr and co-workers4−14 have illustrated this trend. In fact, these authors have established, in contrast to the accepted opinion about the relative character of the experimental electrophilicity/nucleophilicity scales for many reactions in organic and organometallic chemistry, that it would be possible to define nucleophilicity and electrophilicity parameters that are independent of the reaction partners. Mayr et al. proposed that the rates of reactions of carbocations with uncharged nucleophiles obey the linear free energy relationship given by:4−14 log k = s E + N

(1)

where E and N are the electrophilicity and nucleophilicity parameters, respectively, and s is the nucleophile-specific slope parameter. These authors observed that in general the solvent effects on the reaction rates with nucleophiles and hydride donors were small and could be neglected to a first approximation.11 For the determination of the strengths of electrophiles, Mayr et al. performed a correlation analysis of the reactions of the electrophiles with reference nucleophiles that yielded E N and s of the reagents involved.4−14 On this basis, these authors have built up experimental scales of electrophilicity and nucleophilicity for a wide diversity of organic and organometallic species, including diarylcarbenium ions, aryldiazonium ions and alkoxycarbenium ions.4 These scales were built using reactions of cationic carbon electrophiles with nucleophiles in which only one bond is formed in the rate-determining step, and no -bonds in the electrophile are broken.4−14 Other scales of electrophilicity have been proposed on the basis of spectroscopic data based on the relative hydrogen bonding strength of neutral electrophiles in the gas phase.15−17 From the theoretical point of view, the electrophilicity concept has been recently discussed in terms of global reactivity indexes defined for the ground states of atoms and molecules by Roy et al.1819 . In the context of the conceptual density functional theory (DFT), a global electrophilicity index defined in terms of the electronic chemical potential and the global hardness was proposed by Maynard et al.20 in their study of reactivity of the HIV-1 nucleocapsid protein p7 zinc finger domains. Recently, Parr, Szentpály and Liu proposed a formal derivation of the electrophilicity, , from a second-order energy expression developed in terms of the variation in the number of electrons.21 Unfortunately, a quantitative definition of global nucleophilicity cannot be deduced within the same framework, and it remains as an open problem. Therefore, in the present

R. Contreras et al.

141

review, we assume that electrophiles and nucleophiles are opposite ends of the same scale of electrophilicity, yet their inverse relationship has not been proven up to now. By construction, these scales are absolute, in the sense that they are independent of the nucleophile partners, mainly because they are supposed to solely depend on the electronic structure of molecules. This article deals with the application of reactivity indexes defined in the context of the conceptual DFT.2223 The focus is placed on the application of the global electrophilicity,2021 leading to a coherent chemical reactivity model for polar processes in organic chemistry. The electrophilicity index encompasses in its definition both the electronegativity of the electron acceptor and its chemical hardness acting as a resistance to the electronic charge exchange with the environment. The global electrophilicity index leads to a reliable scale of electrophilicity and gives a hierarchy within a unique absolute scale describing the propensity of the electron acceptors to acquire additional electronic charges from the environment, viewed as a sea of electrons.21 The proposed scale has been validated against kinetic and spectroscopic experimental scales for a formidable number of compounds presenting a wide variety of structure and bonding properties, including superelectrophiles. We illustrate in this article how this reactivity index may be successfully used to rationalize reaction mechanisms, substituents effects and reaction rates for a significant number of organic reactions, including polar cycloadditions. A local extension of the electrophilicity index to deal with molecular regions where the additional electronic charge is better accommodated by the electron acceptor was developed in terms of the electrophilic regional Fukui function.2425 The local electrophilicity incorporates the electrophilic Fukui function which projects the global electrophilicity of molecules onto specific atoms or groups in the electron acceptors. This local descriptor successfully explains the regioselectivity of a significant number of organic reactions, including Diels–Alder (DA) reactions, 1,3-dipolar cycloadditions, electrophilic additions to alkenes and other classic polar reactions in organic chemistry.2627 At the end of the present review, we shortly discuss the perspective and future developments that incorporate for instance strain-induced electrophilic activation, an increasingly active field in organic chemistry, and the development of relative scales of nucleofugality for nucleophilic substitution and elimination reactions.

2. Model equations The global electrophilicity index , which measures the stabilization in energy when the system acquires an additional electronic charge N from the environment, has been given the following simple expression:21 =

2 2

(2)

in terms of the electronic chemical potential and the chemical hardness . These quantities may be approached in terms of the one-electron energies of the frontier + L molecular orbital HOMO and LUMO, H and L , as ≈ H and ≈ L − H , 2 2122 respectively. The electrophilicity index encompasses both the propensity of the electrophile to acquire an additional electronic charge driven by 2 (the square of

142


electronegativity, = − ) and the resistance of the system to exchange electronic charge with the environment described by . A good electrophile is in this sense characterized by a high value of and a low value of . Starting from (2), it is possible to define a local (regional) counterpart for the quantity as follows: use the inverse relationship between chemical hardness and the global softness S = 1/ 22 and the additivity rule for S, namely S = sk+ , to rewrite (2). k

There results: =

2 2 + 2 s = k = S= 2

2 2 k k k

(3)

From (3) we may define a semilocal or regional electrophilicity condensed to atom k given by: k =

2 + s 2 k

(4)

Note that within the present model, the maximum electrophilicity within a molecule will be located at the softest site of the system. If we further use the exact relationship between local softness and electrophilic Fukui function, namely sk+ = Sfk+ , then the local electrophilicity given in (4) may be also expressed as:2226 k =

2 + 2 S + f = fk+ s = 2 k 2 k

(5)

thereby showing that the maximum electrophilicity power in a molecule will be developed at the site where the Fukui function for a nucleophilic attack fk+ 22 displays its maximum value, i.e. at the active site of the electrophile. Other local (regional) descriptors of electrophilicity have been previously proposed in the literature.1819

3. Polar cycloaddition reactions 3.1. Polar Diels–Alder reactions DA reactions are the largest family of cycloaddition processes. In a DA reaction, an ethylene derivative, named the dienophile, adds to a 1,3-diene to afford a six-membered carbocyclic product. By varying the nature of the diene and dienophile, many different types of carbocyclic structures can be built up. However, not all possibilities take place easily. For instance, the simplest cycloaddition reaction is the DA reaction between butadiene and ethylene. However, this reaction must be forced to take place: after 17 hours at 165 C and 900 atmospheres, it does give a yield of 78%.2829 An interesting alternative is that the presence of electron-releasing substituents in the diene and electronwithdrawing (EW) substituents in the dienophile or vice versa can drastically accelerate the process. In general, the DA reaction requires opposite electronic features in the substituents at the diene and the dienophile for being reasonably fast. Recent studies point out that this

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type of substitution favours the charge transfer along an asynchronous mechanism. Furthermore, the reaction mechanism changes progressively from a concerted nonpolar to a highly asynchronous polar pathway, with an increasing ability of the electron-deficient reagent to stabilize a negative charge. Under suitable conditions, the intermediates of the stepwise process have been trapped.30−32 At this point, the 1,3-diene and the ethylene systems clearly behave as a nucleophile/electrophile pair. The DA reactions have been widely treated using the frontier molecular orbital (FMO) model.33−35 Within the FMO theory, Sauer and Sustmann have categorized the cycloaddition processes in three different types.36 Type I involves a dominant interaction between the highest occupied molecular orbital of the diene HOMOdiene and the lowest occupied molecular orbital of the dienophile LUMOdienophile . The reactions belonging to this group are classified as normal-electron-demand (NED) DA reactions. Type II involves the interaction between the LUMO of the diene and the HOMO of the dienophile. They are classified as inverse-electron-demand (IED) DA reactions. Type III may be characterized by the similarity of the HOMO and LUMO energies of the diene/dienophile pair. In this case, both HOMOdiene –LUMOdienophile and LUMOdiene –HOMOdienophile interactions may be important in determining the reactivity and regiochemistry of the process. This case involving the DA reaction between electron-deficient reagents is in general not well described by the FMO theory.37 In the case of an extremely large difference of the FMO energy of the cycloaddition partners, this model allows one to predict a mechanism via zwitterionic intermediates. The more general case of DA reactions corresponds to the NED of type I. The activation of the ethylene by substitution with EW groups results in a lowering of the LUMOethylene energy and also results in an enhancement of the rate and regioselectivity of the reaction to a higher degree. Note that in a NED DA reaction, 1,3-butadiene and cyclopentadiene react with appropriate electron-poor ethylenes, while in an IED DA reaction the 1,3-butadiene must be activated by substitution with electron-releasing groups.38

3.1.1. Electrophilicity index and reaction mechanisms of DA Reactions. Polar cycloaddition reactions The global electrophilicity index has been used to classify the electrophilicity of a series of dienes and dienophiles currently present in DA reactions.39 A good correlation between the difference in electrophilicity between the reagents, , and the charge transfer at the corresponding zwitterionic transition structure (TS) was found. This result suggested a possible relationship between the polarity of the process at the TS stage and the difference in electrophilicity of the reagents (see Chart 1). In Table 1, the common reagents, dienes and dienophiles, involved in DA reactions are classified in decreasing order of global electrophilicity.39 The series of dienes and dienophiles involved in DA reactions may be arbitrarily classified into three general groups. Group I of strong electrophiles include compounds with electrophilicity numbers larger than 1.50 eV. A second group of moderate electrophiles (group II) include compounds with electrophilicity values comprised within the range 149 eV−090 eV. A third group of marginal electrophiles (group III) includes compounds with electrophilicity values lesser than 0.90 eV. The first

144

Electrophilicity index in organic chemistry Table 1 Global electrophilicity scale and global propertiesa for some common dienes and dienophiles involved in Diels–Alder reactions Molecule 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

−04348 −02586 −02122 −02083 −01837 −02074 −01870 −01958 −01831 −01610 −01683 −01726 −01509 −01624 −01586 −01423 −01410 −01261 −01259 −01270 −01212 −01182 −01135 −01137 −01259 −01106 −01028 −01024 −01239 −01024 −00867 −01148 −00894 −00753 −00680

02867 01529 01285 01821 01516 02075 01740 02001 02007 01922 02135 02329 01929 02363 02268 01882 01933 01912 01957 02083 02120 02037 01999 02048 02504 02016 01875 01958 02855 02441 01801 03344 02564 02525 02390

897 596 477 324 320 282 273 261 227 184 180 174 161 152 151 146 140 113 110 105 094 093 088 086 086 083 077 073 073 059 057 054 042 031 027

a and in a.u., in eV. All quantities were evaluated at the B3LYP/ 6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

group of strong electrophiles have electrophilicity numbers ranging from = 897 eV (N -methylmethyleneammonium cation, 1) to = 151 eV (methyl acrylate, 15). A second group of reagents that display a moderate electrophilicity power goes from = 146 eV (cyclohexenone, 16) to = 093 eV (1-methyl-1,3-butadiene, 22). Finally, the third group goes from = 088 eV (2-trimethylsilyloxy-1,3-butadiene, 23) to ( = 027 eV (dimethylvinylamine, 35) with marginal electrophilicity numbers, so that they

R. Contreras et al. CH3

145 NC

CN

S

O

N H

NC

1

CN

F3C

O

4

3

H N

BH3

O

COH

CF3

2

CN

O

5

COH

NO2

O

CO2Me

MeO2C

CN 6

7

CO2CH3

9

8

10

COMe

CN

CO2Me CO2Me

CO2CH3 11

12

O

O

16

17

13

14

15

OAc

18

19

20

TMSO

CH3 N

21

22

23

OCH3

OTMS

24

25

O

26 N(CH3)2

27

28

29

OCH3

31

32

33

30

NH

34

N(CH3)2

35

Chart 1 (from Reference 39) can tentatively be classified as nucleophiles, in a global sense. Within this absolute scale, several DA reagents like nitroethylene (8) and furane (30) are correctly classified as strong electrophile and nucleophile, respectively, yet they can act as dienes or dienophiles in polar DA cycloadditions. The classification based on the global electrophilicity scale presented in Table 1, correctly describing activating and deactivating effects promoted by chemical substitution on the diene/dienophile pair, may be complemented with a static analysis of the charge transfer pattern expected for transition state structures involved in DA reactions. This aspect is useful to discuss since the difference in polar character of these structures

146


helps to rationalize the reaction mechanisms. Consider for instance the ethylene/1,3butadiene interaction as a reference. Since both ethylene (29) and 1,3-butadiene (20) are classified as marginal and moderate electrophiles, respectively, the reaction will present a charge transfer pattern mostly consistent with a nonpolar process. This reaction constitutes the prototype of a pericyclic cycloaddition reaction,40 with a very unfavourable barrier. According to this classification, the acrolein/1,3-butadiene interaction on the other hand is expected to show a slightly different picture that follows from their more higher difference in electrophilicity: acrolein (10) is expected to act as electrophile and 1,3-butadiene (20) as nucleophile, and the reaction mechanism is expected to be more polar in character. This prognosis based on the difference in is further reinforced by the corresponding values of their electronic chemical potentials quoted in Table 1: it may be seen that the electronic chemical potential in acrolein (10) ( = −01610 au) is larger (in absolute value) than the electronic chemical potential of 1,3-butadiene (20) ( = −01270 au), thereby indicating that the net charge transfer will take place from the diene towards the dienophile. Note that 1,3-butadiene (20) and ethylene (29) have similar electronic chemical potentials ( = −01270 and 0.1239 au, respectively), and, therefore, neither of them provides charge to the other, leading to a nonpolar cycloaddition reaction. The polarity expected at the transition state structure predicted from the difference in electrophilicity has been confirmed by suitable population analysis.39 The more polar character of the acrolein (10)/1,3-butadiene (20) interaction with respect to the ethylene (29)/1,3-butadiene (20) can also be seen from a static charge transfer model, using for instance Pearson’s equation to estimate the amount of charge transfer.41 Within this model, the charge transfer from the nucleophile towards the electrophile may be easily represented in terms of an Ohm’s-like equation that uses the electronic chemical potential and chemical hardness of the isolated interacting pairs, namely N = Nu− − E+ / Nu− + E+ . Using the values of and quoted in Table 1, we obtain N = 008 e for the acrolein(10)/1,3-butadiene (20) interaction, compared to N = 001 e for the ethylene (29)/1,3-butadiene (20) interaction, thereby predicting a slightly more polar character for the former. The apparent relationship between static polarity of the electrophile/nucleophile interaction, N , and the difference in global electrophilicity, , for a series of electron-poor dienophiles used in DA reactions are depicted in Table 2 and Figure 1a. Furthermore, Lewis acid (LA)-catalysed DA reactions, represented by the acrolein–BH3 complex (5) /1,3-butadiene (22) interaction (entry 6 in Table 2), have been found to react through a more polar mechanism as compared with the uncatalysed process (see entry 2 in Table 2). The charge transfer pattern for the LA-catalysed processes may also be estimated by Pearson’s equation, using the values of electronic chemical potential and chemical hardness for the acrolein–BH3 complex (5) and 1,3-butadiene (22) quoted in Table 1. Thus, for the catalysed cycloaddition process, entry 6 in Table 2, the estimated charge transfer is N = 016 e. Note that the enhancement in charge transfer for the LA-catalysed process results both from an increase in the chemical potential difference from = 087 eV in the acrolein (10)/1,3-butadiene (22) interaction to = 149 eV in the acrolein–BH3 complex (5)/1,3-butadiene (22) interaction and from a simultaneous lowering in the chemical hardness of the acrolein–BH3 complex (5) (see Table 1). The DA reactions of 1,3-butadiene (20) with the nine dienophiles presented in Table 2 illustrate well how to establish a qualitative relationship between the difference in electrophilicity of the dienophile/diene pair, , and the polarity of the process, N .

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Table 2 Relationship between polaritya and the difference in electrophilicity powerb for some diene/dienophile pairs involved in the Diels–Alder reactions between several dienophiles and 1,3-butadiene (20) Entry 1 2 3 4 5 6 7 8 9

Dienophile

Static polarity N

29 10 12 11 6 5 3 2 1

001 008 010 010 019 016 026 037 063

Polarity at TS N

−032 079 069 088 177 215 372 491 792

000 011 012 021 024 021 023 039 056

a Polarity values (N , in e) are the static values calculated from Pearson’s equation (see the text for details), polarity at the TS (N, in e) from NBO population analysis (see the text). All quantities were evaluated at the B3LYP/6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163. b The electrophilicity difference (, in eV) is evaluated with reference to the electrophilicity power of 1,3-butadiene (20) ( = 114 eV, see Table 1).

(a)

(b)

ΔN °

ΔN °

0.6

0.6

0.4

0.4

0.2

0.2

2.0

4.0

6.0

8.0

Δω

0.2

0.4

0.6

ΔN

Figure 1 (a) Correlation between the static polarity (N , in e) evaluated from Pearson’s equation and the electrophilicity difference (, in eV); (b) correlation between the static polarity (N , in e) evaluated from Pearson’s equation and polarity calculated from an NBO population analysis (N , in e) at the corresponding transition structures

The difference in electronic chemical potential between the electrophile/nucleophile pair in the dimethyl 2-methylenemalonate (11)/1,3-butadiene (20) interaction, entry 4 in Table 2, becomes 1.13 eV with a total resistance to the charge transfer = 1094 eV. These figures yield an expected charge transfer of about N = 010 e for the interaction between 1,3-butadiene and dimethyl 2-methylenemalonate compared to the value N = 016 e for the acrolein–BH3 complex (5)/1,3-butadiene (20), and with the value N = 001 e for ethylene (29)/1,3-butadiene (20) interactions. Moreover, the increase of the polarity within the series of the mono-, di- and tetracyanoethylene

148


derivatives 12, 6, and 2, N = 010 e 019 e and 0.37e, respectively (see Table 2), is in complete agreement with the increase in electrophilicity of the corresponding dienophile, = 174 eV, 2.82 eV and 5.96 eV, respectively (see Table 1),42 and also with the increase of the rate reaction.43 The extreme case of N -methylmethyleneammonium cation (1) /1,3-butadiene (20) interaction, entry 9, presents the highest polarity (N = 063 e) that corresponds with highest value in the difference of electrophilicity ( = 897 eV, see Table 1). Finally, the reliability of the model based on the global electrophilicity difference to anticipate the polarity of a transition state structure for a cycloaddition reaction has been tested by comparing the predicted values, N , with those obtained from a natural population analysis4445 (NPA) at the corresponding transition state structures, N (see Table 2). A comparison of the static polarities calculated and those obtained from the corresponding transition state structures shows a good agreement (see Figure 1b). Note that for this short series of electron-poor dienophiles used in NED–DA reactions, the order relationship in the expected charge transfer, acrolein (10) < acrylonitrile (12) ≈ dimethyl 2-methylenemalonate (11) < acrolein−BH3 complex (5) < 1 1 − dicyanoethylene 6 < hexafluorothioketone 3 < tetracyanoetylene (2) < N -methylmethylene ammonium (1), is approximately consistent with the difference in electrophilicity, , displayed by the diene/dienophile pair (see Table 2 and Figure 1a). Therefore, the relationship between the electrophilicity difference of the dienophile/diene pair and the static polarity may be a useful tool to describe the electronic pattern expected for the transition state structures involved in DA reactions, describing nonpolar ( small) or polar ( big) mechanisms.39 On the other hand, the stepwise vs concerted nature of the DA cycloaddition reactions has been previously analysed by Sauer and Sustmann.36 These authors rationalize the mechanistic aspects of DA reactions based on the basis of the FMO Klopman’s equation,46 which introduces the coefficients and the frontier energy levels of the diene/dienophile pair. The approach based on the DFT indexes is quite different, in the sense that it does not need the electronic properties of the interacting diene/dienophile pair to set up an absolute scale of electrophilicity in these systems.39 This approach is a static reactivity picture developed around the frontier energy levels defining the electronic chemical potential. It correctly describes the direction of the charge transfer in NED and IED processes on one hand and introduces a unique absolute electrophilicity scale that correctly predicts the reactivity in many DA reactions where one or both reagents are widely functionalized. The electrophilicity scale provides a simple way to assess the more or less polar character of the interaction, on the basis of the electrophilicity gap between the reacting partners. This is achieved within a simple model where the information encompassed in the MO coefficients is implicitly self-contained in the electron density according to the density functional formalism.

3.1.2. Normal- and inverse-electron-demand DA reactions Most of the structural and electronic features induced by chemical substitution are often reflected as responses in the global reactivity indexes.47 We propose that the global electrophilicity index encompasses most of the relevant information that may roughly account for the global reactivity pattern observed in the DA reactions. For

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149

instance, in a NED DA reaction, the ethylene component (dienophile) usually bears one or more EW groups that enhance both the reaction rate and the yield of the kinetic control products.354849 Taking again ethylene/butadiene system as reference, we expect that increasing the substitution by EW groups in the dienophile should be reflected in an increase in the electrophilicity of the ethylenic moiety as measured by the index. For example (with reference to Table 1), substitution of one hydrogen atom in ethylene (29) by the EW -CHO group, brings the global electrophilicity of ethylene from 0.73 eV to 1.84 eV in acrolein (10). Note that the reaction of 1,3-butadiene with acrolein takes place within half an hour in quantitative yield, compared to the 78% yield obtained in the ethylene/1,3-butadiene reaction under more extreme external conditions.28 The dimerization of 1,3-butadiene on the other hand (85% yield after 10 days at 150 C) is much slower than its reaction with acrolein. It is interesting to note that the electrophilicity of 1,3-butadiene (20) = 105 eV is lower than that shown by acrolein, according to the classification given in Table 1. Methyl acrylate (15) and methyl vinyl ketone (13) that bear EW substituent of comparable power and that react at a similar rate display very similar values of global electrophilicity ( = 151 and 1.65 eV, respectively, see Table 1). Nitroethylene (8), a well-known powerful dienophile containing one of the most powerful EW groups,50 shows one of the highest value in electrophilicity = 261 eV among the neutral dienophiles, as well as dimethyl 2-methylenemalonate (11) = 193 eV. The thioketone (3) that shows a high electrophilicity = 477 eV readily reacts with 1,3-butadiene (20) in almost quantitative yield at low temperature −78 C.51 In 1964, J. Sauer et al.43 reported the reactivity of cyclopentadiene (26) with the cyanoethylene series in the DA reaction (see Scheme 1 and Table 3). An analysis of the relative rates shows that the rate increases with the cyano substitution on ethylene. However, it is easy to see that the successive substitution on the C1 and C2 carbon atoms has a different incidence. For instance, while 1,1-dicyanoethylene (6) reacts more than 10.000 times faster than acrylonitrile (12), cis and trans 1,2-dicyanoethylene (37 and 38) are only ca. 100 times faster. In addition, tetracyanoethylene (2), the most reactive dienophile within the series, reacts only 100 times faster than tricyanoethylene (36) (see Table 3). These results indicate that the symmetric substitution at the C1 and C2 carbon atom of ethylene produces a loss of effectiveness of the EW effect of the substitution. The transition state structures for the series of DA reactions between cyclopentadiene (26) and the mono-, di-, tri- and tetracyanoethylene derivatives were studied using DFT R1

R2

R1 R2

R4

R3

R3 R4

+

26

12 6 37 38 36 2

R1 = CN, R2, R3, R4 = H R1, R2 = CN, R3, R4 = H R1, R3 = CN, R2, R4 = H R1, R4 = CN, R2, R3 = H R1, R2, R3, = CN, R4 = H R1, R2, R3, R4 = CN

Scheme 1

150

Electrophilicity index in organic chemistry Table 3 Rate constants and activation energies (in kcal/mol) for the Diels–Alder reaction of dienophiles with cyclopentadiene 105 kM−1 s−1

Dienophile 2 36 6 37 38 12

Tetracyanoethylene Tricyanoethylene 1,1-dicyanoethylene Maleonitrile Fumaronitrile Acrylonitrile

Ea

43 × 107 48 × 105 45 × 104 91 81 1.04

ca. 8.0 11.4 12.6 15.2

methods.42 These polar cycloadditions can be considered as the nucleophilic attack of 26 to the electron-poor substituted ethylene. The increase of the EW substitution on ethylene (the electrophile) increases the rate of these polar cycloadditions. However, the symmetric arrangement of the substituents on the cis and trans 1,2-di and tetracyanoethylene decreases the effectiveness of the substitution. The analysis of the transition state geometries associated with the cycloaddition of these symmetrically substituted ethylenes shows that they correspond to synchronous bond-formation processes. The very low activation energy computed for the reaction with tetracyanoethylene 2 together with the large charge transfer found at the corresponding transition state structure indicates that this synchronous process is associated with polar cycloaddition, where 26 and the symmetrically substituted ethylene 2 act as nucleophile and electrophile, respectively, instead of a dienophile/diene pair in a pericyclic process.42 An analysis of the global electrophilicity of this series of cyano ethylenes reveals the increase of the electrophilic character of the cyano ethylene with increasing substitution (see Table 4). Finally, the analysis of the local electrophilicity for the symmetrically substituted ethylenes explains the synchronicity on the bond formation on these cycloadditions, as a consequence of the symmetric distribution of the global electrophilicity on the C1 and C2 electrophilic carbon atoms of the symmetrically substituted ethylene.42 The electrophile/nucleophile interaction in NED-DA reactions may also be enhanced by chemical substitution on the diene by an electron-releasing group. It is experimentally known that even a weak electron-releasing methyl group makes this diene more Table 4 Global and local propertiesa of the six cyano-substituted ethylenes Molecule 2 36 37 38 6 12

f1+

1

f2+

2

596 438 301 308 282 174

02570 02359 03058 02998 02087 02657

153 103 092 092 059 046

02570 03331 03058 02998 04989 04686

153 146 092 092 141 082

and k in eV. All quantities were evaluated at the B3LYP/6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

a

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151

reactive than 1,3-butadiene (20). A methoxy group on the diene makes them still more reactive. For instance, it has been experimentally established that acrolein (10) reacts with 1,3-pentadiene (22) by heating up to 130 C with 80% yield after 6 hours. Replacement of the methyl group by methoxy at the same position enhances the reaction rate, keeping approximately the same yield after two hours. Substitution of one methyl by the trimethylsilyloxy group, compound 28, lowers the reaction rate, maintaining a similar yield (81%) after 24 hours. Other electrophilic activating/deactivating effects induced by chemical substitution may be easily deduced from Table 1. The above examples correspond to NED-DA reactions, where the charge transfer takes place from the electron-rich diene (the nucleophile) to the electron-poor dienophile (the electrophile). However, it is also possible to have the electron-releasing substituent on the dienophile and the EW substituent on the diene. Such reactions are said to have an IED. They are much less common, because having the substituent with the inverse inductive effects order is not as effective as having them in the usual way.52 This is for instance the case of the interaction of nitroethylene (8) = 261 eV with either methyl vinyl ether (33) = 042 eV or dimethylvinylamine (35) = 027 eV53 . Nitroethylene (8), which was classified as a strong electrophile, may also be considered as a heterodiene, where the C1 and C2 carbon atoms of 1,3-butadiene have been substituted by oxygen and nitrogen atoms, respectively. This heterodiene will act as a dienophile (in a NED-DA reaction, the diene is usually the nucleophile), while towards electron-rich ethylenes it will act as heterodiene in an IED-DA reaction.54 Note that in both cases nitroethylene acts as a strong electrophile (see Table 1); it is located above 1,3-butadiene (20) and methyl vinyl ether (33) or dimethylvinylamine (35) in the electrophilicity scale. The DA reactions between the electron-deficient diene 39 and the electron-rich ethylenes 33 and 40 and the electron-deficient ethylenes 11 and 15 have been reported by Spino et al. (see Scheme 2).37 The FMO theory was used to predict the reactivity of these reagents in DA reactions. These authors conclude that in the NED-DA reaction, the FMO theory could predict the relative reactivity, while in the case of the IED one, it could not.37 The high reactivity of the electron-deficient ethylene 11 with the electron-deficient diene 39 was studied using both the analysis of the potential energy surface for these cycloadditions and analysis of the global and local reactivity indexes.38 The analysis of X X

MeO2C

OMe

OMe MeO2C

40 or 33

MeO2C

+ X

MeO2C

CO2Me

MeO2C

CO2Me

39 15 or 11

X

MeO2C

40 X=OMe 15 X=H 33 X=H 11 X=CO2Me

Scheme 2

152


the geometry and electronic structure of the transition state structures of the DA reactions involving these electron-deficient reagents provided an explanation of the participation for the diene 39, = 158 eV, as a nucleophile against powerful electrophiles as 11, = 180 eV. This fact, which requires the nonparticipation of the EW substituents along the nucleophilic attack of the system of the electron-deficient diene 39, can be anticipated by the relative position of the diene / ethylene pair in the electrophilicity scale. A strong electrophile located at the top of the scale determines the nucleophilic character of the other reagent located below it.38 This DFT analysis allows to characterize the HOMONU− –LUMOE+ interactions along a polar DA reaction. Note that in this series of DA reactions the electron-deficient diene 39 acts as electrophile towards strong nucleophiles as 40, whereas it acts a nucleophile towards strong electrophiles as 11. 3.1.3. Lewis acid-catalysed DA reactions Lewis acid (LA) catalysis has a relevant role in the DA reactions. It is well known that the presence of LAs increases both rate and regioselectivity. In a catalysed NEDDA reaction, the LA is coordinated to the electron-poor ethylene, thereby markedly decreasing the LUMOethylene energy and increasing the ionicity of the process. The acroleine–BH3 complex (5)/cyclopentadiene (26) is one of most studied LA-catalysed DA reaction.55 It may be seen in Table 1 that the presence of BH3 coordinated to acroleine, acroleine–BH3 complex (5), increases the electrophilicity in 1.36 eV relative to acroleine (10), in agreement with the more polar character of the LA-catalysed cycloaddition. A greater increase in electrophilicity is found for the protonated iminium cation (1) = 897 eV relative to the imine (25) = 086 eV. The imine shows an electrophilicity 10 times lesser than its protonated parent. Thus, if we consider the proton as the strongest LA in nature, this enhancement of electrophilicity is in agreement with the large acceleration of the DA reaction of imines in an acidic medium.56 Therefore, this enhancement in the reaction rate may be again explained by the strong increase of the electrophilicity of the dienophile induced by protonation at the nitrogen atom.57 The DA reaction of 1-aza-1,3-butadienes is a valuable methodology for synthesis of nitrogen heterocycles. The participation of simple 1-aza-1,3-butadiene in an IED-DA is rarely observed; however, this problem can be circumvented by introducing either EW or electron-donating substituents on the nitrogen atom. Fowler et al.5859 and Boger et al.6061 have shown that the inclusion of a strong EW acyl o sulfonyl group on the nitrogen atom favours considerably the IED-DA reactivity of the desired azabutadienes. Boruah et al.62 described the IED-DA reaction of the N -acyl-4-chloro-1-azabutadiene 41 with a series of vinylamines 42 to give the derivatives 44 (see Scheme 3). Ac

Ac

N

N +

N

Cl

Cl 41

N

N

42

43

Scheme 3

44

N

R. Contreras et al.

153 O

C

O

CH3

CH3 N

N

CH3

+ 45

CH3

C N

BH3

N(CH3)2

35

Scheme 4 These reactions are initiated by a DA reaction between 41 and 42 to give the intermediate 43, which affords in the reaction conditions the final derivative 44.62 In addition, this reaction was manifestly catalysed with the presence of TiCl4 or BF3 · OEt2 LA. The DA reaction between 1-acyl-1-aza-1,3-butadiene (45) and dimethylvinylamine (35) in the absence and in the presence of BH3 , as a LA catalyst, was theoretically studied (see Scheme 4).63 1-Aza-1,3-butadiene (48) has a larger electrophilicity, = 147 eV, than 1,3-butadiene (20), = 105 eV (see Table 5 and Chart 2). In addition, dimethylvinylamine (35) has a lower electrophilicity, = 027 eV, than ethylene (29), = 073 eV, the former being classified as a strong nucleophile. Thus, the electrophilicity difference between 1-aza-1,3-butadiene (48) and dimethylvinylamine (35), = 12 eV, suggests a larger polar character for this cycloaddition as compared with that obtained for the butadiene + ethylene one, = 032 eV, a process classically classified as pericyclic.40 Acylation of the nitrogen atom of the 1-aza-1,3-butadiene (48) markedly increases its electrophilic character, in agreement with the lowering of the activation energy.63 The electrophilicity of 1-acyl-1-aza-1,3-butadiene (45) is = 214 eV. Furthermore, the presence of the BH3 coordinated to the N -acyl-1-aza1,3-butadiene increases the electrophilicity of the BH3 complex 46 to 3.24 eV. As a Table 5 Global propertiesa of same dienes involved in IED Diels–Alder reactions of 1-azabutadiene derivatives Molecule 46 47 45 48

1-Acyl-1-aza-1,3-butadiene-BH3 1-Sulfonyl-1-aza-1,3-butadiene 1-Acyl-1-aza-1,3-butadiene 1-Aza-1,3-butadiene

−01855 −01825 −01625 −01505

01446 01946 01680 02100

a

324 233 214 147

and in a.u., in eV. All quantities were evaluated at the B3LYP/6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

BH3 CH3

O

SO3H

O

CH3

N

N

N

N

46

47

45

48

Chart 2

154


result, the LA-catalysed DA reaction between the acetyl derivative 46 and the vinylamine 35 presents a large = 297 eV, thereby indicating a large polar character for the process.63 The N -sulfonyl-1-aza-1,3-butadiene (47) + methyl vinyl ether (33) cycloaddition reaction presents a similar electrophilicity difference, = 191 eV, to that for the reaction between 1-acyl-1-aza-1,3-butadiene (45) and dimethylvinylamine (35). Although the vinyl ethers are less nucleophilic than vinylamines, the electrophilicity for dimethylvinylamine is lower than that for methyl vinyl ether (see Table 1). The larger electrophilicity for the N -sulfonyl-1-aza-1,3-butadiene (47) than for the N -acyl1-aza-derivative (45) explains why both reactions present similar reactivities. Some DA reactions have been reported with aldehydes as dienophiles, and usually they need long reaction times, high temperatures and sometimes high pressure, restricting the practical synthetic utility of this chemistry. This hindrance can also be overcome with the use of EW substituents on the dienophile, very reactive dienes or LA catalysis.60−66 =O double bond of the ketones is less reactive than that in aldehydes. There As the C= are very few reports of DA reactions using simple ketones.646768 Recently, Huang and Rawal69 have reported the hydrogen bond-promoted HDA reactions of unactivated ketones (see Scheme 5). These authors concluded that the increased reaction rate in chloroform could arise from a C-H· · ·O=C interaction, which would render the carbonyl group a stronger heterodienophile.69 Calculations at the B3LYP/6-31G* level were used to show how hydrogen bond formation influences the chemical reactivity of ketones.70 The effect of the chloroform on the activation energies was modelled by means of discrete-continuum models. Explicit hydrogen bond formation to chloroform lowers the gas-phase activation barrier. A DFT analysis of the global electrophilicity of the reagents provided a sound explanation of the catalytic effects of chloroform (see Table 6 and Chart 3). The electrophilicity of acetone O TBSO

1) Chloroform +

2) AcCl, –78 °C

O O

NMe2 49

50

51

Scheme 5 Table 6 Global properties and global electrophilicitya of acetone and acetone hydrogen bonded to one and two discrete chloroforms Molecule 52 53 54 55 56

−01669 −01655 −01503 −01385 −01277

01962 02362 02325 02334 02330

193 158 132 112 095

a and in a.u., in eV. All quantities were evaluated at the B3LYP/ 6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

R. Contreras et al.

O CH3

52

155 BH3

Cl3CH

CH3

CH3

O

HCCl3

CH3

53

O

CH3

HCCl3

CH3

54

O

O H CH3

CH3

55

CH3

56

Chart 3 (56), 0.95 eV, is lower than that for acetaldehyde (55), 1.12 eV, in agreement with the enhanced reactivity of the latter as an electrophile. Coordination of the oxygen atom of acetone by a LA modelled by BH3 (see 52 in Chart 3) enhances the electrophilicity of the acetone to 1.93 eV. On the other hand, explicit solvation of acetone, 53 and 54, increases the electrophilicity to 1.58 and 1.32 eV, respectively (see Table 6). In both cases, the solvated acetone has a larger electrophilicity than acetaldehyde. As a consequence, a larger reactivity towards a nucleophile is expected. Therefore, the role of chloroform can be considered to be comparable to that produced by a Lewis acid catalyst; it increases the electrophilicity of the electron-acceptor reagent, thereby increasing its reactivity in a polar DA reaction.70 In summary, the effect of the LA catalyst on the DA reactions can be explained by an increase of the electrophilicity of the electron-poor DA component which is the dienophile on a NED-DA reaction and the diene in an IED-DA reaction. The enhancement in electrophilicity entails an increase of the ionicity of the process, which is usually accompanied by a decrease of the activation energy in the cycloaddition. 3.1.4. Regioselectivity of polar DA reactions. The local electrophilicity index. In the preceding section, we have shown that the classification of the diene/dienophile pair within a unique electrophilicity scale is a useful tool for predicting the polar character of the cycloaddition, and therefore the feasibility of a cycloaddition process.39 A diene/dienophile pair located at the ends of this scale will display a polar reactivity characterized by a large charge transfer at the transition state structures involved in the mechanism of this particular cycloaddition reaction. For these polar cycloadditions, the more favourable regiosiomeric pathways can be associated with the bond formation at the more electrophilic and nucleophilic sites of unsymmetrical diene and dienophile reagents, respectively. The interaction between unsymmetrical dienes and dienophiles can give two isomeric adducts, depending upon the relative position of the substituent in the cycloadduct, head-to-head or head-to-tail (see Scheme 6). Regioselectivity has been previously described in terms of a local hard and soft acid and base (HSAB) principle, and some empirical rules have been proposed to rationalize the experimental regioselectivity pattern observed in some DA reactions.7172 There is not a single criterion, however, to explain most of the experimental evidence accumulated in cycloaddition processes involving four centre interactions. An excellent source for the discussion of regioselectivity in concerted pericyclic reactions is given in reference.73 The local electrophilicity/nucleophilicity character of reagents on the other hand may also be of significant utility to predict the regioselectivity patterns that can be expected for a given reaction and to quantitatively assess the effects of electron-releasing and EW substituents in the electrophile/nucleophile interacting pair. A useful simplification for

156

Electrophilicity index in organic chemistry (a) 1-substituted butadienes

D A

Head-to-head

ortho

D A D

meta Head-to-tail

A

(b) 2-substituted butadienes D Head-to-head

A meta

D

A D para Head-to-tail

A

Regioisomeric pathways for a Diels–Alder cycloaddition. The more electrophilic and nucleophilic sites are remarked.

Scheme 6 (a) 1-substituted butadienes D A

Larger fk (lower ωk)

D

D A

A

Larger wk

ortho

(b) 2-substituted butadienes (lower ωk) Larger fk Larger ωk D D

D A

A

A para

The most favorable reactive channels along the nucleophile/electrophile interaction.

Scheme 7 the study of regioselectivity in DA reactions may be obtained by looking at those processes having a markedly polar character, where the transition state structure associated with the rate-determining step mostly involves the formation of a single bond between the most electrophilic and nucleophilic sites in the DA reagent pair (see Scheme 7).74 The most nucleophilic site at the electron-donor moiety will be the one presenting the highest value of the Fukui function for an electrophilic attack fk− ,22 while the most electrophilic site at the electron-acceptor moiety will be the one presenting the

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157

highest value of the Fukui function for an nucleophilic attack fk+ , the more or less polar character of the global interaction being described, according to our proposed model, by the difference in the absolute electrophilicity .39 Table 7 summarizes the values of electrophilic and nucleophilic Fukui functions, together with the global and local electrophilicity index for some dienophiles and dienes involved in a series of DA reactions. Since most of the dienophiles considered are substituted ethylenes, we can take ethylene as a reference to discuss the variations in local reactivity induced by chemical substitution. Ethylene (29) presents a local electrophilicity value k = 037 eV at the equivalent carbon atoms C1 and C2. Note that acetylene (32), having equivalent Fukui functions for both electrophilic and nucleophilic attacks, presents a lower electrophilicity pattern as compared with that of ethylene (k = 027 eV at the equivalent carbon centres of structure 29). Consider for instance the interaction between acrolein (10) and 1-methoxy-1,3butadiene (27), which according to our classification will show an electrophilicity difference = 107 eV. This interaction corresponds to a NED-DA reaction in which an electron-poor ethylene, the electrophile, reacts with an electron-rich diene, the nucleophile. The electrophilic site in acrolein (10) is the C2 carbon, with a local electrophilicity value k = 069 eV. The highest value of fk− in 1-methoxy-1,3-butadiene (27) is located at the carbon atom C4. Therefore, the most favorable interaction will take place between the C2 center of 10 and the C4 center of 27, leading to the formation of the ortho adduct (see Scheme 7). The addition of 1-methoxy-1,3-butadiene (27) to acrolein (10) is experimentally known to preferentially afford the ortho regioisomer (80% yield, at 100 C after 2 hours).48 Other interactions between electrophile/nucleophile pairs showing higher difference in absolute electrophilicity are therefore predicted to present a large regioselectivity. Note that the local electrophilicity descriptor contains, according to (5), a global contribution as a factor of the genuine local electrophilic index fk+ , i.e. the Fukui function for a nucleophilic attack at site k. From Table 7, it may be seen that on the basis of this local descriptor of electrophilicity, the C2 site is still three times more favorable than the C1 site in acrolein (10), (k = 069 eV and 0.25 eV, respectively). Another interesting result follows from the comparison of site reactivity of acrolein (10) and the acrolein–BH3 complex (5), representing the LA-catalysed processes. On the basis of the Fukui function alone, the C2 carbon in acrolein is predicted to be slightly more electrophilic than in the LA coordinated species, in contrast with the significant enhancement in the rate constant experimentally observed for the LA-catalyzed process (fk+ = 0372 and 0.357 for 10 and 5, respectively). However, on the basis of the local electrophilicity index k , the C2 carbon in acrolein–BH3 complex (5), k = 114 eV, is approximately twice more electrophilic than the corresponding site in acrolein (10), k = 069 eV. It is also interesting to examine the effect that the electron-releasing groups may have on the local electrophilicity pattern in ethylene 29. These systems have some importance in the IED processes. Consider for instance the electron-rich ethylenes 33 and 35. On the basis of the Fukui function alone, methyl vinyl ether 33 shows nucleophilic activation at the C2 carbon, fk− = 0470. Dimethylvinylamine 35, on the other hand, displays a similar local reactivity pattern, fk− = 0411. Note that on the basis of the index, both compounds are predicted as marginally electrophilic. Both compounds 33 and 35 are

158

Electrophilicity index in organic chemistry Table 7 Local propertiesa of common dienes and dienophiles involved in Diels–Alder reactions

Site(k)

fk−

fk+

1

897

5

320

6

282

8

261

10

184

12

174

13

165

15

151

20

105

21

104

22

093

23

088

24

086

27

077

28

073

29

073

31

057

32

054

33

042

35

027

C N C1 C2 C1 C2 C1 C2 C1 C2 C1 C2 C1 C2 C1 C2 C1 C4 C1 C4 C1 C4 C1 C4 C1 C4 C1 C4 C1 C4 C1 C2 C1 C4 C1 C2 C1 C2 C1 C2

0235 0274 0014 0056 0198 0336 0010 0006 0096 0011 0268 0367 0091 0011 0060 0013 0338 0338 0380 0289 0296 0309 0465 0212 0277 0273 0217 0290 0218 0264 0500 0500 0117 0230 0500 0500 0203 0470 0093 0411

0671 0299 0079 0357 0209 0499 0077 0279 0137 0372 0265 0469 0152 0351 0199 0409 0332 0332 0304 0341 0302 0321 0274 0359 0273 0322 0312 0326 0297 0318 0500 0500 0293 0304 0500 0500 0463 0435 0442 0399

Molecule

k 6022 2680 0253 1144 0589 1407 0200 0726 0253 0685 0461 0816 0250 0579 0300 0617 0355 0355 0316 0354 0282 0300 0240 0315 0234 0277 0240 0251 0217 0232 0365 0365 0162 0173 0268 0268 0194 0183 0119 0108

and k in eV. All quantities were evaluated at the B3LYP/6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

a

R. Contreras et al.

159

therefore predicted to react with electron-poor dienes, as for instance nitroethylene 8, to afford the ortho regioisomer in an IED-DA reaction.54 Other well-known regioselectivity patterns observed in DA cycloadditions may also be deduced from Table 7.74 Therefore, the local electrophilicity index encompassing the effect of global electrophilicity seems to work better than the Fukui functions alone to describe the regioselectivity in DA processes.

3.2. 1,3-Dipolar cycloadditions 3.2.1. Global reactivity in 1,3-dipolar cycloadditions 1,3-Dipolar cycloadditions (1,3-DC) constitute together with the DA reactions one of the more important class of cycloaddition reactions. The general concept of 1,3-DCs was introduced by Huisgen and coworkers in the early 1960s.75 These authors stated the basis for the understanding of the mechanism of concerted cycloaddition reactions. The development of the 1,3-DC reactions has in recent years entered a new stage as the control of the stereochemistry in the addition step is now the major challenge. The stereochemistry of these reactions may be controlled either by choosing the appropriate substrates or by controlling the reaction using a metal complex acting as catalyst.76 A dipole is a system of three atoms over which there are distributed four -electrons. There are a wide variety of dipoles that include a combination of carbon, oxygen and nitrogen atoms within their structures. A picture of the most common dipoles used in 1,3-DC reactions is given in Chart 4. On the other hand, the dipolarophiles can be substituted alkenes or alkynes. The nature of the 1,3-DC reaction mechanism is still an open problem in physical organic chemistry. For instance, the mechanism proposed by Huisgen is that of a singlestep, four-centre cycloaddition, in which two new bonds are both partially formed at the transition state structure, although not necessarily to the same extent.7778 For nitrile oxide (69) cycloadditions, experimental data were interpreted either as being consistent with a concerted mechanism7778 or in favour of a stepwise mechanism with diradical intermediates.7980 The 1,3-DCs appear to be controlled by FMO interactions. In the case of an extremely large difference of the FMO energy of the cycloaddition partners, this model allows one to predict a mechanism via zwitterionic intermediates.81 For instance, Huisgen reported in 1986 the first experimental evidence for the two-step 1,3-DC reaction between a thiocarbonyl ylide (75) and tetracyanoethylene (2) (see Scheme 8).82 Recent studies by these authors for the same 1,3-DC processes have shown that they are examples of a borderline case from a concerted to a two-step mechanism.81 The large HOMO energy of the thiocarbonyl ylides and the low LUMO energy of the tetracyanoethylene are responsible for the large zwitterionic character of these cycloaddition reactions.82 Even though the presence of significant steric hindrance effects at least at one end of the dipole is an additional requirement for the Huisgen’s stepwise mechanisms,82 Fokin, Sharpless et al. have recently reported the stepwise copper-catalysed 1,3-DC reaction of nonhindered azides with terminal alkenes.83 As in the case of the DA reactions, these cycloadditions are classified as belonging to types I, II and III according to Sustmann model.8485 Several theoretical treatments have been devoted to the study of the 1,3-DC reactions of nitrones with substituted alkenes. When electron-deficient alkenes such as -unsaturated carbonyl compounds

160

Electrophilicity index in organic chemistry O

O

O

HN

O

O

O

O

O

H

H N

HN

O

H 2C

N N O

64

H

H

N

N

H2C

66

O

62

N N

63

NH

NH

59

61 NH

H2C

O

N

60

HN

HN

58

57

H2C

O

65

O

H2C

67

O CH2

68

H H2C

HC N O

N

HN N N

NH

71

70

69

H HC N CH2

H2C

N

HC N NH

CH

74

73

72

Chart 4

S

CH2

NC

CN

+ CN

NC

O 75

45 °C 8 h 84%

S O

(CN)2 (CN)2

2

Scheme 8 are activated by coordination to a LA catalyst, the C-C double bond of the alkene is highly polarized, and the electrophilicity of the -carbon increases. In 1,3-DC reactions of -unsaturated carbonyl acceptors activated by a LA, the nucleophilic attack of the dipole is kinetically favoured.86 As a result, it is to be expected that the bond formation at the -carbon of dipolarophile would take place preferentially at the -carbon.87 For instance, 1,3-DC reaction of the nitrone 67e with acrolein in the presence of BF3 (76) leads to the formation of the corresponding Michael adduct complex intermediate that cyclizes to the final cycloaduct (see Scheme 9).87 On the other hand, the IED 1,3-DC reaction of nitrones is also feasible. It requires a dominant LUMOdipole –HOMOdipolarophile interaction.76 These reactions need the activation of the nitrone by a LA. There exist very few examples of reactions between these activated nitrones and electron-rich alkene.8889 Jorgensen et al. have reported

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161

Ph N

H O

O

+

Ph 67e

Ph

0 °C 8 h

LA

Ph

Ph

N

N

Ph

O

O

CHO

OHC 76

77 LA catalyst None ATPH

78

Yield % 77 : 78 20:80 5 >99:1 quant

Scheme 9 Ph

Ph N

O

AlMe3

r.t. + OtBu

Ph

Ph

N O

99% OtBu

79

Scheme 10 the IED 1,3-DC reaction of activated nitrones by chiral LAs with electron-rich alkene (see Scheme 10).90 The reaction presents a total ortho regioselectivity, while the exo stereoselectivity depends on the bulky chiral LA used as catalyst.90 The effect of the LA catalyst on the IED 1,3-DC reactions of nitrones with methyl vinyl ether has been theoretically studied.91 The presence of the LA coordinated to the nitrone increases the asynchronicity of the bond formation and the charge transfer for the ortho cycloaddition. This behaviour may be understood by a change in the mechanism for the catalysed IED 1,3-DC reactions: the inclusion of the LA causes nitrone and methyl vinyl ether to behave as an electrophile /nucleophile, rather than a dipole/dipolarophile pair.91 In Section 3.3.1, we have shown the reliability of the index to predict the reactivity of polar DA reactions.39 The model based on , used to characterize the DA cycloadditions, may be extensible to the 1,3-DC reactions.9293 Thus, depending on the electrophilicity potential displayed by dipole/dipolarophile pairs, the mechanisms for these 1,3-DC reactions will have a more or less marked polar character. The global electrophilicity values of the simplest dipole reagents (see Chart 4) used in the 1,3-DC reactions quoted in Table 8 were evaluated using (2). As in the case of the DA reagents, the series of dipoles may be classified into three general groups. Group I of strong electrophiles include compounds 57 to 63, with electrophilicity numbers larger than 1.50 eV. A second group of moderate electrophiles (group II) include compounds 64 to 68, with electrophilicity values comprised within the range 1.40–0.93 eV. A third group III of marginal electrophiles include compounds 69 to 74, with electrophilicity values smaller than 0.93 eV. Ozone (57) is a well-known strong and stable electrophile.48 Ylid (73), azomethine (72) and nitrile imine (74) that have been classified as marginal electrophiles are known as reactive intermediates that have to be produced in situ.48 However, suitable substitution on dipoles like azides, nitrones and nitrile oxides enhances both their stability and their electrophilicity patterns. The global electrophilicity and its variation induced by chemical substitution of common dipolarophiles has been quantitatively described and classified in the literature.

162

Electrophilicity index in organic chemistry Table 8 Global properties and global electrophilicitya scale for common dipole models involved in 1,3-dipolar cycloaddition reactions Molecule 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

−02599 −02207 −01813 −01655 −02034 −01682 −01329 −01339 −01807 −01361 −01260 −00983 −01249 −00990 −01199 −00868 −00667 −00742

01505 01585 01552 01530 02363 02261 01452 01737 03231 02064 02038 01407 02919 01844 02972 01586 01644 02661

610 418 288 243 238 170 165 140 137 122 106 093 073 072 066 065 037 028

a

and in a.u., in eV. All quantities were evaluated at the B3LYP/ 6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

The electrophilicity of the dipoles may be drastically changed by suitable substitution. In Table 9, we summarize the enhanced/diminished electrophilicity pattern induced by substituent effect for compounds nitrone (67a–i), nitrile oxide (69a–c) and azide (71a–c) that were classified in Table 7 as moderate, 67, and marginal electrophiles, 69 and 71 (see Chart 5). Nitrone (67) when substituted at position R1 or R2 by an electronreleasing methyl decreases its global electrophilicity from 1.06 eV (67) to 0.87 eV (67a) at the R1 position and to 0.81 eV (67b) at the R2 position. However, substitution by a Ph group at R1 and R2 results in a slight enhancement in electrophilicity (1.45 and 1.67 eV in 67c and 67e, respectively). Note that substitution by the strong EW-CHO group on nitrone at R1 results in a strong electrophilic activation in compound 67f, but substitution by CHO at R2 produces an effect even greater: the electrophilicity of nitrone (67) is increased up to four times in compound 67g. Substitution at R1 and R2 by CHO and Me (67h) or by Me and CHO (67i) results in a global electrophilic activation to a lesser extent. Nitrile oxide (69) which was evaluated as a marginal electrophile in Table 8 ( = 073 eV) may be further converted into a poorer electrophile (and probably as a good nucleophile) by substitution at the carbon site by a Me group (69a). However, substitution at the same site by a Ph group renders nitrile oxide a moderate electrophile (69b), but substitution with a CHO group renders the nitrile oxide a strong electrophile ( = 268 eV, 69c). Therefore, it is expected that they will easily react with electron-rich dipolarophiles. Finally, azide (71) displays a quite different activation pattern induced

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163

Table 9 Global properties and global electrophilicitya for some substituted dipole models involved in 1,3-dipolar cycloaddition reactions Molecule

R1

Nitrone derivatives 67a Me 67b H 67c Ph 67d Ph 67e Ph 67f CHO 67g H 67h CHO 67i Me

R2

H Me H Me Ph H CHO Me CHO

−01135 −01080 −01289 −01244 −01312 −01797 −02006 −01573 −01675

02005 01948 01562 01558 01404 01638 01397 01427 01637

087 081 145 135 167 268 392 236 233

−01064 −01406 −01949

02816 01847 01927

055 146 268

−01417 −01302 −01812

02275 01908 02240

120 121 199

Nitrile oxide derivatives Me Ph CHO

69a 69b 69c

Azide derivatives Me Ph CHO

71a 71b 71c

a and in a.u., in eV. All quantities were evaluated at the B3LYP/6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

R2 R1HC

N

O

RC N

O

67a R1 = Me, R2 = H

69a R = Me

67b R1 = H, R2 = Me 67c R1 = Ph, R2 = H

69b R = Ph 69c R = CHO

RN N N 71a R = Me 71b R = Ph 71c R = CHO

67d R1 = Ph, R2 = Me 67e R1 = Ph, R2 = Ph 67f R1 = CHO, R2 = H 67g R1 = H, R2 = CHO 67h R1 = CHO, R2 = Me 67i R1 = Me, R2 = CHO

Chart 5 by chemical substitution. Replacement of the hydrogen atom by Me, Ph and CHO groups always results in an electrophilic activation from = 066 eV in compound 71 to = 120 eV, 1.21 eV and 1.99 eV, respectively (compounds 71a, 71b and 71c in Table 9). Thus, it seems that the electrophilicity enhancement induced by the CHO

164


group results from a cooperative effect of the enhancement in chemical softness, and the increase in the absolute value of the electronic chemical potential (compare the values of compounds 67, 69 and 71 in Table 8, with their corresponding values for the CHO derivatives in Table 9). These aspects introduce different mechanistic patterns expected for the 1,3-DC reactions, as compared with DA cycloadditions (concerted vs stepwise with some zwitterionic character). This result may again be traced to the electrophilicity difference at the ground states of the reacting pairs.39 These results suggest that the description of the reactivity and the reaction mechanism involved in the 1,3-DC processes can be systematized as in the case of the DA cycloadditions. Such a model should be able to determine the charge transfer pattern and to decide which of the partners is acting as nucleophile/electrophile in a polar process, or even to anticipate a concerted pathway in those cases where the electrophilicity/nucleophilicity difference is small. In order to test this model of reactivity, some 1,3-DC reactions, which are shown in Schemes 8–11, will be analysed. In Table 10, the electrophilicity for a series of dipoles given in these schemes is presented. The thiocarbonyl ylid 75 has an electrophilicity value = 144 eV. Therefore, it may be classified as a moderate electrophile. It can react with strong electrophiles like thetracyanoethylene (2), = 596 eV, in a NED 1,3-DC reaction. The large predicted for this reaction, 4.52 eV, points out to a large charge transfer at the corresponding transition state structure,93 in agreement with Huisgen’ proposal.81 N -phenyl-phenylnitrone (67e), on the other hand, with an electrophilicity value = 167 eV, can react with strong electrophiles like acrolein (see Scheme 9) or with marginal electrophiles (nucleophiles) like butyl vinyl ether (see Scheme 10) in NED or IED 1,3-DC reactions, respectively. The rate of these cycloadditions is clearly enhanced by the presence of a LA. Thus, coordination of acrolein to AlMe3 , 76, as a model of ATPH,86 increases the electrophilicity of the -unsaturated carbonyl compound from 1.84 to 3.61 eV. Therefore, the large increase in for the catalysed reactions, 1.77 eV, accounts for the acceleration of the cycloaddition.86 Furthermore, coordination of nitrone 67e to AlMe3 increases the electrophilicity of this reagent from 1.67 to 2.74 eV (compound 79). This large increase in the electrophilicity of nitrone enhances the predicted value for the IED 1,3-DC reaction, in agreement with the increase of the reaction rate91 and the charge transfer pattern91 observed for the process.

Table 10 Global properties and global electrophilicitya for some dipole examples

80 79 75 81 82

−01775 −01490 −01168 −01141 −01014

01542 01102 01285 01589 02803

278 274 144 111 050

a and in a.u., in eV. All quantities were evaluated at the B3LYP/6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

R. Contreras et al.

165

N i

110 °C 21 h

+

O

CO2CH3

Ph

ii Ph

CO2CH3 70 °C 48 h

+

O

OBu

O

99% OBu

N N

25 °C 5 days

N

+

CO2CH3 15

71b

O2N

N N N

+

N

Ph

N

77% CO2CH3

O

C2H5

80 v

N

Ph

67d

iii

iv

O

99%

15

67d

N

N

Ph

20 °C 11 days

N

Ar

N

99% OEt 0 °C 12–16 h

+

N N

CO2CH3

N N

98%

81

vi

CO2CH3

C N

O

CHO

82

N

N O CHO

1h C

Ph

83 %

10

69b

vii

0 °C 2 days

+

O

+

CO2CH3 15

93%

N O CO2CH3

Scheme 11 N -methyl-phenylnitrone (67d) has an electrophilicity value of = 135 eV. Therefore, it can react also with strong electrophiles like methyl acrylate (Scheme 11, entry i), or with marginal electrophiles (nucleophiles) like butyl vinyl ether (Scheme 11 entry ii), in NED or IED 1,3-DC reactions, respectively. Arylazide (71b), which has an intrinsic electrophilicity value of 1.21 eV, is expected to react with electron-poor dipolarophiles, as for instance methyl acrylate (15) (Scheme 11, entry iii). However, the presence of a strong EW group at the para position significantly increases the electrophilicity of azide to = 278 eV in compound 80, thereby activating this molecule to react in an IED 1,3-DC process with ethyl vinyl ether (Scheme 11, entry iv). Diazomethane (64) on the other hand presents an intrinsic electrophilicity of 1.40 eV, and it is accordingly classified as a moderate electrophile. However, the presence of two electron-releasing methyl groups on the diazopropane 81 decreases the electrophilicity to 1.11 eV. This compound is expected to react with electron-poor alkenes in a NED 1,3-DC (see Scheme 11, entry v). Finally, the nitrile oxide derivatives

166


69b and 82, which present a low value of electrophilicity, react with acrolein 10 and methyl acrylate 15 through NED 1,3-DCs also (see Scheme 11, entries vi and vii). These examples allow to anticipate that nitrones that behave as moderate electrophiles, and azides and nitrile oxides that behave as marginal electrophiles, will likely react with electron-poor dipolarophiles in a NED 1,3-DC. However, the presence of strong EW group on the dipole or coordination of the dipole with LAs, results in a large increase in the global electrophilicity of the dipole, thereby activating these molecules to undergo IED 1,3-DC reactions. 3.2.2. Regioselectivity in 1,3 cycloaddition reactions The analysis of the nucleophilic and electrophilic Fukui functions at the more nucleophilic and electrophilic reagent involved in a 13DC allows the prediction of the regioselectivity in these polar reactions.27 A series of regioselective 1,3-DC processes for which experimental data are available were chosen as model reactions (see Scheme 12). Including in this series are reactions between the nitrile ylides 8394 and 8595 with methyl acrylate (15), and the 1,3-DC reactions of the nitrile imine 8496 and the azomethine ylide 8697 with methyl propiolate (14). A comparative analysis was made on the basis of the relative energies, asynchronicity and charge transfer at the transition state structures on one hand, complemented by an analysis based on the reactivity indexes defined in the context of DFT evaluated at the ground state of reactants.74

R

R R CF3 Ph

Ph

C N C 1

2

3

CF3

15

CF3

CF3 N

CF3

Ph

N

CF3

Major product

Minor product

83 R

R R Ph

N N C Ph 1

2

14

3

Ph

N

CH3 1

2

3

CH3

CH3

CH3

C N C

Ph

15

85

N

CH3

Major product

Ph

N

1

N2

CH2

R

3

CH3

CH3

14

86 R=CO2CH3

N

CH3 Minor product

Scheme 12

CH3

Minor product R

R CH3CH

Ph

N

R

R

R

N

Ph

Minor product

Major product

84

Ph

Ph

N

CH3

N

CH3 Major product

R. Contreras et al.

167

These 1,3-DC reactions take place along asynchronous concerted processes. The analysis of the energetic results showed that while for the reactions of the dipoles 83, 84 and 85, the B3LYP/6-31G* calculations predict the regioselectivity experimentally observed, for the dipole 86 the calculations fail to predict the experimental results.27 Previous theoretical studies on 13DC reactions have pointed out that the regioselectivity of this type of cycloadditions is strongly dependent on the computational level used.9899 The electronic chemical potential , chemical hardness , and global electrophilicity for the dipoles 83–86 are displayed in Table 11. Also included in Table 11 are the values of local electrophilicity and the values of the Fukui function for an electrophilic attack fk− and for a nucleophilic attack fk+ at sites k for these dipoles. The two dipolarophiles present similar electrophilicity values, 1.52 eV (14) and 1.49 eV (15) (see Table 1). According to the absolute scale of electrophilicity based on the index,39 these compounds may be classified as strong electrophiles. On the other hand, the four dipoles present electrophilicity values in a wider range: 1.95 eV for 83, 1.43 eV for 84, 1.15 eV for 83 and 0.31 eV for 86. Thus, while 85 is classified as a strong electrophile, 86 is classified as a marginal electrophile (and probably a good nucleophile). Note that even though 83 presents a larger electrophilicity value than methyl acrylate (15), the latter has a larger absolute value in chemical potential (see Table 1), which is the index that determines the direction of the electronic flux along the cycloaddition. Therefore, along this series of 1,3-DC reactions, the more favorable interaction will take place between the less electrophilic species, 86, namely the dipoles in the present case, and the electrophilic dipolarophile methyl propiolate (14). This analysis is in agreement with the lower activation enthalpy and larger charge transfer pattern found for the reaction model 85, and the reaction model 86. In addition, the lower charge transfer found for models 83 and 84 can be related to the similar electrophilicity values displayed by the dipoles and dipolarophiles. In these cases, along the cycloaddition pathway, any of them have the same trend to supply or accept electron density from each other. In Section 4.1.4., we have shown that the analysis of the local electrophilicity,74 k , at the electrophile together with the analysis of the nucleophilic Fukui functions, fk− ,22 at the nucleophile, allowed the prediction of the regioselectivity in DA competitive cycloadditions. For the two electron-deficient ethylenes 14 and 15, classified as strong electrophiles, the C2 carbon atom (the -position) presents a larger local electrophilicity Table 11 Global and local propertiesa of dipoles 83–86

k

f−

f+

83

−0141

0138

195

84

−0117

0131

143

85

−0108

0138

115

86

−0059

0151

031

C1 C3 N1 C3 C1 C3 C1 C3

02232 04488 02685 01613 05339 02909 04379 04937

01819 00714 00759 00662 02100 00089 03165 03460

Molecule

a

k 036 014 011 009 024 001 010 011

and in a.u., and k in eV. All quantities were evaluated at the B3LYP/6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163. k defines the site in the molecule where the property is being evaluated.

168


value than the C1 site. Therefore, the C2 will be the preferred position for a nucleophilic attack by a dipole. This result is in agreement with the asynchronicity shown at all the transition state structures. This behaviour, that has also been observed in other polar cycloaddition reactions, suggests that the most electrophilic reagents control the asynchronicity of the process by a larger bond formation process at the most electrophilic site of the molecule.27 A different local picture is found for the four dipoles where the values of the nucleophilic Fukui functions, fk− , depend on both the structure of the dipole and the substitution pattern. For instance, for the nitrile ylide 83, the C3 carbon atom has a larger fk− value than the C1 carbon atom, 0.4488 and 0.2232, respectively, while for the nitrile ylide 85, there is a change in the local activation: now the C1 carbon atom presents the larger fk− value (see Table 11). These results explain the change of regioselectivity for these 1,3-DC reactions. Substitution of the two electron-releasing methyl groups present in 85 by two EW CF3 groups produces a change in the polarization of the HOMOdipole of 83, which acts as a nucleophile, thereby reversing the regioselectivity pattern. For the azomethyne ylide 86, the presence of the electron-releasing methyl group on the C1 position polarizes the HOMOdipole through the C3 carbon atom. As a result, the unsubstituted C3 carbon atom presents a larger fk− value compared to that at the C1 site. Therefore, along the cycloaddition reaction, the more favourable reactive channel takes place through the C3-C4 bond formation by the nucleophilic attack of the C3 carbon atom of 86 to the more electrophilic C4 site of 14, according to the experiment results.

3.3. Other cycloaddition reactions The analysis performed on the basis of the global and local electrophilicity of reagents involved in DA and 13-DC cycloadditions has been extended to other cycloaddition reactions as 4 + 3100101 and 2 + 2102 cycloadditions. The direct construction of seven-membered rings via 4+3 cycloadditions is the most attractive strategy for preparing this frequently observed natural product substructure. A great amount of effort has been focused on methods to synthetize the less accessible three-atom component of these reactions. Oxyallyl cations are the most employed intermediates to generate this moiety (see Scheme 13). Alternatively, the use of 2(silyloxy)acroleins and related compounds in the presence of a LA catalyst as the three-atom component in 4 + 3 cycloadditions has received much interest in the last years (see Scheme 14). The 2-hydroxyallyl cation 87 has an electrophilicity index of = 215 eV (see Table 12).100 This very high value indicates that it will participate as a strong electrophile in reactions with a large ionic character. The transition state structure associated with the electrophilic attack of 2-hydroxyallyl cation 87 to 1,3-butadiene (20) is located −89 kcal/mol (MP2/6-31G*) lower than reactants.103 On the other hand, the 2-silyloxyacrolein (88) has an electrophilicity of = 176 eV.100 This value is slightly lower than that for acrolein (10), = 184 eV. Coordination of LA, AlCl3 , to 2-silyloxyacrolein increases the electrophilicity of 2-(trimethylsilyloxy)acrolein LA complex (89) to 4.09 eV, being classified as a strong electrophile. On the other hand, furan (30) has a very low electrophilicity value, 0.58 eV, thereby suggesting that it could react as a good nucleophile. The difference in global electrophilicity between the reagent

R. Contreras et al.

169 O

OH +

87

20

Scheme 13 O O

TMS O

OTMS + AlCl3

89

O

AlCl3

O 30

Scheme 14 Table 12 Global properties and global electrophilicitya for some reagents involved in 4 + 3 and 2 + 2 cycloadditions

87 89 88 90 91 93 92

−04383 −02120 −01535 −01643 −01550 −00712 −00671

01214 01493 01821 01883 01763 02682 02532

2150 409 176 195 185 026 024

a

and in a.u., in eV. All quantities were evaluated at the B3LYP/6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

pair, = 351 eV, points out to a large polar character for this process, in clear agreement with the large charge transfer found along this stepwise 4 + 3 cycloaddition.100 This cycloaddition is initiated by the nucleophilic attack of furan (30) to the -conjugated position of the substituted acrolein 89 to give a zwitterionic intermediate. The presence of the electron-releasing 2-silyloxy group polarizes the HOMO electron density of this intermediate located mainly in the acrolein moiety, thereby favouring the ring closure at the carbonyl carbon, with the formation of the seven-membered ring.100 Regioselective reactions of substituted benzynes are of theoretical and synthetic interest. The 2 + 2 cycloaddition of the benzyne 91 possessing a fused four-membered ring to the ketene silyl acetal 92 yields the cycloadducts 94 and 95 with a high regioselectivity (22:1) (see Scheme 15).104 The electrophilicity of benzyne (90) is 1.95 eV, a value that falls within the range of strong electrophiles in the scale.102 This value, which is larger than that evaluated for acetylene (32), = 054 eV, allows to explain the reactivity of the benzyne derivatives towards nucleophilic additions. The electrophilicity of the fused four-membered

170

Electrophilicity index in organic chemistry OSiR3 92 (93)

OEt + OEt

91

22:1

94 EtO

OSiR3

92

95

OSiR3

O

O

93

Scheme 15 benzyne 91, = 185 eV, is smaller than that evaluated for benzyne (90), as a consequence of the electron-releasing effect induced by the alkyl groups. On the other hand, the ketene acetals 92 and 93 have very low electrophilicity values, = 024 and 0.26 eV, respectively, and they behave as strong nucleophiles. The difference in electrophilicity, ∼ 17 eV, indicates that this process will have a large polar character. This fact is in agreement with the large charge transfer found at the transition state structure and zwitterionic intermediate found along the stepwise 2 + 2 cycloaddition.102 It is noteworthy that the feasibility for this 2 + 2 cycloadditions is not only a consequence of the large absolute electrophilicity of the benzyne but also by the strong nucleophilic character of the ketene acetal 92. Both factors contribute to the enhancement of , and as a result, to the feasibility of these 2 + 2 cycloadditions along a polar process. The analysis of the local electrophilicity at the fused four-membered benzyne derivative 91 indicated that the strained substituent present on the C3 position breaks the symmetry of benzyne (90), and the C1 carbon atom becomes slightly more electrophilic than the C2 one. This result is in agreement with the regioselectivity observed in the nucleophilic additions to 91 (see Scheme 15).

4. Empirical relationships between electrophilicity and rate coefficients 4.1. Electrophilicity of carbenium ions In order to further validate the theoretical scale of electrophilicity based on the global electrophilicity index,105 we first compared the theoretical scale with the experimental electrophilicity determined from kinetic data by Mayr et al. for a series of 28 carbenium ions.106107 They are displayed in Chart 6. Included in these series are the tritylium, benzhydrylium and benzylium ions. These charged electrophiles are expected to show enhanced electrophilicity patterns as compared with the neutral electron donors we will discuss later. The electrophilicity values are in the range [9.0–14.0] eV (see Table 13, second column). The local electrophilicity pattern at the carbocation site is displayed in Table 13, fourth column. They are obtained by projecting the global electrophilicity with the electrophilic Fukui function fk+ , using (5). These values have been used to

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171

Table 13 Global electrophilicity,a electrophilic Fukui function at the carbocation site, local electrophilicity at the carbocation site and log of the rate coefficients for hydrolysis of carbenium ions, kw bc Carbenium ions 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

fC+

C

log kw

111 107 102 101 97 94 93 91 79 130 126 123 121 118 115 106 101 99 94 94 137 128 125 122 116 112 104 95

03363 03207 03105 03015 02956 03044 02853 02924 02781 03550 03456 03376 03311 03252 03161 02903 02790 02845 02845 02799 04376 04358 04172 03925 03948 04163 04054 03743

372 344 317 307 288 286 264 267 219 461 436 415 399 384 362 307 281 280 263 263 600 557 522 478 457 464 422 354

518 300 198 −202 −234 117 −368 −264 −471 848 808 751 632 596 511 058b −048b −159b −225b −266b 1100 1023 960 830 770 930 770 685

and C in eV. All quantities were evaluated at the B3LYP/6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163. b Log kw values (in s−1 ) from references [4],[14] and [108]. c Log kw values (in s−1 ) from reference [106]. a

obtain linear relationships with the experimental rate coefficients. Figure 2 shows the comparison between the local electrophilicity index evaluated at the carbocation site, C , and the logarithm of the rate coefficient for the hydrolysis for the whole series of 28 carbenium ions depicted in Chart 6. The resulting regression equation is: log kw = 4714C − 13827

(6)

The linear relationship between both variables is qualitatively acceptable (regression coefficient R ≈ 094) if one considers that this series of carbenium ions comprises a large variety of different structures, thereby suggesting that the regional electrophilicity patterns at a carbocation site, imbedded in different chemical environments may be used

172

Electrophilicity index in organic chemistry 12 10 8

log kw

6 4 2 0

R = 0.938 N = 28 P < 0.0001

–2 –4 –6 2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

ωC [eV] Figure 2 Comparison between experimental log kw for the neutral hydrolysis of 28 carbenium ions and the local electrophilicity index at the carbocation site, C . R is the regression coefficient; N is the number of points and P is the probability that the observed correlation was randomly obtained

to correctly assess the effect of chemical substitution on the electrophilic potential of molecules, and therefore it can be further considered as a reliable descriptor of reactivity. The deviations of compounds 116–118 that show the highest values of electrophilicity at the carbocation site may be at least partially due to the fact that the rate constants with water are close to the diffusion control limit. For this reason, the increase of C cannot be reflected in these rate constants.105106 Despite the deviations observed, we may still validate the linear relationship given in (6) by using it to predict the rate coefficient for the neutral hydrolysis of other carbenium ions not included in the present data base. An immediate application of (6) is the evaluation of the rate coefficient for the p-CH3 tritylium ion not included in the regression analysis shown in Figure 2, for which C = 360 eV. Application of (6) to this compound yields a predicted log kw = 314. The experimental electrophilicity parameter E = −013 for this compound has been recently reported by Minegishi Mayr.106 It is upper bounded by the E parameter for tritylium ion (E = 051 log kw = 518, compound 96 in Table 13) and lower bounded by (p-OCH3 tritylium ion (E = −187, logkw = 300, compound 97 in Table 13). Since the E parameter shows a linear relationship with log kw ,8 the order relationship log kw (p-OCH3 tritylium ion) < logkw (p-CH3 tritylium ion) < log kw (tritylium ion) is also satisfied. Another pertinent prediction concerns the rate coefficient for the hydrolysis of the tris-(o,p-OCH3 2 , tritylium ion experimentally evaluated by Ritchie108 and not included in the present database. Using its local electrophilicity value C = 265 eV, the predicted rate coefficient obtained from (6) yields log kw = −133 which is again correctly upper and lower bounded by those of tritylium ion (log kw = 518, compound 96 in Table 13) and (p-NMe2 , p-OCH3 ) tritylium ion (log kw = −234, compound 100 in Table 13).

R. Contreras et al.

173

The analysis of the effect that the different substituents may have on the rate coefficients in this case is particularly hard to perform by using simple inductive effects, as described for instance by the Hammett substituent constants. Note that, despite the fact that most of the 28 carbenium ions exhibit para substitution, the general structure on top of Chart 6, shows a complex substitution pattern at the carbocation centre. In order to asses the effect of multiple substitution at this site, we first considered those compounds that have two fixed hydrogen atoms at the p-position of the phenyl group, which according to Hammett classification have p H = 00, and the third phenyl group substituted at p-position with H (compound 96 in Chart 6) OCH3 (compound 97 in

R1 R2

Carbenium ions 96

R1

+

R3

R2

σ p(ω C)

R3

0.01 97 OCH3

–0.34

98 OCH3

OCH3

–0.68

99 N(CH3)2

–0.81

100 N(CH3)2

OCH3

OCH3

OCH3

N(CH3)2

N(CH3)2

OCH3

OCH3

N(CH3)2

N(CH3)2

N(CH3)2

N(CH3)2

–1.05

101 OCH3

–1.07

102 –1.35

103 –1.31

104 105

H

106

H

–1.92 1.14

CH3

0.82 107

H

CH3

CH3

0.56

108

H OCH3

109 110

H

0.35

OCH3

CH3

OCH3

OCH3

0.16

H

Chart 6

–0.11

174

Electrophilicity index in organic chemistry R1 R2

Carbenium ions

R1

111

H

112

H

113

H

+

R3

R2

σp(ωC)

R3 N(CH3)CH2CF3

N(CH3)CH2CF3

–0.81 N

N

O

O

–1.13 N(CH3)2

N(CH3)2

–1.15 114

H N

N CH3

115

CH3

–1.36

H N

N

–1.36 116

CH3

H

117

CH3

CH3

118

CH3

H

2.89 2.35 CH3

1.90 119

H

H OCH3

1.35 120

CH3

H OCH3

1.08 121

H

OCH3

122

CH3

OCH3

123

CH3

OCH3

1.17 0.64 OCH3

–0.21

Chart 6

Continued

Chart 6), NMe2 (compound 99 in Chart 6) plus the p-CH3 substituted compound not shown in Chart 6 (for which C = 360 eV). The comparison between the p values and the local electrophilicity index C is shown in Figure 3. It for this short series the p increases linearly with the local electrophilicity index C (regression coefficient R ≈ 099). The resulting empirical equation is: p = 1261C − 4676

(7)

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175

0,0

σp

– 0,2

– 0,4

– 0,6

R = 0.991 N=4 P < 0.009

– 0,8

3,0

3,1

3,2

3,3

3,4

3,5

3,6

3,7

3,8

ωC [eV] Figure 3 Comparison between Hammett substituent p constants for singly substituted carbenium ions and the local electrophilicity index C . R is the regression coefficient, N is the number of points and P is the probability that the observed correlation was randomly obtained

From this equation, we define a new substituent constant p C which is uniquely determined by the knowledge of the C index. The p C values for the whole series of carbenium ions considered in this work are shown in Chart 6, last column. The following results are relevant: in the tritylium subseries (compounds 96–104 in Chart 6), multiple substitution at the carbocation site with p-OCH3 and p-NMe2 substituted phenyl groups results in an electrophilic deactivation p C < 0, thereby indicating that the net effect of these groups is to act as electron-releasing substituents. Note that the rate coefficients are consistently predicted to be lesser than the reference compound 96 (see Table 13). While the -OCH3 group seems to make an approximately additive contribution to p C of c.a. 0.34 units, in the case of increasing substitution with NMe2 this rule is less clear. However, a more important result is found in the subseries of benzhydrylium ions (compounds 105–115 in Chart 6). For this series, the p C values indicate that while for the double substitution with phenyl groups at R2 and R3 and some combination of p-CH3 and p-OCH3 , the global effect is activating (p C > 0 for compounds 105–109 in Chart 6), some other combinations involving p-OCH3 , p-NMe2 , mfa2 , mor2 , thq2 and pyr2 9 result in a global substituent effect that becomes deactivating (p C < 0 for compounds 110–115 in Chart 6). Note that the highest activating effect is shown by the Ph2 CH+ ion (p C = 114, compound 105 in Chart 6), thereby suggesting that in this compound, two adjacent Ph groups cooperatively stabilize the carbocation by resonance. Increasing substitution by one and two methyl groups at p-position significantly attenuate this activation pattern (see compounds 106 and 107 in Chart 6). Other combinations including mixed substitution with (Ph and p-OCH3 ) and (p-OCH3 and p-CH3 ) result in a marginal electrophilic activation at the carbocation site (see for instance compounds 108 and 109 in Chart 6). Any combination of substitution involving p-OCH3 , p-NMe2 and mfa2 , mor2 , thq2 and pyr2 ,106 on the other

176


hand, systematically results in increasing electrophilic deactivation at the carbocation site (p C < 0, for compounds 110–115 in Chart 6). Finally, for the subseries of substituted benzyl ions (compounds 116–123 in Chart 6), we may observe that for some combination of p-CH3 or p-OCH3 with CH3 and OCH3 groups at R1 and R3 , the global effect is activating (p C > 0 for compounds 116–122 in Chart 6), with the only exception of compound 123, for which p C < 0. Note that the only difference with respect to compound 122 for which p C > 0 is the p-OCH3 substitution at R2 . The effect observed in the cases of p-OCH3 in compounds 119 and 123 may be again traced to resonance effects. For instance, while in compound 119 there is the possibility to form one oxonium structure at the p-OCH3 position by resonance, structure 123 offers an additional oxonium resonant structure at R3 .

4.2. Electrophilicity of benzylating and alkylating reagents in Friedel–Crafts reactions Another pertinent application of the global electrophilicity index concerns the relation found between this reactivity index and the substrate selectivity in Friedel–Crafts alkylation and benzylation reactions.109 One of the factors determining the reactivity in the electrophilic aromatic substitution (EAS) reactions is the electrophilicity of the attacking group. For instance, it has been shown that substituents may affect the substrate selectivity as measured by the kTolueno /kBenzene kT /kB hereafter) ratio. kT and kB are the rate constants for the acylation reaction of toluene and benzene, with respect to the same reference acylating agent, respectively. Thus, while electron-donor substituents at ortho or para positions with respect to the benzylic centre increase the kT /kB ratio, EW groups at these positions decrease it.110111 On the other hand, the acylation of toluene and benzene clearly proved the importance of substituents on the electrophilicity of the substituting agent, which was reflected by high kT /kB ratios. The effect of the nucleophilicity of the aromatic substrate was observed to cause similar effects,112 so that with increasingly more basic aromatics, even relatively weak electrophiles resulted in early transition states resembling more starting materials than intermediates.110−112 This result prompted us to look at the substrate selectivity in EAS reactions using static reactivity models developed around the ground states of reactants. The global electrophilicity of benzylating and alkylating agents participating in Friedel–Crafts EAS reactions were ranked within an absolute scale. This theoretical scale correctly accounted for the electrophilic activation/deactivation promoted by electron-releasing and EW substituents. The comparison between the global electrophilicity index and the experimental substrate selectivity index kT /kB showed a linear relationship. With the resulting empirical regression equation, the values of the kT /kB index predicted, from the knowledge of the index, resulted to be accurate to within 10% and less.113

4.3. Electrophilicity of the carbonyl group The electrophilicity index has also been used to address fundamental aspects on the chemistry of carbonyl compounds. The electrophilicity of the carbonyl carbon is one

R. Contreras et al.

177

of the determining factors of the chemical reactivity exhibited by fundamental compounds, such as aldehydes, ketones or carboxylic acid derivatives. The substituent effects on nucleophilic reactions of a series of acyl-substituted phenyl acetates and phenyl-substituted phenyl acetates have been studied recently by Neuvonen et al.114 Good correlations with negative slopes for the plots of log k vs the 13 C NMR chemical =O have been found. The upfield shift of the carbonyl carbon signal in shift C C= 13 the C NMR spectra was related with the EW ability of the phenyl substituent of phenyl dichloroacetates or benzoyl substituent in methyl benzoates.114 Since an upfield chemical shift is usually associated with an increase of the electron density at the site, Neuvonen et al. interpreted this positive variation of the electron density at the carbonyl carbon as a decrease of the electrophilicity at this site. These authors114 proposed that the destabilization of the ground state (GS) induced by EW substitution could account for the increase of the reaction rate by means of a decrease in the activation energy: E = = ETS − EGS . Isodesmic reactions were used to evaluate the GS destabilization for a series of X-substituted phenyl trifluoroacetates, phenyl dichloroacetates and phenyl acetates. We studied the effects of the EW substitution on the activation energies by computing the barriers for nucleophilic attack by the hydroxide ion on the same series of substituted phenyl acetates.115 The energy study was complemented with the calculation of the global electrophilicity index. The results obtained for the GSs of carbonyl compounds show that the electrophilicity does increases with the EW substituent effects in these systems and that despite the observed electron density accumulation at the carbonyl group, the enhanced electrophilicity induced by the EW groups in the substrates drives the nucleophilic attack at the carbonyl carbon. The analysis based on thermodynamic, Eiso , and kinetic, E = , energetic aspects shows that the effect of the EW substitution on the corresponding TS largely outweighs the energy destabilization at the GS. The present interpretation of the EW substitution effect on the carbonyl compounds based on the electrophilicity index is completely consistent with the kinetic data reported for these systems.115

4.4. Reactivity of the carbon–carbon double bond towards nucleophilic additions =C double bond is a reactive site on The soft electron density present on the C= the hydrocarbon skeleton. In the case of unsubstituted alkenes, the two carbon atoms participating in the double bond present a negative charge because of the larger elec=C−H). The more tronegativity of the carbon atom relative to the hydrogen atom (C= electronegative character of the sp2 hybrid respect to the sp3 one is an additional source for the better stabilization of a negative charge in the former case. Therefore, =C functionality is the attack by electrophiles. the expected reactivity pattern of the C= However, this behaviour can be drastically modified by suitable substitutions. For =C bond modifies the reactivinstance, the presence of an EW substituent at the C= ity pattern of an adjacent carbonyl group, thereby leading to the well-known result that the -unsaturated carbonyl compounds usually undergo conjugate nucleophilic additions, named Michael additions. The nucleophilic activation of the -unsaturated carbonyl compounds is a challenging problem involving intramolecular selectivity in polyfunctional systems, which may be conveniently treated in terms of local (regional)

178


reactivity indexes. For instance, -unsaturated carbonyl compounds are commonly used as Michael acceptors. Extensive kinetic work on the nucleophilic additions to activated olefins have been reported by Bernasconi et al.116−119 They include the thiolate ion addition to substituted -nitrostilbenes116 and morpholine and piperidine additions to substituted benzylidenemalononitriles.117 Recently, Mayr et al.120 presented a kinetic study on the reactivity of a series of benzylidenemalononitriles towards a wide variety of nucleophiles. These authors used an experimental scale of electrophilicity to predict the reactivity of these systems. Both, Mayr’s and Bernasconi’s groups have provided experimental rate coefficients that will be used to test the predictive capability of the theoretical electrophilicity scale.39 In Chart 7 we included a short series of benzylidenemalononitriles as well as nitrostilbenes.116 Note that the most common functionalities associated with the chem=C bond, namely aldehydes, ketones, esters, istry of the nucleophilic addition to the C= anhydrides and nitriles, are EW substituents present in dienophiles and dipolarophiles used in NED DA and 1,3 dipolar cycloadditions, and they are included in Table 1.121 In Table 14, we summarize the global properties and local electrophilicity values for the series of benzylidenemalononitriles and -nitrostilbenes. Both series present ethylene derivatives with large electrophilicity values, > 23 eV, they being classified as strong electrophiles within the theoretical electrophilicity scale.39 The benzylidenemalononitriles and the -nitrostilbenes substituted at the -phenyl ring present electrophilic activation when the hydrogen atom at the para-position of the aromatic ring in compounds 129 and 135 are replaced by the strong EW group −NO2 to give compounds 124 and 130, respectively. Note that the presence of a -Me group at the same position in compound 136 results in electrophilic deactivation of 135. A quantitative relationship between Hammett substituent constant for substituted ethylene and the global electrophilicity index has been found.122 Therefore, it is not surprising to find a good correlation between ln k for the addition of HOCH2 CH2 S− to substituted -nitrostilbenes reported by Bernasconi et al.116 and the global electrophilicity index , as shown in Figure 4.

O2N

CN

NC

CN CN Cl

CN 124

Ph

129

127

MeO

CN

NO2

CN

Ph 135

133

NO2

Ph

CN

CH3 136

Chart 7

CN

Ph NO2 132

131

130

NO2

Br 134

CN 128

CN

Me2N Ph

CN

CN

126

CH3

H

NO2

Cl

NO2

NO2

CN

CN

CN

125

CN

Br

CN

CN 137

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179

Table 14 Global properties and local electrophilicitiesa of the series of electrophilically activated benzylidenemalononitriles and -nitrostilbenes 124 – 137 Molecules 124 125 126 127 128 129 130 131 132 133 134 135 136 137

k (C1)

−02112 −02030 −01907 −01860 −01874 −01832 −01815 −01774 −01788 −01672 −01681 −01647 −01614 −01489

01451 01454 01503 01447 01472 01529 01501 01490 01536 01419 01463 01504 01474 01279

418 385 327 325 324 299 299 287 283 268 263 245 241 236

058 061 054 052 052 048 025 045 024 041 022 022 021 033

k (C2) 074 090 093 092 092 088 067 085 063 081 059 053 052 070

and in a.u., and k in eV. All quantities were evaluated at the B3LYP/6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

a

13.5

ln(k) = 3.70 (ω ) + 1.83 R 2 = 0.95

13.0

ln(k)

12.5 12.0 11.5 11.0 10.5 10.0 2.4

2.6

2.8

3.0

ω Figure 4 Plot of ln(k) vs the electrophilicity index for the reaction of the addition of HOCH2 CH2 S− to substituted -nitrostilbenes. Rate coefficients k from reference 116

The nitrile derivatives subseries is particularly interesting, since it contains a short series of benzylidenemalononitrile compounds (129, 133 and 137), which have been kinetically evaluated by Mayr. For these compounds, there exist data for both the rate coefficients and the electrophilicity numbers (E).123 This subseries will give us the opportunity to test the predictive value of our model. Consider for instance compounds 129, 133 and 137. The experimental order of electrophilicity (E) is 137 < 133 < 129. Note that our predicted electrophilicity is in good qualitative agreement with this order. Unfortunately, the quantitative comparison is difficult since the experimental rate

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coefficients and electrophilicity values in comparable conditions (20 C H2 O/DMSO 1/1 v/v, piperidine as reference nucleophile) are only available for these three benzylidenemalononitriles (compounds 1a–1c in reference 123 ). Despite this limitation we compare in Figure 5 our predicted electrophilicity values with the experimental reaction rate coefficients reported by Mayr et al. It may be seen that the agreement is reasonably good. From the regression (8) lnk = 555 − 409

(8)

we predict that the expected reaction rate coefficient for the reaction of compound 124 with piperidine would be about 1000 times faster than the reaction of piperidine with compound 129 (compound 1a in reference 124). However, due to symmetry considerations,42 such an enhancement should be significantly lesser than this figure. The predicted electrophilicity value for compound 124, not evaluated in Mayr et al.’s database, is E = −188 using the regression equation in Figure 6. However, this extrapolation is weak in view of the small number of experimental points available. A more reliable

13.0

ln(k) = 5.55 (ω) – 4.09 R 2 = 0.89

12.5 12.0 11.5

ln(k )

11.0 10.5 10.0 9.5 9.0 8.5 8.0 2.2

2.4

2.6

ω

2.8

3.0

3.2

Figure 5 Plot of ln(k) vs the electrophilicity index for the reaction of benzylidenemalononitriles series with piperidine. ∗ Predicted value. Rate coefficients k from reference 123

–8

E = 6.18 (ω) – 27.71 R 2 = 0.98

–9

E

– 10 – 11 – 12 – 13 – 14 2.2

2.4

2.6

2.8

3.0

3.2

ω Figure 6 Plot of experimental electrophilic parameter E vs the electrophilicity index for the benzylidenemalononitriles series. ∗ Predicted value. Rate coefficients k from reference 123

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quantitative comparison may be obtained by an interpolation procedure for a compound that is expected to be bound, both in the experimental E scale and in the reaction rate coefficients. This is the case of compound 131, whose predicted electrophilicity in the theoretical scale is = 287 eV. Using the regression equation in Figure 6, this theoretical electrophilicity leads to the prediction that in the experimental scale compound 131 should show an E number around −994. Furthermore, the predicted rate coefficient for the reaction of this compound with piperidine in the same conditions should be bound by the rate coefficients of compounds 133 and 129 k ≈ 14∗ 105 M−1 s−1 , obtained from the regression equation (8)). Regarding the incorporation of the experimental rate coefficients evaluated for compound 131 by Bernasconi’s group in comparable experimental conditions,119 the experimental value for the reaction of compound 36 with piperidine in a solvent mixture 50% Me2 SO-50% H2 O is k = 214∗ 105 M−1 s−1 , thereby showing the predictive capability of the global electrophilicity index. Finally, five new substituted benzylidenemalononitriles were incorporated118 to the Mayr et al.’s series. The result of the comparison between ln k and the global electrophilicity index is displayed in Figure 7. Note that this time the quantitative comparison between both quantities is not as good as the previously discussed correlation R2 = 075. The main factor that may account for this deviation may be probably traced to solvent effects (not incorporated in the evaluation of the global electrophilicity index), preferentially affecting the highly polar electron-releasing –NMe2 group (compound 137) and the highly polar EW –NO2 and –CN groups (compounds 124 and 125).42 Thus, if we exclude compound 137 of this series, the correlation improves considerably R2 = 090. 4.4.1. Regioselectivity in the nucleophilic addition to carbon–carbon double bond The regioselectivity is well represented by a local reactivity picture in asymmetrically substituted ethylenes.121 The local electrophilicity values, k , for the two carbon =C1CN2 and atom belonging to the C=C double bond, named as C1 and C2 C2= =C1NO2 , of the series of benzylidenemalononitriles and the -nitrostilbenes 124– C2= 137 are given in Table 14, while local electrophilicity values of the electron-poor 16

ln(k ) = 2.59 (ω) + 4.28 R 2 = 0.75

15 14

ln(k)

13 12 11 10 9 8 2.2

2.7

3.2

ω

3.7

4.2

Figure 7 Plot of ln(k) vs the electrophilicity index for the reaction of substituted benzylidenemalononitriles series with piperidine. Rate coefficients k from reference 117

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substituted ethylenes used as dienophiles in DA reactions are given in Table 7. A joint analysis of the global electrophilicity of these molecules, and the projected local electrophilicity at the sites C1 and C2, allows to obtain some additional conclusions: (i) in most of the cases the local electrophilicity value at the C2 position is larger than that at the C1 position. In general, the k value at the C2 is c.a. 2.4 times the one localized at the C1 position. (ii) The inclusion of a phenyl group at the C2 position decreases this relation to c.a. 1.9 (see Mayr’s subseries including compounds 129, 133 and 137). Note that for compound 124, whose global electrophilicity is predicted to be drastically enhanced with respect to the parent compounds 129, 133 and 137 by the presence of the −NO2 group at the para position of the phenyl substituent, the C2 /C1 ratio decreases to 1.3. This result may suggest a lost of effectiveness of this molecule as a potential strong Michael acceptor in the sense that despite its high global electrophilicity it becomes at the same time less regioselective. This result is reminiscent of that obtained in the analysis of the cyanoethylene subseries.42 Therein, the tetracyano derivative was shown to display the highest global electrophilicity, yet its reaction mechanism with cyclopentadiene was consistently predicted to proceed via a polar concerted synchronous pathway, not a stepwise one with a first step corresponding to a Michael addition.121 (iii) The distribution of the local electrophilicity at the carbon atoms belonging to the =C bond represents in most of the cases c.a. 50% of the global electrophilicity of C= the molecule. Only in compound 134 of the series considered here, this distribution represents less than 30% of the global electrophilicity. This fact together with the large activation of the C1 position respect the C1 one allows to conclude that the C2 site is the most electrophilically activated centre of these electron-deficient substituted ethylenes, in complete agreement with experimentally observed regioselectivity shown by these molecules in the Michael addition reactions.

5. Substituent effects on electrophilicity. The electronic contribution to the p parameter of the Hammett equation One of the main goals in physical organic chemistry is the systematic description of the influence of chemical substitution in the reactivity pattern of molecules.124 The major difficulty to achieve this objective is that substituent effects are experimentally assessed as global responses, and therefore steric and solvent effects may mask the intrinsic electronic contributions. Many detailed linear relationships between substituent groups and chemical properties have been developed to date.71014125 In many cases, such relationships can be expressed quantitatively, thereby providing useful clues for interpreting reaction mechanisms and to predict reaction rates and equilibria. The most widely applied of these relationships is the Hammett equation,126 which relates rates and equilibria of many reactions of compounds containing substituted phenyl groups, Ph-X. It is expressed by the following linear equation: log k/ko = p

(9)

where ko is the rate (or equilibrium) constant for X = H, and k is the rate (or equilibrium) constant for the group X. The slope is a constant for a given reaction under a given set of conditions, and p is a constant characteristic of the group X. With the

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value calculated for a given reaction at hand, and with known p values for several groups, the reaction rates for processes that have not been run can be predicted. The p values embody the total electronic effects. These effects have their origin in four important contributions according to Topsom classification.129 They are: (a) the substituent dipole leading to a field effect; (b) the electronegativity difference between the substituent and the atom directly attached to it, leading to an inductive or electronegativity effect; (c) charge transfer between suitable orbitals of the substituent and the group to which it is attached, leading to a resonance or hyperconjugative effects; and (d) polarizability effects.127 This last contribution appears to be relatively small, except for large hydrocarbon substituents.127 A positive value of p indicates an EW group, while a negative value is associated with an electron-releasing group. However, due to the fact that solvent effects may contribute to the p parameter, it becomes very difficult to evaluate the intrinsic electronic contributions separately. The p values are derived from experiments in solvents of high polarity, usually water or water/methanol mixtures,128 so they do not provide parameters only containing the intrinsic electronic contributions. These contributions are of interest for gas-phase reactions, or reactions that take place in solvents of very low polarity. Theoretical models provide interesting alternatives to evaluate intrinsic electronic substituent effects. This can be done for instance by means of the response functions defined as global or local reactivity indices. The variations of a reactivity index for a set of functional groups attached to a common molecular frame may also be taken as a measure of the influence that the different substituents may have on the reactivity pattern of molecules. We have shown that the intrinsic electronic contribution to the Hammett substituent constant, e , may be obtained from a statistical analysis that follows from the comparison of the experimental p values and the electronic electrophilicity index evaluated for isolated molecules.127 The analysis was performed for the electrophilic=CH2 , and the p values reported by ity of a series of substituted ethylenes, X-CH= Hansch et al.128 for a wide list of functional groups (FG), X-, commonly present in organic compounds. For this study, we selected 42 representative FGs from a list of more than 500 substituents compiled by Hansch et al.128 The electrophilicity index for the whole series of ethylene derivatives, together with the experimental p given by Hansch et al.,128 is listed in Table 15. The procedure to obtain estimates of the intrinsic electronic substituent effects from the reactivity index was as follows: we first compared the computed values with the experimental p (see Figure 8). The analysis revealed a poor linear correlation between both quantities, with a regression coefficient R2 = 053. A better correlation was found when all the 42 points were fitted to a logarithm curve (R2 = 084, see Figure 8). The poor linear correlation observed for the whole series of molecules can be attributed in part to bulk and solvent effects encompassed in the experimental p values. Thus, when the data for 20 selected FGs of the series marked as ∗ in Table 15 were analysed, the logarithmic correlation was significantly improved (R2 = 099 see Figure 9). This analysis allowed us to find a valuable logarithmic correlation between the experimental Hammett substituent parameter p and the electrophilicity index given by: e = 143 ∗ log − 020

(10)

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Table 15 Electrophilicity for the substituted ethylenes X-CH = CH2 , Hammett constants, p for the substituents-X and computed e

138 139 140 141∗ 142∗ 143∗ 144 145 146 147 148∗ 149 150∗ 151∗ 152∗ 153∗ 154∗ 155∗ 156∗ 157 158∗ 159 160 161 162∗ 163 164 165 166 167 168 169∗ 170 171 172∗ 173∗ 174∗ 175∗ 176∗ 177∗ 178 179

SMe2 + PMe3 + NMe3 + NO NO2 COCF3 CBr 3 COPh CHO CN COMe C6 F5 CO2 H COCHMe2 COEt CCl3 CO2 Me NC CO2 Et SiH3 CONH2 CF3 N3 CCF3 3 CONHMe Ph OCOMe Cl SMe Br SiMe3 H F SH Me Et Pr OSiMe3 OH OMe NH2 NMe2

p

e

601 543 500 330 261 245 198 196 183 174 165 162 161 160 157 157 151 148 148 133 132 131 130 126 120 113 101 091 090 089 083 073 068 065 060 058 058 045 044 042 030 027

090 073 082 091 078 080 029 043 042 066 050 027 045 047 048 046 045 049 045 010 036 054 008 055 036 −001 031 023 000 023 −007 000 006 015 −017 −015 −013 −027 −037 −027 −066 −082

131 125 120 094 079 076 062 062 057 054 051 050 050 048 048 048 046 044 044 038 037 037 036 034 031 028 021 014 013 013 008 000 −004 −007 −012 −014 −014 −030 −031 −034 −055 −061

in eV. All quantities were evaluated at the B3LYP/6-31G∗ level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163. ∗ Selected data for regression in Figure 9. a

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185 1.5

R 2 = 0.53 R 2 = 0.84

1.0

σp

0.5

0.0 0

2

4

6

ω

– 0.5

– 1.0

Figure 8 Plots of p vs the electrophilicity for the substituted alkene series

σe(ω) = 1.43 log (ω) – 0.20 R 2 = 0.99

σp

1.0

0.5

0.0

–1

1

2

3

4

ω – 0.5

Figure 9 Plots of p vs the electrophilicity for the selected subseries of substituted alkenes

where e represents the theoretical electronic contribution to the experimental Hammett substituent constant p values and is the electrophilicity index for the corresponding substituted ethylene. Note that this subseries contains FGs classified from strong EW NO2 to strong electron-releasing OCH3 . Values of the computed e for the whole series of the 42 compounds are compared with the p values in the last two columns of Table 15. An analysis of p vs e values for the whole series given in Table 15 is displayed in Figure 10 R2 = 084. Deviation of the proposed e values from the experimental p ones can be mainly traced to the presence of bulk and solvent effects encompassed in the p Hammett parameter. The consistency of the theoretical e scale has been illustrated as follows:122 the series SMe2 + PMe3 + and NMe3 + , which have the largest e values, 1.31, 1.25 and 1.20, correspond to the most EW FGs within the series. These values are larger than the experimental p values: 0.90, 0.73 and 0.82, respectively. This significant deviation of e from the experimental p values can be traced to large solvent effects due to the

186

Electrophilicity index in organic chemistry 1.5

R 2 = 0.84

1.0

σp 0.5

0.0 –1.0

–0.5

0.0

0.5

1.0

1.5

σe(ω) –0.5

–1.0

Figure 10 Plots of p vs the computed e for the complete series

positive charge present in these groups. On the other hand, the CCl3 and COEt, which have mainly inductive and resonance effects, respectively, display similar e values, ≈ 048, in agreement with the similar p values found experimentally (0.48 and 0.46, respectively). The e values predicted for the CHO and COMe, 0.57 and 0.51, are consistent with the larger electrophilic character expected for aldehydes respect to ketones. Note however that the p = 042 quoted for the CHO in Hansch’s scale,128 is lower than that measured for the COMe one p = 050. For the carbonyl, COR, and carboxyl, COX, subseries there is good correlation between the p and e values, which are also consistent with the electrophilicity pattern predicted for the corresponding ethylene derivatives. The EW substitution at the carbonyl group increase both p and e . Note that for the FGs located in the middle of the series displayed in Table 15 the p and e have approximately the same values. The electron-releasing groups located at the end of the Table 15 present also a good correlation with the p values. The e values predicted for the OH and OMe, −031 and −034, are consistent with the larger nucleophilic character expected for the methoxy group with respect to the hydroxyl one. Note however that the p = −027 quoted for the OMe in Hansch’s scale,128 is lower than that measured for the OH one p = 037. In summary, the calculated electrophilicity index, , for a series of substituted ethylenes may be used to make reliable estimates of the intrinsic electronic contributions to the p constants of Hammett equation for a series including 42 functional groups commonly present in organic compounds. The computed e parameters account for the intrinsic electronic substituent effects, which are contained in the experimental values of the p substituent constants. This e scale is expected to be a useful predictive tool to assess the reactivity pattern of gas-phase reactions or those reactions that take place in solvents of very low polarity.

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6. The philicity concept in the chemistry of carbenes Carbenes belong to a class of highly reactive carbon intermediates where the carbon atom has two nonbonding electrons. Methylene CH2 is the reference structure giving rise to the general nomenclature that characterizes divalent carbon species. These structures are simply named as substituted derivatives of methylene.129 Carbenes are usually thought of as being sp2 -hybrized structures, yet their spin multiplicity may correspond to a singlet or triplet ground state (GS). The singlet GS leaves a vacant p-orbital, thereby conferring a high electrophilic character to this structure. The triplet GS on the other hand has two nonbonding sp2 and p-orbitals containing one electron each.129 Triplet carbenes are therefore characterized by biradical rather than ionic properties. The reactivity pattern of carbenes is rather wide and includes cycloaddition reactions to alkenes, cycloaddition to 1,3-dienes, cycloaddition to arenes and alkynes and insertion to C-H and X-H bonds.130131 Singlet carbenes being highly electron deficient react with nucleophiles including tertiary amines, phosphines, ethers, sulphides and sulphoxides.132133 The nature of the substituents has an important effect on the electronic properties of carbenes. For example, as the carbene substituents R in :CR2 become a better -donor, the GS changes from triplet to singlet state. Substitution by halogens results in singlet GS. For instance, dichlorocarbene is a singlet GS due to electron donation from chlorine to the vacant p-orbital of carbon that stabilizes the singlet state through dipolar resonance structures, although because of the relative electronegativities there is a strong -polarization in the opposite direction.129130 Therefore singlet carbenes are in general highly electrophilic species.129−132 Electrophilicity of carbenes was first experimentally evaluated by Moss et al., using a kinetic model based on the addition reaction of carbenes to simple alkenes to yield cyclopropanes.125134135 The proposed classification incorporates within a unique scale species having electrophilic, ambiphilic and nucleophilic properties generically named philicity.125134−137 The philicity scale is defined by a unique empirical index mCXY , where CXY are the carbene species, measured from the least-square slope of log ki /ko CXY vs logki /ko CCl2 plot. The quantity ki is the rate constant for the reaction towards a particular alkene pair, measuring the relative response, or selectivity of carbene CXY, to changes in the alkene structure. The quantity ko is the rate constant adjusted to a standard alkene. The experimental scale has been the subject of successive modifications, despite the predictive semiquantitave theory involved, and the significant body of congruent experimental data. This database is an excellent source for the validation of theoretical models of global electrophilicity based on (2) (see Table 16). According to Moss et al.125 experimental classification, the carbenes having mCXY values lower than 1.50 were classified as electrophilic carbenes (E); those having mCXY values greater than 2.2 were classified as nucleophilic carbenes (N) and those having mCXY values between 1.5 and 2.2 were classified as ambiphilic (A) species. Note that the theoretical scale follows an inverse order with respect to experimental scale. Here, a high value of means a high electrophilicity. Within the theoretical scale, the electrophilic carbenes are characterized by an electrophilicity value ranging from 1.21 eV for MeCF to 2.40 eV for PhCCN. Ambiphilic species on the other hand show values around unity. Nucleophilic carbenes are characterized within the theoretical classification by values lower than unity. Note that the MeCOMe species that was classified as

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ambiphilic by Moss125134−137 appears within our theoretical scale in the frontier of the ambiphilic/nucleophilic carbenes. The comparison of the global electrophilicity pattern in both scales is excellent. It is also interesting to note that the carbenes predicted as electrophilic species show in general values of closer or to greater than −50 eV, the highest values of electronegativity within the series. These species quoted as group I in Table 16 are consistently classified as electrophilic carbenes, in agreement with Moss et al.’s scale.125

Table 16 Global theoretical and experimental philicity values of singlet carbenes Itheo (eV)

Atheo (eV)

(eV)

Group I. Electrophilic carbenes CH2 -T 1045 023 −534 MeCF 971 000 −485 MeCCl 916 068 −492 CF2 1231 −002 −615 CHF 1070 047 −559 MeCBr 899 091 −495 PhCF 859 111 −485 PhCCl 818 151 −485 PhCBr 806 167 −486 CCl2 1016 146 −581 MeCCN 944 212 −578 PhCCN 828 237 −533 Group II. Ambiphilic carbenes MeCOMe 834 −067 −384 MeOCF 1038 −056 −491 PhCOMe 780 033 −407 MeOCCl 960 −018 −471 MeOCBr 932 −008 −462 PhOCF 938 −009 −464 PhOCCl 910 044 −477 PhOCBr 896 060 −478 Group III. Nucleophilic carbenes PhCNMe2 698 −037 −331 MeOCOPh 853 −044 −404 COMe2 898 −052 −423 MeCNMe2 740 −045 −347 COH2 985 −061 −462 a

(eV)

a (eV)

b (eV)

1022 971 849 1234 1022 808 747 666 640 870 733 592

140 121 143 153 153 152 157 176 185 194 228 240

156

901 1094 747 978 941 947 866 836

082 110 111 113 113 114 131 136

735 897 950 785 1046

074 091 094 077 102

149 147

193

mCXY (Moss scale)c

Observedc Moss philicity

058 147

E E

096 071 064 100

E E E E

121 185 134 159

A A A

174 149

A A

211 222 291 271

N N N N

Predicted philicity by

E E E E E E E E E E E E N A A A A A A A N N N N N

values (in eV) were obtained from (2) using theoretical values of ionization potentials, I (in eV) and electron affinities, A (in eV). The electronic chemical potential (in eV) and chemical hardness (in eV) have been approached by ≈ −I + A/2 and ≈ I − A, in terms of the vertical I and A values, for the singlet GS of the carbenes at the B3LYP/6-311++G(d,p) level of theory using the Gaussian98 suite of programs ref 163. b values were calculated from experimental values of I and A for some electrophilic carbenes for which this information is available.138 c Observed Moss philicity values from references 125, 134–137.

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The ambiphilic species (group II in Table 16) on the other hand are less electronegative than the first group of electrophilic carbenes and more electronegative than the carbenes classified as nucleophiles (group III in Table 16). The variation in chemical hardness is not regular within the whole series of carbenes. However, PhCCN, the most electrophilic species evaluated in the theoretical scale, has a relative high value of electronegativity and at the same time the lowest value in chemical hardness. The MeCNMe2 species on the other hand, which is predicted in both scales as the most nucleophilic carbene, has the lowest value of electronegativity and a moderately high value in chemical hardness. The comparison between the experimental and theoretical scales of electrophilicity is however not quantitative, a result probably traced to the problems in predicting electron affinity values that determine both the electronegativity and hardness patterns according to Parr’s model of electrophilicity. Another aspect related to the global electrophilicity pattern of singlet carbenes is the analysis of substituent effects inducing electrophilic activation/deactivation. The activation/deactivation pattern will be discussed with respect to the global electrophilicity of the triplet ground state of methylene CH2 T as reference and described by the quantity = carbene − CH2 T. Methylene in its GS (triplet) exhibits a global electrophilicity pattern of 1.40 eV. The results are summarized in Table 17, last column. Within the series of electrophilic carbenes (group I in Table 17), with the only exception of MeCF, chemical substitution at the carbon site results in electrophilic activation ( > 0) for the whole series. For the ambiphilic and nucleophilic series on the other hand (groups II and III in Table 17) chemical substitution results in electrophilic deactivation ( < 0), without exceptions.139

7. Beyond the electrophilicity concept: superelectrophilicity The development of the superacidic media to obtain stable cations permitted the isolation of different intermediate species like nitronium, oxonium and carboxonium ions. They were found to participate in a wide variety of reactions.140−142 Olah et al.’s studies on superacidic systems concluded that superacids, apart from being highly ionizing, low nucleophilicity media, in some cases were capable of producing electrophilic activation by further protolytic (or electrophilic) interaction (coordination and/or solvation).143 In a recent account, Olah and Klumpp discuss the superelectrophilic solvation involving the interaction of electron-donating groups (ligands) of overall electron-deficient species (electrophiles) with strongly electron-acceptor superacids.144 This electrophilic activation has been observed in liquid superacids, on solid acids, and even in enzymatic biological systems.143 Electrophiles capable of further interactions by coordination or solvation with strong Brönsted or Lewis acids can be activated in this way. The resulting enhancement of reactivity is very significant compared to that of their parent compounds under conventional conditions and indicates the formation of a new species named superelectrophiles.143 On the other hand, 13 C NMR spectroscopy has extensively been used to study the structure of oxonium, carboxonium and oxycarbenium ions and diprotonated carboxylic acids,144−146 since it allows the direct monitoring of the cationic centre and since the chemical shifts and coupling constants can be correlated with the geometry and hybridization of the cation. This technique has also been used by Olah et al. to provide

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Table 17 Global and local electrophilicitya , electrophilic Fukui function at C atom of the carbene and relative global electrophilicity values of singlet carbenes (eV)

C (eV)

fC+

(eV)

140 103 113 134 141 118 072 081 085 148 138 098

1000 0850 0793 0875 0922 0780 0455 0462 0458 0764 0606 0409

000 −019 003 013 013 012 017 036 045 054 088 100

068 092 043 091 091 078 090 094

0836 0835 0392 0802 0798 0683 0689 0685

−058 −030 −029 −027 −027 −026 −009 −004

035 032 048 073 086

0477 0422 0526 0777 0845

−066 −063 −049 −046 −038

Group I. Electrophilic carbenes CH2 -T MeCF MeCCl CF2 CHF MeCBr PhCF PhCCl PhCBr CCl2 MeCCN PhCCN

140 121 143 153 153 152 157 176 185 194 228 240

Group II. Ambiphilic carbenes MeCOMe MeOCF PhCOMe MeOCCl MeOCBr PhOCF PhOCCl PhOCBr

082 110 111 113 113 114 131 136

Group III. Nucleophilic carbenes PhCNMe2 MeCNMe2 MeOCOPh COMe2 COH2

074 077 091 094 102

a and c in eV. All quantities were evaluated at the B3LYP/6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 163.

evidence about the intermediacy of pentaphenyl and heptaphenyl cations in the reaction of triphenylmethyl cation with diphenyldiazomethane or diphenylketene.145146 Olah et al.146 have applied 17 O NMR spectroscopy to study a series of oxonium and carboxonium ions. These authors observed a 240–250 ppm 17 O-shielding effect for ketones upon protonation, showing a decrease in the carbon–oxygen bond order by about 40%. A similar effect was observed for protonated carboxylic acids.145 Protosolvation (further protonation of electrophiles) of oxonium and carboxonium ions leading to doubly electron-deficient or dicationic species results in an enhancement of reactivity with respect to their oxonium or carboxonium ions.147−150

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From a theoretical point of view, superelectrophilic species have been studied in terms of their electronic structures, energies and gas-phase proton affinities.140 Lammerstma et al. presented an excellent review about organic dications.151 The relationships between gas-phase experiments and ab initio molecular orbital studies (on isolated species) are emphasized to provide an understanding of the structures and energies of these doubly charged species. An important theoretical result concerns the thermodynamic stability of dications in superacid media. For instance, the stability of the H4 O2+ was computationally studied by Olah et al.140 These authors concluded that H4 O2+ is thermodynamically unstable towards deprotonation to H3 O+ , yet the dication has a significant kinetic stability.140 Despite the kinetic stability, calculations performed on the isolated systems showed that the protonation of H3 O+ was thermodynamically unfavourable. However, it must be recognized that these computational data may not be related to the condensed phase conditions.142 Based on thermodynamic arguments, Lammertsma and Schleyer showed that not all the small organic dications are stable in superacid media.152 High-level calculations on the other hand are particularly useful to theoretically compute NMR chemical shifts. For instance, Olah et al.153−157 have reported numerous studies on ab initio calculations of NMR, GIAO156 and IGLO157 indexes. Both GIAO and IGLO methods are used for the calculation of magnetic susceptibility and chemical shifts tensors. While GIAO (gauge including atomic orbital) method156 transforms the gauge of the basis set functions to the position of their nuclei, in the IGLO (individual gauges for localized orbitals) method,157 the gauges of the final wavefunctions are transformed to their centres of charge. Good comparisons between the experimental NMR chemical shifts and the ab initio ones for both electrophilic cations and dications have been reported; 153155 relationship between electrophilicity and NMR chemical shift is based on a model where a downfield NMR shift is interpreted as a decrease of the electron density at the electrophilic (positively charged) site. However, we have previously shown that the relationship between electrophilicity and electronic charge deficiency may not be as general as believed. In Table 18 are displayed the global electrophilicity values for a series of alkyloxonium and carboxonium dications and diprotonated carboxylic acids. The values are presented in the order of increasing electrophilicity for each group, with reference to the neutral parent compounds (see last column in Table 18). It may be seen that the whole series of dications present a dramatic enhancement in electrophilicity. Note that their enhanced electrophilicity mainly result from their remarkable high electronegativity (the negative value of the electronic chemical potential, = − ). Note also that, despite the fact that these species show very high values of hardness (an effect that according to (2) plays against the ability of the chemical species to accept further electronic charge from the environment), they still remain as powerful electron acceptors. The analysis within the individual groups shows that in the alkyloxonium series, the highest electrophilicity value is that associated with the H4 O2+ dication. Experimental studies have evidenced the presence of H4 O2+ (protohydronium dication) from hydrogen/deuterium exchange in isotopic hydronium ions with strong superacids.142 Theoretical calculations in gas phase have concluded that the diprotonation of water leads to an intermediate thermodynamically unstable with a considerable barrier to dissociation.140142 However, in superacidic solutions, clustering of these dications with

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Electrophilicity index in organic chemistry Table 18 Global properties and local electrophilicitiesa

H2 O H3 O+ H4 O2+

−311 −1386 −2582

963 1303 1886

050 737 1767

CH3 OH CH3 OH2 + CH4 OH2 2+

−257 −1173 −1922

924 1064 1165

036 647 1585

CH3 OCH3 CH3 OHCH3 + CH3 3 O+ CH3 3 OH2+

−217 −1042 −980 −1785

936 960 998 977

025 566 481 1631

Alkyloxonium mono- and dications

Carboxonium mono- and dications H2 CO H2 COH+ H2 COH2 2+

−423 −1429 −2331

616 804 912

145 1270 2979

CH3 COH CH3 CHOH+ CH3 CHOH2 2+

−377 −1296 −1992

635 803 725

112 1046 2737

CH3 COCH3 CH3 COHCH3 + CH3 COH2 CH3 2+

−348 −1218 −1865

630 803 703

096 924 2474

Mono- and diprotonated carboxylic acids HCOOH HCOH2 + HCOHOH2 2+

−398 −1290 −2078

792 902 946

100 922 2282

CH3 COOH CH3 COH2 + CH3 COHOH2 2+

−362 −1209 −1869

774 908 812

084 805 2151

a

and in a.u., in eV. All quantities were evaluated at the B3LYP/6-31G* level of theory for the ground-state optimized geometries, using the GAUSSIAN98 package of programs ref 165.

their precursors could stabilize and delocalize the excess of charge.140142 For the methanol series, the charged derivatives are predicted to be less electrophilic than the water cation and dication. The dimethyl ether series shows an even lower electrophilic character as compared with the water and methanol series. Note that within this series, protonation of dimethyl ether produces a more effective electrophilic activation than methylation ( = 566 eV and = 481 eV, respectively). The enhanced electrophilicity pattern in the series of carboxonium ions may reflect the stabilizing effects of the alkyl groups. Probably, the most common of these is the

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inductive effects, yet another source of alkyl group stabilization involves rehybridization energies, hyperconjugation and other polarizability effects.158 The increasing inductive effect by methyl substitution results in a progressive electrophilic deactivation. Compare for instance the series of formaldehyde, acetaldehyde and acetone in Table 18. Finally, the series of carboxylic acids also show electrophilic deactivation induced by methyl substitution. In Table 19 are depicted the electrophilic Fukui function at site k fk+ , and the local electrophilicity k . The relevant electrophilic sites are the carbon atom in the series of the alkyloxonium ions and the carbonyl carbon atom in the carboxonium and diprotonated carboxylic acids. Also, we have incorporated in Table 19 the theoretical computed 13 C NMR and 17 O NMR chemical shifts available from the literature154155 to compare our results. For the series of alkyloxonium ions, the enhanced local electrophilicity predicted for the hydrogen atoms in H4 O2+ (H = 442 eV, see Table 19) may be indicative of the cluster formation in superacid media: under this condition the H4 O2+ protons might be shared by more than one H3 O+ by hydrogen bonding.142 This interpretation also is in agreement with previous results suggesting that this associative mechanism minimizes the charge–charge repulsion.26154 For the methanol subseries, the second H+ added to form the dication is bound to the CH3 group. This substitution pattern causes the local electrophilicity of carbon to dramatically decrease from 2.01 eV in the monocation to 0.67 eV in the dication. Note that this prediction is in agreement with the theoretical NMR chemical shifts as described by the IGLO and GIAO-MP2 indexes. Unfortunately, the experimental NMR chemical shift for these compounds is only available for methanol. However, for the first series of alkyloxonium ions, our predicted electrophilicity enhancement is in agreement with the experimental chemical shift for water and hydronium ion. It also interesting to mention that while the carbon atom is electrophilically deactivated, one of the hydrogen atom bound to the carbon atom of the CH3 group notably increases its electrophilicity to H = 539 eV (not shown in Table 19). Here again, the high value in local electrophilicity predicted for the hydrogen atom is consistent with the explanation offered by Olah142143 to account for the extra stabilization of the dication due to the possibility of forming clusters in superacidic media. For the dimethyl ether subseries, a qualitative good correlation between local electrophilicity and the theoretical 13 C NMR chemical shifts described by the IGLO and GIAO-MP2 indexes was obtained. Note further that for the species for which the experimental NMR chemical shift is available (dimethyl ether and the trimethyl oxonium cation), the enhancement in electrophilicity is again correlated with the enhancement in the experimental chemical shift at the carbon atom. Diprotonated carbonyl compounds like aldehydes and ketones are highly stabilized by resonance as compared with alkyl cations.26144 They have been extensively examined in the past using a wide variety of techniques.145159−161 These compounds can behave like oxonium (positive charge on the oxygen atom of the carbonyl group) or carbenium (positive charge on the carbon atom) ions. Despite the numerous theoretical works concerned with the prediction of their existence, diprotonated species have not been experimentally observed by NMR techniques due to the low concentration of these extremely reactive electrophiles, even in superacid media. From Table 19, it may be seen that the diprotonated formaldehyde exhibits the highest local electrophilicity C = 2596 eV in agreement with the strong theoretical 13 C NMR chemical shifts deshielding

194


Table 19 Electrophilic Fukui function fk+ , local superelectrophilicity k , calculated and experimental NMR chemical shifts fH +

H

NMR

IGLO(II) a

GIAO-MP2 (tzp/dz) a

Exp.a

Alkyloxonium mono- and dications H2 O H3 O+ H4 O2+

17

05000 03333 02500

025 246 442

O O 17 O

fC+

C

CH3 OH CH3 OH2 + CH4 OH2 2+

01369 03104 00425

005 201 067

13

C C 13 C

517 784 499

540 802 514

502

13

CH3 OCH3 CH3 OHCH3 + CH3 3 O+ CH3 3 OH2+

00582 02721 03246 03022

001 154 156 493

13

585

624

594

766 1281

819 1331

793

17

NMR

C C 13 C 13 C

00 −22 77 IGLO(II) a

00 282 473 GIAO-MP2 (tzp/dz) a

00 102 Exp. a

13

Carboxonium mono- and dications H2 CO H2 COH+ H2 COH2 2+

06508 07725 08715

095 981 2596

13

Ccarbonyl Ccarbonyl 13 Ccarbonyl

2031 2400 2664

1853 2288 2653

CH3 COH CH3 CHOH+ CH3 CHOH2 2+

06031 07084 07217

067 741 1975

13

2085b 2248b 2547b

1908b 2116b 2496b

CH3 COCH3 CH3 COHCH3 + CH3 COH2 CH3 2+

05353 06540 06728

051 604 1664

13

2684 2917

1986 2563 2788

2051 2487

Ccarbonyl

13

Ccarbonyl Ccarbonyl 13 Ccarbonyl 13

Ccarbonyl Ccarbonyl 13 Ccarbonyl 13

2238

Mono- and diprotonated carboxylic acids HCOOH

06222

062

13

1730

1602

1676

HCOH+ 2

625 1603

13

Ccarbonyl 13 Ccarbonyl

1926 1897

1833 1805

1776

HCOHOH2 2+

06767 07022

CH3 COOH CH3 COH+ 2

05360 06226

045 501

13

Ccarbonyl Ccarbonyl

1843 2113

1719 2020

1769 1930

CH3 COHOH2 2+

06393

1375

13

Ccarbonyl

2137

2034

a b

13

k in eV. From ref 153 and references therein. Predicted values from correlation equation of Figure 11a and 11b.

predicted from the IGLO and GIAO-MP2 indexes (compare third, seventh and eighth columns in Table 19). For the acetone subseries, both the IGLO and GIAO-MP2 indexes predict an increasing deshielding of about of 23 ppm at the carbonyl carbon atom in CH3 COH2 CH3 2+ as compared with the protonated acetone, CH3 COHCH3 + .159 Note that this pattern is again in qualitative agreement with the local electrophilicity variations

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at the carbonyl carbon atom (C = 604 eV and 16.64 eV, respectively). Finally, for the carboxylic acid series, the comparison of the local electrophilicity index with the theoretical 13 C NMR chemical shifts obtained by IGLO and GIAO-MP2 methods and the available experimental data is again in good qualitative agreement. The carbonyl carbon atom of dications is predicted to be shifted to downfield, and this result is consistent with an electrophilicity enhancement described by the local electrophilicity index.162 However a definitive testing about the reliability and usefulness of the present model based on global and local electrophilicity indexes must be proved on a more quantitative basis. Figure 11a shows the comparison between IGLO and C quantities for the formaldehyde series. The resulting regression equation showed in Figure 11a was used to predict the IGLO chemical shifts for the acetaldehyde series included in Table 19, seventh column.162 A similar comparison was made between GIAO and C indexes for the series of formaldehyde. The results are displayed in Figure 4b. From the resulting regression equation showed in Figure 11b, the GIAO values for the acetaldehyde series were predicted. They are displayed in Table 19, eighth column.162 Note that the predicted IGLO and GIAO values of the acetaldehyde series are bound by those of formaldehyde (lower bound) and acetone (upper bound) series.162 The definitive testing of these predictions are subject to experimental confirmation.

8. Concluding remarks and perspectives In this chapter, we have reviewed the usefulness of the global and local electrophilicity indexes to quantitatively account for the reactivity and selectivity patterns observed in a large series of classical organic reactions. The global electrophilicity index, , categorizes within an unique absolute scale the propensity of the electron acceptors to acquire additional electronic charge from the environment. This classification allowed an impressive number of systems in DA reactions to be rationalized in terms of their reaction mechanisms in polar and nonpolar processes. The global electrophilicity scale provides a simple way to assess the more or less polar character of a process on the 290

δ IGLO (II) ppm

270 250 230 210 190

δ = 2.4216ωC + 206.86

170

R = 0.97

150 0

10

ωC

20

30

Figure 11a Comparison between 13 C NMR/IGLO(II) chemical shifts and the local electrophilicity index, C , for the monocation-, dication- and the neutral formaldehyde species. R is the regression coefficient

196

Electrophilicity index in organic chemistry 300

δ GIAO-MP2 ppm

250 200 150 100

δ = 3.0842ωC + 188.72

50

R = 0.98 0 0

5

10

15

ωC

20

25

30

Figure 11b Comparison between 13 C NMR/GIAO-MP2 chemical shifts and the local electrophilicity index, C , for the monocation-, dication- and the neutral formaldehyde species. R is the regression coefficient

basis of the electrophilicity gap between the interacting partners. This is achieved within a simple model where the information encompassed in the MO coefficients is implicitly self-contained in the electron density. The reaction mechanism may be anticipated from the relative position of the reagents within the electrophilicity scale. Two main results follow from this rule: (a) A strong electrophile located at the top of the scale determines the nucleophilic character of the other reagent located below it. Limiting cases in IED DA processes are consistently accounted for by this empirical rule. (b) The relationship between the electrophilicity difference of the dienophile/diene pair and the static polarity may be an useful tool to describe the electronic pattern expected at the transition state structures involved in DA reactions, describing nonpolar ( small) or polar ( big) mechanisms. An additional important result follows from rule (a). It seems that in polar cycloaddition reactions the most electrophilic reagent control the asynchronicity of the process by a larger bond formation process at the most electrophilic site of the electron acceptor. This result may be traced to the more prevalent role of the electron acceptor in the electrophile/nucleophile interaction, which follows from the extra work associated with the redistribution of the electronic charge that it takes from the nucleophilic partner. The local electrophilicity counterpart, on the other hand, consistently predicts the regioselectivity expected in DA and 1,3-dipolar cycloadditions. The electrophilicity index also accounts for the electrophilic activation/deactivation effects promoted by EW and electron-releasing substituents even beyond the case of cycloaddition processes. These effects are assessed as responses at the active site of the molecules. The empirical Hammett-like relationships found between the global and local electrophilicity indexes and the reaction rate coefficients correctly account for the substrate selectivity in Friedel–Crafts reactions, the reactivity of carbenium ions, the hydrolysis of esters, the reactivity at the carbon–carbon double bonds in conjugated Michael additions, the philicity pattern of carbenes and the superelectrophilicity of nitronium, oxonium and carboxonium ions. This last application is a very promising area of application. The enhanced electrophilicity pattern in these series results from

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the stabilizing effects of the alkyl groups associated with rehybridization energies, hyperconjugation and other polarizability effects. Furthermore, it is rather remarkable that the electrophilicity index may explain the enhanced electrophilicity shown by organic dications which results from intrinsic electronic effects that do not consider the effect of superacidic media and that show good correlation with the NMR-based IGLO and GIAO indexes. Beyond the present applications of the electrophilicity index in organic chemistry, there are some others that are being considered in our group. For instance, based on the encouraging results that follow from the comparison between the electrophilicity index and the reaction rate coefficients, it is expected that quantitative comparisons between the electrophilcity index and the Hammett substituent constants may open the possibility of having new theoretically predicted values that have not been yet experimentally determined. Consider for instance the case of multiple substitutions where the additivity rules may not apply. Another interesting area of application of the electrophilicity index concerns the chemistry of small cycloalkynes that posses a strain-induced electrophilicity related to the bending of the sp3 -sp-sp bond angle. This ongoing study offers an interesting way of looking at electrophilic activation promoted by a less classical substituent effect induced for instance by molecular strain. From a theoretical point of view, this is the first example of global or local activation in molecules induced by effects that are directly related to changes in the external potential. Finally, the global and local electrophilicity indexes may be also used to describe the nucleofugality of classical leaving groups in organic chemistry. This potential application incorporates the important families of nucleophilic substitution and elimination reactions. This study is however a bit more complex than the cases presented in this review, because the systematization of nucleofugality within an absolute scale requires an important number of requisites that must be fulfilled, most of them regarding the different reaction mechanisms involved in these complexe reactions.

Acknowledgements This work was supported by Fondecyt, grants 1030548 and 1060961, Millennium Nucleus for Applied Quantum Mechanics and Computational Chemistry, grant N P02-004-F (MIDEPLAN-CONICYT, Chile), projects UTFSM 130223 and 130423, the Ministerio de Ciencia y Tecnología of the Spanish Government by DGICYT, project BQU2002-01032, and the Agencia Valenciana de Ciencia y Tecnología of the Generalitat Valenciana, reference GRUPOS03/176.

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Clovis, J.S.; Eckell, A.; Huisgen, R.; Sustmann, R. Chem. Ber. 1967, 100, 60. Padwa, A.; Chen, Y.-Y.; Dent, W.; Nimmesgern, H. J. Org. Chem. 1985, 50, 4006. Cossío, F.P.; Morao, I.; Jiao, H.; Schleyer, P.v.R. J. Am. Chem. Soc. 1999, 121, 6737. Carda, M.; Portolés, R.; Murga, J.; Uriel, S.; Marco, J.A.; Domingo, L.R.; Zaragozá, R.J.; Röper, H. J. Org. Chem. 2000, 65, 7000. Sáez, J.A.; Arnó, M.; Domingo, L.A. Org. Lett. 2003, 5, 4117. Arno, M.; Picher, M.T.; Domingo, L. R.; Andrés, J. Chem. Eur. J. 2004, 10, 4742. Domingo, L.R.; Pérez, P.; Contreras, R. Lett. Org. Chem. 2005, 2, 68. Cramer, C.J.; Barrow, S.E. J. Org. Chem. 1998, 63, 5523. Hamura, T.; Ibusuki, Y.; Sato, K.; Matsumoto, T.; Osamura, Y.; Suzuki, K. Org. Lett. 2003, 5, 3551. Aizman, A., Contreras, R., Pérez, P. Tetrahedron, 2005, 61, 889. Minegishi, S.; Mayr, H. J. Am. Chem. Soc. 2003, 125, 286. Lucius, R.; Loos, R.; Mayr, H. Angew. Chem. Int. Ed. Engl. 2002, 41, 91 Ritchie, C.D. Can. J. Chem. 1986, 64, 2239. Olah, G.A. in Friedel-Crafts and Related Reactions, Wiley: New York, 1964; Vol I–IV. Olah, G.A.; Tashiro, M.; Kobayashi, S. J. Am. Chem. Soc. 1970, 92, 7448. Olah, G.A.; Kobayashi, S. J. Am. Chem. Soc. 1971, 93, 6964. Taylor, R. Electrophilic Aromatic Substitutions, Wiley: Chichester, 1990. Meneses, L.; Fuentealba, P; Contreras, R. Tetrahedron, 2005, 61, 831. a) Neuvonen, H.; Neuvonen, K. J. Chem. Soc. Perkin Trans. 1999, 2, 1497. b) Neuvonen, H.; Neuvonen, K.; Koch, A.; Kleinpeter, E.; Pasanen, P. J. Org. Chem. 2002, 67, 6995. Contreras, R.; Andrés, J.; Domingo, L.R.; Castillo, R.; Pérez, P. Tetrahedron 2005, 61, 417. Bernasconi, C.F.; Killion Jr, R.B. J. Am. Chem. Soc. 1988, 110, 7506. Bernasconi, C.F.; Killion Jr, R.B. J. Org. Chem. 1989, 54, 2878. Bernasconi, C.F.; Leyes, A.E.; Rappoport, Z. J. Org. Chem. 1999, 64, 2897. Bernasconi, C.F.; Ketner, R.J.; Chen, X.; Rappoport, Z. J. Am. Chem. Soc. 1998, 120, 7461. Mayr, H.; Kempf, B.; Ofial, A.R. Acc. Chem. Res. 2003, 36, 66. Domingo, L.R.; Pérez, P.; Contreras, R. Tetrahedron. 2004, 60, 6585. Domingo, L.R.; Pérez, P.; Contreras, R. J. Org. Chem. 2003, 68, 6060. Lemek, T.; Mayr, H. J. Org. Chem. 2003, 68, 6880. Exner, O. in Advances in Linear Free Energy Relationships, Chapman N.B.; Shorter, J. Eds; Plenum: London, 1972. Moss, R.A. Acc. Chem. Res. 1989, 22, 15. For a review see Jaffé, H.H. Chem. Rev. 1953, 53, 191. a) Topsom, R.D. Acc. Chem. Res. 1983, 16, 292; Marriot, S.; Topsom, R.D. J. Am. Chem. Soc. 1984, 106, 7. b) Marriot, S.; Reynolds, W.F.; Taft, R.W.; Topsom, R.D. J. Org. Chem. 1984, 49, 959. Hansch, C.; Leo, A.; Taft, R.W. Chem. Rev. 1991, 91, 165. Moody, C.J.; Whitham, G.H. Reactive Intermediates; Oxford Chemistry Primers: Oxford University Press, 1997. Kirmse, W. Carbene Chemistry, 2a Ed., Academic Press, New York, 1971. Jones, M.; Carbenes Moss, R.A. Ed,; Wiley, New York, Vol. II. 1973 and Vol II, 1975. Wentrup, C. Reactive Molecules: the Neutral Reactives Intermediates in Organic chemistry; Wiley: New York, 1984. Arduengo III, A.J. Acc. Chem. Res. 1999, 32, 913. Moss, R.A.; Shen, S.; Hadel, L.M.; Kmiecik-Lawrynowicz, G.; Wlostowska, J.; KroghJespersen, K. J. Am. Chem. Soc. 1987, 109, 4341. Moss, R.A. Acc. Chem. Res, 1980, 13, 58. Moss, R.A.; Wlostowska, J.; Terpinski, J.; Kmiecik-Lawrynowicz, G.; Krogh-Jespersen, K. J. Am. Chem. Soc. 1987, 109, 3811.

100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127.

128. 129. 130. 131. 132. 133. 134. 135. 136.

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137. Moss, R.A.; Pérez, L.A.; Wlostowska, J.; Guo, W.; Krogh-Jespersen, K. J. Org. Chem. 1982, 47, 4177. 138. Mallard, W.G. Linstrom, P.J. NIST Chemistry Webbook, NIST Standard Reference Database Number 69. Available: http://webbook.nist.gov. National Institute of Standards and Technology: Gaithersburg, MD, 1998. 139. Pérez, P. J. Phys. Chem. A. 2003, 107, 522. 140. Olah, G.A.; Prakash, G.K.S.; Barzaghi, M.; Lammerstma, K.; Schleyer, P. von R.; Pople, J.A. J. Am. Chem. Soc. 1986, 108, 1062. 141. Hartz, N.; Golam, R.; Olah, G.A. J. Am. Chem. Soc. 1993, 115, 1277. 142. Olah, G.A.; White, A.M. O’Brien, D.H. Chem. Rev. 1970, 70, 561. 143. Olah, G.A.; Angew. Chem. Int. Ed. 1993, 32, 767. 144. Olah, G.A.; Klumpp, D.A. Acc. Chem. Res. 2004, 37, 211. 145. a) Olah, G.A.; Prakash, G.K.S.; Sommer, J. Superacids, Wiley-Interscience, New York, 1985 and references therein; b) Perst, H. Oxonium Ions in Organic Chemistry, Verlag, Chemie, Weinheim, 1973. 146. Olah, G.A.; Berrier, A.L.; Prakash, G.K.S. J. Am. Chem. Soc. 1982, 104, 2373. 147. Olah, G.A.; Germain, A.; Lin, H.C.; Forsyth, D.A. J. Am. Chem. Soc. 1975, 97, 2928. 148. Weiske, T.; Koch, W.; Schwarz, H. J. Am. Chem. Soc. 1993, 115, 6312. 149. Palm, V.A.; Haldna, U.L.; Talvik, A.J. The Chemistry of the Carbonyl Group, Patai, S. Ed.; Interscience New York, 1971, 641. 150. Friedel-Crafts and Related Reactions, Olah, G.A. Ed., Interscience New York, 1966, p. 421. 151. Lammertsma, K.; von R. Schleyer, P.; Schwarz, H. Angew. Chem. Int. Ed. Engl. 1989, 28, 1321. 152. Lammertsma, K.; Schleyer, P. von R. J. Am. Chem. Soc. 1990, 112, 7935. 153. Olah, G.A.; Burritcher, A.; Rasul, G.; Gnann, R.; Christe, K.O.; Prakash, G.K.S. J. Am. Chem. Soc. 1997, 119, 8035. 154. Rasul, G.; Prakash, R.; Zduneck, L.Z.; Prakash, G.K.S.; Olah, G.A. J. Am. Chem. Soc. 1993, 115, 2236. 155. Prakash, G.K.S.; Rasul, G.; Liang, G.; Olah, G.A. J. Phys. Chem. 1996, 100, 15805. 156. Olah, G.A.; Herges, R.; Laali, K.; Segal, G.A. J. Am. Chem. Soc. 1986, 108, 2054. 157. Rasul, G.; Prakash, S.; Olah, G.A. J. Org. Chem. 2000, 65, 8786. 158. a) Brauman, J.I.; Riveros, J.M.; Blair, L.K. J. Am. Chem. Soc. 1971, 93, 3914; b) Brauman, J.I.; Blair, L.K. J. Am. Chem. Soc. 1971, 93, 3914; c) Brauman, J. I.; Blair, L.K. J. Am. Chem. Soc. 1971, 93, 3911. 159. Olah, G.A.; Schleyer, P. von R. (Eds), Carbonium Ions; Wiley Interscience: New York, Vol. I–V, 1968–1976, and reviews therein. 160. Bethell, D.; Gold, V. Carbonium Ions; Academic Press: London; New York, 1967. 161. Yokoyama, A.; Ohwada, T.; Shudo, K. J. Org. Chem. 1999, 64, 611. 162. Pérez, P. J. Org. Chem. 2004, 69, 5048. 163. Frisch, M.J. et al., Gaussian 98, Revision A.6 Gaussian, Inc.: Pittsburgh PA, 1998.



Chapter 10

Electronic structure and reactivity of aromatic metal clusters a

Paulina González, b Jordi Poater, ac Gabriel Merino, c Thomas Heine, b Miquel Solà, and a Juvencio Robles a

Facultad de Química, Universidad de Guanajuato, Noria Alta s/n, Guanajuato, Gto 36050, México, b Institut de Química Computacional and Departament de Química, Universitat de Girona, E–17071 Girona, Catalonia, Spain and c Institut für Physikalische Chemie und Elektrochemie, TU Dresden, D–01062 Dresden, Germany

Abstract It has been argued that some small newly discovered all-metal clusters may be considered as aromatic. This would make them the first aromatic chemical systems with no carbon content at all. The initial report was on the ionic aluminum cluster, Al4 2− , which exhibits a planar square structure and has two delocalized -electrons, thus satisfying Hückel’s rule for aromatic systems. It also shows a relatively high chemical and structural stability. By means of quantum chemical calculations, Kuznetsov et al. have shown that the square structure is preserved when the Al cluster reacts to form some bimetallic clusters with various chemical compositions: MAl4 − and M2 Al4 (where M = Li+ Na+ , and Cu+ ) and that the aromatic nature is apparently preserved. In this paper, we report a density functional theory (DFT) study, using the hybrid functional B3LYP and a 6–31+Gd basis set, which allows us to rationalize and to quantify the aromatic nature of MAl4 − species and to extend our study to other bimetallic clusters which have not yet been reported. These clusters have the general formula MAl4 n M = Li+ Na+ K + Be2+ Mg2+ Ca2+ Sc3+ Al3+ B3+ Ga3+ and Ti4+ where n = −1 0 +1 +2. We also evaluate the stability as a function of charge and discuss different properties to assess the aromaticity of these systems, namely through reactivity indices such as the DFT absolute hardness, nucleus-independent chemical shift (NICS), and delocalization indexes (DIs). 203

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1. Introduction Aromaticity is the simplest way to explain the stability of unsaturated cyclic hydrocarbons with 4n + 2 electrons delocalized in the -orbitals perpendicular to the ring plane.1 Even though the introduction of the aromaticity concept in chemistry is quite old, its definition is still controversial. It is not surprising to find many attempts to define this term depending on different approaches to describe the electronic structure. In view of these problems of subjectivity, it is remarkable that aromaticity is useful to rationalize and understand the structure and reactivity of many organic molecules. As a result, the concept of aromaticity is truly a cornerstone in organic chemistry. In 1971, Wade proposed a similar concept to describe delocalized -bonding in closed-shell boron deltahedra.2−4 However, stability based on aromaticity had not been confirmed for any metallic moiety until Li et al. published their seminal paper entitled “Observation of all-metal aromatic molecules.”5 A series of compounds consisting of a square planar Al4 2− , face-capped by an M+ cation (M = Li, Na, and Cu), were produced by laser vaporization, and their electronic spectra were obtained using negative-ion photoelectron spectroscopy. Li et al. found that theoretical vertical detachment energies of the pyramidal structures are in excellent agreement with the experimental spectra, thereby demonstrating that C4v structures are the global minima for the MAl4 − species (see Figure 1). Ab initio calculations show that Al4 2− has two electrons residing in a -orbital, satisfying the Hückel rule for aromatic compounds. Li et al.5 concluded that this orbital holds “the key to understanding the structure and bonding of MAl4 − species.” However, electron delocalization in Al4 2− is not so simple. There are two delocalized -bonding orbitals (HOMO-1 and HOMO-2) spread across all four Al atoms. Therefore, a doubly aromatic behavior is observed (- and -aromaticity), which is different from hydrocarbon aromatic molecules.6 Indeed, the system of four-membered hydrocarbons is found to be antiaromatic.7 It should be noted that Al4 2− has four valence electrons less than the corresponding aromatic hydrocarbon C4 H4 2+ , and consequently, it is an electron-deficient aromatic system. Besides, the symmetry of these molecular orbitals is not quite similar to the one found in the corresponding aromatic hydrocarbons. Immediately, the all-metal aromaticity concept was extended from Al4 2− into a series of molecules containing XAl3 − X = Si, Ge, Sn, and Pb), Hg4 6− Al3− , and Ga3 − units.8−11 In 2001, Kuznetsov et al. provided experimental and theoretical evidence that Ga4 2− and In4 2− also have geometrical and electronic properties that may be considered as aromatic.10 Inspired by these conclusions, they proposed the existence of a Ga4 2− aromatic unit in K2 Ga4 −C6 H3 −2 6−Trip2 2 Trip = C6 H2 −246−i Pr 3 .12 In this

Figure 1 Structures of Al4 2− MAl4 − , and M2 Al4 M = Li, Na, Cu)

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sense, gas-phase studies offer unique electronic and structural information, “which is valuable in providing a conceptual framework for understanding the structure and bonding of new materials and compounds, as well as guiding the discoveries of new ones.”10 We return to the question of - and -aromaticity in Al4 2− . Fowler et al. evaluated the ring current in Al4 2− and MAl4 − M = Li Na and Cu and concluded that -electrons are responsible for the delocalized diamagnetic current induced by a perpendicular magnetic field.1314 On the basis of the analysis of aromatic ring current shielding calculations (ARCS),15 Juselius et al. concluded that -electrons contribute to the diatropic ring current, and thus Al4 2− is both - and -aromatic.16 This is opposite to the magnetic character of annulenes, which is essentially determined by the subsystem. Santos et al. have studied both the total and the – separated electron localization function (ELF)1718 of several molecules.19 They found that Al4 2− has a surprisingly high ELF bifurcation value of 0.99, which is even higher than the value associated with benzene. At the same time, Al4 2− shows a high bifurcation value of ELF (0.88), which suggests strong -delocalization. Further evidence of the -delocalization in Al4 2− can be given by the analysis of the individual canonical molecular orbital contributions to nucleus independent chemical shift (NICS) (MO-NICS). Within gradient-corrected DFT, the six -orbitals contribute more than 50 percent of the diatropicity of Al4 2− , while the sum of the MO-NICS contributions2021 of the -orbitals in both benzene and D2h cyclobutadiene is positive (paratropic).22 Al4 2− has also been studied employing the gauge-independent magnetic induced currents (GIMIC).23 With GIMIC, integrated current densities can easily be produced, which is a clear advantage over current density maps, as it provides quantifiable results. In addition, it is possible to subtract disturbing effects coming from surrounding Li cations. The GIMIC method clearly shows that the Al4 2− moiety is a diamagnetic molecule as no paramagnetic current is observed. Boldyrev and Kuznetsov obtained a rough evaluation of the resonance energies for Na2 Al4 and Na2 Ga4 .24 The resonance energies are high: 125 kJ mol−1 (B3LYP/6– 311+G∗ ) and 200 kJ mol−1 (CCSD(T)/6–311 + G(2df)) for Na2 Al4 and 88 kJ mol−1 (B3LYP/6–311+G∗ ) for Na2 Ga4 compared to 83 kJ mol−1 in benzene. However, it should be noted that it is hard to accurately evaluate the resonance energy in these clusters due to two factors: the interaction between Na+ and Al4 2− and the problem of identifying a reference molecule with an Al−Al double bond. In the same vein, Zhan et al. concluded that in terms of the magnitude of Dewar resonance energy,6 the aromaticity of the Al4 2− is multiple-fold as compared with the usual “1-fold” aromaticity of benzene.6 Al4 2− can be represented by 64 potentially resonating Kekulé-like structures; each Kekulé-like structure has three localized chemical bonds, compared to only two Kekulé structures of benzene. Consequently, the resonance energy of Al4 2− (∼304 kJ mol−1 as the upper limit and ∼220 kJ mol−1 as the lower limit) is at least 2.5 times that of benzene. Therefore, Al4 2− could be considered as a “3-fold” aromatic system. Is it possible to build new materials using an Al4 2− fragment? Seo and Corbett argued that perhaps in a solid, the cluster surrounded by cations cannot preserve the equivalent Al4 2− unit, because “the surrounding cations will probably stabilize empty lone-pair-like orbitals through electrostatic interactions” and “if the -bonding is not strong enough, it may break down and all p-orbitals may be filled to become lone pairs because of the energy gained through cation–anion interactions”.25 Recently, Mercero and Ugalde26 found a minimum with a sandwich-type structure: Al4 2 Ti2− . This cluster is a result

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of the interaction of two Al4 2− fragments with Ti2+ (which is similar to metallocenes). The fragmentation of this molecule Al4 2 Ti2− → 2Al4 2− + Ti2+ will not occur easily, because the binding energy is ∼2550 kJ mol−1 , which is similar to the experimental binding energy of ferrocene (∼2650 kJ mol−1 ). In recent years, many other important works on similar systems, and their properties have been published. Some of them are studies on the structure and aromaticity of other MAl4 n systems (M = Li, Na, Mg, Ca, Sc, Mn, Co, and Si), S4 2+ Se4 2+ P4 2− P4 Mn Si4 2+ Si4 Si4 2− , and NaSi4− , or a valence-bond study on the /-aromaticity in Al4 2− , or aromaticity and antiaromaticity studies in Lix Al4 n clusters.27−32 However, as far as we know, DFT reactivity and stability descriptors, such as the absolute or global hardness ( ), have not been employed to describe or assess the aromaticity and its trends in these all-metal aromatic clusters. This is important to be done since hardness has been an excellent descriptor of traditional aromaticy in organic cyclic hydrocarbons,33 and one may wonder whether this descriptor is useful to reproduce this extended concept of “all-metal aromaticity.” In DFT, global hardness is formally defined as34 1 2 E (1)

= 2 N 2 where E is the exact energy density functional as a function of the system number of electrons. It can be shown that a finite-differences approximation to this equation yields 1

= I − A 2

(2)

where I is the ionization potential and A is the electron affinity of the species. These properties are sometimes computed by use of the Koopmans’s theorem, although its reliability is contested when applied to open-shell systems. Therefore, we choose to compute both I and A “exactly”, i.e. performing two calculations for each case, for both the neutral and cationic/anionic species. In order to compare the global hardness as an all-metal aromaticity descriptor, we determine other quantities employed to evaluate aromaticity. The delocalization index (DI, in Equation 3)3536 is derived from Bader’s atoms-in-molecules (AIM) theory37 from the exchange-correlation density matrix xc : A B = −2

XC r1 r2 dr1 dr2

(3)

A B

where A and B are two given atoms within the molecule in study. The DI provides a quantitative measure of the number of delocalized or shared electrons among two given atoms A and B in a molecule. DI (given in electrons) is close to 1.0 for one equally shared pair of electrons in the A–B bond. For instance, the H2 molecule has a DI value of 1.0, and this value diminishes when ionic contributions to binding increase, such

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as in LiF, where DI = 02. This quantity has been previously computed to understand aromaticity trends in a number of systems.363839 The protons of traditional hydrocarbon aromatic systems show unusually large chemical shifts inside or above the rings. The NICS index measures the absolute magnetic shielding of the ring at a given position (ring center, above ring, etc).40 Aromatic systems yield negative NICS values, while positive NICS values are associated with anti-aromatic systems. In this work, we discuss the results of DFT calculations for some all-metal clusters with the general formula MAl4 n at a validated level of theory and numerical precision and compute a number of accepted properties to describe aromaticity, such as , geometrical parameters, the DI, and the NICS indexes. Hereby, we pursue the evaluation of DFT calculations and reactivity descriptors to explain and assess aromaticity in the anionic all-metal clusters derived from the Al4 2− unit. We determine the effect of different charges and multiplicities on the geometry of Al4 n n = −2 −1 0 1 and calculate the structures of new complexes MAl4 n where M = Li+ Na+ K + Be+2 Mg+2 Ca+2 Sc+3 Ti+4 , and B+3 Al+3 Ga+3 . In order to compare the DFT reactivity descriptors, we compute other parameters (NICS and DIs) and study periodic trends.

2. Objectives • Evaluate the ability of the DFT calculations and reactivity descriptors to explain and assess aromaticity in the anionic all-metal cluster, Al4 2− . • Determine the effect of different charges and multiplicities on the aromaticity and geometry of Al4 n n = −2 −1 0 1. • Study the stability and aromaticity and possible periodic trends in new complexes with the general formula MAl4 n formed from Al4 2− and M, where M = Li+ Na+ K + Be+2 Mg+2 Ca+2 Sc+3 Ti+4 , and B+3 Al+3 Ga+3 .

3. Methodology • Use the previously studied Al4 2− unit to compare, select, and validate an adequate DFT functional and basis set for the rest of this work. All calculations are performed with Gaussian 98.41 • Compute the lower energy structures for the Al4 2− unit. • Calculate the lower energy structures for the Al4 n unit with (n = −2 −1 0, and 1) to consider charge and multiplicity effects. • Determine the lower energy structures for the MAl4 n complexes. • Compute the different aromaticity and stability descriptors.

4. Results 4.1. Validation and selection of the DFT method and basis set The different geometrical parameters obtained from full optimizations (no symmetry restrictions) for the Al4 2− unit at different levels of theory are given in Table 1. The DFT

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Electronic structure and reactivity of aromatic metal clusters Table 1 Geometrical parameters obtained for Al4 2− using different methods of calculation and basis sets Model

Base

Al–Al

pBPa B3LYP B3LYP CCSD(T)

DN∗∗ 6-31+G∗ 6-311+G∗ 6-311++G∗

2.599 2.593 2.592 2.580

a

Becke–Perdew functional with nonlocal corrections through perturbation theory. DN∗∗ basis set is equivalent in size to 6–31G∗∗ .45 Bond lengths are given in Å.

calculations were carried out using Gaussian 98.41 Independently of the methodology, the global minimum on the potential energy surface has a D4h symmetry. It was verified that our calculations employing the nonlocal GGA hybrid density functional B3LYP4243 with the 6-31+G(d) basis set44 reproduces quite closely the parameters obtained with CCSD(T)/6-311+G(d) reported by Li et al.5 Therefore, hereby we adopt the B3LYP/6-31+G(d) level for all our further calculations.

4.2. Results for the bare Al4 unit Let us briefly analyze the molecular orbitals (MOs) of Al4 2− . It can be seen that, as expected, these MOs are symmetrically quite different from the analogous “traditional” aromatic cyclic hydrocarbons (Figure 2). This shows that the concept of aromaticity in the all-metal clusters is certainly an extension of the original concept and is not exactly similar. This has been already discussed in the introductory section, but it is important to pinpoint that although other aromaticity descriptors (hardness, DI, and NICS) may go

Figure 2 Occupied molecular orbitals of Al4 2− , HOMO-4 is degenerated

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(as we will show next) in the same trends in all-metallic systems as in traditional cyclic hydrocarbons, caution must be exercised when using the same aromaticity concept to encompass the all-metal clusters as well. The Al4 n unit with different total charge n (n = −2 −1 0, and 1) has been fully optimized for each case to assess electronic charge and spin effects on the geometry. The detailed geometrical parameters are summarized in Table 2. A frequency analysis is done to confirm that all obtained geometries are at absolute minima. It can be noted that only when the total charge is −2, a stable planar structure is obtained. For other charges, rhomboids are favored. In order to assess the ability of the DFT hardness as an all-metal aromaticity and global stability descriptor, we compute it from calculation of the neutral, anionic, and cationic species for each lower energy cluster structure to determine the vertical I and A (Table 2). We employ (2) to compute the hardness. It can be seen in Figure 3 that, with the exception of the cluster with total charge zero (the Hückel rule shows this is an anti-aromatic system, although convergence was not fully attained in this case), there is a smooth decreasing behavior of the hardness following the increase in charge. What are the factors stabilizing these molecules? One of them is, of course, the electron delocalization, but there are other effects influencing the stability as Coulombic repulsions and hardness include all of them.

4.3. Results for the MAl4 n complexes Kuznetsov et al. have shown that the basic aluminum square structure is preserved when the Al cluster reacts to form some bimetallic systems with chemical compositions: MAl4 − and M2 Al4 (where M = Li+ Na+ , and Cu+ ), and the Table 2 Geometrical parameters of lower energy optimized structures of Al4 n unit n

State

Al–Al

−2 −1 0 +1

Singlet Doublet Triplet Sextuplet

2593 2570 2585 2635

900 755 689 1114

900 1055 1110 686

4216 4105 4280 3755

See Scheme 1 for angles notation. Bond lengths are given in angstroms. Angles are given in degrees. Hardness values are given in eV.

A

β

α

Scheme 1

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Electronic structure and reactivity of aromatic metal clusters 4.4

Hardness (eV)

4.3 4.2 4.1 4 3.9 3.8 3.7 –3

–2

–1

0

1

Charge

Figure 3 Hardness of lower energy structures for Al4 n unit with (n = −2 −1 0, and 1) computed from (2)

aromatic nature is apparently preserved. Hereby we report our calculations at the B3LYP/6-31 + Gd level which allow us to rationalize and to quantify the possible aromatic nature of the abovementioned MAl4 − species and to extend our study to other bimetallic systems that had not been previously reported with general composition MAl4 n M = Li+ Na+ K + Be2+ Mg2+ Ca2+ Sc3+ Al3+ B3+ Ga3+ Ti4+ where n = −1 0 +1 +2. In general, the pyramidal structure turns out to be always the lower energy minimum, and usually the planar structure appears as the next one (second) in energy.

4.4. Results for the complexes with general formula MAl4 n where M is an alkaline cation In Table 3, bond lengths of fully optimized pyramidal (lower minima) MAl4 −1 complexes, where M is an alkaline cation, are shown. In all cases, the aluminum square unit is preserved. For comparison, geometrical parameters of the square planar structures are also given in Table 3. Both pyramidal and square structures are singlet states. We also carried out the lowest energy triplet state but found it to be significantly higher in energy. In the pyramidal case, the square unit is preserved, even though it interacts with alkaline Table 3 Lower energy structures for Al4 − complexes where M is an alkaline cation Pyramidal

Square Planar

M

Ediss

Al–Al

Al–M

Al1 –Al2

Al2 –Al3

Al1 –Al4

Al1 –M

Li Na K

21117 19852 17249

2612 2607 2600

2870 3018 3532

2515 2520 2529

2560 2577 2583

2643 2663 2648

2633 2960 3397

See Scheme 2 for atoms notation in square planar case. Ediss are in kcal mol−1 and bond lengths are given in Å.

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211 2

1 M

3

4

Scheme 2 cations. However, in the latter case, the Al square units are deformed to trapezoids. This lowers the initial aromaticity. Obviously, the distance from aluminum unit to the M cation increases as one goes down in the alkaline family, being longest for K. From the values given in Table 3, one can see that the Al−Al bond lengths within the pyramidal MAl4 −1 complexes are always longer than the equivalent in the bare Al4 2− cluster (2.593 Å). This can be rationalized considering the -aromatic system charge transfer (especially from the HOMO) to the M cation through the interaction of its empty ns cation orbital with the Al4 2− fragment HOMO. This charge transfer is significant, almost one electron (Mulliken charges in the cation M M = Li Na K) become −008, −013, and 0.07 respectively). Therefore, the original charge in the Al fragment HOMO is reduced, and as a consequence the relative Al–Al bond lengths increase. Also the dissociation energies Ediss for the reactions MAl4 −1 → Al4 −2 + M+ for the pyramidal MAL−1 4 complexes are shown in table 3. This provides an estimation of the energetic stability of these complexes toward dissociation. From values in Table 3, it is clear that the complexes are quite stable, and the stability decreases as one descend in the periodic family. In Table 3, one can also see that the Al–Al bond lengths within the square planar MAl4 −1 complexes (in comparison with the bare Al4 2− cluster (2.593 Å)), decrease for Al1 –Al2 and Al2 –Al2 but increase for Al1 –Al1 . This can be explained considering the interaction between the M cation ns orbital with the Al unit HOMO-4 (see Figure 2). Obviously there is also an interaction between the cation ns and the HOMO-1 and HOMO-2, but the observed effect in Al–Al distances is basically explained through the interaction with the HOMO-4. In Table 4, the energy differences E between pyramidal and square isomers for MAl4 −1 complexes are shown. All of them show pyramidal structures lower in energy than the square planar and the E lie in the range of 3.5 to 6 kcal mol−1 . Hardness values are computed from (2) for isomers of MAl4 n systems (Figure 4 and Table 4). In agreement with the energy differences, the pyramidal structures are always Table 4 Energy differences (E) between pyramidal and square isomers in kcal mol−1 and hardness in eV for MAl4 n complexes where M is an alkaline cation M

E

(pyramidal)

(planar)

Li Na K

362 579 548

3872 3631 3067

3581 3394 2851

212

Electronic structure and reactivity of aromatic metal clusters 4.1

Hardness (eV)

3.9 3.7 3.5 3.3 3.1 2.9 2.7

Li

Na

K

M

Figure 4 Hardness of lower energy structures of MAl4 n complexes where M is an alkaline cation. Rhomboids and squares represent pyramidal and square structures, respectively

harder than the square isomers. As shown in Figure 4, the hardness values decrease when going down a family in the periodic table. To better understand the interaction between Al4 2− and an alkaline cation, let us examine the Walsh diagram for a given case, NaAl4 − (Figure 5). From this diagram, it is clear that the cation interaction with the Al cluster unit is essentially electrostatic, and there are no new MOs formed. A Mulliken charges calculation shows that there is almost one electron transferred from the whole Al unit to the Na cation. The same holds for all other alkaline cations. The orbital distribution is similar in the MAl4 − complex to the bare Al4 2− fragment, but its HOMO delocalized -orbital is stabilized (lower energy) as a consequence of the electrostatic interaction with the cation. This interaction is the principal factor for the higher stability of the MAl4 − complexes. However, the covalent contribution is certainly not negligible, since the Al fragment HOMO is greatly stabilized through the M cation ns LUMO interaction, and this should have both electrostatic and covalent components. Al42–

NaAl4–

Figure 5 MO Walsh diagram for the formation of pyramidal NaAl4 −

J. Robles et al.

213

4.5. Results for the complexes with general formula MAl4 n where M is an alkaline-earth (group II) cation In Figure 6, the optimized geometries obtained for MAl4 n complexes with M from group II cations are displayed. The optimized geometrical parameters are given in Table 5. For Mg and Ca, the interaction between the cation and the aluminum unit is similar to that of alkaline cations (large electrostatic component), and the Al cluster unit is preserved as a square. Mulliken charges calculation for these MAl4 n complexes show that there are almost two electrons transferred from the whole Al unit toward the cation. However, the interaction with beryllium yields a quite distorted Al unit. This may be explained in view of the fact that Be2+ is a hard Lewis acid, so it strongly interacts with the square unit, which behaves as a Lewis base. Covalent bond formation from this unit to beryllium seems to be stronger (shorter distances) but distorts the Al unit, reducing the overall system aromaticity. Evidently, the longest distances from the Al unit to the cation are for Ca. Comparing the results for Be with those for Mg and Ca cations provides us with another indication that covalent interaction may be important besides the electrostatic one. Be+2 is the least metallic of the three cations in this family. The explanation for the larger distortion and stronger bond formation in the Be complex may be that although the Al4 2− HOMO and the Be+2 LUMO-ns interaction is still important, in this case through the geometry distortion, other interactions take an important role, namely the Be+2 LUMO with HOMO-1, with HOMO-2, and especially with the HOMO-4 of the Al fragment. The latter interaction would induce an increase in two of the Al−Al distances in the Be complex and a decrease in the other two, and this provides a rationale for the predicted distortion. The energy differences between pyramidal and square isomers for MAl4 n complexes with M from group II are given in Table 6. All of them are within 5.0 to 9.5 kcal/mol, which is a larger range compared to alkaline cations (3.5 to 6 kcal/mol).

Mg+2

Be+2

Ca+2

Figure 6 Lower energy structures for MAl4 n complex systems with M from group II cations

Table 5 Some geometrical parameters of lower energy structures (pyramids) for MAl4 n complex systems with M from group II cations M

Multiplicity

Be Mg Ca

Singlet Singlet Singlet

Distances are given in Å.

Al1 –Al2

Al2 –Al3

Al1 –M

Al2 –M

Al3 –M

2723 2681 2649

2642 2681 2649

2459 2889 3232

2153 2889 3232

3087 2889 3232

214

Electronic structure and reactivity of aromatic metal clusters Table 6 Energy differences between pyramidal and square isomers (kcal/mol) for MAl4 n complexes where M is an alkaline–earth cation M

E

pyramidal

planar

Be Mg Ca

940 498 586

4775 4691 4275

4185 4257 3801

Hardness values are given in eV.

5.2

Hardness (aV)

4.7

4.2

3.7

3.2

2.7

Be

Mg

Ca

M

Figure 7 Hardness (in eV) for MAl4 n complexes, M = Be, Mg, Ca. Rhomboids and squares represent pyramidal and square structures, respectively

As in the alkaline complexes, for the group II cation complexes, the pyramidal isomer is always harder than the square (Table 6 and Figure 7). In Figure 7, it can be seen that in both isomers, the Be complex (both isomers) is somewhat softer (than expected from the general trend). This may imply that its aromatic character is decreased due to the strong interaction-induced distortion of the Al unit.

4.6. Results for the complexes with general formula MAl4 n where M is a cation from group I and II: Other aromaticity descriptors—DI and NICS The DI and NICS indexes were computed for the MAl4 n complexes with alkaline and group II cations M. These results are summarized in Table 7. The higher the DI value, the higher the delocalization in the ring, thus providing an enhanced aromatic character to the system. The meta-DI (m-DI) values in Table 7 decrease as one descends in a periodic family. The NICS values are all negative, suggesting that all systems are aromatic. The largest absolute value both above ring (NICS (1)) and at the ring center (NICS (0)) is found for the Al4 −2 cluster.

J. Robles et al.

215 Table 7 NICS (in ppm) and DI (in electrons) computed for MAl4 n complexes with alkaline and group II cations Cluster

NICS (1)

NICS (0)

m-DI

Al4 2− LiAl4 − NaAl4 − KAl4 − BeAl4 MgAl4 CaAl4

−341 −188 −238 −267 −148 −51 −152

−285 −211 −254 −265 −157 −107 −184

0809 0759 0711 0706 – 0601 0579

Level of calculation is B3LYP/6–31+G(d). The m-DI of BeAl4 could not be obtained by technical reasons.

The NICS trends, although different from the DI ones, are presumed to be correct. Why? When the Al unit geometry within the MAl4 n complex is close to the bare Al4 2− cluster geometry, the system should preserve its electron delocalization. For instance, when a cation is placed on the same Al unit plane, close to one side (square structures) or when a harder cation (such as Be2+ ) is placed above the Al-cluster plane, this unit is deformed and its aromaticity decreases. The observed trend in MAl4 − complexes for M = Li, Na and K, indicates that the Li complex is relatively more stable (harder), but it is also the alkaline cation producing the largest deformation on its Al unit (see Table 3). Therefore, this is the complex in the family where the interaction between cation and aromatic fragment is most enhanced, as also seen in its Walsh Diagram (Figure 5). This deformation certainly is an indication of covalent contributions to binding. Thus, even when this is a “harder” complex, it is also the least aromatic in the family. On the other hand, the hardness values represent a more general “global” complex stability, where other effects (Coulombic repulsions in the Al unit decreased by the fragments electrostatic interaction) that contribute to overall stability, besides the electronic delocalization, are also included. In traditional hydrocarbon aromatic systems, hardness is a better overall descriptor of aromaticity because in these systems the dominant factor toward stability is essentially the -electron delocalization. We believe that this is reflected in the obtained aromatic descriptor values.

4.7. Results for the complexes with general formula MAl4 n where M is a cation from IIIA group and some transition metal cations In Figure 8, minimum energy structures are displayed for MAl4 n where M is a cation from IIIA group (B+3 Al+3 Ga+3 ) and some transition metal cations (Sc+3 Ti+4 ). It is interesting that for these cases all Al-cluster units are distorted, except the one with the Sc cation, where the square Al unit is preserved. Also displayed in this Figure are the Mulliken charges. In all systems, a large electron transfer from the Al cluster unit toward the cation is apparent. These results indicate strong covalent contributions besides the electrostatic interaction between cation and Al-cluster, and thus new bonds are formed, causing the initial geometries to become quite changed.

216

Electronic structure and reactivity of aromatic metal clusters Al+3

0.50

–0.31

0.19

B+3

0.27

0.27

0.27

Sc+3

0.18

Ga+3

0.26 0.19

–0.04

0.31

0.31

0.21

0.18

0.44

0.38

0.21

Ti+4 0.31

0.74 0.14 0.14

0.14 0.14

0.18 0.38

Figure 8 Lower energy structures and Mulliken charges for MAl4 n systems with M from IIIA group and some transition metal cations

5. Conclusions We have shown that DFT calculations at the B3LYP/6-31+G(d) level are suitable for evaluating the geometrical and electronic parameters of clusters derived from Al4 2− . This holds in particular for our optimized geometries, which are in very good agreement with CCSD(T) / 6-311++G(d) calculations (see Table 1). Our calculations confirm the all-metal aromatic character of MAl4 n clusters. Several indexes of aromaticity known from unsaturated hydrocarbons were applied to these metal clusters, apparently succeeding in describing its all-metal aromaticity: The geometry of the Al4 2− unit preserves a square structure (bond length equalization), the global hardness is in the same magnitude as in traditional aromatic molecules, and NICS and DI indexes also support the aromatic character of these molecules. However, it is clear that the electron delocalization, which is the central factor for stabilization of these molecules, is not only restricted to the system in the Al4 2− unit, but also to an even larger extent to the framework. In some sense, these clusters show a “double aromatic” behavior. Our calculations also yield higher aromatic character to the Al4 2− unit if the interaction with cations is essentially electrostatic, although we have argued that the covalent contributions during the fragments interactions may be quite important, as in the Be and Li cases. In the former, the interaction deforms the Al unit, decreases the aromaticity but increases the overall stability (as measured by its global hardness). All indexes show a general agreement with the fact that as the electronegativity of the metal cation increases, the aromaticity of the cluster decreases. This picture is also confirmed by a simple charge model, which demonstrates that the aromatic character of Al4 n decreases when n is increasing from −2 to +1.

Acknowledgements We are grateful to Jesus Hernández–Trujillo from Facultad de Química, UNAM, for very helpful comments and fruitful discussion. JR acknowledges support from

J. Robles et al.

217

CONACYT through project SEP-2003-C02-43453 from Fondo Sectorial SEPCONACYT. We are also grateful for SGI Series 2000 computer time from Depto. de Supercómputo, DGSCA, UNAM. PG is grateful for a scholarship from CONACYT during her Masters Thesis work, this work partially fulfilling requirements for this degree. MS and JP thank the Spanish MCyT for financial support through projects nos. BQU2002-0412-C02-02 and BQU2002-03334, and the DURSI of the Generalitat de Catalunya through project No. 2001SGR-00290. MS is indebted to the DURSI for financial support through the Distinguished University Research Promotion awarded in 2001. JP also acknowledges the DURSI for the postdoctoral fellowship 2004BE00028.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Schleyer, P. v. R., Chem. Rev. 2001, 101, 1115. Wade, K., J. Chem. Soc. D: Chemical Communications, 1971, 792–793. Wade, K., Inorg. Nucl. Chem. Lett. 11, 8, 559. Wade, K., Chem. Brit. 1975, 11, 177. Li, X.; Kuznetsov, A. E.; Zhang, H. F.; Boldyrev, A. I.; Wang, L. S., Science 2001, 291, 859. Zhan, C. G.; Zheng, F.; Dixon, D. A., J. Am. Chem. Soc. 2002, 124, 14795. Moran, D.; Manoharan, M.; Heine, T.; Schleyer, P. v. R., Org. Lett. 2003, 5, 23. Kuznetsov, A. E.; Corbett, J. D.; Wang, L. S.; Boldyrev, A. I., Angew. Chem. Int. Edit. 2001, 40, 3369. Li, X.; Zhang, H. F.; Wang, L. S.; Kuznetsov, A. E.; Cannon, N. A.; Boldyrev, A. I., Angew. Chem. Int. Edit. 2001, 40, 1867. Kuznetsov, A. E.; Boldyrev, A. I.; Li, X.; Wang, L. S., J. Am. Chem. Soc. 2001, 123, 8825. Kuznetsov, A. E.; Boldyrev, A. I., Struct. Chem. 2002, 13, 141. Twamley, B.; Power, P. P., Angew. Chem. Int. Edit. 2000, 39, 3500. Fowler, P. W.; Havenith, R. W. A.; Steiner, E., Chem. Phys. Lett. 2001, 342, 85. Fowler, P. W.; Havenith, R. W. A.; Steiner, E., Chem. Phys. Lett. 2002, 359, 530. Juselius, J.; Sundholm, D., Phys. Chem. Chem. Phys. 1999, 1, 3429. Juselius, J.; Straka, M.; Sundholm, D., J. Phys. Chem. A 2001, 105, 9939. Becke, A. D.; Edgecombe, K. E., J. Chem. Phys. 1990, 92, 5397. Silvi, B.; Savin, A., Nature 1994, 371, 683. Santos, J. C.; Tiznado, W.; Contreras, R.; Fuentealba, P., J. Chem. Phys. 2004, 120, 1670. Heine, T.; Schleyer, P. v. R.; Corminboeuf, C.; Seifert, G.; Reviakine, R.; Weber, J., J. Phys. Chem. A 2003, 107, 6470. Corminboeuf, C.; Heine, T.; Weber, J., Phys. Chem. Chem. Phys. 2003, 5, 246. Chen, Z. F.; Corminboeuf, C.; Heine, T.; Bohmann, J.; Schleyer, P. v. R., J. Am. Chem. Soc. 2003, 125, 13930. Juselius, J.; Sundholm, D.; Gauss, J., J. Chem. Phys. 2004, 121, 3952. Boldyrev, A. I.; Kuznetsov, A. E., Inorg. Chem. 2002, 41, 532. Seo, D. K.; Corbett, J. D., Science 2001, 291, 841. Mercero, J. M.; Ugalde, J. M., J. Am. Chem. Soc. 2004, 126, 3380. Hu, X. B.; Li, H. R.; Liang, W. C.; Han, S. J., Chem. Phys. Lett. 2005, 402, 539. Zhai, H. J.; Kuznetsov, A. E.; Boldyrev, A. I.; Wang, L. S., Chem. Phys. Chem. 2004, 5, 1885. Jin, Q.; Jin, B.; Xu, W. G., Chem. Phys. Lett. 2004, 396, 398. De Proft, F.; Fowler, P. W.; Havenith, R. W. A.; Schleyer, P. v. R.; Van Lier, G.; Geerlings, P., Chem.-Eur. J. 2004, 10, 940. Havenith, R. W. A.; Fowler, P. W.; Steiner, E.; Shetty, S.; Kanhere, D.; Pal, S., Phys. Chem. Chem. Phys. 2004, 6, 285.

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32. Havenith, R. W. A.; van Lenthe, J. H., Chem. Phys. Lett. 2004, 385, 198. 33. Zhou, Z. X.; Parr, R. G.; Garst, J. F., Tetrahedron Lett. 1988, 29, 4843; Zhou, Z. X.; Parr, R. G., J. Am. Chem. Soc. 1989, 111, 7371. 34. Parr, R. G.; Yang, W., Density-Functional Theory of Atoms and Molecules; Oxford University Press; Clarendon Press: Oxford, 1989. 35. Fradera, X.; Austen, M. A.; Bader, R. F. W., J. Phys. Chem. A 1999, 103, 304. 36. Poater, J.; Fradera, X.; Duran, M.; Sola, M., Chem.-Eur. J. 2003, 9, 400. 37. Bader, R. F. W., Atoms in Molecules: A Quantum Theory; Clarendon Press: Oxford; New York, 1990. 38. Poater, J.; Fradera, X.; Duran, M.; Sola, M., Chem.-Eur. J. 2003, 9, 1113. 39. Poater, J.; Sola, M.; Viglione, R. G.; Zanasi, R., J. Org. Chem. 2004, 69, 7537. 40. Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H. J.; Hommes, N. J. R. V., J. Am. Chem. Soc. 1996, 118, 6317. 41. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millan, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Baboul, A. G.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzales, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Andreas, J. L.; Head-Gordon, M.; Reploge, E. S.; Pople, J. A. Gaussian 98, Gaussian 98 revision A7; Gaussian Inc: Pittsburg, PA, 1998. 42. Becke, A. D., J. Chem. Phys. 1993, 98, 5648. 43. Lee, C. T.; Yang, W. T.; Parr, R. G., Phys. Rev. B 1988, 37, 785. 44. Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A., J. Chem. Phys. 1980, 72, 650. 45. Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A., Ab initio Molecular Orbital Theory. ed.; Wiley: New York, 1986.


Chapter 11

Small gold clusters form nonconventional hydrogen bonds X-H· · · Au: gold–water clusters as example ab

E. S. Kryachko and

1a

F. Remacle

a

Department of Chemistry, Bât. B6c, University of Liège, Sart-Tilman, B-4000 Liège 1, Belgium and b Bogoliubov Institute for Theoretical Physics, Kiev, 03143 Ukraine

1. Introduction: golden gate to nano-dimensions Gold is the noblest among the seven “coinage” metals which also include copper, iron, mercury, lead, silver, and tin. The ancients related them to certain gods and to certain stellar objects and to the weekdays as well. Gold was naturally linked to the Sun due to its bright yellow color2 and was associated with Sunday. It has been known for at least 7000 BC. Gold is not subject to rust, verdigris, or emanation, it steadily resists the corrosive action of salt and vinegar. This was apparently the reason why gold was used as a medium of exchange for nearly 3000 years. The first gold coins appeared in Lydia in 700 BC. The word “gold” is believed to originate from Sanskrit meaning “to shine” [1]. In the periodic table of the elements, the atom of gold Au occupies the 79th place with the ground-state electronic configuration Xe4f 14 5d10 6s1 . Gold has always been considered as the noblest element, likely because gold was first found in some river sands in its native metallic form and attracted the man’s attention due to its color and luster, and its resilience to tarnish and corrosion. Gold is the least reactive of the coinage metals. In its bulk form, gold is essentially inert [2–5] and has a yellowish color. When the dimensions of the particles are shrinked from the bulk to a scale of the order of the Fermi wavelength of the electron, that is to a nanometer size, their behavior dominantly obeys the principles of quantum mechanics [6]. One may therefore expect the emergence of new properties which may be entirely different from those of the bulk and sometimes are completely unexpected. It is believed that they are due to the socalled quantum “size effect” that modifies the electronic structure and increases the ratio 219

220

Small gold clusters form nonconventional hydrogen bonds

of atoms located on the surface relative to the total number of atoms composing the nanoparticles (NPs) [7,8]. Indeed, gold NPs of 2–3 nm bridge the “material gap” [9,10] and exhibit novel shape-dependent electronic, optical, and magnetic properties. In its nanodimensions, gold can be of a ruby-red, purple, or even blue color depending on the NP size [2,11]. In the NP regime, gold drastically changes its catalytic activity compared to its inert bulk form. For example, oxide-supported gold clusters, particularly gold dispersed on various metal oxides as well as nanosized islands on titania oxide, demonstrate an enhanced chemical reactivity [3,5,12,13] that makes them very well suited for use as chemical catalysts in many reactions like combustion of hydrocarbons [14], reduction of nitrogen oxide [15], propylene epoxidation [16] (see also [17]), and low-temperature oxidation of carbon monoxide [3,5,12,13,18–21]. The carbon monoxide oxidation encompasses numerous practical applications. In these, the activity of the gold NPs critically depends on the NP size, the nature of the support, and the detailed synthetic procedure [2,13,21,22]. The STM experiments by Valden et al. [3,20] demonstrate that the efficiency of CO oxidation on TiO2 -supported 2D Au islands depends on the island thickness, which is interpreted in terms of a quantum size effect [3,20]. As recently shown by Goodman and Chen [23], the bilayer gold catalyst exhibits an activity for CO oxidation more than 10 times larger than the monolayer and the reaction proceeds ca. 50 times faster on the bilayer gold catalyst than on Au/TiO2 catalysts. On the other hand, Freund and co-workers [24] has recently reported the first experimental evidence that thin islands of gold have the same CO adsorption behavior as large gold NPs and extended gold surfaces. This finding implies that the gold reactivity arises from the presence of highly non- or low-coordinated gold atoms (see also [25]). To summarize, the experimental studies emphasize the quantum-size effect, lower coordination [24,26,27], as well as the charge, shape [25,28], and support interface [29] effects as playing important roles in the exceptional activity of gold NPs. This has attracted a theoretical interest and motivated further research aiming at providing insights into the molecular origins of such phenomena [3,5,24,27,30–40] (see also [41–43] for current reviews), focusing particularly on the exceptional catalytic properties of small gold aggregates [12a-b]. First-principles calculations demonstrate a clear correlation between particle size and chemical activity, which can be explained in terms of an enhanced density of low-coordination sites with decreasing cluster size [27,44]. It is suggested that the Au–Au coordination number has a larger effect on the reactivity of the Au particles than the electronic structure, support, or strain, and thus, the presence of low-coordinated gold atoms is the major factor that determines the catalytic activity of the Au NPs [25]. Other studies [5,45–47] also show that the oxide provides an excess charge to the Au cluster, which is important for the ability to bind and activate O2 . The most reactive sites [45] occur on gold–oxide support interfaces, where the precise interface structure depends on the cluster size and geometry. Recent investigations on size-selected small gold clusters, Au2≤n≤20 , soft-landed on a well-characterized metal oxide support (specifically, an MgO(001) surface with and without oxygen vacancies or F centers [30,40]), reveal that gold octamers bound to F centers of the magnesia surface are the smallest known gold heterogeneous catalysts that can oxidize CO into CO2 at temperatures as low as 140 K.

E. S. Kryachko and F. Remacle

221

Let us recall that the gold atom has a single 6s electron in the valence shell. Hence, the highest occupied molecular orbitals (HOMOs) of even-numbered open-shell Au2k cluster anions consist of a hole and an unpaired electron, whereas the odd-numbered Au2k+1 - are occupied by two paired electrons, i.e., their HOMOs are closed-shell. The even-numbered Au2k - clusters with open-shell electronic configuration provide lower electron affinities (EAs) than the odd-numbered Au2k+1 - ones with closed valence shells [48]. The closed-shell Au clusters are therefore less reactive and the open-shell cluster anions with lower EAs react in general with O2 efficiently. For instance, Au2 - , Au4 - , and Au6 - clusters are highly reactive toward oxidation of CO to CO2 [34,37]. In contrast, the cationic ones are inert toward O2 chemisorption [49]. In the gas phase, the energy gaps between the HOMO and the lowest unoccupied molecular orbital (LUMO) of the Au clusters become negligibly small, i.e., 1). These two criteria are the most unequivocal and powerful signs of that the hydrogen bond is formed; (vi) Proton nuclear magnetic resonance 1 H NMR) chemical shifts in the A-H · · · B hydrogen bond are shifted downfield A-HB A -H compared to the monomer, that is, iso H ≡ iso H − iso H n n an O DA2 > an O D2A . The formation of the H-bond in this system induces a flow of the electron density that can be traced by the changes in the Mulliken atomic and the natural population analysis (NPA) charges: qM O1 = −0070, qM O2 = −0040, qM H3 = +0060, and qM H5 = qM H6 = +0020 and qNPA O1 = −0031, qNPA O2 = −0012, qNPA H3 = +0022, and qNPA H5 = qNPA H6 = +0013e. This comparison ends up the introductory part of the review. In the next sections, we discuss the main features of the nonconventional H-bond forming between gold and water clusters. In particular, the following questions are addressed in the next section: What is the fate of the intramolecular H-bond of the water dimer when the latter interacts with a gold cluster, say, a triangular gold cluster Au3 [101], chosen as the simplest catalytic model of Au particles (see [103] and references therein)? Is the water dimer capable of anchoring Au3 , by itself an important issue in molecular electronics (see [104] and references therein)?

226


Table 2 The NMR chemical shifts of the oxygen atoms and bridging hydrons of water oligomers H2 O1≤n≤6 evaluated at the B3LYP/6-311++G(2d,2p) computational level H2 O1≤n≤6

iso O iso O

an O an O

iso H iso H

an H n H

H2 O1

324.8

54.3

31.3

19.0

H2 O2

D: 321.0 −38 A: 316.3 −85

66.7 +134 43.1 −112

28.1 −32

30.2 +112

H2 O3

DA: 307.9 −84 308.1 −82 308.7 −76

60.4 +273 60.9 +278 62.9 +298

26.7 −14 27.0 −11 26.7 −14

27.9 +92 27.2 +85 28.1 +94

H2 O4

DA: 301.6 −147

52.5 +94

24.8 −33

32.9 +27

H2 O5

DA: 299.4–300.9

48.5–51.0

24.2–24.5

34.0–34.8

ring H2 O6 prism H2 O6

DA: 299.8

49.1

24.1

35.5

DA2 : 304.1 294.1 302.8 D2 A: 295.8

59.3 52.4 57.7 28.8

285.9

25.3

298.0

41.5

26.2 22.5 24.8 27.9 29.0 28.6 26.0 27.9 28.7

28.0 35.3 32.0 22.6 19.1 19.7 25.9 25.3 21.7

DA: 301.6 303.7 DA2 : 293.8 301.2 D2 A: 295.5

53.3 62.0 53.0 53.5 32.5

290.1

37.3

25.6 25.2 23.1 24.8 28.6 27.8 26.0 28.3

30.4 31.3 33.4 31.6 21.0 24.4 28.3 22.5

cage

H2 O6

The changes in the NMR chemical shifts of H2 O3≤n≤4 are referred to those of the water dimer, whereas the changes in the chemical shifts of the water dimer to those of the water monomer. D stands for proton donor, A for proton acceptor.

3. Where gold meets water: nonconventional hydrogen bond in Au3 –water dimer complex Before answering the above questions, we consider the simplest case of interaction between a single water molecule H2 O and a triangular gold cluster Au3 . This interaction leads to the formation of the nonplanar complex Au3 –H2 O, shown in Figure 2, with a binding energy of 102 kcal · mol−1 . As can be seen from Figure 2, the complexation between Au3 and H2 O occurs via the “anchoring” bond Au1 -O2 , at the upper edge of the Au3 cluster, where its spin-up and spin-down LUMOs, LUMO and LUMO , with the corresponding orbital energies of −4055 and −3858 eV, mostly protrude (see Figure 3; also the rules and Figure 3 in Ref. [17]). The MO picture of the Au1 -O2 anchoring in the complex Au3 -H2 O given in terms of the HOMO , HOMO , and HOMO-1 is presented in Figure 4.


227

0.287 0.964

H6

H5 0.290

O2 0.964

–0.512

2.330

Au1 0.275

2.721

2.630

Au4 –0.145

2.833

Au3 –0.194

Figure 2 The Au3 –H2 O complex. Mulliken charges are indicated in italics. The energy of the Au3 –H2 O complex amounts to −484388571 hartree and the ZPVE to 1569 kcal · mol−1

LUMO α ε = –4.055 eV

LUMO β ε = –3.858 eV

HOMO α ε = –5.266 eV

HOMO β ε = –7.232 eV

Figure 3 The HOMOs and LUMOs of the triangular cluster of gold

228



HOMO–1 α ε = –6.783 eV

HOMO β ε = –6.601 eV

Figure 4 The MO picture of the Au1 –O2 anchoring in the complex Au3 –H2 O

The Au1 -O2 anchoring naturally activates the O-H bonds of the water molecule causing its hydrogen atoms to partially lose the electron charge and to become more positive (the differences in the corresponding Mulliken charges qM H5 = +006e and qM H6 = +005e). As a consequence, the electron charge on the negatively charged oxygen atom partly increases qM O2 = −004e and partly flows, through the anchored gold atom Au2 qM Au1 = +015e, to the lower-edge atoms of gold qM Au3 = −008e and qM Au4 = −019e) accumulating an additional negative charge. The length of the anchoring Au-O bond in the complex Au3 -H2 O is 2330 Å. The reference system Au-H2 O which adopts either planar or nonplanar very weakly bound conformers is characterized by longer Au-O bonds, i.e., 2.880 and 3075 Å, respectively [53].5 In some sense, the shortening of the anchoring bond can be treated as a simple evidence of a quantum-size effect inherent to gold clusters for the trivial case of the Au → Au3 transition. A higher stabilization, by a factor of 10, of the complex Au3 -H2 O relative to Au-H2 O (the corresponding binding energies of the conformers amount to only 0.73 and 111 kcal · mol−1 ; this value is to be compared with the adsorption energy ≈ 3 kcal · mol−1 of water on the Au(111) surface [60a]) is another and, obviously, related manifestation of a quantum-size effect, merely meaning that more “room” is available within the three-gold cluster, Au3 , to redistribute the total electron density when it anchors to the oxygen atom O2 of H2 O than within a single atom of gold. In this respect, it is also worth noticing that the anchoring Au-O bond in Au3 -H2 O is shorter by 023 Å than the Au-F one in Au3 -FH, so that the binding energy of the latter is smaller by 6 kcal · mol−1 [56d]. This is apparently accounted for a smaller electron charge transfer from the hydrogen fluoride molecule to the cluster of gold since the fluorine atom has a larger EA than the oxygen. Let us now turn to the case when water dimer interacts with Au3 . It is natural to expect that they form with each other the anchoring bond. It appears however that the water dimer anchors the triangular Au3 cluster much stronger than the water monomer – the binding energy increases by 7 kcal · mol−1 relative to the Au3 –H2 O complex. As shown in Figure 5, the anchoring bond in Au3 -H2 O2 is shorter by 0072 Å than in Au3 -H2 O which likely facilitates a further electron donation to the gold cluster, which plays the


229 H6 0.962 O2 –0.615

0.410

0.989

H5 1.736 162.8

2.258

0.962 O7 –0.525

179.3

H9

0.976

Au1

H8 0.232

0.297 164.1 2.601

Au4 –0.131

2.789

2.832

2.591

Au3 –0.210

Figure 5 The Au3 –H2 O2 complex. The distance rO2 -O7 = 2697 Å. Mulliken charges are indicated in italics. The energy of the Au3 –H2 O2 complex amounts to −560870257 hartree and the ZPVE to 3160 kcal · mol−1 . The PES of Au3 -H2 O2 also includes the complex H2 O–Au3 –H2 O which is less stable

role of an electron acceptor. It is also shorter by 0015 Å the anchoring Au-F bond in Au3 -HF2 [56d] (see Figure 6). The Mulliken population analysis indicates a significant loss of the electron charge on the bridging hydron H5+041 qM H5 = +011e that is involved in the intramolecular hydrogen bonding in the water dimer and a partial charge transfer of −008 e to O2 and of −026 e to the lower-edge gold atoms Au3 and Au4 through the Au-Au bonds. Less charge is transferred via the route going through the O2 -H5 · · · O7 hydrogen bond and the O7 -H8 Au3 contact qM O7 = −002 e and qM H8 = −003 e, it is donated to Au3 and Au4 . The net charge incomes to the latter atoms amount to qM Au4 = −013 and qM Au3 = −015 e. Summarizing, the total redistribution of the electron charge within the entire complex Au3 -H2 O2 is a consequence of two concomitant bonding effects (the MO picture of bonding is presented in Figure 7). One of them is a familiar gold–oxygen anchoring that apparently dominates and governs the major part of the charge transfer. The other is the H-bonding that involves the O7 -H8 group and the gold atom, Au3 . Its existence is evidently predetermined by the anchoring.6 This bonding, operating through the contact O7 - H8 · · · Au3 , causes multiple and marked changes in the properties of the water dimer which we summarize in Table 3 (see also Table 4). Among them are an elongation of the O7 -H8 bond of the water dimer, RO7 -H8 = 0014 Å, and a red shift of the corresponding stretch, O7 -H8 = −254 cm−1 . In addition, the IR activity of the latter is enhanced by a factor of ≈ 8. Moreover, the NMR chemical shift of the hydrogen atom H8 that bridges the water

230


Figure 6 The Au3 –HF2 complex


HOMO-1 α ε = –6.859 eV HOMO β ε = –6.673 eV

Figure 7 The MO picture of bonding in the complex Au3 -H2 O2

dimer and gold cluster downshifts by iso H7 = −31 ppm. A similar downshift takes place for the bridging hydron in the water dimer (see Table 2). We also notice that the contact distance, rH7 · · · Au3 = 2591 Å, in this bonding obeys the van der Waals cut-off criterion (iv). Therefore, all the necessary and sufficient conditions (i)–(vi) of the definition of a conventional hydrogen bond (Section 2) are satisfied for the bond O7 -H8 · · · Au3 . Comparing Tables 2 and 3, we conclude that the latter resembles a weak conventional hydrogen bond where the gold atom, Au3 , acts as a (“nonconventional”) proton acceptor, through its lone-pair-like 5d±2 and 6s orbitals (see Figure 7), to the conventional proton donor O7 -H8 . We relate this weak hydrogen bond to the class of


231

Table 3 Selected features of the most stable complexes Au3 –H2 O2 and Au3 –HF2 (Figures 5 and 6) relevant to prove that conditions (ii)–(vi) are satisfied Au3 –H2 O2 376 D

Au3 –HF2 141 D

∠O7 H8 Au3 = 1641

∠F6 H7 Au3 = 1683

(iii)

RO7 -H8 = 0014 Å

RF6 -H7 = 0037 Å

(iv)

rH8 · · · Au3 = 2591 Å

Condition (ii)

rH7 · · · Au3 = 2250 Å

−1

(v)

O7 -H8 = −254 cm AIR O7 -H8 · · · Au3 /AIR O7 -H8 = 79

F6 -H7 = −844 cm−1 AIR F6 -H7 · · · Au3 /AIR F6 -H7 = 122

(vi)

iso H8 = −31 ppm iso O7 = −157 ppm iso Au3 = −250 ppm an H8 = 127 ppm an O7 = 172 ppm an Au3 = −403 ppm

iso H7 = −44 ppm iso F6 = −589 ppm iso Au3 = 186 ppm an H7 = 183 ppm an F6 = 489 ppm an Au3 = −395 ppm

A-H is the shift of the stretching mode A-H taken with respect to the corresponding monomer and AIR stands for the IR activity. iso and an are taken with respect to the corresponding monomers.

Table 4 The key features of the nonconventional H-bond O-H · · · Au of the most stable complexes Au3 −H2 O2≤n≤6 and Au4 −H2 On=26 . RO-H O-H, and iso H (in ppm) are defined with respect to the corresponding monomer Complex Dipole/ Figure

RO-H RO-H H

rH · · · Au O-H AIR O-H

iso H iso H

Au3 −H2 O2 3.76 D Figure 5

RO7 -H8 = 0976 Å RO7 -H8 = 0014 Å H = 1641

2591 Å

O7 -H8 = 3562 cm−1 AIR = 661 km · mol−1 − O7 -H8 = 254 cm−1 = 79

iso H8 = 275 iso H8 = −31

Au4 − H2 O2 3.19 D Figure 16

RO7 -H8 = 0979 Å RO7 -H8 = 0017 Å H = 1676

2536 Å


iso H8 = 264 iso H8 = −42

Au3 − H2 O3 3.57 D Figure 9

RO10 -H11 = 0979 Å RO10 -H11 = 0004 Å H = 1728

2452 Å

O10 -H11 = 3504 cm−1

iso H11 = 277 AIR = 754 km · mol−1 iso H11 = +10 − O10 -H11 = 385 cm−1 = 157

Au3 − H2 OI4 3.36 D Figure 10

RO13 -H14 = 0976 Å 2453 Å RO13 -H14 = −0007 Å H = 1736

O13 -H14 = 3558 cm−1

iso H14 = 272 AIR = 427 km · mol−1 iso H14 = +24 − O13 -H14 = 325 cm−1 = 48

Au3 − H2 OII4 1.85 D Figure 10

RO10 -H11 = 0974 Å 2541 Å RO10 -H11 = −0009 Å H = 1545

O13 -H14 = 3611 cm−1

iso H14 = 285 AIR = 470 km · mol−1 iso H14 = +37 − O13 -H14 = 270 cm−1 = 53 (Continued)

232


Table 4 Continued Complex Dipole/ RO-H RO-H H Figure

rH · · · Au O-H AIR O-H

iso H iso H

RO10 -H11 = 0970 Å 2989 Å RO10 −H11 = −0013 Å H = 1695

O10 -H11 = 3680 cm−1

iso H11 = 286 AIR = 322 km · mol−1 iso H11 = +38 − O10 −H11 = 201 cm−1 = 36

RO13 -H14 = 0971 Å RO13 -H14 = −0012 Å H = 1630

2818 Å


Au3 − H2 OIV 4 4.10 D Figure 11

RO13 -H14 = 0967 Å RO13 -H14 = −0016 Å H = 1489

3013 Å

Au3 − H2 OV4 1.68 D Figure 12

RO7 −H8 = 0971 Å 2774 Å RO13 −H14 = −0012 Å H = 1614

O7 −H8 = 3666 cm−1 AIR = 378 km · mol−1 − O7 −H8 = 215 cm−1 = 42

RO13 −H14 = 0974 Å 2556 Å RO13 −H14 = −0009 Å H = 1744

O13 −H14 = 3603 cm−1

iso H14 = 279 AIR = 429 km · mol−1 −30 − O13 −H14 = 278 cm−1 = 48

RO10 −H11 = 0974 Å 2589 Å RO10 −H11 = −0014 Å H = 1747

O10 −H11 = 3602 cm−1 AIR = 724 km · mol−1 − O10 −H11 = 285 cm−1 = 102 O16 -H17 = 3608 cm−1 AIR = 339 km · mol−1 − O13 -H14 = 279 cm−1 = 48

iso H11 = 282 iso H11 = −27

Au3 − H2 OIII 4 3.58 D Figure 11

Au3 − H2 OI5 2.38 D Figure 13

iso H14 = 286 iso H14 = +38

O13 -H14 = 3749 cm−1

iso H14 = 287 AIR = 345 km · mol−1 iso H14 = +39 − O13 −H14 = 132 cm−1 = 39

iso H8 = 286 iso H8 = −23

RO16 -H17 = 0974 Å RO16 -H17 = −0014 Å H = 1726

2604 Å

RO16 -H17 = 0970 Å RO10 -H11 = −0010 Å H = 1594

2756 Å


iso H17 = 289 iso H17 = −20

ring

RO7 -H8 = 0966 Å RO7 -H8 = −0006 Å H = 1489

3076 Å


iso H8 = 289 iso H8 = −20

cagel

RO13 -H14 = 0974 Å RO13 -H14 = −0014 Å H = 1774

2499 Å


iso H14 = 279 iso H14 = −30

ring

RO7 -H8 = 0972 Å RO7 -H8 = −0012 Å H = 1733

2687 Å


iso H8 = 270 iso H8 = −39

Au3 − H2 OII5 3.91 D Figure 13 Au3 − H2 O6 4.28 D Figure 14 Au3 − H2 O6 4.25 D Figure 15

Au4 − H2 O6 4.95 D Figure 17

iso H17 = 284 iso H17 = −25

The highest asymmetric O-H stretch of the cyclic water cluster is taken as the reference mode. ≡ AIR O-H · · · Au/AIR O-H.


233

“nonconventional”7 ones, by analogy with the others formed between small gold clusters and formamide and formic acid [56c], DNA bases [56a] and DNA duplexes [56b], alanine [56e], and hydrogen fluoride clusters [56d]. Structurally, the oxygen atom O7 in the nonconventional H-bond O7 -H8 · · · Au3 in the complex Au3 –H2 O2 is also taking part in the conventional H-bond, O2 -H5 · · · O7 , in such a manner that the oxygen atom O7 donates the hydron H8+023 to the gold atom Au3 and simultaneously accepts the hydron H5+041 from the anchored donor atom O2 . The proton donor–acceptor bifunctionality of O7 explains the difference mentioned above in its Mulliken charge, i.e., qM O7 = −002e. The gross additional charge, qM H5 = +011e, that H5 gains under the Au1 – O2 anchoring and the accompanied activation of the O2 -H5 donor group attract more strongly the proton acceptor O7 . As a consequence, the H-bond O2 -H5 · · · O7 is strengthened: RO2 -H5 = 002 Å O2 -H5 = −375 cm−1 iso H5 = −48 ppm rH5 · · · O7 = 1736 Å rO2 · · · O7 = 2697 Å rH5 · · · O7 = −0221 Å, and rO2 · · · O7 = −0222 Å (see Table 5). Formally, this H-bond can thus be characterized as belonging to the category of moderate hydrogen bonds. A similar strengthening of the intramolecular conventional F2 -H5 F6 H-bond also occurs in the complex Au3 –HF2 (Figure 6), although the strengthening mechanism is somewhat different from that of Au3 –H2 O2 : a weaker Au1 -F2 anchoring RAu1 -F2 = 2414 Å and a lower activation of the donor F2 –H5 group are compensated by a larger proton-acceptor ability of the fluorine atom relative to the oxygen. To further pursue the comparison between the complexes Au3 –H2 O2 and Au3 –HF2 , we focus on their nonconventional hydrogen bonds O7 -H8 · · · Au3 (Figure 5) and F6 -H7 · · · Au3 (Figure 6). According to Table 3, the latter is certainly a much stronger nonconventional hydrogen bond and, by all features (ii)–(vi), belongs to the moderateto-strong hydrogen bond (see Table1).8 The stronger character of the F6 -H7 · · · Au3 bond relative to the O7 -H8 · · · Au3 one is partly accounted for by a larger proton acceptor ability of the fluorine atom than that of the oxygen. Although this is a well-known fact [73,76], we nevertheless make some comparison between the conventional intramolecular hydrogen bonds that are formed within the H2 O and HF dimers in these complexes. When Table 5 The key features of the activated H-bond Au1 -O2 -H5 · · · O7 of the some stable complexes Au3 −H2 O2≤n≤6 and Au4 −H2 O6 Complex

RO2 -H5 , H5 rH5 · · · O7 rO2 · · · O7 O2 −H5 AIR O2 −H5 iso H5

Au3 − H2 O2 Au3 − H2 O3 Au3 − H2 O4 I Au3 − H2 O4 II Au3 − H2 O4 IV Au3 − H2 O5 II Au3 − H2 O6 ring Au4 − H2 O6 ring

0989 1000 1002 1004 1019 1011 1024 1039

1628 1776 1787 1778 1719 1778 1728 1765

1736 1627 1612 1602 1555 1573 1526 1481

2697 2627 2614 2606 2568 2584 2545 2519

3332 (412) 3097 (1098) 3056 (1652) 3028 (1394) 2762 (1080) 2902 (2183) 2678 (1535) 2439 (1264)

941 1096 1104 1112 1229 1164 1235 1290

233 216 217 211 191 205 185 171

The structures of these complexes are reported in Figures 5, 6, and 8–10. RO-H, O-H, and iso H (in ppm) are defined with respect to the corresponding monomer. The highest asymmetric O-H stretch of the cyclic water cluster is taken as the reference mode. O-H is the out-of-plane bending vibrational mode (in cm−1 ).

234


compared with those of the corresponding isolated dimers, the F2 -H5 and O2 -H5 bonds are elongated by 0.024 and 0020 Å, respectively. The H-bond distance in F2 -H5 · · · F6 is shorter by 0129 Å than in the O2 -H5 · · · O7 . The stretching vibrational mode F2 -H5 is shifted by 472 cm−1 to lower wave numbers, whereas the O2 -H5 by only 375 cm−1 . Therefore, the hydrogen bond F2 -H5 · · · F6 is more strongly perturbed by Au3 than the O2 -H5 · · · O7 one, despite the fact that, as noticed above, the donor group F2 -H5 is less affected by the weaker Au1 –F2 anchoring bond in Au3 -HF2 than the O2 -H5 group by the stronger Au1 –O2 anchoring in Au3 -H2 O2 . This means that the stronger perturbation of the F2 -H5 · · · F6 bond mostly arises from the larger proton-acceptor ability of the F6 atom that simultaneously weakens the donor group F6 -H7 . As a result, H7 becomes more strongly shared with the unanchored gold atom Au3 . In conclusion, our computational results suggest that two basic interactions, anchoring and nonconventional hydrogen bonding, underlie the mechanism of complexation between H2 O2 and a Au3 gold cluster. On the one hand, the former prevails and, through a charge redistribution within the entire system, strengthens the propensity of the unanchored gold atom to act as a proton acceptor via its lone-pair-like 5d±2 and 6s orbitals. This effect is demonstrated in Figure 8 where the structures of the protonated complexes Au3 H+ and Au3 H3 O+ are reported. The proton affinity PABH+ , simply defined as the ZPVE-corrected difference EBH+ − EB, serves as the well-known indicator of the proton-acceptor ability of B. The computed difference PAAu3 H3 O+I − PAAu3 H+ = 146 kcal · mol−1 provides a clear evidence that the anchoring of the water molecule at the three-gold cluster raises the proton-acceptor ability of the latter. On the other hand, the nonconventional hydrogen bonding, acting cooperatively through the Au–Au bonding within a three-gold cluster as well as through the conventional intramolecular H-bonded bridge within the water dimer, reinforces the anchoring. The nonconventional H-bond also strongly activates the adjacent O2 -H5 · · · O7 hydrogen bond within the water dimer in such a way that this, initially a weak bond in the isolated dimer, becomes the moderate one.

4. Larger gold–water complexes Au3 –H2 O3≤n≤5 The most stable water oligomers H2 O3≤n≤5 are structurally characterized by the cyclic H-bonded pattern with a simple DA motif [109]. This feature emerges when they interact with a triangular gold cluster. The resulting, low-energy complexes Au3 − H2 O3≤n≤5 that are formed under this interaction are shown in Figures 9–13. A glance across these figures allows us to classify them into three structural classes: I: open-water clusters with Au–O anchoring at the terminal oxygen atom within a given cluster of water; II: open-water clusters with Au–O anchoring at an internal oxygen atom; and III: cyclic and closed-water clusters. V Class II is represented by the structures Au3 –H2 OIII 4 (Figure 11), Au3 –H2 O4 I (Figure 12), and Au3 –H2 O5 (Figure 13) where the entire gold cluster functions as a nonconventional double proton acceptor and forms two weak nonconventional Hbonds. 9 It also strongly anchors at the O2 oxygen atom of the water chain, giving rise to the shortest anchoring Au1 -O2 bonds of 2.201, 2.204, and 2179 Å, respectively, among all the reported complexes. This strong gold–oxygen anchoring results in a more negative I Mulliken charge on O2 , e.g., of −070e in Au3 –H2 OIII 4 and −074e in Au3 -H2 O5 ,


235

Figure 8 The Au3 H+ and Au3 H3 O+ complexes. The complex Au3 H3 O+ adopts two stable structures, Au3 H3 O+I and Au3 H3 O+II . The latter, although less stable by 132 kcal · mol−1 than the former, is rather interesting because the Au3 cluster forms an ionic nonconventional hydrogen bond with the water monomer. The stretch mode Au-H+ is equal to 20717 km · mol−1 20914 km · mol−1 , and 1626 cm−1 177 km · mol−1 in the complexes Au3 H+ Au3 H3 O+I , and Au3 H3 O+II , respectively. The other stretches sym O5 -H67 and asym O5 -H67 amount to 3763 (184) and 3857 cm−1 189 km · mol−1 in Au3 H3 O+I and 3712 (196) and 3815 cm−1 168 km · mol−1 in Au3 H3 O+II . The proton affinity of the Au3 gold cluster is equal to PAAu3 H+ = 18769 kcal · mol−1 PAAu3 H3 O+I = 20234 kcal · mol−1

compared, for instance, to the corresponding one of −062e in the complex of Au3 with water dimer. The adjacent water molecule O2 H5 H6 acts as a double proton donor in mediating of the closed water chain. V The Au3 –H2 OIII 4 and Au3 –H2 O4 isomers are rather different from each other. In the former, the two lower-edged gold atoms act as single proton acceptors, whereas in the latter, one only of the unanchored gold atom is a double proton acceptor (interestingly, the Au4 -Au5 bond length is the same, 2819 Å, in both clusters). A marginal energy difference of −055 kcal · mol−1 between these two structures shows a small preference for a single atom within the triangular gold cluster to function as a double proton acceptor. This preference appears in a clearer way upon comparison of the nonconventional

236

Small gold clusters form nonconventional hydrogen bonds H9

H6 0.962

O2 –0.654

1.000

1.627

H5

177.6

0.437

O7 0.961 –0.624

110.6

0.985 H8 0.364

2.228

175.5 1.734

163.0

Au1 110.9

0.323

O10 –0.529

0.962 H12

0.979 172.8 2.631

H11

2.757

0.244

2.452 2.797

Au4

Au3

–0.147

–0.224

Figure 9 The most stable Au3 –H2 O3 complex. The distance rO2 -O7 = 2627 Å. Mulliken charges are indicated in italics. The energy of the Au3 –H2 O3 complex amounts to −637350529 hartree and its ZPVE to 4722 kcal · mol−1 H9

0.961

Au3–(H2O)4I

O7

0.962 1.002 H6 H –0.634 178.7 5 O2 0.373 H8 0.429 –0.638 178.4

0.962

0.989

–0.624 H12

H6 O2

–0.637

2.219

0.983

H11 0.366

1.602

1.004 H5 0.434

2.217

Au1 0.311

Au1 0.318

158.6

1.756

2.751

–0.531 O13

2.637

0.962

2.453 H14 0.247

Au3 –0.219

2.800

Au4 –0.169

–0.585 O10 0.978 H11 164.7 0.346

H12

2.634

1.845 O13 0.231 0.974 –0.508 H14 0.962

2.736

2.541 Au4 –0.137

0.991 H8 0.393 175.4 1.699

0.965

0.976 173.6

H15

O7

177.8 –0.649

1.698

O10 0.960

H9 0.960

Au3–(H2O)4II

1.612

2.786

Au3 –0.251

H15

154.5

Figure 10 The complexes Au3 –H2 OI4 and Au3 –H2 OII4 . Mulliken charges are indicated in italics. The total electronic energy E (in hartree), ZPVE (in kcal · mol−1 ), enthalpy H (in hartree), and entropy S (in cal · mol−1 · T−1 ) of the complexes Au3 − H2 O4I–V are summarized in the following table:

Au3 –H2 OI4 Au3 –H2 OII4 Au3 –H2 OIII 4 Au3 –H2 OIV 4 Au3 –H2 OV4

E + 713

ZPVE + 60

H + 713

−0826014 −0824251 −0824043 −0822789 −0825144

280 280 262 344 276

−0708390 −0706631 −0706245 −0704728 −0707347

S 15061 15070 15089 14583 14745


237

Au3–(H2O)4III 0.961 O7

Au3–(H2O)4IV

H9

0.408 1.721 –0.609 H5 0.401 0.980 –0.703 169.7 H6 H8 0.343 O2 0.988 H15 0.962

172.4

1.834 159.2 0.980 O13 –0.509

1.814

2.201

0.971 H14 163.0 0.226

0.299 Au1

0.970 O10 –0.492 H11 H12 0.962 0.209 169.5

H12

0.330 H11

–0.558 O13 0.961 H9 1.827 0.971 H80.316 165.5 160.7 1.964

0.978 0.960 O 10 –0.590

H6 0.963 1.019 O2 –0.710 165.1 1.759 H 0.985 171.9 5 H15 O7 1.555 0.446 2.269 0.368 –0.610 0.967

H14 0.255

Au1 0.267

148.9 2.818

2.773

–0.202 Au3

2.616 2.819

2.989

Au4 –0.157

3.013

2.601

2.781 Au3 –0.137

2.834

Au4 –0.210

IV Figure 11 The complexes Au3 –H2 OIII 4 and Au3 –H2 O4 . Mulliken charges are given in italics

Figure 12 The complex Au3 –H2 OV4 with a nonconventional double proton-acceptor character of the gold atom Au3

H-bonded patterns of these two structures (see Table 4): the O7 -H8 · · · Au3 and O13 -H14 · · · Au3 H-bond of Au3 -H2 OV4 have shorter H-separations (2.774 and 2556 Å, respectively) compared to those in Au3 -H2 OIII 4 (i.e., 2.818 and 2989 Å), and their stretching vibrational modes O7 -H8 ) and O13 -H14 ) are shifted lower (at most by 66 cm−1 ) than the corresponding ones in Au3 –H2 OIII 4 . It is worth noticing that the nonconventional hydrogen bonds O13 -H14 · · · Au3 in Au3 –H2 OV4 O10 -H11 · · · Au4 and O16 -H17 · · · Au3 in Au3 -H2 OI5 are weaker than in Au3 -H2 O2 (cf. Tables 3 and 4). Furthermore, as indicated by the bond lengths (viz., rAu1 -Au4 = 2616 Å and V rAu1 -Au3 = 2773 Å in Au3 -H2 OIII 4 , by rAu1 -Au3 = 2799 Å in Au3 -H2 O4 , and

238


Au3–(H2O)5I

0.961

H15

–0.606 O13

1.718

0.980 H14 0.350

174.3

–0.613 0.405 1.711 O7 H5

0.439 H6

171.1

H9 0.961

–0.738 172.7 O2 0.988 0.989

H18

0.974

1.789

2.698

2.252 –0.683

0.974 H11 H12 0.219 174.7

2.788 Au3

0.960 –0.611 O 13 H 15 0.983 0.307 H 18

173.7

0.968

2.609

H 14

1.758 0.355

2.068

O16 –0.566

0.970

Au 1

2.589

2.604 –0.195

175.7 1.723

H5 0.439 1.011 O2

0.326 Au1 2.689

172.6

H6 0.962

O10 –0.510

H17 0.237

1.573

177.8

0.352

2.179

O16 –0.518

–0.636 0.961 H 9

0.981

174.2

1.792 0.962

Au3–(H2O)5II

0.960 H 12 0.395 1.690 O10 –0.621 H8 0.992 H 11 175.2 0.987 O7 0.361

H 17

0.274 2.774 2.756

Au4 –0.186 –0.143 Au 4

2.821 Au 3 –0.209

Figure 13 Two most stable Au3 –H2 O5 complexes. Mulliken charges are indicated in italics. The total electronic energy E (in hartree), ZPVE (in kcal · mol−1 ), enthalpy H (in hartree), and entropy S (in cal · mol−1 · T−1 ) of the complexes Au3 –H2 OIII 5 are summarized in the following table:

Au3 –H2 OI5 Au3 –H2 OII5

E + 790

ZPVE + 78

H + 790

−0304023 −0300380

027 097

−0158604 −0154644

S 16574 16204

also by rAu1 -Au3 = 2689 Å and rAu1 -Au4 = 2698 Å in Au3 -H2 OI5 , a strengthening of the proton-acceptor ability of the unanchored gold atom weakens its bonding with the anchor one. Class I includes the structures Au3 –H2 O3 (Figure 9), Au3 –H2 OI4 , and Au3 –H2 OII4 (Figure 10). The binding energy of the former structure, taken relative to the triangular gold cluster and the cyclic water trimer, amounts to 195 kcal · mol−1 , that is 3 33 kcal · mol−1 larger that of Au3 –H2 O2 . This increment Eb ≡ Eb Au3 -H2 O3 − Eb Au3 -H2 O2 is partially related to a strengthening of almost all bonds of Au3 -H2 O3 . In particular, the nonconventional hydrogen bonding in Au3 -H2 O3 is stronger than in Au3 -H2 O2 , which is markedly indicated by a substantially shorter H-bond separation (2.452 vs. 2591 Å, respectively), a slightly more elongated O-H bond 0979 vs 0976 Å), and a larger red shift (385 vs 254 cm−1 ; see Table 4). Another representative of class I, the complex Au3 –H2 OI4 is the global minimum on the potential energy surface of Au3 –H2 O4 , although, as follows from the legend to Figure 10, the latter PES is rather flat. For instance, the complex Au3 –H2 OIII 4 , which belongs to class II, lies only 106 kcal · mol−1 higher Au3 –H2 OI4 . The binding 4 energy Eb Au3 -H2 OI4 = 166 kcal · mol−1 and results in a negative difference Eb = −29 kcal · mol−1 , that is, the complex Au3 –H2 O3 is more stable than Au3 –H2 OI4 . In contrast, Eb Au3 -H2 OI5 = 179 kcal · mol−1 , i.e., slightly larger by 07 kcal · mol−1 5 than Eb Au3 -H2 O2 , and hence Eb = 13 kcal · mol−1 . The main features of Au3 –H2 O3 Au3 –H2 OI4 , and Au3 –H2 OII4 include (see Table 4): (a) the gold-anchored oxygen atom O2 acts as a single H-donor. The anchoring bond is longer than that in Au3 -H2 OIII 4 by ≈ 002 Å; (b) the repulsion between O2 and the hydron H5 increases with n (RO2 -H5 = 0989 n = 2 1000 n = 3,


239

and 1002 Å n = 4) and a shorter moderate–strong hydrogen bond is formed with the neighboring water molecule (rH5 · · · O7 = 1736 n = 2 1627 n = 3, and 1612 Å n = 4); (c) the nonconventional O13 -H14 · · · Au3 hydrogen bonding in the complex Au3 -H2 OI4 is stronger than in Au3 -H2 OV4 and features as a weak–moderate one. II Two less stable structures Au3 –H2 OIV 4 (Figure 11) and Au3 –H2 O5 (Figure 13) IV belong to class III. The energy of Au3 –H2 O4 is higher than that of Au3 –H2 OI4 by 266 kcal · mol−1 , whereas the energy of Au3 –H2 OII5 is by 299 kcal · mol−1 higher than that of Au3 –H2 OI5 . The nonconventional O-H · · · Au H-bonds formed by the cyclic water oligomers, H2 O4 and H2 O5 , respectively are much weaker than those of the global minimum structures Au3 -H2 OI4 and Au3 -H2 OI5 (the H-bond distance rH · · · Au in Au3 -H2 OIV 4 is even beyond the van der Waals cut-off by 0015 Å; see however the remark in the condition (iv)). On the other hand, both complexes II Au3 –H2 OIV 4 and Au3 –H2 O5 exhibit a substantially elongated O2 -H5 bond reaching 1.019 and 1011 Å, respectively, and a short H-bond O2 -H5 · · · O of 1.555 and 1573 Å. This implies that the anchoring Au1−O2 bond perturbs so strongly the adjacent water molecule that the weak intramolecular hydrogen bond that is formed within the isolated cyclic water oligomer is transformed in the Au3 –cyclic water oligomer complex into a strong, mostly ionic H-bond.

5. Gold–water hexamer complexes Water hexamer is a very particular cluster among oligomers of water molecules because its 2D conformers, such as boat, chair, or ring, coexist at the bottom of the PES of H2 O6 with the 3D ones, prism and cage [110]. A similar behavior occurs when a gold cluster prism Au3 interacts with the water hexamer: the complexes Au3 –H2 Oring 6 Au3 –H2 O6 cageI−II (Figure 14), and Au3 –H2 O6 (Figure 15) are present at the bottom of the PES of Au3 –H2 O6 and lie in energy rather close to each other, within the interisomer, with val of 378 kcal · mol−1 at 0 K.10 Among them, the Au3 –H2 OcageI 6 ring −1 = 183 kcal · mol has the lowest energy, below the Au Eb Au3 -H2 OcageI 3 –H2 O6 6 prism −1 −1 by 215 kcal · mol and below the Au3 –H2 O6 by 378 kcal · mol . Another goldlies 310 kcal · mol−1 higher the Au3 -H2 OcageI . cage structure Au3 –H2 OcageII 6 6 isomer is due We suggest that the slight preferential stability of the Au3 –H2 OcageI 6 from the fact that, among all four complexes, it is the only one having a nonconventional hydrogen bond O19 -H21 · · · Au3 .11 This bond is rather strong and characterized by a comparatively short H-bond of 2499 Å (see Table 4 for the associated red shift of O19 -H21 and downfield shift of the iso H21 ). Moreover, its formation entirely destroys the cage pattern of the initial water hexamer. In this respect, the Au3 –H2 O6 cageI isomer is different from the other three complexes Au3 –H2 O6 whose corresponding hexamer structures are largely preserved. In parallel, as discussed above in the case of the cyclic Au3 –H2 O23 complexes, the nonconventional H-bond formation reinforces the anchoring Au1 –O2 bond, which contracts to 2217 Å and therefore belongs to the category of the shortest ones (compare with Au3 –H2 O4 II and Au3 –H2 O4 III ). However, the O2 -H5 bond with RO2 -H5 = 1006 Å adjacent to Au1 -O2 is less activated in Au3 –H2 O6 cageI than in Au3 –H2 O6 ring RO2 -H5 = 1024 Å and O2 -H5 = 2678 cm−1 ) where it is much weaker, with rAu1 -O2 = 2265 Å.

240

Small gold clusters form nonconventional hydrogen bonds 0.975

Au3–(H2O)6 prism H9 H12 0.960 Au3–(H2O)6 ring O10 0.980 1.751 H14 1.780 H11 O13 179.5 179.1 1.728 H9 0.973 0.983 178.8 H8 1.686 H15 H17 0.961 178.6 H6 H18 0.985 0.960 O2 0.990 H5 1.526 1.686 O16 1.024 172.8 O19 H 178.8 20 2.265 0.966 Au1

2.830 Au4

0.960

O10 1.746

H11

O7 0.970

152.7

H14 H15 0.967 0.960 H12 0.988 O13

H8

1.681 2.106 164.5 1.973 0.973 H20 H6 2.003H 18 0.974 H17 O 0.994 O2 19 0.961 H 1.754 0.988 5 159.0 H21 2.267 156.3

148.9 H 21 2.777

2.603

1.875

Au1

3.076

2.758

2.616

Au3 Au4 ring

2.819 Au3

prism

Figure 14 The complexes Au3 –H2 O6 and Au3 –H2 O6 . The total electronic energy E (in hartree), ZPVE (in kcal · mol−1 ), enthalpy H (in hartree), and entropy S (in cal · mol−1 · T−1 ) of the complexes Au3 -H2 O6 are summarized in the following table:

ring

Au3 –H2 O6 prism Au3 –H2 O6 cagel Au3 –H2 O6 cageII Au3 –H2 O6

E + 886

ZPVE + 94

H + 886

−0775009 −0773790 −0779689 −0775069

057 144 136 156

−0601396 −0599110 −0605621 −0600305

S 18030 17910 17118 17558

Physically, the weaker activation of the O2 -H5 bond in Au3 –H2 O6 cageI than in Au3 –H2 O6 ring partly stems out from a different proton functionality of the corresponding water molecule O2 H5 H6 in both complexes. In the former complex, it only donates the proton H5 to the cage hexamer. In the latter one, the O2 H5 H6 also functions as a proton acceptor forming a rather “nice,” almost linear, and quite short H-bond O7 -H8 · · · O2 and is also shared between two H-bonds. This causes a natural weakening of the O2 -H5 bond.

6. Enlarging gold clusters toward lower coordination Is the propensity of gold to participate, as proton acceptor, in forming a nonconventional hydrogen bond discussed above a feature of three-gold clusters only? Or do larger gold clusters behave alike? In this regard, let us recall that, as mentioned in Introduction, recent studies [24,26,27] unambiguously demonstrate that lower-coordinated gold atoms determine the exceptional reactivity of gold NPs. Is this conclusion—the less the gold atom is coordinated within a given cluster, the stronger its nonconventional protonacceptor ability—also valid for the proton-acceptor ability of gold? In Sections 3-5, we only dealt with the triangular cluster of gold where each atom is two-coordinated.


241

Figure 15 The complexes Au3 –H2 O6 cageI and Au3 –H2 O6 cageII between a triangular gold cluster and the cage water hexamer

The smallest and stable gold cluster that possesses a single-coordinated atom is the T-shape Au4 cluster [111]. How does it interact with water oligomers, particularly with water dimer and hexamer? Figures 16 and 17 display the complexes Au4 –H2 O2 and Au4 –H2 O6 ring . A binding energy Eb Au4 –H2 O2 = 223 kcal · mol−1 , larger than Eb Au3 –H2 O2 , indicates that the complex Au4 –H2 O2 is more strongly bound than Au3 –H2 O2 . This stronger bonding is manifested in the following features (see Table 4 and Figure 16 in particular): (a) a strengthening of the Au1 –O2 anchoring bond. The difference in the anchoring bond length between Au4 –H2 O2 and Au3 –H2 O2 , RAu1 -O2 , is equal to −0027 Å; (b) a higher activation of the adjacent O2 -H5 bond RO2 -H5 = 0009 Å and a strengthening of the intramolecular dimeric hydrogen bond (rH5 · · · O7 = −0071 Å and RO2 -O7 = −0038 Å) implying that the latter is actually a strong hydrogen bond. Relative to the water dimer, the stretching vibrational mode is downshifted by O2 -H5 = 537 cm−1 ; (c) a strengthening of the nonconventional hydrogen bond O7 -H8 · · · Au3 (rH8 · · · Au3 = −0055 Å and RO7 -H8 = 0003 Å, O7 -H8 = −44 cm−1 ). The complex Au4 –H2 O6 ring (see Figure 17 and Table 4) is also more strongly bound than Au3 –H2 O6 ring : Eb Au4 – H2 O6 ring = 179 kcal · mol−1 . The most noticeable changes in the bonding characteristics of Au4 –H2 O6 ring with respect to Au3 –H2 O6 ring take place for the O2 -H5 bond that undergoes an enormous elongation to 1039 Å (see Table 5) and for the nonconventional one O19 -H21 · · · Au3 . In contrast to that in Au3 –H2 O6 ring where it is so weak that it is hardly considered as a hydrogen bond, the O19 -H21 · · · Au3 bond in Au4 –H2 O6 ring nicely obeys the van der Waals cut-off condition (iv) (see Table 4). Relative to the ring water hexamer, the formation of the O19 -H21 · · · Au3 H-bond is accompanied by an elongation of the O19 -H21 bond by 0012 Å

242


Figure 16 The complex Au4 –H2 O2 . Its total electronic energy amounts to −696892443 hartree and its ZPVE to 3216 kcal · mol−1

Figure 17 The complex Au4 –H2 O6 ring characterized by a total electronic energy of −1002794330 hartree and a ZPVE equal to 9478 kcal · mol−1


243

and red shift of its stretch by 252 cm−1 . To conclude, the two Au4 complexes shown in Figures 16 and 17 respectively definitely demonstrate that the single-coordinated gold atom of the T-shape Au4 cluster is a stronger nonconventional proton acceptor with water oligomers than the two-coordinated one of Au3 .

7. Summary: gold clusters as nonconventional proton acceptors To recapitulate what we have discussed so far, we emphasize that three major bonding effects underlie the interaction between the gold clusters Au3 and Au4 and water oligomers. They are: the anchor Au-O bonding, the nonconventional O-H · · · Au hydrogen bonding, and the activation of the adjacent O-H bond. We provide a computational evidence that the formation of the donor–acceptor coordinative (anchor) bond Au-O in the studied complexes is a necessary prerequisite (but not sufficient) to the formation of the nonconventional hydrogen bond and to the activation of the adjacent O-H bond, and therefore this bond dominates the interaction mechanism. One intriguing question emerging from the present study is how the gold cluster acquires its proton-acceptor ability? We suggest the following mechanism. Under the formation of the anchoring gold–oxygen bond, the oxygen atom donates electron charge to the LUMO of the gold cluster, thereby inducing a through-bond charge transfer over the gold cluster. As a result, additional negative charges accumulate on the unanchored lower-edged gold atoms. This charge transfer makes the lone-pair-like 5d±2 and 6s orbitals of the unanchored gold atoms available to the proton-donor groups of the water oligomer. It is cooperatively balanced with another charge transfer that occurs through the conventional water–water intramolecular H-bonded bridges. Altogether, the charge reorganization results in a total flow of charge from the proton-donor group to the acceptor gold atom, leading to the nonconventional O-H · · · Au hydrogen bonding. When this bond is formed, it donates backward to the anchoring one that is thereby reinforced. A common point of view on the hydrogen bond conditions (iii)–(v) is that they may provide an estimation of the strength of the hydrogen bond. The latter is typically expressed in terms of the energy EHB of the H-bond formation (see Section 2). This is not however the case here since the anchoring is a prerequisite for the nonconventional hydrogen bonding. Nevertheless, a reasonable upper bound to EHB can be roughly estimated as EAu3 -H2 O2 − EAu3 -H2 O + EH2 O2 ≈ −4 –5 kcal · mol−1 . This value is consistent with the early ones obtained for the complex between Au3 and formic acid [57]c. Physically, the mechanism of the third major bonding effect, i.e., the activation of the adjacent O-H bond in the complexes Au3≤m≤4 -H2 O2≤n≤6 is the following. While the anchoring gold–oxygen bond is formed, the lobe of the LUMO of the gold cluster Au3 or Au4 (see Figure 3) deeply protrudes into the interior of the anchored water molecule and markedly activates the adjacent O2 -H56 groups. The activation of the O2 -H5 bond weakens the H-bond within the water oligomer and enhances their ability to share their hydrons with the neighboring water molecules. As a result, they acquire all features inherent to the moderate-to-strong hydrogen bonds. This is demonstrated in Table 5. If one of these water molecules is also involved in the nonconventional hydrogen bond with the unanchored gold atom, the nonconventional H-bond reinforces the intramolecular

244


hydrogen one and the latter becomes practically ionic, as in the case of the complex where the elongation of the O2 -H5 bond reaches 1.039 Åand the O2 -O7 Au4 –H2 Oring 6 distance shortens to 2.519 Å. We assume that the activation of the O-H bond adjacent to the anchoring gold–oxygen one takes place for those gold clusters whose LUMOs are sufficiently protruded. This is, for example, the case of the Au8 gold cluster of D4h symmetry (see [100] and references therein) which interacts with the ring water hexamer (see Figure 18). The formation of the anchoring gold–oxygen bond, characterized by the length of 2.341 Å, activates the adjacent O2 -H5 bond which elongates to 1.008 Å, while its stretching mode is downshifted by 350 cm−1 . To conclude, we anticipate that the estimates of the red shifts of the O-H stretching vibrational modes and of the energetics of the formation of the Au–O anchor and nonconventional O-H · · · Au hydrogen bondings might be relevant for controlling the complexation of water clusters on gold particles. We also suggest that the nonconventional hydrogen bonding that occur when water clusters are adsorbed on gold

ring

Figure 18 The complex Au8 –H2 O6 and a ZPVE equal to 9605 kcal · mol−1

has a total electronic energy of −1548201103 hartree


245

surfaces could be characterized by IR spectroscopy, which therefore makes this bonding particularly useful as a recognition pattern in a surface catalysis involving gold particles.

Acknowledgments This work was supported by the Region Wallonne (RW. 115012). The computational facilities were provided by NIC (University of Liége) and by F.R.F.C. 9.4545.03F (Belgium). E.S.K. thanks F.R.F.C. 2.4562.03F (Belgium) for a fellowship. We gratefully thank Profs. Alfred Karpfen, Pekka Pyykkö, and Camille Sandorfy for interesting discussions and Prof. Alejandro Toro-Labbé for his kind invitation to contribute to this book “Theoretical Aspects of Chemical Reactivity.”

Notes 1

Directeur de Recherches, FNRS, Belgium.

2

Some alchemists even considered gold as “condensed sunbeams” [1] and used to represent it by the symbol of a circle, the hallmark of mathematical perfection. Let us recall in this respect, e.g., the “Golden Section” in geometry, the concept of a “gold number” in colloidal chemistry after R. A. Zsigmondy (Z. Anal. Chem. XL, 697 (1901)) and J. B. Richter and M. Faraday (Phil. Trans. CXLVII, 145 (1857), and the “Golden Rule” in quantum theory.

3

Actually, the idea of a weak specific interaction involving the hydrogen atom goes back to Nernst [65] who discussed a dimeric association of molecules with hydroxyl groups in 1891 and Werner [66] who introduced 10 years later the concept of a minor valence (“Nebenvalenz”) as a proper description of hydrogen bonding (see also the related works by Oddo and Puxeddu [67] and Pfeiffer [68]). The hydrogen bond was apparently quoted for the first time by Lewis [69] in 1923 and later by Bernal and Megaw [70], and Huggins [71].

4

Actually, this definition relies on the definitions of a local bond and a proton donor. It also partly resembles that given by Pauling [64] (“ under certain conditions an atom of hydrogen is attracted by rather strong forces to two atoms, instead of only one, so that it may be considered to be acting as a bond between them. This is called the hydrogen bond.”), by Lippert [93] (“A H-bond exists if one hydrogen atom H is bonded to more than one other atom, for instance, to two atoms named X and Y.”), and by Perrin and Nielson [94] (“A hydrogen bond is an attractive interaction between a hydron donor A-H and hydron acceptor B.”).

5

Notice that the complex Au+ -H2 O has a shorter (by 004–025 Å) Au-O bond that is also indicated by its larger binding energy equal to 25–40 kcal · mol−1 (a variance depends on the computational level, see [57]).

6

Such bonding motif of a prevailing role of the anchoring contrasts with that established for the Au3 –HF complex where both bonding interactions may separately exist and originate two different conformers, either with a Au-F anchoring or with a F-H · · · Au contact—for the latter being slightly stronger bonded [56d].

7

In fact, as noticed by Hobza and Havlas [105], “the published definitions of the H- bond are not unambiguous and many exist”. Nowadays, the concept of a hydrogen bond is much broader than it was expected nearly a century ago [80,82], and that is why it permits, together with the conventional H-bonds, an existence of a wide class of those, say “nonconventional” ones, identified during the last three decades experimentally and theoretically, which do not partially

246


satisfy either the traditional view on the proton donor and proton acceptors or the conditions (i)–(vi). It might be worth to revisit the latter and do not treat them in particular as the necessary and sufficient conditions. A class of “nonconventional” hydrogen bonds particularly include the A-H · · · M ones, formed between conventional hydrogen bond donors and electron-rich transition metals M, such as Co, Ru, Rh, Os, Ir, and Pt functioning as weak “nonconventional” proton acceptors [106–108]. 8

This is the first example reported so far of a very strong, nearly ionic nonconventional hydrogen bond formed between a metal and a conventional proton donor.

9

The proof of this statement is straightforward and is therefore omitted. The interesting reader may proceed by analogy with the proof for the complex Au3 –H2 O2 .

10

By analogy with the work on water hexamers [110g], the order of stability of the lower-energy ring prism cage clusters Au3 –H2 O6 Au3 –H2 O6 , and Au3 –H2 O6 changes with T due to large entropy cageII effects (see Table 3 in [110g]). The complex Au3 –H2 O6 characterized by a large total dipole moment of 664 D can prevail in polar solvents.

11

An extremely weak nonconventional hydrogen bond likely exists in the complex Au3 –H2 O6 (Figure 14), although the van der Waals cut-off criterion (iv) is not fully obeyed in this case.

ring

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Small gold clusters form nonconventional hydrogen bonds and Physics, edited by J.-P. Julien, J. Maruani, D. Mayou, S. Wilson, and G. Delgado-Barrio (Springer, Dordrecht, 2006). pp. 433–450. (a) J. Hrusák, D. Scröder, and H. Schwarz, Chem. Phys. Lett. 225, 416 (1994). (b) H. Hertwig, J. Hrusák, D. Scröder, W. Kock, and H. Schwarz, Ibid. 236, 194 (1995). (c) D. Feller, E. D. Glendening, and W. A. de Jong, J. Chem. Phys. 110, 1475 (1999). (d) H. M. Lee, S. K. Min, E.C. Lee, J.-H. Min, S. Odde, and K. S. Kim, Ibid. 122, 064314 (2005). L. Poisson, F. Lepetit, J.-M. Mestdagh, and J.-P. Visticot, J. Phys. Chem. A 106, 5455 (2002). (a) R. S. Smith, C. Huang, E. K. L. Wong, and B. D. Kay, Surf. Sci. 367, L13 (1996). (b) P. Löfgren, P. Ahlström, D. V. Chakarov, J. Lausmaa, and B. Kasemo, Ibid. 367, L19 (1996). (c) G. Pirug and H. P. Bonzel, Ibid. 405, 87 (1998). (d) F. Boccuzzi and A. Chiorino, J. Phys. Chem. B 104, 5414 (2000). (e) S. Meng, E. G. Wang, and S. Gao, J. Chem. Phys. 119, 7617 (2003). (f) M. E. Abdelsalam, P. N. Bartlett, T. Kelf, and J. Baumberg, Langmuir 21, 1753 (2005). (g) S. Wang, H. Yao, S. Sato, and K. Kimura, J. Am. Chem. Soc. 126, 7438 (2004). (a) A. Michaelides, V. A. Ranea, P. L. de Andres, and D. A. King, Phys. Rev. Lett. 90, 216102 (2003). (b) S. Meng, E. G. Wang, and S. Gao, J. Chem. Phys. 119, 7617 (2003). (c) S. Meng, E. G. Wang, and S. Gao, Phys. Rev. B 69, 195404 (2004). (d) S.-P. Ju, J.-G. Chang, J.-S. Lin, and Y.-S. Lin, J. Chem. Phys. 122, 154707 (2005). T. S. Moore and T. F. Winmill, J. Chem. Soc. 101, 1635 (1912). M. Huggins, Thesis (University of California, 1919). See also: C. M. Huggins, Angew. Chem., Int. Ed. Engl. 10, 147 (1971). W. M. Latimer and W. H. Rodebush, J. Am. Chem. Soc. 42, 1419 (1920). L. Pauling, The Nature of the Chemical Bond (Cornell University Press, Ithaca, 1939). See also L. Pauling, Proc. Natl. Acad. Sci. U.S.A. 14, 359 (1928). W. Nernst, Z. Phys. Chem. 8, 110 (1891). A. Werner, Liebigs Ann. Chem. 322, 261 (1902). G. Oddo, E. Puxeddu, Gazz. (ii) 36, 1 (1906). P. Pfeiffer, Ann. 398, 137 (1913). G. N. Lewis, Valence and Structure of Atoms and Molecules (Chemical Catalog, New York, 1923). Ch. 12. J. D. Bernal, H. D. Megaw, Proc. Roy. Soc. London A 151, 384 (1935). M. L. Huggins, J. Org. Chem. 1, 407 (1936). D. Hadži and H. W. Thompson (Eds.), Hydrogen Bonding (Pergamon Press, London, 1959). C. G. Pimentel and A. L. McClellan, The Hydrogen Bond (Freeman, San Francisco, 1960). W. C. Hamilton and J. A. Ibers, Hydrogen Bonding in Solids (Benjamin, New York, 1968). P. A. Kollman and L. C. Allen, Chem. Rev. 72, 283 (1972). P. Schuster, G. Zundel, and C. Sandorfy (Eds.), The Hydrogen Bond. Recent Developments in Theory and Experiments (North-Holland, Amsterdam, 1976). G. A. Jeffrey and W. Saenger, Hydrogen Bonding in Biological Structures, 2nd edition (Springer, Berlin, 1994). G. A. Jeffrey, An Introduction to Hydrogen Bonding (Oxford University Press, Oxford, 1997). S. Scheiner, Hydrogen Bonding. A Theoretical Perspective (Oxford University Press, Oxford, 1997). G. R. Desiraju and T. Steiner, The Weak Hydrogen Bond in Structural Chemistry and Biology (Oxford University Press, Oxford, 1999). P. Schuster, in Intermolecular Interactions: From Diatomics to Biopolymers, edited by B. Pullman (Wiley, Chichester, 1978). p. 363. T. Steiner, Angew. Chem. Int. Ed. 41, 48 (2002). T. Steiner and G. R. Desiraju, Chem. Commun. 891 (1998). M. S. Gordon and J. H. Jensen, Acc. Chem. Res. 29, 536 (1996).


249

85. C. Sandorfy, Top. Curr. Chem. 120, 41 (1984). 86. S. Scheiner, in Pauling’s Legacy – Modern Modelling of the Chemical Bond, Vol. 6, edited by Z. B. Maksic and W. J. Orville-Thomas (Elsevier, Amsterdam, 1977). p. 571. 87. (a) J. E. Del Bene, in The Encyclopedia of Computational Chemistry, edited by P. v. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III, and P. R. Schreiner (Wiley, Chichester, 1998). Vol. 2, p. 1263; (b) J. E. Del Bene and M. J. T. Jordan, Int. Rev. Phys. Chem. 18, 119 (1999); (c) J. E. Del Bene, in Recent Theoretical and Experimental Advances in Hydrogen Bonded Clusters, NATO ASI Series C, Vol. 561, edited by S. S. Xantheas (Kluwer, Dordrecht, 2000). p. 309. (d) E. S. Kryachko, in Hydrogen Bonding-New Insights, edited by S. Grabowski (Springer, Dordrecht, 2006). pp. 293–336. 88. E. Kryachko and S. Scheiner, J. Phys. Chem. A 108, 2527 (2004). 89. (a) J. F. Hinton and K. Wolinski, in Theoretical Treatments of Hydrogen Bonding, edited by D. Hadži (Wiley, Chichester, 1997). p. 75; (b) E. D. Becker, in Encyclopedia of Nuclear Magnetic Resonance, edited by D. M. Grant and R. K. Harris (Wiley, New York, 1996). p. 2409. 90. P. Schuster, In Ref. [76], Vol. I. Theory, Ch. 2. 91. I. G. Kaplan, Theory of Molecular Interactions. Studies in Physical and Theoretical Chemistry, Vol. 42 (Elsevier, Amsterdam, 1986). 92. M. Meot-Ner (Mautner), Chem. Rev. 105, 213 (2005). 93. E. Lippert, In Ref. [76], Vol. I. Theory. Ch. 1. 94. C. L. Perrin and J. B. Nielson, Annu. Rev. Phys. Chem. 48, 511 (1997). 95. S. Scheiner, Annu. Rev. Phys. Chem. 45, 23 (1994). 96. All computations reported in the present work were conducted with the GAUSSIAN 03 package of quantum chemical programs [97]. The Kohn–Sham self-consistent field formalism with the hybrid density functional B3LYP potential was used together with relativistic effective core potential (RECP) for gold and the basis sets 6-311 + +G2d 2p for oxygen and hydrogen. The energy-consistent 19- 5s2 5p6 5d10 6s1 valence electron RECP for gold atoms developed by Ermler, Christiansen, and co-workers with the primitive basis set 5s5p4d [98] was used. For a recent review on small gold clusters see Ref. [99] and also Ref. [100] for the 2D–3D coexistence in small neutral and charged gold clusters. The main features of the triangular Au3 cluster are gathered in note [101]. All geometrical optimizations were performed with the keywords “tight” and “Int=UltraFine”. The unscaled harmonic vibrational frequencies, zero-point vibrational energies (ZPVE), and enthalpies were also calculated. The binding energy Eb AB of the complex AB is defined as the energy difference, Eb AB ≡ EAB − EA + EB. The ZPVEcorrected binding energies Eb are reported throughout this work. By analogy with Refs. [99,100], we use a so-called density functional margin for gold clusters equal approximately to 4 kcal · mol−1 . For other complexes treated in the present work, a similar computational error margin amounts to ≈ 1–2 kcal · mol−1 . 97. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, GAUSSIAN 03 (Revision A.1), Gaussian, Inc., Pittsburgh, PA, 2003.

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98. R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen, J. Chem. Phys. 93, 6654 (1990). 99. F. Remacle and E. S. Kryachko, Adv. Quant. Chem. 47, 423 (2004). 100. F. Remacle and E. S. Kryachko, J. Chem. Phys. 122, 044304 (2005). 101. Within the present computational approach, the triangular conformer of Au3 gold cluster is characterized by the electronic energy of −407907290 hartree, ZPVE = 0418 kcal · mol−1 , and enthalpy equal to −407900617 hartree. Its geometry is determined by rAu1 -Au2 = rAu2 -Au3 = 2654 Å rAu1 -Au3 = 2992 Å, and the bond angle ∠Au1 Au2 Au3 = 686 , implying a so-called “geometrical frustration” or asymmetry due to the Jahn–Teller distortion of the ground electronic state of the triangular conformation of Au3 [102]. The chain structure Auch 3 is characterized by the electronic energy of −407911124 hartree, ZPVE = 0427 kcal · mol−1 , and the enthalpy equal to −407904441 hartree. Its bond lengths rAu1 -Au2 = rAu1 -Au3 = 2619 Å and the bond angle ∠Au2 Au1 Au3 = 1152 . The chain structure is the most stable conformer of Au3 lying below the triangle structure by 24 kcal · mol−1 , after ZPVE, which is consistent with the value of 23 kcal · mol−1 recently reported by H. M. Lee, M. Ge, B. R. Sahu, P. Tarakeshwar, and K. S. Kim, J. Phys. Chem. B 107, 9994 (2003). Throughout the present work, Au3 is identified with the triangular gold cluster since as shown in Ref. [56]d, the chain cluster Auch 3 is not relevant for binding large clusters and forming nonconventional hydrogen bonds. 102. (a) G. Bravo-Pérez, I. L. Garzón, and O. Novaro, J. Mol. Struct. (Theochem) 493, 225 (1999). (b) H. Grönbeck and W. Andreoni, Chem. Phys. 262, 1 (2000). 103. (a) D. H. Wells, Jr., W. N. Delgass, and K. T. Thomson, J. Catal. 225, 69 (2004). (b) Z.-P. Liu, S. J. Jenkins, and D. A. King, Phys. Rev. Lett. 94, 196102 (2005). 104. (a) J. R. Heath and M. A. Ratner, Phys. Today 56, 43 (2003). (b) J. M. Tour, Molecular Electronics (World Scientific Publishing, Singapore, 2003). (c) C. Joachim, J. K. Gimzewski, and A. Aviram, Nature (London) 408, 541 (2000). (d) A. Nitzan, Annu. Rev. Phys. Chem. 52, 681 (2001). 105. P. Hobza and Z. Havlas, Chem. Rev. 100, 4253 (2000). 106. (a) L. Brammer, M. C. McCann, R. M. Bullock, R. K. McMullan, and P. Sherwood, Organometallics 11, 2339 (1992); (b) S. G. Kazarian, P. A. Hanley, and M. Poliakoff, J. Am. Chem. Soc. 115, 9069 (1993); (c) A. Albinati, F. Lianza, P. S. Pregosin, and B. Müller, Inorg. Chem. 33, 2522 (1994); (d) Y. Gao, O. Eisenstein, R. H. Crabtree, Inorg. Chim. Acta 254, 105 (1997); (e) L. Brammer, D. Zhao, F. T. Lapido, and J. Braddock-Wilking, Acta Crystallogr. Sect. B 53, 680 (1995); (f) D. Braga, F. Grepioni, and G. R. Desiraju, Chem. Rev. 98, 1375 (1998). 107. (a) E. S. Shubina, N. V. Belkova, and L. M. Epstein, J. Organomet. Chem. 17, 536 (1997); (b) G. Orlova and S. Scheiner, Organometallics 17, 4362 (1998); (c) L. M. Epstein and E. S. Shubina, Ber. Bunsenges. Phys. Chem. 102, 359 (1998). 108. (a) L. M. Epstein and E. S. Shubina, Coord. Chem. Rev. 231, 165 (2002); (b) L. Brammer, Dalton Trans. 3145 (2003); and references therein. 109. (a) K. S. Kim, M. Dupuis, G. C. Lie, and E. Clementi, Chem. Phys. Lett. 131, 451 (1986). (b) S. S. Xantheas, T. H. Dunning, Jr., J. Chem. Phys. 99, 8774 (1993). (c) P. N. Krishnan, J. O. Jensen, and L. A. Burke, Chem. Phys. Lett. 217, 311 (1994). 110. K. Kim, K. D. Jordan, and T. S. Zwier, J. Am. Chem. Soc. 116, 11568 (1994). (b) K. Liti, M. G. Brown, C. Carter, R. J. Saykally, J. K. Gregory, and D. C. Clary, Nature 381, 501 (1996). (c) J. K. Gregory and D. C. Clary, J. Phys. Chem. 100, 18014 (1996). (d) D. A. Estrin, L. Paglieri, G. Corongiu, and E. Clementi, Ibid. 100, 8701 (1996). (e) J. Kim and K. S. Kim, J. Chem. Phys. 109, 5886 (1998). (f) E. S. Kryachko, Int. J. Quant. Chem. 70, 831 (1998). (g) E. S. Kryachko, Chem. Phys. Lett. 314, 353 (1999). 111. The T-shape four-gold cluster Au4 C2v has the following bond lengths rAu1 -Au10 = rAu4 -Au10 = 2759 rAu1 -Au4 = 2626, and rAu3 -Au10 = 2573 Å (see Figure 13). Its electronic energy amounts to −543921072 hartree, ZPVE = 0788 kcal · mol−1 .


Chapter 12

Theoretical design of electronically stabilized molecules containing planar tetracoordinate carbons a

Alberto Vela, b Miguel A. Méndez-Rojas, and c Gabriel Merino

a

Departamento de Química, Cinvestav, A. P. 14-740 México, D.F., 07000, México, Departamento de Química y Biología, Universidad de las Américas, Puebla, Ex-Hda. de Sta. Catarina Martir, Cholula 72820, Puebla, México and c Facultad de Química, Universidad de Guanajuato. Col. Noria Alta s/n C.P. 36050, Guanajuato, Gto., México b

1. Brief historical background In the early days of chemistry, Edward Frankland suggested the idea of valence to explain the building of several organometallic compounds1 . It was the first step toward understanding the structure of organic molecules. In 1856, Couper2 and Kekule3 came to the conclusion that carbon is tetravalent. Later, van’t Hoff 4 and LeBel5 established a relationship between optical activity and the atomic spatial arregment. These proposals are milestones of organic chemistry. A tetrahedral molecule with a central carbon atom and four different ligands has two different arrangements in space which are mirror images of each other. Thus, this “simple” three-dimensional (3D) vision of molecular structure paved the way to the fundamental concept of chirality in chemistry. In 1848, Pasteur resolved the two optically active forms of tartaric acid, and a few years later he realized that the handedness of the carbon atom had a paramount importance in the biological activity of compounds. Nowadays, it is solidly established that the absolute configuration of molecules crucially determines its biological activity. During the twentieth century, new molecular species with carbon atoms in unusual environments were proposed, opening what can be called the inorganic side of carbon. For example, carbon can be surrounded by more than four ligands.6 The fascinating chemistry of hypercarbons is very nicely presented in Olah’s Nobel lecture.7 In 1970, almost a century after van’t Hoff and LeBel tetrahedral model was published, Hoffmann, Alder, and Wilcox suggested an idea, challenging the structural foundations of organic 251

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Theoretical design of electronically stabilized molecules

chemistry.8 In their now classic paper, they presented a discussion, based on the molecular orbital structure of planar methane, about the electronic requirements necessary to stabilize a planar tetracoordinate carbon (ptC) atom. Inspired on this idea, several groups have successfully predicted, and even experimentally characterized, molecules containing a ptC.9−14 It is not pretentious to say that these discoveries have opened a new age for carbon chemistry. In this chapter, we summarize our recent contributions in this fascinating quest to find new molecules containing ptCs and to provide a rationale of their stability.

2. How to confine a ptC into a molecule? Two general strategies have been used to stabilize a ptC in a molecule: one of them is purely electronic and the other is a steric enforcement of the planar orientation of the bonds, the so-called mechanical approach. Let us start with the simplest molecule containing a tetracoordinate carbon atom, methane (CH4 ). If one forces it to acquire a planar D4 h structure, the central carbon atom adopts a sp2 hybridization, with one lone pair perpendicular to the molecular plane, the a2u orbital depicted in Figure 1. By lowering the symmetry of CH4 from Td to D4 h , only six electrons occupy bonding orbitals (orbitals eu and a1 g in Figure 1), contrasting with the four bonding pairs present in Td methane. The Hoffmann–Alder–Wilcox strategy to stabilize a ptC is to include the lone pair in the bonding framework by replacing one or more hydrogen atoms with good -donor/-acceptor ligands, or by incorporating the lone pair into a 4n + 2 delocalized system.8 Several molecules containing a ptC were predicted, and even experimentally detected, using the electronic approach (Figure 2).15−21 Among them, a beautiful series of pentaatomic molecules including a ptC were proposed by Boldyrev, Schleyer, and Simons and were experimentally detected by Wang and Boldyrev.22−28 Interestingly, all these structures share a common feature—they have at least one atom, different from carbon, which is bonded to the ptC. In the mechanical approach, geometrical constraints that force the central carbon atom and its nearest neighbors to be in a plane are imposed. The bonds can be constrained by rings and cages, as has been done in the family of molecules known as alkaplanes.1329−34 However, the experimental efforts done to isolate one of these mechanically stabilized molecules containing a ptC have been unfruitful. The lack of success may be due to the fact that the p-orbital normal to the ptC plane, generally the HOMO, is strongly

Figure 1 Molecular orbitals of methane with D4h symmetry

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Li

Li C

H

H H

H

H

H B

B

Si

R1

Ga

C C

B H

C Ga

B H

Cp2Zr

C

R2

Si M X M = AlR2, GaMe2, BR2, ZrCp2

Figure 2 Selected examples of molecules containing a ptC

localized on the central carbon atom, resulting in a cage that is held together by very weak forces that prevent its stabilization and, hence, its experimental isolation. Despite all these in vitro and in silico efforts, the statement by Keese was still challenging the ptC’s research community: “ no structures with a planar tetracoordinate C(C)4 have been found ”35 Two years later, Rasmussen and Radom30 designed the first stable ptC surrounded only by carbons, [ptCC4 ], the dimethanospiro[2,2]octaplane (Figure 3).30 Wang and Schleyer found a set of boron spiroalkanes with a ptCC4 through substitution of carbon by boron atoms.3637 In 2003, a novel family of molecules based on a C2− 5 moiety was proposed by us, which constituted the simplest and smaller set of molecules containing a ptCC4 , and more importantly, the first and, at that time, the only one stabilized purely by electronic factors.38

Figure 3 Alkaplane proposed by Rasmussen and Radom30

254


n− 2.1. C2− 5 and C5 M2−n

Obviously, the smallest possible cluster containing a CC4 skeleton is C5 . Experimental and theoretical studies showed that the global minimum of the neutral species is the linear structure, and there was no evidence of planar isomers of C5 containing a ptC.3940 We considered the possibility of extracting or adding electrons to this moiety and to search stable isomers on the potential energy surface (PES), aiming to find a C5 n cluster containing a ptC.3841 –2

+2 C

C

C

C

C

C

C

2

C

+2e–

–2e–

C

C

C

C

C

1

C

C

3

After removing two electrons one finds 2 as a stationary point with two imaginary frequencies. On the other hand, and to our pleasant surprise, the addition of two electrons to 1, generates the local minimum 3 (C5 2− ) having the desired ptCC4 atom. The D2h -optimized geometry of C5 2− is shown in Figure 4. Note that the largest C-C bond length (150.9 pm) is comparable with those calculated for dimethanospiro[2.2]octaplane (150.4 pm)14 and several [4.4.4.5]fenestrene derivatives (149.3–152.9 pm).42 The experimental observation of C5 2− strongly depends on the topography of the PES that drives the dynamics of the cluster and thus, its mean lifetime. To gain a better idea about the stability or metastability of the parental skeleton C5 2− , an extensive exploration of its PES was performed (see Figure 5). Among all the stationary points located on the PES, only four were local minima, including 3. Two of them (1A and 1B) are lower in energy than 3 (202.4 and 1848 kJ mol−1 , respectively), whereas the 3D D3 h structure, 3C, is 1795 kJ mol−1 higher in energy. Structure 1D is the transition state involved in the rearrangement that isomerizes 1B into 1, and vice versa. It should be

Figure 4 B3LYP/6-311++G(2d)-optimized geometry of the parental skeleton C2− 5 .

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255

126.7 124.9

129.3

136.7 169.7

128.9

152.5

140.0 149.0 173.0

152.9 136.5

142.6

142.0

139.8

140.0

143.9 1A, C2v NIMAG = 0 ΔE = –202.4 kJ mol–1

1B, C2v NIMAG = 0 ΔE = –184.8 kJ mol–1

1C, D3h NIMAG = 0 ΔE = 179.5 kJ mol–1

1D, C1 NIMAG = 1 ΔE = 12.6 kJ mol–1

128.6

168.8 105.6

127.3 135.6

141.7

131.6

147.1

88.7

145.1 127.7

137.0

157.4

130.0 1E, C1 NIMAG = 1 ΔE = 511.7 kJ mol–1

157.8

124.9

169.5 142.2

148.5

180.8

1F, D∞h NIMAG = 1 ΔE = –176.1 kJ mol–1

1G, D3h NIMAG = 1 ΔE = 181.6 kJ mol–1

1H, Cs NIMAG = 1 ΔE = 92.7 kJ mol–1

Figure 5 Stationary points of the C5 2− skeleton calculated with B3LYP/6-311++G(2d). NIMAG is the number of imaginary harmonic frequencies obtained for each structure using the same methodology, and E is the energy difference of the corresponding structure minus that of 3, including the scaled ZPE (0.9806). All distances are in picometers and angles in degrees

noted that structure 1D is not planar and lies very close in energy to 1 126 kJ mol−1 . Structures 1E-1H have one imaginary frequency, and they are not connected to the ptC-containing molecule. The MP2 and CCSD(T) calculations of the local minimum 3 confirm the stability of the dianion, supporting the results obtained with the hybrid exchange-correlation density functional. is a local minimum? Consider a neutral spiropentadiene as a starting Why C2− 5 is structure (Figure 6). After removing four protons from this latter compound, C4− 5 obtained. This process generates four lone pairs, one on each terminal carbon and pointing outward, roughly in the direction where the hydrogen nuclei were located in the original structure. Now, after rotating one cyclopropenyl ring against the other, all five carbon atoms are in the same plane, and one ends up with the planar C4− 5 . The central -type lone pair, a characteristic feature of square planar carbon systems, for which there are will be surely destabilized. Vacating this MO will result in C2− 5

256


H

H –4H+

–2e–

On rotation

H

H

C54–

C4H4

planar C54–

planar C52–

Figure 6 Simplified picture of the construction of the MOs of C5 2− from C5 4− derived from C5 H4

four lone pairs pointing outward and a -system of two occupied orbitals. A careful shows that the four localized pairs of 1 transform examination of the MOs for C2− 5 as ag + b1u + b2u + b3g . Their delocalized equivalents are to be found in 1b2u beg 2ag , and 2b1u (Figure 7). Two very low-lying MOs (ag and b1u ) with mainly s character C42–

C C52–(D2h)

–8

b1g b1g

b2u

2b1u

–10 b1u ag

–12

.

p

b1u

2ag

2b2u

b2g b3u b2g

ag –14

1b1u

b3u –16 b3g b3g b2u –18

1ag

1b2u

–20

s

Figure 7 Correlation diagram between the fragment orbitals of the four “outside” carbon atoms and the central ptC. The bold and thin lines distinguish between filled and empty molecular orbitals

A. Vela et al.

257

are omitted in the diagram. The destabilization of 2ag and 2b1u (mostly bonding) over 1b2u and b3g (mostly antibonding) is due to the interaction with the set of MOs. Since these radial MOs mix with the tangential MOs (1ag 1b1u , and 2b2u ) due to their common symmetry, the orientation of the canonical orbitals differs slightly from those shown schematically in Figure 7. The p-orbitals perpendicular to the plane, which are involved in the formation of MOs, are shown as circles, symbolizing their “top” (above the plane) phase. The two MOs are identified as b2g and b2g in Figure 7. The corresponding antibonding combination lies above the LUMO b1g . Note that the porbital of the central carbon atom is the lowest-lying p-orbital of the carbon framework, covering completely the CC4 skeleton and contributing to the double-bond character of the C2 -C2 bond, which turns out to be the fundamental reason to understand the stability of the C2− 5 structure. It is worth noting that the HOMO and four other occupied MOs of 1 have positive energies, a situation that has been also found in other dianions such as CAl2− 4 4344 and CB2− To have any chance of experimentally detecting this structure, it is 6 . mandatory to stabilize these orbitals. One possibility is to introduce positive charges into the parent dianion. This idea was used to detect CAl2− 4 , which after “dressing” it with a counter cation, the resulting NaCAl− 4 anion was studied by photoelectron spectroscopy. In the present case, we explored the influence of adding several metals Li+ Na+ K+ Cu2+ Be2+ Mg2+ Ca2+ , and Zn2+ to the parental structure C2− 5 (Table 1 and Figure 8). Interestingly, the addition of the cations does not destroy the planarity of the molecules. The bare C2− 5 anion, 1, and 3 have D2h symmetry, while 2 belongs to a C2 point group. The harmonic analysis of these molecular structures reveals that they are local minima. As it can be seen in Table 1, adding one counterion with a formal charge of +1M = Li, Na, K, and Cu) does not change the planarity of Table 1 Selected bond lengths (in picometers), angles (in degrees), and smallest frequencies, Freq (in cm−1 ) of 1–3.

1 2

3

M

n

C1 -C2

C1 -C2

C2 -C2

C2 -M

C2 -C1 -C2

C2 -C1 -C2

Freq

– Li Na K Cu Be Mg Ca Zn Li Na K Cu Be Mg Ca Zn

2− 1− 1− 1− 1− 0 0 0 0 0 0 0 0 2 2 2 2

1509 1517 1525 1526 1508 1547 1553 1542 1559 1501 1500 1506 1489 1485 1482 1499 1461

– 1506 1487 1490 1503 1492 1461 1482 1449 – – – – – – – –

1334 1330 1328 1328 1330 1328 1321 1329 1316 1328 1327 1327 1334 1333 1326 1323 1325

– 1930 2305 2614 2.002 1624 2053 2250 2036 2006 2351 2668 2078 1770 2154 2397 2141

1275 1324 1307 1282 1418 1413 1368 1296 1420 1275 1275 1277 1268 1266 1269 1276 1261

– 1231 1247 1273 1134 1151 1195 1263 1146 – – – – – – – –

1683 2188 1574 1200 2015 2354 1869 1743 1617 1723 817 529 856 1954 1314 1038 654

258


Figure 8 Schematic representation of 2 and 3

the structure. The geometrical deformations are more pronounced when one dication is incorporated in the structure. The extreme situation corresponds to C5 Zn (see Table 1), where the C1 -C2 distance increases 5.0 pm and the C1 -C2 distance decreases 6.0 pm. The inclusion of two metal cations produces a reduction of the C1 -C2 bond lengths from 03C5 K2 to 48 pmC5 Zn22 . Furthermore, the C2 -C2 bond lengths remain practically unchanged after the insertion of one or two metal ions. In order to gain further insight into the bonding mechanism prevailing in these systems, we perform a topological analysis of the electron density45 and the electron localization function,46 as well as a study of their magnetic response to an applied external magnetic field given by the locally induced magnetic field, Bind r.47 Any interested reader on this analysis may consult directly our work.41 From this study, we find that the interaction of the parental C2− 5 skeleton with the alkaline and alkaline earth atoms is basically ionic, with a remarkable transferability of properties from the isolated dianion to the C5 M2 salts. The study of the magnetic response shows that, indeed, the electron delocalization plays a very significant role in stabilizing these ptC-containing compounds. All the theoretical evidence indicate that C5 Li2 is a good candidate for experimental isolation. This molecule has several interesting properties: it is planar, like the parent C5 2− anion, it is the hardest, and one of the most diatropic. However, the experimental observation of C5 Li2 also depends on the possible rearrangements to more stable isomers. For C5 2− , the rearrangement barrier is only 126 kJ mol−1 , including the zero-point energy correction, but the inclusion of two lithium cations increases this barrier to 583 kJ mol−1 (Figure 9). Furthermore, the energy difference between the planar structure of C5 2− and the closest isomer is −1840 kJ mol−1 , while for C5 Li2 , this value is appreciably smaller −174 kJ mol−1 . Therefore, the isomerization barrier in C5 Li2 is sufficiently high to support our optimism that this molecule can be experimentally detected. Is it possible to build C5 Li2 polymers? The design of a fragment containing two C5 2− units (a dimer) would require including four monocations keeping the electroneutrality of the system. The relative stabilities of the monomers give an idea on how one can arrange the metal ions around the C5 unit (Figure 10). The most stable isomer in the C5 Li2 series was 4, suggesting two alternative arrangements for a C10 Li− 3 dimer, one where both C5 2− units are coplanar, 5, and another, where the units are perpendicular to each other, 7 (Figure 10). Structure 7 is a minimum on the PES, while 5 has two imaginary frequencies. In the optimized structure of C10 Li4 , 6, two of the Li+ ions occupy positions between the terminal carbons which are not linked to each other. To gain further insight into the preferred position of the metal, a trimer C15 Li2− 6 , 8, was

A. Vela et al.

259 C

100

C

80

Li C

40 Li

20

C C

Li Li

C

0

E, kJ mol–1

C

C C

C

Li

C C

60

2–

C

–20

C C

C

–40

C

Li

C C

C

2–

C

C

–60

C C

–80

C

–100 –120

2–

C

–140

C

–160

C C

–180

C

–200 –5

–4

–3

–2

–1

0

1

2

3

4

5

6

7

8

9

10

Reaction coordinate

Figure 9 Energy profiles for the isomerization of C5 2− and C5 Li2 . The origin of both reaction paths is the corresponding transition state

also constructed. Both 7 and 8 are minima and continue the bonding motif found for C5 Li2+ 4 , 9. These results provided clues for designing extended systems in 2D and 3D containing a ptC.

2.2. ptCs in extended systems48 The idea of having an extended network based on structures containing a ptC, surrounded by diverse elements, has been explored in the past by several authors (Figure 11). One of the first works in this field was again from the mind behind the idea of ptCs. Merschrod, Tang, and Hoffmann built an unusual nickel carbide network, Ni3 C5 2− , with repetitive units CNi4 4− , featuring infinite 1D vertex-sharing chains on Ni squares. Each square was centered by a carbon and flanked by C2 units. The orbital interaction schemes revealed that there is little Ni-Ni bonding and essentially no Ni to C2 backdonation. However, in this case, the tetrahedral alternative is favored over the planar building block.49 Later, Li et al. discussed the possibility of designing new materials containing pentaatomic ptC species as building blocks for bulk solid materials, based on CAl4 2− . Their findings pointed out that bulk solid materials with the composition 26 M+ 2 CAl2− 4 may be prepared. Geske and Boldyrev performed ab initio calculations on the Na2 CAl4 2 dimer in order to test if the two CAl4 2− groups react to form a more stable dimeric structure, or if the two CAl2− 4 groups remain separated in a true dimeric structure. They established that structures with the C-C bond are higher in energy than the structures with two isolated structural CAl− 4 units separated by more than 5 Å, with their structural and electronic integrity preserved. However, alternative structures 4− involving reaction between two CAl2− 4 groups forming a C2 Al8 cluster without the C-C bond are higher in energy, but they are still competitive with the true dimeric structure.

260

Theoretical design of electronically stabilized molecules 132.8 150.1

133.2 200.6

196.0

233.8

150.8 4 NIMAG = 0 132.4 198.2

5 NIMAG = 2 133.0

203.4

148.3

151.6

195.7

150.8

149.2

7 NIMAG = 0

6 NIMAG = 0 135.0

221.8

202.4

132.0

154.7 150.6

152.0

202.0

148.9 201.9

134.5 8 NIMAG = 0

202.4 9 NIMAG = 0

2− Figure 10 Optimized structures of C5 Li2 , C10 Li− 3 , C10 Li4 , and C15 Li6

While they found alternative structures of Na4 C2 Al8 with the energy comparable to that of the true dimer, they claimed that a solid ionic salt with the pentaatomic tetracoordinate planar carbon CAl4 2− building block may have good chances to be synthesized, making it the first solid containing a pentaatomic ptC.50 None of these attempts were successful enough to stabilize a ptC surrounded only by carbon atoms. Based on our previous calculations on the stability of dimeric and trimeric structures, different 1D and 2D networks were built. The stable dimers 6 and 7, as well as structure 5, lend themselves to extension to 1D chains of C5 units. These polymeric chains are depicted in Figure 12 as I, II, and III. Since the ratio of a C5 unit to the metal in a unit cell of I and III is 1:1, a divalent metal ion is needed to compensate for the −2 charge on C5 . In the case of II, the 1:2 C5 to metal ratio requires a singly charged cation like Li+ . If Li+ is replaced by a divalent tetracoordinate metal ion, II can be converted into a 2D network IV, as shown in Figure 12. Structure I requires a tetrahedral coordination around the metal and a divalent metal ion (Zn2+ or Be2+ , for example, as these ions have been found before in similar systems such as ZnCN2 n and CBe2 n ). Structure II may use Li+ ions, meanwhile Pt 2+ may be the right choice for structures III and IV, as it often exists in a square planar arrangement. In fact, the theoretical analysis showed that compounds I-Zn2+ and II-Li+ were stable, having a tetragonal and orthorhombic unit cells, respectively (Figure 13),

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261 Ni

Ni

C Ni

Ni Ni

C

Ni Ni

C

Ni

Ni Ni

Ni

Al

Ni

Al

Al Na

C

Al Na

Al

Al Na

C Al

Al

C

Na Al

Na

Al

Al

Al

Al C

Na

Ni

Al

Al

C

Al

Al

Al C

Al

Al

Figure 11 Several extended network based on structures containing a ptC

but not I-Be2+ , which even failed to converge. The optimized lattice parameters of the different C5 Mx n systems studied are given in Table 2. Calculations of the band structure and total density of states (DOS) for C5 Zn n and C5 Li2 n show large band gaps, suggesting that these solids have semiconducting or insulating behavior. The same situation happens for all other extended network systems. For C5 Pt n , it was found that III in Figure 14 was preferred over IV, which agrees with the relative stabilities of the isomeric forms of C5 Li2+ 4 . Thus, this study shows that C2− 5 is a building block that generates interesting crystal structures having ptCs and semiconducting isolating properties.

2.3. ptCs in cyclic hydrocarbons51 As the electronic strategy to stabilize a ptC derived from the C2− 5 unit was shown to be successful, the next logic step was to start building around this unit a hydrocarbon skeleton that could enhance its stability. The main goal in this point is to avoid the isomerization process which is energetically favored toward finding linear carbon chains. Very recently, we started to explore a series of cyclic hydrocarbons containing a ptCC4 .51 Candidates were obtained by combining the parental C2− 5 anion with an unsaturated fragment: two hydrogens from ethane and 1,3-dibutadiene were removed, providing the corresponding dications which interact with C2− 5 , yielding the five- and seven-membered ring systems C7 H2 , 10, and C9 H4 , 12, respectively. The same strategy was used to build the six- and eight-membered anionic rings C8 H3− , 11, and C10 H5− , 13, from allyl and pentadienyl anions. Their structures are depicted in Figure 15. The harmonic analysis shows that all of them are local minima on their respective PES with an appreciable positive lowest vibrational frequencies >100 cm−1 .

262

Theoretical design of electronically stabilized molecules M M

M M

M

M

M M M

M

M M

M

M M

M M

M M

M

(II)

(III)

(I)

M

M

M

M

(IV)

Figure 12 A schematic diagram for a 1D (I, II, III) and a 2D (IV) pattern of C2− 5 units bridged by appropriate metal ions. The repeating units are shown by dashed lines

I

II

Figure 13 Optimized lattice of (I) C5 Zn and (II) C5 Li2

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263

Table 2 Cell parameters (a, b, c in Å) and V (volume in Å3 ) of the different C5 Mx systems are given System C5 ZnI C5 Li2 C5 Pt C5 ZnV C5 Be C5 ZnVII C5 Ni

III

a

b

c

V

990 533 384 476 445 378 382

990 991 446 476 445 378 382

933 997 522 914 882 1053 1044

91514 52680 8932 20656 17483 15046 15204

IV

Figure 14 The 3D unit cells of the III and IV forms of C5 Pt are shown

Figure 15 Optimized structures of cyclic hydrocarbons containing a ptC. Bond lengths are given in picometers

As it was mentioned before, the experimental detection of any of these molecules depends on their mean lifetimes. To have an estimate of these lifetimes, a set of Born– Oppenheimer molecular dynamic (BO–MD) trajectories were done. As one can see in Table 3, and as expected, ring-opening is accompanied by a small activation barriers, in the range of 10–40 kJ mol−1 . As it can be seen in Figure 16, a typical MD trajectory of 13, which has the lowest activation barrier, the planar structure preserves its geometry for about 6.0 ps at 300K. This is a very short lifetime but, considering that it corresponds to the hydrocarbon with the smallest activation energy, one can be optimistic that the lifetimes of 10–12 can be large enough for its experimental detection.

264

Theoretical design of electronically stabilized molecules Table 3 Smallest frequencies (Freq) in cm−1 , energy difference between the ptC molecule and its closest isomer E, and its corresponding activation energy, Ea .

Freq E Ea

10

11

12

2463 −424 380

1690 −289 231

1020 −521 307

13 1501 −1107 107

All energies are in kJ mol−1 .

0.03

0.02

ΔEpot

0.01

0

–0.01

–0.02

–0.03 0

2

4

6

8

10

Time

Figure 16 BO–MD simulation of molecule 13. Relative potential energy Epot (in a.u.) vs. time (in ps)

The MOs of these molecules that contribute more to their stability are shown in Figure 17. Clearly, the lowest-lying -orbitals (b1 symmetry) have an important contribution from the central -type lone pair, and it is essentially distributed over the entire CC4 skeleton. The HOMOs of the neutral species 10 and 12 are in-plane lone pairs of the carbon atoms labeled as C3 . In contrast, in the anions 11 and 13, the HOMOs are -orbitals. The number of electrons in each molecule is six in 10, eight in 11 and 12, and ten in 13. It should be noted that cyclic hydrocarbons containing 4n + 2 electrons (10 and 13) preserve the C2− 5 fragment almost intact. We have also explored the electron delocalization and magnetic response of these molecules. All have strong diatropic contributions inside the three-membered rings that constitute the C5 skeleton. For the 4n + 2-electron species (10 and 13), the main cycles of the molecules have an aromatic response, while for the 8-electron cycles (11 and 12), they show an antiaromatic response. Isosurfaces of the z-component of Bind Bind z of the molecules are shown in Figure 18. The aromatic molecules have a strongly shielding region close to the carbons inside the ring, which resembles the form

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Figure 17 HOMOs and -MOs = 005 au for 10–13

Figure 18 Isosurfaces of the z-component of the induced magnetic field, Bind z . Bind z = 90 T and Bext = 10 T, perpendicular to the molecular plane. Blue and red indicate shielding Bind z < 0 and deshielding areas, respectively

of the carbon -orbitals. The antiaromatic molecules show a deshielding cone outside the ring, with the carbon atoms just inside the deshielding region. We are currently exploring the electronic, magnetic, and structural characteristics of several new members of this cyclic hydrocarbon family. Symmetric and asymmetric systems, with rings of different size at each side, are showing an increased stability due to the size and symmetry of these hydrocarbons.

3. Future perspectives on ptCs derived from C2− 5 (and its experimental detection) The most important messages that we have learned from our studies about ptC-containing molecules are that the delocalization of the lone pair located in the central carbon atom and the size of the energy barriers that prevent the isomerization of the molecule

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are among the most important factors to be considered in the molecular engineering of these species. Taking into account these crucial factors, we are presently studying systems where these rearrangements are decreased, or even better, stopped completely. By learning these theoretical “know-hows”, we are optimistic that we will see in vitro, and in our lifetimes, the experimental detection of some of these in silico designed molecules that challenge one of the paradigms of organic chemistry.

Acknowledgments We are grateful to many colleagues for their collaboration in this project: Thomas Heine, Gotthard Seifert, and Robert Barthel (Technische Universität Dresden), Pattath D. Pancharatna (Cornell University), Clemence Corminboueuf (University of Geneva), Hiram I. Beltrán (Instituto Mexicano del Petróleo), and Nancy Perez (Universidad de las Américas). We owe special thanks to Prof. Roald Hoffmann for his thorough and thoughtful inspiration of this work. We gratefully acknowledge the support from CONACYT (Projects G34037-E and G32710-E), Dutsche Forschungsgemeinschaft (DFG), and Decanatura de Investigación y Posgrado (DIP-UDLA).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Frankland, E., Phil. Trans. Roy. Soc. 1852, 142, 417. Couper, A. S., Ann. Chem. 1859, 110, 46. Kekule, A., Bull. Soc. Chim. Fr. 1965, 3, 98. van’t Hoff, J. H., Arch. Neerl. Sci. Exactes Nat. 1874, 445. LeBel, J. A., Bull. Soc. Chim. Fr. 1874, 22. Tal’roze, V. L.; Lyubimova, A. K., J. Mass Spectr. 1998, 33, 502. http://nobelprize.org/chemistry/laureates/1994/. Hoffmann, R.; Alder, R. W.; Wilcox, C. F., J. Am. Chem. Soc. 1970, 92, 4992. Siebert, W.; Gunale, A., Chem. Soc. Rev. 1999, 28, 367. Erker, G., Chem. Soc. Rev. 1999, 28, 307. Collins, J. B.; Dill, J. D.; Jemmis, E. D.; Apeloig, Y.; Schleyer, P. v. R.; Seeger, R.; Pople, J. A., J. Am. Chem. Soc. 1976, 98, 5419. Erker, G.; Rottger, D., Angew. Chem. Int. Edit. Engl. 1993, 32, 1623. Radom, L.; Rasmussen, D. R., Pure Appl. Chem. 1998, 70, 1977. Sorger, K.; Schleyer, P. v. R., Theochem-J. Mol. Struct. 1995, 338, 317. Priyakumar, U. D.; Reddy, A. S.; Sastry, G. N., Tetrahedron Lett. 2004, 45, 2495. Priyakumar, U. D.; Sastry, G. N., Tetrahedron Lett. 2004, 45, 1515. Li, S.-D.; Ren, G.-M.; Miao, C.-Q.; Jin, Z.-H., Angew. Chem. Int. Edit. 2004, 43, 1371. Sahin, Y.; Prasang, C.; Hofmann, M.; Subramanian, G.; Geiseler, G.; Massa, W.; Berndt, A., Angew. Chem. Int. Edit. 2003, 42, 671. Minyaev, R. M.; Minkin, V. I.; Gribanova, T. N.; Starikov, A. G.; Hoffmann, R., J. Org. Chem. 2003, 68, 8588. Huang, J. H.; Luci, J. J.; Lee, T. Y.; Swenson, D. C.; Jensen, J. H.; Messerle, L., J. Am. Chem. Soc. 2003, 125, 1688. Minkin, V. I.; Minyaev, R. M.; Hoffmann, R., Uspekhi Khimii 2002, 71, 989. Schleyer, P. v. R.; Boldyrev, A. I., J. Chem. Soc. Chem. Commun. 1991, 1536. Boldyrev, A. I.; Simons, J., J. Am. Chem. Soc. 1998, 120, 7967.

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24. Li, X.; Wang, L. S.; Boldyrev, A. I.; Simons, J., J. Am. Chem. Soc. 1999, 121, 6033. 25. Boldyrev, A. I.; Li, X.; Wang, L. S., Angew. Chem. Int. Edit. 2000, 39, 3307. 26. Li, X.; Zhang, H. F.; Wang, L. S.; Geske, G. D.; Boldyrev, A. I., Angew. Chem. Int. Edit. 2000, 39, 3630. 27. Wang, L. S.; Boldyrev, A. I.; Li, X.; Simons, J., J. Am. Chem. Soc. 2000, 122, 7681. 28. Boldyrev, A. I.; Wang, L. S., J. Phys. Chem. A 2001, 105, 10759. 29. McGrath, M. P.; Radom, L., J. Am. Chem. Soc. 1993, 115, 3320. 30. Rasmussen, D. R.; Radom, L., Angew. Chem. Int. Edit. 1999, 38, 2876. 31. Dodziuk, H., J. Mol. Struct. 1990, 239, 167. 32. Dodziuk, H.; Leszczynski, J.; Nowinski, K. S., J. Org. Chem. 1995, 60, 6860. 33. Dodziuk, H.; Leszczynski, J.; Nowinski, K. S., Theochem-J. Mol. Struct. 1997, 391, 201. 34. Dodziuk, H.; Nowinski, K. S., Theochem-J. Mol. Struct. 1994, 117, 97. 35. Thommen, M.; Keese, R., Synlett 1997, 231. 36. Wang, Z. X.; Schleyer, P. v. R., J. Am. Chem. Soc. 2001, 123, 994. 37. Wang, Z. X.; Schleyer, P. v. R., J. Am. Chem. Soc. 2002, 124, 11979. 38. Merino, G.; Mendez-Rojas, M. A.; Vela, A., J. Am. Chem. Soc. 2003, 125, 6026. 39. Bernath, P. F.; Hinkle, K. H.; Keady, J. J., Science 1989, 244, 562. 40. Dua, S.; Bowie, J. H., J. Phys. Chem. A 2002, 106, 1374. 41. Merino, G.; Mendez-Rojas, M. A.; Beltran, H. I.; Corminboeuf, C.; Heine, T.; Vela, A., J. Am. Chem. Soc. 2004, 126, 16160. 42. Rao, V. B.; George, C. F.; Wolff, S.; Agosta, W. C., J. Am. Chem. Soc. 1985, 107, 5732. 43. Li, X.; Zhai, H. J.; Wang, L. S., Chem. Phys. Lett. 2002, 357, 415. 44. Exner, K.; Schleyer, P. v. R., Science 2000, 290, 1937. 45. Bader, R. F. W., Atoms in Molecules. A Quantum Theory. ed.; Oxford University Press: Oxford, 1990. 46. Silvi, B.; Savin, A., Nature 1994, 371, 683. 47. Merino, G.; Heine, T.; Seifert, G., Chem. Eur. J. 2004, 10, 4367. 48. Pancharatna, P. D.; Mendez-Rojas, M. A.; Merino, G.; Vela, A.; Hoffmann, R., J. Am. Chem. Soc. 2004, 126, 15309. 49. Merschrod, E. F.; Tang, S. H.; Hoffmann, R., Z. Naturforsch. B 1998, 53, 322. 50. Geske, G. D.; Boldyrev, A. I., Inorg. Chem. 2002, 41, 2795. 51. Perez, N.; Heine, T.; Barthel, R.; Seifert, G.; Vela, A.; Mendez-Rojas, M. A.; Merino, G., Org. Lett. 2005, 7, 1509.



Chapter 13

Chemical reactivity dynamics in ground and excited electronic states P. K. Chattaraj and U. Sarkar Department of Chemistry, Indian Institute of Technology, Kharagpur-721302, India

1. Introduction Density functional theory (DFT) [1–5] has been quite successful in providing strong theoretical bases for popular qualitative chemical concepts like electronegativity [6–8] and hardness [9–11]. Pauling [6] introduced the concept of electronegativity as “the power of an atom in a molecule to attract electrons to itself.” The hardness concept was provided by Pearson [9] in his famous hard and soft acid base (HSAB) principle. These concepts are better appreciated in terms of the associated electronic structure principles. Sanderson’s [12] electronegativity equalization principle states that, “the electronegativities of all the constituent atoms in a molecule have the same value and that can be expressed as the geometric mean of the electronegativity values of the associated isolated atoms.” Two hardness-related principles are the HSAB principle [9] and the maximum hardness principle [13]. The statement of the former is “hard acids prefer to coordinate with hard bases and soft acids with soft bases for both thermodynamic and kinetic considerations” and that of the latter is “there seems to be a rule of nature that molecules arrange themselves to be as hard as possible.” The wavefunction of an N -electron system is completely characterized by N and the external potential, r because these two quantities fix the Hamiltonian of the system. The electronegativity and the hardness measure the response of the system when N changes at fixed r . Within DFT, they are defined as the following first-order [14] and the second-order [15] derivatives,

E = − = − N 269

(1) r

270

Chemical reactivity dynamics in ground and excited electronic states

and =

1 2

2 E N 2

= r

1 2

N

(2) r

where E is the total energy and is the chemical potential (negative of the electronegativity) appearing as the normalization constraint in DFT. A related quantity is the electrophilicity index given by [16], 2 2 = (3) 2 2 The linear response function [3], Rr r = r /r N , is used to study the effect of varying r at constant N . If the system is acted upon by a weak electric field, polarizability may be used as a measure of the corresponding response. A minimum polarizability principle [17] may be stated as, “the natural direction of evolution of any system is towards a state of minimum polarizability.” Another important principle is that of maximum entropy [18] which states that, “the most probable distribution is associated with the maximum value of the Shannon entropy of the information theory.” Attempts have been made to provide formal proofs of these principles [19–21]. The application of these concepts and related principles vis-à-vis their validity has been studied in the contexts of molecular vibrations and internal rotations [22], chemical reactions [23], hydrogen bonded complexes [24], electronic excitations [25], ion–atom collision [26], atom-field interaction [27], chaotic ionization [28], conservation of orbital symmetry [29], atomic shell structure [30], solvent effects [31], confined systems [32], electric field effects [33], and toxicity [34]. In the present chapter, will restrict ourselves to mostly the work done by us. For an elegant review which showcases the contributions from active researchers in the field, see [4]. Atomic units are used throughout this chapter unless otherwise specified. As opposed to the global reactivity descriptors described above, the analysis of site selectivity in a molecule demands the local descriptors like the Fukui function defined as [35,36],

=

fr =

r N

= r

r

(4)

N

Owing to the discontinuity in the r versus N plot, three different approximate versions but well suited for different varieties of chemical reactions have been proposed as follows [36], +

fr = −

fr = fr0 =

r N r N

+ N +1 r − N r LUMO r for nucleophilic attack

(5a)

N r − N −1 r HOMO r for electrophilic attack

(5b)

r

−

r

N +1 r − N −1 r 2

for radical attack

(5c)

P. K. Chattaraj and U. Sarkar

271

The condensed-to-atom variants to the above equations may be written as [37], fk+ = qk N + 1 − qk N

(6a)

fk− = qk N − qk N − 1

(6b)

fk0 =

qk N + 1 − qk N − 1 2

(6c)

where qk is the electron population of the atom k in a molecule. Two related local quantities are local softness [38], sr , and philicity [39], r , defined as sr = S˜ · fr

(7a)

r = · fr

(7b)

and

where S˜ = 1/2 is the global softness of the system. The corresponding condensed quantities can also be easily defined. The reliability of these descriptors vis-à-vis other know descriptors is analyzed [40]. A variational procedure [41] and a gradient expansion technique [42] for the determination of fr are also known. These local quantities ought to be used in a judicious way [43]. Time evolution of these quantities and the dynamic generalizations of the associated electronic structure principles may be made use of in developing an alternative but complementary chemical reaction dynamics through dynamic reactivity profiles. Very little progress has been made so far in extending these studies to excited electronic states especially in a time-dependent (TD) situation. In the present chapter, quantum fluid DFT is employed to study a chemical reaction mimicked as a collision process between an ion and an atom in its ground and excited electronic states as well as the interaction of the atom with an external laser field and to monitor the time evolution of various reactivity parameters in order to gain insights into the related structure principles in a dynamical context, involving ground as well as excited states. The basic aspects of quantum fluid DFT is introduced in Section 2. Section 3 analyzes the nature of the chemical reactivity in the excited states. Sections 4 and 5 respectively describe ion–atom collision and atom–field interaction processes. Chaotic ionization and regioselectivity are presented in Sections 6 and 7, respectively. Finally, Section 8 contains some concluding remarks.

2. Quantum fluid DFT The DFT [1,2] rests on two theorems proved by Hohenberg and Kohn, essentially for the ground state. It asserts that the single particle density r contains all the information of a system, and the total energy attains the minimum value for the true density. A TD version of this DFT is also provided [44] which shows that the mapping between the time-dependent external potential, r t, and the density, r t, is uniquely invertible up to an additive trivial TD function in the potential, implying that all the properties of a

272


system are functionals of r t and the current density j r t. This allows us to study a dynamical process in case we have an equation to obtain r t and j r t at all times. An excited state version of DFT is, however, not straightforward to obtain [45,46]. Also this TDDFT strengthens the quantum fluid dynamics (QFD), which describes the dynamics of a quantum system in terms of the flow of a probability fluid associated with the probability density r t and the current density j r t. According to TDDFT [44], the time evolution of these two quantities for a many-electron system is governed by two basic QFD equations, viz., the equation of continuity + •j = 0 t and an Euler-type equation of motion

(8a)

j = P r t j r t (8b) t where P is a functional whose exact form is not known. In order to have an approximate form for P an amalgamation of TDDFT and QFD is made to obtain the quantum fluid density functional theory (QFDFT) [17,47] wherein the equations (8) take the forms,

and

=0 + • t

(9a)

1 2 G r t dr + ext r t = 0 + + + t 2 r − r

(9b)

where is the velocity potential. The universal functional G comprises kinetic and exchange-correlation energy functionals and ext r t is the external potential. The TDDFT [44] legitimately allows us to write the above equations in 3D space, and a 3D complex valued hydrodynamical function r t may be defined in the following polar form within an irrotational approximation as √ (10a) r t = r t1/2 expir t i = −1 r t = r t2

(10b)

im − im re = j r t = re

(10c)

Although r t is the wave function for one-electron systems only, it may provide r t and j r t as above, for many-electron systems as well. A generalized nonlinear Schrödinger equation (GNLSE) may be obtained [17,47] as follows, as the backbone of QFDFT, by combining equations (9a) and (9b), r t 1 (11a) − 2 + eff r t r t = i 2 t where the effective potential may be written as r t T E eff r t = NW + xc + dr + ext r t r − r

(11b)


273

where TNW and Exc are the non-Weizsäcker part of the kinetic energy and the exchangecorrelation energy functionals respectively.

3. Chemical reactivity in the excited electronic states An extension of the Hohenberg–Kohn theorems to an arbitrary excited electronic state has not been possible till date. It has been possible only for the lowest state of a given symmetry [45] and for the ensemble of states [46]. It may be anticipated from the principles of maximum hardness and minimum polarizability that a system would become softer and more polarizable on electronic excitation since it is generally more reactive in its excited state than in the ground state. Global softness, polarizability, and − 2 fr , electrostatic potential, and several local reactivity parameters r t quantum potential have been calculated [25] for different atoms, ions, and molecules for the lowest energy state of a particular symmetry and various complexions of a twostate ensemble. It has been observed that a system is harder and less polarizable in its ground state than in its excited states, and an increase in the excited state contribution in a two-state ensemble makes the system softer and more polarizable. Surface plots of different local quantities reveal an increase in reactivity with electronic excitation.

4. Ion–atom collisions In the collision process between an atom and a proton, the whole scattering system has been considered [17,47] as a supermolecule. The form for the external potential becomes ext r t = −

Z 1 t − r R

−

1 2 t − r R

(12)

1 and R 2 are radius vectors of the target (nucleus of the atom with atomic where R number Z) and the projectile H+ respectively. The origin of the coordinate system is fixed on the target nucleus, and the position of the projectile is determined through the solution of the associated classical equation of motion. The TD chemical potential may be written as t =

Et 1 2 T r t Exc dr + = + + + ext r t 2 r − r

(13)

This quantity is equal to the total electrostatic potential at a point rc where the sum of functional derivatives of total kinetic and exchange-correlation energies is zero, i.e., −t = t = and

r t dr + ext rc t rc − r

2 1 T Exc + =0 + 2 r =rc

(14a)

(14b)

274


This point rc coincides at t = 0 with that given by Politzer et al. [48] as a measure of the covalent radius of an atom. Hardness is calculated as [49] =

1 r r fr r dr dr N

(15a)

where the hardness kernel is given in terms of the Hohenberg–Kohn universal functional, F , as [49] r r =

1 2 F 2 r r

(15b)

and the Fukui function, fr , is modeled as prescribed by Fuentealba [50]. It is important to note that the calculations of the electronegativity and hardness described above do not require any a priori knowledge of the total or orbital energy values of the system as is done in the most of the researches involving their computation. As a matter of fact, we may completely bypass the Schrödinger equation if the electron density is obtained from some other source, say, from an experiment or as the direct solution of a single-density equation [51]. The high projectile velocity used in this work allows to legitimately assume the cylindrical symmetry of the charge density about an axis − ≤ z˜ ≤ passing through the target nucleus. Polarizability dynamics is studied in terms of the time evolution of the diagonal component of the polarizability tensor along the z˜ axis, designed as t as follows,

z˜ t z˜ t

t = Dind

(16)

z˜ where Dind t is the electronic part of the induced dipole moment and z˜ t is the z˜ component of the external Coulomb field. The entropy, St, associated with the electron cloud can be defined as [17,47]

St =

5 3 k − k ln r t + k lnkr t/2 r tdr 2 2

(17a)

where k is the Boltzmann constant and r t is a space- and time-dependent temperature defined in terms of the kinetic energy density as

j r t2 3 r tkr t + ts r r t = 2 2 r t

(17b)

The numerical solution of the GNLSE is performed [17] using a leap frog-type finite difference scheme starting from a near Hartree–Fock density of the target atom in its ground√and excited electronic states. This method is stable [52] because of the presence of i = −1. Figure 1 depicts the time evolution of various quantities. The TD chemical potential profile helps [17] dividing the whole collision process into three distinct


275

regimes: approach, encounter, and departure. In the encounter regime, the actual charge exchange process takes place, and the condition (14b) is nowhere met. Hardness and entropy maximize and polarizability minimizes for both the states in the encounter regime where the chemical reaction occurs, validating [17] the principles of maximum hardness, minimum polarizability, and maximum entropy in a dynamical context. The maximum and the maximum S values are larger, and the minimum value is smaller in the ground state than the corresponding values in the excited state.

(a)

(b)

40

30

30

20

20

μ

μ

40

10

10

0

0

–10

–10 –20

–20 0

5

10

15

0

20

5

15

10

t

20

t

Figure 1a Time evolution of chemical potential during a collision process between an atom and a proton: (a) ground state, (b) excited state

1.00E+008

1.00E+008

(a)

6.00E+007

6.00E+007

η

8.00E+007

η

8.00E+007

(b)

4.00E+007

4.00E+007

2.00E+007

2.00E+007

0.00E+000

0.00E+000 8

9

10

t

11

12

8

9

10

11

12

t

Figure 1b Time evolution of hardness during a collision process between an atom and a proton: (a) ground state, (b) excited state

276

Chemical reactivity dynamics in ground and excited electronic states 0.000008

0.000008

(a)

0.000007

(b)

0.000007

0.000005

0.000005

0.000004

0.000004

α

0.000006

α

0.000006

0.000003

0.000003

0.000002

0.000002

0.000001

0.000001

0.000000

0.000000 –0.000001

–0.000001 9

8

10

11

12

9

8

10

t

11

12

t

Figure 1c Time evolution of polarizability during a collision process between an atom and a proton: (a) ground state, (b) excited state 120

120

(b)

100

100

80

80

60

60

S

S

(a)

40

40

20

20

0

0 8

9

10

11

t

12

8

9

10

11

12

t

Figure 1d Time evolution of entropy S during a collision process between an atom and a proton: (a) ground state, (b) excited state

5. Atom–field interactions An atom in its ground and excited electronic states is interacted with a laser field of varying colors and intensities and linearly polarized in z˜ direction. The associated TD external potential is given by Z + 1˜z for a monochromatic pulse r Z = − + 2˜z for a bichromatic pulse r

ext r t = −

(18a)


277

where and

1 = sin 0 t 2 = 05sin 0 t + sin 1 t

(18b) (18c)

In order to have slow oscillations during and immediately after the laser source being switched on, is written in terms of the maximum amplitude 0 and the switch-on time t as = 0 t/t = 0

for 0 ≤ t ≤ t otherwise

(18d)

A leap frog-type finite difference scheme is adopted to numerically solve [27] the associated GNLSE for three different field intensities. Figure 2 presents the time evolution of the external field as well as that of TD chemical potential and hardness. Both t and t oscillate in phase with the external field. But the former being the first order variation exhibits this behavior at lower intensity than that of t. Other quantities like t St, and t also show (not shown here) characteristic oscillations. The overall dynamics may be understood as follows. The spherical symmetry of the nuclear Coulomb field forces the electron density distribution to be spherical in the absence of the external field. When the external z˜ -polarized laser pulse is switched on, there starts a tug-of-war between the two to govern the electron density distribution, viz., the latter would try to make it cylindrically symmetric due to the axial nature of the external field. For the very small-intensity laser field, effective an atomic density with slight pulsation results. There forms an oscillating dipole with an increase in the external field intensity and that emits radiation including higher-order harmonics. The calculated harmonic spectra look like those obtained by Erhard and Gross [53]. There is no conspicuous effect of the external field with 0 = 10−6 on the dynamics. Still competition persists for 0 = 001 and starts oscillating in-phase but not . The dynamics is totally governed by the external field at 0 = 100 where the in-phase oscillations of as well are clearly manifested for both the electronic states and for both monochromatic and bichromatic laser pulses.

6. Chaotic ionization in Rydberg atoms Hydrogen [54] and helium [55] atoms are known to exhibit regular/chaotic dynamics in the presence of external field of different colors and intensities. Chaotic ionization from the Rydberg states of the atoms [54,55] has been very intriguing for the experimentalists. Both QFD [56,57] and quantum theory of motion (QTM) [58,59] have been able to explain the quantum domain behavior of the classically chaotic systems [60]. In QFD, the quantum dynamics is mapped onto that of a probability fluid of density and current density r t and j r t respectively obtainable as solutions to the QFD equations. The fluid moves under the influence of the classical Coulomb potential augmented by a quantum potential defined as qu = −

2 1/2 1 r t 2 1/2 r t

(19)

ε0 = 0.01

278

ε0 = 1.0e–06

ε0 = 100

0.0000012

ε2

ε2

0.0000004

0.0000000

0.008

80

0.004

40

ε2

0.0000008

0.000

–0.004

–40

–0.0000008

–0.008

–80

5

10

15

20

0

5

t

10

15

0

20

0.0000005

0.005

50

ε1

100

ε1

0.010

0.0000000

0.000

–0.005

–50

–0.0000010

–0.010

–100

10

t

15

20

0

5

10

t

15

20

15

20

0

–0.0000005

5

10

t

0.0000010

0

5

t

15

20

0

5

10

t

Figure 2a Time evolution of the external electric field: 1 (—) monochromatic pulse; 2• • • bichromatic pulse. Maximum amplitudes: 0 = 10−6 001 100 0 = 1 = 2 0


–0.0000004

0

ε1

0

150

–12.712 –12.727005

100 –12.720

50

μ

–12.727015

μ

μ

–12.727010

–12.728

ES

0 –50

–12.727020 –12.736

–100


ε0 = 100

ε0 = 0.01

ε0 = 1ε–06 –12.727000

–12.727025 –12.744 2

4

6

8

–150 2

10 12 14 16 18 20

4

6

8

10 12 14 16 18 20

t

5

10

15

20

t

t –0.90

–0.956298

150 100

–0.92

–0.956299

50

μ

–0.956300

μ

μ

–0.94

–0.96

GS

0 –50

–0.956301

–100 –0.98 –150 –1.00

–0.956302

5

10

15

t

20

5

10

15

t

20

5

10

15

20

t

279

Figure 2b Time evolution of chemical potential when an atom is subjected to external electric fields (GS, ground state; ES, excited state): 1 (—) monochromatic pulse; 2• • • bichromatic pulse. Maximum amplitudes: 0 = 10−6 001 100

0 = 1 = 2 0

4

4

4

3

3.32140

4.0 3.5

2

η

2

3.32130

3.0 2.5

2

4

6

8

12

10

14

16

18

2.0

1

3.32120 20

2

4

6

2

5

10

15

0

20

5

10

15

20

0 8

7

7

7

6

6

5

4

7.274096

4

η

7.274098

η

η

η

5

3

7.274092

2

7.27400 5

15

10

5

10

15

5

20

t

1

10

t

15

20

15

10

20

t

1

0

0 5

GS

7.2740

2

20

t

1

0

7.2742

7.274094

3

7.27404

2

20

7.274100

7.27408

3

15

6

7.27420

4

10

t

8

7.27412

5

t

8

7.27416

10 12 14 16 18 20

8

0

t

5

6

t

0 0

4

t

0

η

1

8 10 12 14 16 18 20

t

η

1

0 0

5

10

t

15

20

0

5

10

15

20

t

Figure 2c Time evolution of hardness when an atom is subjected to external electric fields (GS, ground state; ES, excited state): 1 (—) monochromatic pulse; 2• • • bichromatic pulse. Maximum amplitudes: 0 = 10−6 , 0.01, 100; 0 = , 1 = 2 0


3.32125

ES

η

η

η

3.32135

η

3.321290 3.321289 3.321288 3.321287 3.321286 3.321285 3.321284 3.321283 3.321282 3.321281 3.321280

3

4.5

η

3

2

ε0 = 100

ε0 = 0.01

280

ε0 = 1ε−6

1

1

0.01

|dω|2

0.01

0

10

20

30

0

40

10

20

30

40

0

1

0.1

1

ε0 = 0.01

20

Harmonic order

30

40

40

ε0 = 100

GS 0.01

1E-3 10

30

|dω|2 0.01

1E-3

20

0.1

|dω|2

|dω|2

0.1

0.01

10

Harmonic order

Harmonic order

ε0 = 1e–6

0

ES

0.01

Harmonic order 1

ε0 = 100

0.1

|dω|2

0.1

|dω|2

0.1

1

ε0 = 0.01


ε0 = 1e–6

1E-3 0

10

20

Harmonic order

30

40

0

10

20

30

40

Harmonic order

281

Figure 2d Harmonic spectra of an atom for various laser frequencies and intensities (GS, ground state; ES, excited state): 1 (—) monochromatic pulse; 2• • • bichromatic pulse. Maximum amplitudes: 0 = 10−6 , 0.01,100; 0 = , 1 = 2 0

282


Another important quantum potential based theory is QTM where a “wave and particle” picture is assumed for describing a physical system. The wave motion is governed by the solution of the TD Schrödinger equation, and the particle motion is followed by solving the pertinent Newton equation of motion with forces originating from both classical and quantum potentials. Chaotic dynamics is analyzed through [57] the “canonically conjugate” r t and −r t in QFD and through [59] the distance between two initially close Bohmian trajectories and the associated Kolmogorov–Sinai entropy in QTM. Other nonlinear dynamical concepts like solitons [61] and fractals [62] also provide important insights into the dynamical behavior of the system. Apart from the reactivity descriptors discussed so far, an important diagnostic of the quantum signature of classical chaos [63], viz., the uncertainty product is also calculated [28] which is defined as below, Vps =

2 2 1/2 2 2 pz˜ − < pz˜ > ˜ − < ˜ > z˜ − < z˜ > p ˜ − < p ˜ >

(20)

A sharp increase in Vps implies a chaotic motion since it is a measure of associated quantum fluctuations [63]. This quantity has been shown [28] to be very large for the highly excited states of hydrogen and helium atoms in the presence of both monochromatic and bichromatic laser pulses when compared with the corresponding quantities for the ground states. Ground state hardness, on the other hand, is always larger than that in the excited state, as expected from the maximum hardness principle, and it appears that a relatively smaller value signals a possible chaotic dynamics.

7. HSAB principle in action and regioselectivity in chemical reactions A chemical reaction between an atom X = He Li+ Be2+ B3+ C4+ in various electronic states and a proton to form an XH+ molecule has been mimicked by a collision between X and H+ . Maximum hardness and minimum polarizability values characterize a favorable dynamical process. Since a system becomes more reactive on electronic excitation, it becomes softer and more polarizable. A proton being a hard acid would prefer the most to react with X in its ground state according to HSAB principle in a dynamical context, and the preference will gradually decrease with electronic excitation. This fact is corroborated in this QFDFT [17,47] study on protonation [64] through a decrease in the maximum hardness value and an increase in the minimum polarizability value on excitation. Reactivity of noble gas elements toward protonation increases as He < Ne < Ar < Kr < Xe. For the proton–molecule reaction: AB + H+ → ABH+ + BAH+ , the regioselectivity is clearly manifested [64]. For both homo-nuclear diatomics H2 N2 , and F2 and heteronuclear diatomics (HF, BF, and CO) hardness maximizes max and polarizability minimizes min in the encounter regime, and the max value decreases and min value increases with excitation implying an increase in reactivity. The preference of attack of H+ through the A-end or through the B-end is easily understood in terms of relative magnitudes of max and min values for the attack on A and B sites.


283

8. Concluding remarks Time evolution of different reactivity parameters like electronegativity, hardness, polarizability, Shannon entropy, and uncertainty product is monitored for the collision process of an atom in its ground and excited electronic states with a proton as well as the interaction of the atom with an external laser field of different colors and intensities, within a quantum fluid density functional framework. Dynamical variants of the associated electronic structure principles, viz., electronegativity equalization principle, maximum hardness principle, HSAB principle, minimum polarizability principle, and maximum entropy principle, reveal themselves. An atom or a molecule gets softer and more polarizable with electronic excitation. Important insights into the chaotic ionization in Rydberg atoms as well as the regioselectivity in chemical reactions are obtained through these studies. The nature of the atom–field interaction is also now clearly understood.

Acknowledgments We are grateful to CSIR, New Delhi and BRNS, Mumbai for financial assistance. One of us (PKC) would like to thank his friend Professor Alejandro Toro-Labbé (QTC@PUC) for inviting him to contribute an article in this volume. He is also thankful to his students Drs. Somdatta Nath, Santanu Sengupta, Abhijit Poddar, Buddhadev Maiti, and Mr. Debesh Ranjan Roy for their help in various ways.

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Chapter 14

Quantum chemical topology and reactivity: A comparative static and dynamic study on a SN 2 reaction Laurent Joubert, Ilaria Ciofini, and Carlo Adamo Laboratoire d’Electrochimie et de Chimie Analytique, UMR CNRS 7575, Ecole Nationale Supérieure de Chimie de Paris, 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France

Abstract Ab initio molecular dynamic simulations, using density functional theory (DFT) and the recent atom-centered density-matrix propagation (ADMP) method, were employed to study the bond breaking and formation for a case-study SN 2 reaction. Using the real space partition scheme of Bader’s quantum chemical topology (QCT), we performed a population analysis to examine intra- and intermolecular electronic charge transfer along the ADMP trajectory. These results were compared to a static approach, which is performed along the intrinsic reaction coordinate (IRC) path. Although similar features are found for both static and dynamic approaches, the QCT analysis allows to rationalize the differences observed during the formation of the ion–molecule complex. In particular, the dynamic approach suggests a stronger electron exchange tending to spontaneously maximize both covalent and noncovalent (i.e. electrostatic) interactions.

1. Introduction The interpretation of quantum chemical calculations in terms of classical chemical concepts is not an easy task. In particular, electron transfer is strongly related to the definition of atoms and bonds in a molecular system. Beyond the simplistic picture of Lewis1 based on intuitive concepts, two different ways, trying to give quantitative information on the nature of chemical bond, cohabit. On the one hand, classical localization procedures of the wave function give different partitions of the Hilbert’s space 287

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leading to atomic charges, such as Mulliken or Löwdin ones,23 or to hybrid orbitals.4−6 More recently, topological analyses based on the electron density, such as Bader’s quantum chemical topology (QCT)78 or on derived functions, such as the electron localization function (ELF),9 have proven their reliability as robust methods, crafted to put in evidence the subtle properties of bonding and reaction mechanisms.10−13 Nowadays, a wide choice of tools are available to chemists for the theoretical analysis of any kind of chemical reactions. These tools are even more powerful when coupled with efficient algorithms for the exploration of the potential energy surface, such as the intrinsic reaction coordinate (IRC) algorithm.1415 Calculation of any topological or electronic variable along the reaction path allows for a clear-cut vision of the reaction mechanism and for rationalizing the structure/energies properties of the surface extrema (reactants, transition states, and products).1617 It must be noticed, anyway, that all these approaches have been developed in the framework of a time-independent (i.e. static) view. Only recently Gross and co-workers have formulated a suitable, but rather complex approach including the time evolution of ELF.18 At the same time, ab intio dynamics, such as the Carr–Parrinello approach,19 have recently reached a mature state, and a blooming of reactivity studies can be found in the literature.20−22 Based on an extended Lagrangian molecular dynamics (MD) scheme, this Born–Oppenheimer approach evaluates the potential energy surface at the density functional theory (DFT) level, and both the electronic and nuclear degrees of freedom are propagated as dynamical variables.23 The calculation of the time variation of any chemicophysical properties (observable or not) is then easily computed using structure snapshots extracted from the trajectory and a statistical average.23 A tentative to explore the difference between static and dynamics ab initio approaches has been done,24 but is has been focused only on the thermochemistry, since the choice of unique and meaningful parameters for comparison is not trivial. In the present article, we report a study concerning the reaction mechanism of a prototype reaction using both static and dynamic approaches to explore a DFT potential surface. The static approach is the standard IRC model, while the dynamic one is based on a Carr–Parrinello method performed with localized (Gaussian) orbitals, the so-called atom-centered density matrix propagation (ADMP) model.25 Our aim is to elucidate the differences, and the common aspects, between the two approaches in the analysis of bond breaking/formation. To this end, we have chosen topological quantities as probe molecular descriptors. The so-called Walden inversion, an SN 2 reaction shown in Figure 1, has been chosen as the model, since it is a well-studied reaction, from both a static and a dynamical point of view.26−36 In the present study, we limit our analysis to the gas-phase reaction, and we do not consider solvent effects that are known to strongly modify the potential energy surface (PES).29

2. Computational details All the calculations were carried out with the Gaussian03 program,37 using the hybrid PBE0 functional3038 and the 6-31+G(d,p) basis set.39 Starting from the transition state, we have calculated reaction pathways for both static and dynamic approaches. On the one hand, we followed the minimum energy path between the transition state and

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the stable ion–molecule complex that is, using mass-weighted coordinates, the internal reaction coordinate (IRC).15 On the other hand, the dynamic simulation was performed at 298 K in the canonical ensemble, for a total simulation time of 1.5 ps. The starting point (transition state) was previously optimized with the same basis set and functional used in the simulation. The well-known Velvet algorithm40 was employed for the integration of the equations of motions using a time step of 0.25 fs. The fictitious mass of the electron was set to 0.20 amu, with a scaling for both core and valence electrons as described in Schlegel et al.25 The velocities of the nuclei were scaled each five time steps to ensure a constant temperature within a T = 5 K of tolerance. The stability of the simulations was monitored by checking at each step the idempotency of the density matrix (within a 10−12 threshold) and the so-called adiabaticity index (within a 10−4 threshold, see Shlegel et al. (2001)25 for more details). In practice, we restricted our trajectory analysis to the time frame of interest, scrutinizing the path only during the formation of the stable ion–molecule complex. All along this truncated trajectory (less than 100 fs in time scale), we extracted a number of snapshots, taken each 2 fs, for the subsequent structural and electronic analyses of the reaction. The charge transfer processes were investigated using a QCT approach: the “Atoms in Molecule” (AIM) partition scheme of the electron charge density.78 According to this theory, topological atoms are defined as regions in real space consisting of a bundle of electron density gradient paths attracted to a nucleus. This partition allows evaluating atomic properties, defined as volume integrals over non-overlapping atomic basins. In particular, the population associated with an atom is simply the volume integral of the electronic charge density over the basin. In this work, we calculated and compared the variations of atomic basin populations all along the selected static and dynamic paths. Moreover, we monitored the variations of typical electron density properties calculated at a bond critical point (BCP) to characterize a chemical bond (see for instance Popelier (2000)8 for more details on BCPs). These calculations were performed using the Topmod package41 and a locally developed code.

3. Results The prototypical SN 2 reaction studied in this work (see figure 1) presents in the gas phase a double-well PES with two equivalent local minima corresponding to the formation of a pre- and a post-reaction ion–molecule complex Cl− · · · CH3 Cl and a transition state (TS) of D3h symmetry Cl · · · CH3 · · · Cl− . Owing to the overall symmetry of the reaction, we will restrain our study to a reaction path from the transition state to one of the equivalent minima.

3.1. Tuning the DFT approach The present study relies on the parameter-free exchange-correlation PBE0 functional. Our choice of the functional was guided by its good performance in predicting thermodynamics data for the current SN 2 reaction, using a large basis set (6-311+(3df,3pd)30 ). Since a large number of calculations are needed in order to evaluate the variations of the topological quantities along the trajectories, such large basis is not suitable. For

290


[Cl…CH3…Cl]– ΔEovr ClCH3+ Cl–

Energy

Cl– + CH3Cl

ΔE#

ΔEcomp Cl– … CH3Cl

ClCH3…Cl–

Reaction coordinate

Figure 1 Sketch of energy profile for the SN 2 reaction under study

this reason, we performed some preliminary tests to assess the validity of the PBE0 functional with a medium-sized basis set, 6-31+G(d,p). We first examined the geometrical features of the key structures, i.e. the geometries of the ion–molecule complex and of the transition state. Structural and energetic results are summarized in Tables 1 and 2, respectively. From a structural point of view, our data are very close to the MP2 results32 for the transition state structure, with a deviation of about 0.01 Å for the carbon–chlorine distance. The same trends are observed for the C-Cl2 bonding distance in the ion–molecule complex. In contrast, a larger error (0.20 Å) is observed for the C Cl1 distance, this effect being strictly related to the small basis set considered. In fact, when a larger basis, 6-311+G(2d,p), is considered this distance is significantly augmented (3.23 Å) toward the MP2 value. The most significant thermodynamic quantities are the complexation energy of the ion–molecule complex Ecomp , the activation energy, i.e. the relative energy of the D3 h saddle point with respect to the ion–molecule complex E # , and the overall barrier Eovr , defined as the difference between these two energies. These data are reported in Table 2 and compared with previous theoretical29−35 or experimental results.36 Whereas the computation of initial closed-shell reagents does not generate particular difficulties, the determination of the energy of the charged transition state energy Cl · · · CH3 · · · Cl− by DFT approaches is more involved. In fact, the majority of standard functionals, and in particular those resting on the Generalized Gradient Approximation (GGA),

Table 1 Computed geometry parameters (in Å) for the equilibrium ion–dipole complex and the transition state Complex

HFa /TZ3P + R + 2fd MP2a /TZ3P + R + 2fd DFT-LDAa /TZ3P + R + 2fd DFTb -BP/TZ + 2P DFTc -BP/PAW DFTd -mPW1PW/6-31 + Gdp DFT-PBE0/6-31+G(d,p)

Transition state

RC-Cl1

RC-Cl2

RC-C11 = RC-C12

3 37 3 27 2 98 3 10 3 15 3 01 3 07

1 82 1 81 1 81 1 84 1 89 1 83 1 83

2 39 2 32 2 28 2 34 2 37 2 33 2 33

a: Ref [31], b: Ref [32], c: Ref [34], d: Ref. [29].

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Table 2 Complexation energies, overall energies, and barrier heights (in kcal/mol) for the studied SN 2 reaction Ecomp HFa /TZ3P + R + 2fd; ZPE corrected MP2a /TZ3P + R + 2fd; ZPE corrected MP4b TZ + 2P; ZPE corrected G2c ; ZPE corrected G3c ; ZPE corrected CBS-QB3(+)c ; ZPE corrected DFTb -BP/TZ + 2P; ZPE corrected DFTd -BP/PAW DFTe -B3LYP/6-31G(d); ZPE corrected DFTf -mPW1PW/6-31+G(d,p); ZPE corrected DFT-PBE0/6-31+G(d,p); ZPE corrected experimentg

−8 1 −10 5 −10 6 −10 8 −11 2 −10 7 −10 3 −8 3 −9 5 −9 8 −10 1 −12 ± 2

Eovr

E #

7 6 3 5 1 8 3 1 1 8 2 4 −5 7 −3 0 −0 9 0 7 0 5 3/1 ± 1

15 7 14 0 12 4 13 8 13 0 13 1 4 6 5 4 8 7 10 5 10 6 13 ± 2

a: Ref [31], b: Ref [32], c: Ref. [33], d: Ref [34], e: Ref [35], f: Ref. [29], g: Ref. [36].

fail in determining the energy barriers. This effect has been related to the the selfinteraction error in the exchange part as well as in the correlation part, which implies an exaggerated delocalization of the electron density that overstabilize the transition state (for a discussion on this point see Toulouse et al. (2002)42 ). In contrast, PBE0 calculations provide accurate results, close to the post-HF (Hartree-Fock) calculations and in the range of the experimental estimations. Nevertheless, this result can be due to a simple error compensation between the GGA and HF contributions.

3.2. Static and dynamic topological analyses The energy variations along the intrinsic reaction path (IRP) and the ADMP trajectory are reported in Figures 2 and 3, respectively. We want to underline that the IRP corresponds to a unique and constrained minimum energy path and does not contain any time information, whereas the ADMP profile is computed along a trajectory issued by a dynamics simulation. Therefore, any direct (i.e. point-to-point) comparison is misleading. Nevertheless, their global differences can be discussed. The minimum energy constraint imposes a unique and distinctive IRP that continuously decreases from the transition state to the minimum. In contrast, the energy globally decreases along the ADMP trajectory, but it appears clearly that the reaction does not follow this minimum energy path. Actually, the minimum is rapidly reached after 96 fs, but the curve exhibits some energy risings corresponding to punctual destabilizations of the whole system. In order to rationalize the differences observed between the static approach and dynamic process, we investigated the intra- and intermolecular charge transfers along the different reaction paths. As a first step, we looked at the AIM atomic populations for the two states of interest, i.e. the transition state and ion–molecule complex. The results are gathered in Table 3. At the transition state, each chlorine atom bears a negative charge of −0 70e− . This electron excess corresponds to the sharing of the unit

292


Potential energy (a.u.)

–959.990

–959.995

–960.000

–960.005

–960.010 10

0

20

30

50

40 1/2

MWC step (amu

60

.bohr)

Figure 2 Variations of the potential energy along the IRP from the transition state to the ion–molecule complex

–959.985


–959.990

–959.995

–960.000

–960.005

–960.010 0

12

24

36

48

60

72

84

96

Time (fs)

Figure 3 Variations of the potential energy along an ADMP trajectory from the transition state to ion–molecule complex

charge of the complex increased by a supplementary electron transfer of approximately 0 20 e− from the hydrogens to each of the chlorine atom. Along the IRP, the formation of the ion–dipole molecule corresponds to a global charge transfer of 0 25 e− from the forming chloromethane moiety to the leaving chlorine atom. This charge transfer is slightly increased by 0 01 e− when considering the dynamic process. This global molecule–ion interaction is accompanied by an intramolecular charge transfer in the forming chloromethane molecule. This phenomenon is evidenced by a slight increase of the methyl moiety electronic population, i.e. 0.05 and 0 06 e− at the end of the static

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293 Table 3 Computed AIM basin populations e− for the transition state and the ion–molecule complex Atom

Transition state

Cl1 H1 H2 H3 C Cl2

Ion–molecule complex

17 70 0 83 0 83 0 83 6 11 17 70

17 95 0 84 0 88 0 86 6 10 17 38

and dynamic process, respectively. A deeper insight into the charge transfer processes can be reached by examining the variations of the atomic electronic populations along the IRP and ADMP trajectory. Figures 4 and 5 represent the relative variations of selected AIM basin populations with respect to the initial populations in the transition state configuration for both the static and dynamic approaches, respectively. The global molecule-to-ion charge transfer is materialized by two essential curves. The first one (squares) corresponds to the monotonically increasing atomic population of the leaving chlorine atom Cl1, tending to the population of a chloride anion. In parallel, a second curve (solid), grouping the atomic populations of the entire molecule CH3 Cl, shows the corresponding and decreasing electronic population of the forming molecule. When the ion–dipole complex is formed, the global electronic transfer between the two moieties corresponds to the aforementioned net charge of 0 25e− or 0 26e− , depending on the approach chosen to study the reactivity of the system, i.e. a static or a dynamic

Variations of AIM basin populations (e–)

0.4 0.3 0.2 0.1 0.0 –0.1 –0.2 –0.3 –0.4 0

10

20

30

40

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Figure 4 Variations of AIM populations along the IRP. The superbasin corresponding to the forming CH3 Cl molecule (5 basins) is represented by the solid curve. Other curves with markers: carbon basin (lozenges), Cl1 basin (squares), Cl2 basin (circles), and sum of hydrogen basins (triangles) populations

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one. If we now examine the variations of the individual atomic populations, substantial differences are observed between the static and dynamic models of reactivity. Three supplementary curves are presented on the abovementioned Figures 4 and 5. They correspond to the variations of the atomic populations of (i) the chlorine atom Cl1 of the forming chloromethane molecule, (ii) the three gathered hydrogen atoms, and (iii) the carbon atom. In Figure 4, i.e. along the IRP, the detailed analysis of these variations of the atomic populations allows us to propose a three-step charge transfer mechanism. In the first part of the path, corresponding approximately to 1 0 amu1/2 bohr, the electron excess on the chlorine atom Cl2 is transferred almost totally to the other chlorine through the atomic basins of the hydrogens. This particular point is evidenced by examining the variations of the atomic populations of the two chlorine atoms that almost compensate each other. In other words, the electronic flux coming from the Cl2 chlorine basin and entering the three hydrogen basins corresponds to the exiting flux that penetrates the Cl1 chlorine basin. Besides this main electron transfer, a second one is observed between the carbon and the three hydrogen atoms. In fact, along the path, the carbon–hydrogen bond lengths systematically increase when the methyl group adopts a C3v geometry in the forming molecule and points to the leaving chlorine atom. This phenomenon is here increased by a favorable increasing electrostatic interaction with this chlorine atom. Therefore, the lengthening of these bond lengths explains the local charge transfer observed from carbon to hydrogens. Finally, we note substantial fluctuations in the population variations of the linked carbon and chlorine atoms. Again, these fluctuations compensate each other. They result from two concerted effects. First, a small fraction of the electron flux between chlorine atoms penetrates the carbon basin, resulting in a small increase of the corresponding population that tends to be slightly in excess. Second, the formation of the carbon–chlorine bond counters this flux by inducing a small charge transfer from the carbon to the chlorine atom. In the second step of the charge transfer mechanism, i.e. up to 3 0 amu1/2 bohr, the electronic loss on the chlorine atom Cl2

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becomes stronger that the electronic gain on the leaving chlorine Cl1. Therefore, the electron flux that enters the hydrogen basins becomes larger than the exiting flux. As a result, the electronic population in the hydrogen basins increases. At a lesser extent, the electronic charge transfer from carbon to hydrogens increases as well, reinforcing the electronic population of these basins. Furthermore, we note that the fluctuations observed along the carbon and chlorine (Cl2) curves are amplified when approaching the equilibrium structure of the chloromethane. The last step of the charge transfer mechanism corresponds approximately to the second half of the IRP. Structural changes are minimal in the chloromethane molecule, resulting in equilibrium for carbon and hydrogen basin populations. The chlorine atom Cl1 continues to move away from the molecule implying a strong decrease in the chlorine to chlorine charge transfer, the population of the Cl1 atom approaching those of a chloride anion. In Figure 5, i.e. along the ADMP trajectory, the analysis of atomic population variations suggests a charge transfer mechanism at variance with the static IRC mechanism. We can decompose it in five steps. We first note that the system is destabilized during the first 8 fs of the simulation (see Figure 3a), certainly due to an excess of initial kinetic energy. During that period, population variations are negligible, except for the cyclic fluctuations of the carbon and hydrogen populations corresponding to the C-H stretching vibrational mode. These fluctuations are present all along the trajectory. Between 8 and 16 fs, the charge transfer begins between chlorine atoms. This second dynamic step is very similar to the first part of the IRC path, where an electron excess on the chlorine atom Cl2 is transferred almost totally to the other chlorine through the atomic basins of the hydrogens. The third step corresponds roughly to a frame between 16 and 30 fs and can be compared to the second step of the IRC charge transfer mechanism. In fact, the curve corresponding to the population variations of the Cl2 chlorine atom strongly deviates from the one corresponding to the population variations of the whole chloromethane molecule. Meanwhile, both electronic populations of carbon and hydrogen atoms substantially increase, indicating an intramolecular charge transfer from the chlorine Cl2 basin to the carbon and hydrogen basins. In contrast with the static approach, no fluctuations are observed for the carbon and chlorine Cl2 curves, except for the cyclic variations on the carbon curve corresponding to the C-H stretching vibration mode. At this point, we want to remark that all the obtained results are preserved when different initial conditions (kinetic energy and temperature) are considered, since the excess energy mainly changes the population of the CH vibrational states. This is not surprising since the Cl-CH3 Cl starting ion–complex exhibits a rather poor energy transfer between the inter- and intravibrational modes.43 The third step corresponds to a time frame between 30 and 50 fs. We observe that the electronic population of the hydrogen basins strongly increases due to a substantial electron transfer from the carbon basin. Moreover, the corresponding population variations almost compensate each other. All these subtle variations are related to structural changes during the dynamics. In Figure 6, we isolated the population variations of the carbon and hydrogen basins together with the sum of the variations of C-H distances. During the time frame of interest, we note a substantial lengthening of these distances. This phenomenon was also observed along the static IRP, but with a lower amplitude. Here, the C-H stretching vibrational mode elongates the corresponding bond lengths, strengthening the electrostatic attraction between the leaving chlorine atom and hydrogen atoms and favoring the electron flux

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Figure 6 Comparison between the variations of selected basin populations (C and H) and the sum of C-H distances (solid line) along a dynamic pathway from the transition state to the ion–molecule complex. Curves with markers: carbon basin (lozenges) and sum of hydrogen basins (triangles) populations

between them. This noncovalent effect should favor the stabilization of the system, but we know that the complex is strongly destabilized within this time frame, as evidenced by a sudden potential energy rising in Figure 3. This surprising behavior can also be rationalized by examining the structural evolution of the system. In Figure 7, we represented the variations of the potential energy together with the evolution of the H-C-H valence angles, i.e. we plotted the variations of the sum of these three angles along the trajectory. In the time frame of interest, we can see that the sum of angles decreases

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Figure 7 Comparison between the potential energy variations (in bold) and the sum of the three H-C-H valence angles (solid) in the ion–molecule complex

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Figure 8 Left Y axis: variations along the ADMP trajectory of the C-Cl2 distance (in bold). Right Y axis: variations of the electron density (solid) and the function Lr (dashed) calculated at the C-Cl2 bond critical point r = rc

to only 310 , a global minimum along the whole trajectory. In other words, the strong electrostatic attraction between the Cl1 chlorine atom and hydrogen atoms induces not only a lengthening of the C-H bond lengths but also a substantial closing of the H-C-H valence angles. The latter effect may explain partly the destabilization of the system by increasing electrostatic and Pauli repulsions between the hydrogen atoms. Furthermore, an examination of electron density properties at the C-Cl2 BCP reveals interesting information on the bond creation process that helps to explain this surprising potential energy rising. In Figure 8, we plotted the variations of the C-Cl2 distance along the trajectory, together with the variations of the density r and function Lr = − r calculated at the corresponding BCP. In the time frame of interest, we note a systematic decrease of the C-Cl2 distance down to the global minimum value of 1.682 Å at = 50fs, about 0.15 Å shorter than the equilibrium value. Meanwhile, the BCP electron density strongly increases to finally reach a value of 0 22 e− bohr −3 , characteristic of a strong covalent bond. The Lr function confirms these trends, with an increasing positive value at BCP that indicates a strong charge concentration. In short, the electron exchange is maximized during the dynamic simulation, favoring the penetration of the chlorine atom electron cloud into the one of the methyl moiety. This effect destabilizes the system by strongly enhancing the Pauli repulsion between these atoms. Finally, the two last steps of the dynamic charge transfer mechanism correspond to the second half of the trajectory. First, between 50 and 78 fs, the system, strongly destabilized, tends to reduce quickly its potential energy. Thus, we note a sudden lengthening of the C-Cl2 bond and an opening of the H-C-H valence angles (see Figure 7) coupled with a shortening of the C-H bond lengths (see Figure 6). This last effect induces a strong diminution of the charge density transferred from the carbon to the hydrogen atoms. Moreover, it is important to point out an inverted electron flux from the ion to the chloromethane molecule tempting to equilibrate the destabilized system. Finally, the last part of the dynamic trajectory, beyond 78 fs, corresponds to the effective formation of a

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stable ion–molecule complex and the end of the charge transfer between the molecule and the chlorine atom stabilized by the completion of its external valence shell.

4. Some comments on the static vs. dynamic descriptions Our results show how both ADMP and IRP approaches suggest that the driving force for bonding formation is the charge transfer process. This force, clearly present in the static approach, is so enhanced in the dynamics simulation that it allows for the penetration of the incoming chlorine into electron cloud of the methyl moiety. This results in a very short C-Cl2 distance and in high energy. The subsequent switching on of the Pauli repulsion permits to reach the final state, through a large C-Cl2 amplitude motion. Before this reactive state, the reactants are in a kind of “harmonic reversible” regime where each part still preserves its own electronic identity. This picture is consistent with previous analyses, based on different theoretical models. Among others, we want to recall that similar but maybe more qualitative conclusions were drawn several years ago by Shaik on the basis of molecular orbitals arguments resting on valence bond (VB) calculations.44 In the same philosophy, but more recently, it has been suggested that, within a VB approach, the overlap between the active orbitals of the incoming Cl− and CH3 Cl rules the overall reactive process.43 Moreover, the maximization of the charge transfer strongly reminds the “maximum localization hybrid orbitals” overlap principle suggested more than 30 years ago by Del Re. This criterion has been used as a driving force in a chemical reactivity study based on semi-empirical qualitative approaches.45 Analogous results were also reached by Toro–Labbé using the principle of maximum hardness.46 What is new in our analysis is that the dynamics simulations, based on a ab initio approach using Gaussian orbitals, well underlines this charge transfer phenomenon and its amplitude as driving force. At the same time, we find in a modern and accurate simulation qualitative descriptions and working hypothesis carried out in more approximate schemes. Reassuringly, analyses based on different theoretical backgrounds will converge toward the same description of the same phenomenon, when correctly applied.

5. Conclusion We have analyzed the mechanism of bond breaking and formation in a prototype SN 2 reaction using a static and dynamic approach, based on both DFT and localized (Gaussian) basis sets. Using the real space partition scheme of Bader’s QCT, we performed a population analysis to examine intra- and intermolecular electronic charge transfers along the dynamics trajectory and the static reaction profile. QCT analysis allows the use of well-defined molecular descriptors for the same overall phenomenon (the reaction) in two different frameworks. Our results show that similar features are present in the static and dynamic approaches, but some differences can be observed during the formation of the ion–molecule complex. In particular, the dynamic approach suggests a stronger electron exchange between the different moieties tending to spontaneously maximize both covalent and noncovalent interactions.

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Acknowledgments The authors thank Prof. V. Barone (Naples, Italie) for fruitful discussions and Dr. Michele Pavone (Princeton, USA) for helpful suggestions in the ADMP simulations.

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Chapter 15

A quantitative structure–activity relationship of 1,4-dihydropyridine calcium channel blockers with electronic descriptors produced by quantum chemical topology U. A. Chaudry, N. Singh, and P. L. A. Popelier School of Chemistry, Sackville Site, University of Manchester, Manchester M60 1QD, Great Britain

Abstract Over the years, quantum topological molecular similarity (QTMS) has been developed and used successfully in the framework of quantitative structure–activity relationships (QSARs). In time we accumulated considerable evidence that (geometry) optimised ab initio bond lengths supplemented with bond critical point properties function as reliable descriptors capturing electronic effects. In this article, this assertion is tested in the context of the well-known and most challenging medicinal QSAR of 1,4-dihydropyridine calcium channel blockers. The complexity of this QSAR is due to the varying influence of lipophilic, steric and electronic descriptors depending on the position of the substituent (ortho, meta and para). Four types of chemometric analysis were applied to QTMS descriptors generated at AM1, HF/3-21G∗ or HF/6-31G∗ level of theory: a standard partial least square (PLS) analysis of the whole data set (as well as for the ortho, meta and para subgroups alone), a genetic algorithm (GA) analysis of raw ‘variables’ followed by a PLS regression (GA-PLS), an artificial neural network (ANN) operating on principal components (PCs) of GA-selected variables (PCA-ANN) and a self-organised map (SOM) or Kohonen neural net. Quantum topological descriptors are shown to be sound substitutes of well-established classical empirical parameters such as the Hammett constant, even in convoluted QSARs where nonelectronic parameters feature strongly. It is valuable that modern quantum chemistry can now routinely deliver new well-defined descriptors to be used in complex biological QSARs. 301

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1. Introduction The 1,4-dihydropyridine (DHP) derivatives (Figure 1) are an increasingly important12 class of compounds noted for their contribution in treating cardiovascular disorders such as exertional angina and hypertension. Also known as nifedipine analogues, DHPs act directly on the voltage-dependent calcium channels, which are localised in the cell membrane. They block the flux of Ca2+ ions from the extracellular medium to the cell cytoplasm, thereby controlling many Ca2+ -dependent events and thus treating cardiovascular diseases. Apart from their role as calcium channel blockers, DHP derivatives also act as multidrug resistance reversal agents.3 Nifedipine, for example, was found to enhance the antitumour activity of cisplatin.4 DHP derivatives are often erroneously1 referred to as calcium antagonists, a name coined5 by the researchers who discovered their physiological effects. Our investigation into developing a QSAR for this set of calcium channel blockers was stimulated by an earlier publication comparing QSAR approaches including artificial neural network (ANN) and simple multiple linear regression analysis by Viswanadhan et al.6 They utilised 90 descriptors of graph-theoretic and information-theoretic type, which were compressed to eight descriptors by means of principal component analysis (PCA). Two empirical descriptors, i.e. the Leo-Hansch lipophilic constant and the Hammett electronic parameter , were added to these eight descriptors and subsequently used as inputs to a neural network. Their best model offered a good correlation r 2 = 073 although the cross-validated correlation coefficient q 2 turned out to be only 0.50. The latter coefficient suggests that the model has predictive ability no better than chance. Moreover, the network architecture consisted of more weights than observations,7 considering that 10 input nodes (plus the bias) and eight hidden nodes were used for a set of 46 compounds. Well aware of the pitfall of overtraining, the authors adopted the standard procedure of stopping the training when no improvement in the root mean square error was found for their set.

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Figure 1 Skeletonof2,6-methyl-3,5-dicarbomethoxy-4-(substituted)-phenyl-1,4-dihydropyridine derivatives

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Other qualitative and quantitative structure–activity studies, involving DHP derivatives, were published earlier.8−13 In 1979, Rodenkirchen et al.14 investigated two series of DHP congeners. They determined a significant correlation between activity and steric substituent parameters for ortho-substituted DHPs and established the importance of lipophilicity. In contrast to the studies that followed, no correlation was found between the pharmacological effect and electronic parameters such as the Hammett constant, , the field constant, F , and the resonance constant, R. Ten years later, Mahmoudian and Richards studied8 the binding of 18 DHP analogues to guinea pig ileal preparations, based on biological data taken from the literature.10 Their general conclusion was that the structural requirements for optimal activity are: (i) a hydrogen at the para position, (ii) a bulky and lipophilic group at the ortho position, and (iii) a bulky group with high Hammett constant at the meta position of the phenyl ring. A couple of years later, a substantially extended study involving a larger set of DHP analogues was undertaken by Coburn and co-workers,15 who presented a Hansch-type analysis for a set of 46 compounds. They attributed the biological activity of the DHPs to lipophilicity, an electronic term, and separated steric terms for each position on the aromatic ring. Calcium-entry blockers are a very heterogeneous group of drugs, both chemically and pharmacologically. The DHP class, although consisting of the most potent agents and in spite of the considerable amount of available data, is difficult16 to analyse in terms of valid and useful QSARs. The lowest energy congeners emerging in this study exhibit a molecular geometry consistent with that cited by Rovnyak et al.17 They proposed that the synperiplanar rotamer is the receptor-bound conformation in solution. In other words, the aryl ring is pseudo-axial and oriented in a perpendicular fashion over the DHP ring, which is a flattened boat conformation. Aryl ring substituents exert significant effects both on binding and on pharmacological activity. Para-substitution in the phenyl ring leads to a loss in the activity regardless of the substituent type. However, there is no steadfast evidence to suggest that this is indeed the conformation required for receptor binding. Nevertheless, many papers do support this orientation of the phenyl ring substituents818−20 and ester groups21−23 as shown in Figure 2. Table 1 lists the 45 DHP derivatives studied in this contribution. This selection was based on earlier work of Gaudio et al.,2 who adopted the biological activities as measured by Coburn and co-workers.24 Four compounds present in Gaudio’s work were excluded from the current study for technical reasons (see next section). Gaudio and co-workers performed a thorough examination of 45 DHP analogues with the same activity data and obtained an r 2 of 0.9.

Figure 2 Adopted molecular geometry: the plane of the substituted phenyl ring bisects the 1,4-dihydropyridine ring

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QSAR of calcium channel blockers Table 1 Biological activity log1/ECa50 for the full set of 45 DHP derivatives No.

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3-Br 2-CF3 2-Cl 3-NO2 2-CH=CH2 2-NO2 2-Me 2-Et 2-Br 2-CN 3-Cl 3-F H 3-CN 3-I 2-F 2-I 2-OMe 3-CF3 3-Me 2-OEt 3-OMe 3-NMe2 3-OH 3-NH2 3-OAc 3-OCOPh 2-NH2 4-F 4-Br 4-I 4-NO2 4-NMe2 4-CN 4-Cl 26-Cl2 F5 2-F,6-Cl 23-Cl2 2-Cl, 5-NO2 35-Cl2 2-OH, 5-NO2 25-Me2 24-Cl2 245-OMe3

a

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EC50 is the molar concentration of the drug required to inhibit 50% of the contraction of guinea pig ileum induced by methylfurmethide.15 b 2=ortho, 3=meta, 4=para. According to a strict nomenclature convention, the digits preceding the substituents should be primed.


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A genetic algorithm (GA) combined with partial least square (PLS)25 regression (GA-PLS) has previously been applied to this class of compounds. The GA-PLS method, developed in a series of papers26−30 was applied to 35 DHPs26 in modelling the antagonistic activity using 12 descriptors. The resulting model allowed for better interpretability using six descriptors q 2 = 0685 in contrast to a 12-descriptor PLS model q 2 = 0623, and selected features were found to be consistent with those in the work by Gaudio and co-workers. It is the purpose of this study to subject this notoriously complex QSAR to a newly developed method called quantum topological molecular similarity (QTMS).3132 QTMS enables the comparison of molecules, both in a supervised and in an unsupervised manner, via descriptors defined by the theory of quantum chemical topology (QCT).3334 QCT, originally developed by Bader and co-workers as a generalisation of quantum mechanics to subspaces,35 has been seized by an increasingly large community to extract chemical insight from wave functions3637 and via preliminary studies on transferability38 and computation of atom types39−41 will serve as a paradigm within which to construct a force field from electron density fragments. QTMS started42 with the idea of ‘BCP space’.43 This is an abstract (high-dimensional) space in which the bond of a molecule is represented as a vector of bond critical point (BCP) properties. Typical BCP properties34 are the electron density , the Laplacian of the electron density 2 , the ellipticity and the kinetic electron density K(r), all evaluated at the BCP. In this way, a compact ‘quantum fingerprint’ can be obtained for a given molecule. Molecular similarity was originally43 expressed as a type of Euclidean distance in BCP space, in order to make unsupervised comparisons. Later QTMS appeared only in a supervised context, where it was used to set up QSARs, in conjunction with PLS. QTMS has delivered new QSARs of medicinal,44−46 ecological47−49 and physical organic5051 nature. Over the years, strong evidence has been accumulated that QTMS is able to replace classical electronic descriptors of the Hammett type, but that it fails to supply logP (water/octanol partitioning) and steric descriptors. Of course QTMS’ electronic descriptors can be supplemented with nonelectronic descriptors, obtained outside the QCT context, as we carry out in this study.

2. Method and computational details 2.1. Ab initio data The ab initio program GAUSSIAN9852 optimised, at three different levels of theory, the initial geometries suggested by the program MOLDEN.53 Optimisation refers to finding a local minimum in the potential energy surface, which describes the molecular electronic energy as a function of the nuclear coordinates. We used, in succession, the semi-empirical method AM1, HF/3-21G∗ and HF/6-31G∗ , passing on the optimised geometry of each level as a starting geometry for the next. For the sake of consistency, all past QTMS studies have been carried out using the same five possible levels of theory. The current levels are referred to as A B and C, respectively. Basis sets are not automatically available for iodine, which occurs in three DHPs (compounds 15, 17 and 31 in Table 1). This is why they were omitted from the set of 45 compounds, together

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with compound 34, which suffered from geometry convergence problems. Thus, the dataset subject to the present QTMS analysis contained only 41 compounds.

2.2. Descriptors Once the wave function files have been generated, they are read by (a local version of) the program MORPHY98.54 This program computes the BCP properties, extracts the required bond lengths and exports them into a format convenient for subsequent statistical analysis. Note that AMl wave functions do not yield BCPs because of the absence of core orbitals.55 Viewing the (equilibrium) bond length as a BCP property, we obtain five descriptors for each bond, namely 2 K and Re . Since there are 42 bonds in the common skeleton (Figure 3), there are 42 × 5 = 210 QCT descriptors for each molecule, when described at levels B and C. The CPU time required to compute the BCP properties is marginal compared to that of generating the wave functions. No atomic properties (QCT integration over atomic basins56 ) are included in this work but they can of course be included in a QTMS analysis. To address the complexity in developing a QSAR for a class DHP compounds, descriptors outside the QCT action radius were incorporated, enabling the encoding of useful nonelectronic information. As surveyed in Table 2, a total of 229 (=210+15+1+2+1) descriptors per molecule were used in this study. The additional descriptors, constitutional (charged partial surface area),57 geometrical (shadow)58 and topological,59 were obtained from the Cerius2 package.60 Lipophilicty was encoded by values, which were taken from the literature.2 All final PLS results and graphical outputs were obtained from the program SIMCA-P,61 which constructs the latent variables (LVs) from the original descriptors (or X variables). We examined the so-called variables important to the projection (VIPs),25 also calculated by SIMCA-P, in order to interpret the models obtained. The VIPs give the relative importance of each descriptor contributing to the model. Descriptors with higher VIP scores are considered more 9 p 4

8

5

3

10 m

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2 34 31

30 29 O

33 32

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O

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13

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27 26

39

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Figure 3 Common skeleton and numbering scheme


307

Table 2 Survey of descriptors (‘X-variables’) Descriptor type

Molecular descriptors

QCT Constitutional Topological Geometrical Lipophilic

BCP properties and bond lengths Jurs’ charged partial surface area Wiener indices Shadow indices

Number of descriptors 210 15 1 2 1

relevant in explaining the activity. In order to bundle all descriptors associated with a given bond into one descriptor (or X variable) it is convenient to construct principal components (PCs) from them. The program SPSS62 performs this (unsupervised) data compression and guarantees maximally localised BCP information. The measured biological activity is the tonic contractile response of longitudinal muscle strips of guinea pig ileum. The log1/EC50 values were used as the dependent variable Y , which is the molar concentration of a DHP derivative necessary to inhibit 50% of the concentration of guinea pig ileum induced by methylfurmethide.24 The full dataset (Table 1) spans nearly six log units. We performed four types of analysis, of which the last is unsupervised: (i) a standard PLS analysis of the whole dataset, as well as for the ortho, meta and para subgroups alone. (ii) GA analysis of raw ‘variables’ followed by a PLS regression (GA-PLS).63 (iii) PCs of GA-selected variables introduced to neural network (PCA-ANN). (iv) A self-organised feature map (SOFM or SOM) (Kohonen neural network).6465

2.3. PLS modelling A matrix of 229 X variables (when BCP properties included) and 1 Y variable was introduced to SIMCA-P. Initially, a SAR was developed for the complete set of 41 compounds using all available descriptors. Subsequently, this dataset was split into ortho, meta and para substituents, and the respective models derived. For each subset, classes of descriptors (i.e. electronic, lipophilic and topological) were applied in a stepwise manner. The QCT descriptors incorporated at this stage were those obtained at the level of theory that produced the best PLS model in the previous analysis. The combinations of descriptors that significantly contributed to the PLS model were kept, and descriptors deemed redundant by SIMCA-P were excluded. Building ‘local’ SARs for the three variable positions on the phenyl ring (ortho, meta and para) were expected to reveal which properties were important in explaining the activity at various regions in the DHP structure.

2.4. PLS regression of GA selected variables (GA-PLS) The X-matrix, which consisted of QCT, constitutional, topological, geometrical and lipophilic descriptors was subjected to a GA within the Matlab toolbox.66 We refer

308


to the latter four subsets of descriptors as non-QCT descriptors. The fitness of each chromosome (combination of variables) coincides with the predictivity of the QSAR model, that is, the cross-validated correlation coefficient, q 2 . Model interrogation options were limited within the Matlab toolbox. Hence SIMCA-P performed a PLS analysis on the combination of variables (chromosome) that the GA deemed significant in modelling the activity. Redundant descriptors were discarded, and only the most important X variables entered the PLS analysis, thus improving the predictive ability of the resultant QSAR. Indeed, the best ‘chromosome’ or combination of descriptors reduced the 229 descriptor variables to only 9, a considerable improvement over the PLS analysis alone.

2.5. Neural network analysis of PCs of GA selected variables (PCA-ANN) Owing to the nature of neural network architecture, there are limitations to the number of descriptors that can be introduced to model an activity. Although the GA from the previous analysis vastly reduced the number of X variables to produce an improved PLS model, it is still too high a number to use as inputs here. To this end, PCA was used to assess the intrinsic dimensionality of those X variables selected by the GA described under the previous header. We generated only one model, which incorporated all non-QCT descriptors and the QCT descriptors at level C as this would provide the most promising results. Eight PCs were extracted as linear combinations of descriptors that explained 90% of the variance in the GA output and used as variables for each molecule. This is illustrated in Figure 4.

0

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lative

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%

Figure 4 The individual contribution (%) to the variance in the GA output and cumulative percentage of the eight principal components of the PCA-NN model


309

A feed-forward, back propagation of errors neural network was constructed67 to model any nonlinearity in the dataset. The architecture of the network consisted of two neurons in a single hidden layer. A hyperbolic tangent was chosen as the transfer function that generated the output of a neuron from the weighted sum of inputs from the preceding (input) layer. During one ‘epoch’ the vector (input) of eight descriptors for all compounds in the training set was presented to the network. The network’s output was compared with the experimental ‘target’ value; the rms error between these values was minimised by adjusting the weights in the network. Cross-validation was performed by leave-one-eighth out. The training and test subsets sampled the activity and structural space of the dataset. The unsubstituted DHP (no. 13 in Table 1) was present in all training sets. This third method of analysis applied in the present study combines the seminal work by So and Karplus6869 (involving high-dimensional datasets tackled with their genetic neural network (GNN) approach) and more recently, a comparative study of PCA and NN by Takahata and co-workers.70

2.6. Self-organised feature map (SOFM or just SOM) A Kohonen network constructs a nonlinear projection of high-dimensional pattern to a lower dimensional space, which is particularly useful for data visualisation. The SOM algorithm SOM#2824 simultaneously finds a representative set of reference vectors of the training data and positions them on a regular two-dimensional grid of neurons. Because the net is two-dimensional, it can easily be visualised. The mapping from the input space onto the grid of neurons is learnt from the training data by an iterative process in which the SOM neurons (the reference vectors) are adjusted by small steps with respect to the input vectors. Each bond is described by five descriptors, that is, the four topological descriptors mentioned above, and the equilibrium bond length. We performed a SOM analysis on the full set of compounds, and on each of the monosubstituted (ortho, meta and para) sets separately. Although the Kohonen map can handle multidimensional data, it was thought sensible to reduce the dimensionality via a PCA operating on each bond. The presence of too many input features can heavily burden the training process and can produce a neural network with many more connections than weights. The SPSS program extracted a total of 53 PCs, as some bonds are represented by more than one PC. The total number of variables for the para-substituted set, which contains six molecules, is 318 = 6 × 53. The PCs were fed into the network as input vectors. The default parameters were used in the SOM Toolbox71 apart from the size of the map grid, which was set to 10 by 10 neurons. The number of neurons was varied between 5 × 5 and 10 × 10, where the latter choice turned out to be able to separate neuron clusters clearly. Figure 5 shows how each neuron (i.e. map grid point) is connected by a line to each of its six immediate neighbours. The collection of 10 neurons per row are connected between them by nine lines in total. The two-dimensional SOM arranges similar neurons within a neighbourhood but it does not convey how similar or dissimilar adjacent neurons are. Clearly, the degree of similarity needs to be visualised in order to enhance insight into a neuron map. Amongst the several ways of displaying a trained map we chose the ‘unified distance matrix’ or U-matrix SOM#2824 in short. The U -matrix uses colour to show distances between

310

QSAR of calcium channel blockers Map grid

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Figure 5 The map grid 10 × 10 of the output space of the Kohonen network. Each dot represents a neuron

neighbouring map units. The U -matrix visualises distances between neighbouring map units and helps to see the cluster structure of the map: high values of the U -matrix indicate a cluster border, while uniform areas of low values indicate the clusters themselves. In the U -matrix, certain hexagons should be associated with corresponding map grid points, in the same position. Additional hexagons exist between all pairs of neighbouring grid points on the map grid. As a result, the current U -matrix contains 19= 10 + 9 by 19 little hexagons. For each neuron (i.e. grid point in the map), the average distances between its weight vector and those of its immediate neighbours are computed and plotted.

3. Results and discussion 3.1. PLS modelling A PLS analysis using the entire dataset and the complete set of descriptors was carried out. We constructed three models of progressive strength with both the non-QCT descriptors and the QCT descriptors at each level of theory. Unfortunately, no model yielded a q 2 better than chance, that is, q 2 never exceeded 0.5. At best, a one-component model was derived when incorporating QCT descriptors at level C in conjunction with other descriptors, yielding r 2 = 0617 and q 2 = 0465. The use of bond lengths (level A) offered r 2 = 0534 and q 2 = 0441, while the poorest model obtained with level B QCT descriptors yielded r 2 = 0572 and q 2 = 0358. Given the poor performance for the full set, polysubstituted compounds (36, 38, 39, 40, 41, 42, 43, 44 and 45 in Table 1) were not considered in subsequent analyses. Only the data pertaining to the unsubstituted compound plus ortho-, meta- and para-monosubstituted compounds were considered. PLS models were developed via stepwise additions of classes of descriptors encoding electronic (QCT), lipophilic and size/polar interactions (geometric/topological and constitutional respectively). In this fashion, activity at each of the three monosubstituted positions was modelled by the most important parameters.


311

2.00 1.80 1.60 1.40

VIP[2]

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ell0904

LogP

ell2725

lap1830

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ell1830

rho1830

rho1318

K1318

lap1318

0.00

Figure 6 VIP plot of descriptors for the PLS model constructed for ortho substituents when only QCT descriptors (obtained at level C) and the lipophilic parameter were considered

The ortho-substituted DHPs precipitated PLS models where QCT descriptors alone gave r 2 = 0960 and q 2 = 0642, but when combined with , produced r 2 = 0962 and q 2 = 0649. The contribution by may not seem significant until its VIP ranking 9th and contribution to the model (via regression coefficients) is observed (Figure 6). Descriptors encoding size and polar interactions failed to provide a model at all. Furthermore, compound 28 (2-NH2 ) was found to be a mild outlier via a scores plot. Such anomalous behaviour was noted by Gaudio et al. and could be due to this substituent’s ability to ionise at physiological pH. Meta-substituted DHPs offered a good model with size and polar interactions descriptors, r 2 = 0859 and q 2 = 0790, while QCT descriptors alone afforded r 2 = 0820 and q 2 = 0583. In contrast to ortho-substituted DHPs, was found not to be important at the meta position. Finally, the para-substituted DHPs only produced a model with size and polar descriptors, r 2 = 0965 and q 2 = 0875. It is important to note that no model could be fitted with QCT descriptors alone in spite of the abundance of descriptors (more than four times the number of observations). Based on previous work in our laboratory, we can safely conclude that the current QCT descriptors capture only electronic effects and not steric effects. The fact that no model is produced with QCT descriptors means that electronic descriptors do not feature for the para-substituted compounds. This conclusion is consistent with the work of Gaudio et al.2 It also increases our confidence in the chemical meaning of the correlations we find because the mere abundance of irrelevant descriptors (i.e. electronic ones) does not lead to a model. Figure 7 summarises the types of descriptors that generate a successful model for each of the three possible substituent positions on the phenyl ring. This summary is consistent with that of Gaudio and co-workers.2 The dual action of steric and electronic

312


Figure 7 Localisation of most important descriptor types in terms of substituent position (ortho; meta and para)

effects indicate the complex interaction DHPs have with the receptor site and convey the difficulties in modelling such classes of compounds with a single SAR.

3.2. GA-PLS modelling GA-PLS enabled a description of the activity with a much reduced number of descriptors. The population of descriptors producing the strongest model varied when incorporating QCT descriptors from the three levels (A, B and C). Table 3 summarises the final GA-PLS models. Clear improvements can be noted. Excluding a significant amount of redundant data improves the present models over PLS analysis alone. Secondly, the increase in the size of the descriptor set on going from level A (just bond lengths) to level B does not necessarily improve the predictive capacity, despite a better fit (i.e. higher r 2 ). The latter point reminds us of the importance of cross-validation. Thirdly, level C is superior in both r 2 and q 2 . All GA-PLS models passed the validation criteria upon randomisation of the Y values. Figure 8 shows the observed versus predicted log1/EC50 values for the best GA-PLS model (level C). Figure 9 shows the VIP plot produced by the strongest GA-PLS chromosome consisting of all non-QCT descriptors plus QCT descriptors at level C. The strongest GA-PLS model identified the partial positive surface area (PPSA) and surface weighted charged partial surface area (WPSA) as the two most important descriptors. The former

Table 3 Summary of GA-PLS models Descriptors (non-QCT = constitutional, geometrical, topological and lipophilic) + bond lengths (level A) + QCT (level B) + QCT (level C) a b

Total number of variablesa

LVb

r2

q2

15 50 31

2 2 2

0.775 0.816 0.878

0.666 0.594 0.734

Number of variables present in the fittest chromosome selected by the GA. Number of latent variables (LV) extracted by the PLS method.


313

Figure 8 Observed log1/EC50 values versus values predicted from regression by the strongest GA-PLS chromosome consisting of all non-QCT descriptors plus QCT descriptors at level C

2.00

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ell4112

ell1415

rho1830

K1830

dist0409

ell1314

lap1716

Jurs-WPSA-

Jurs-PPSA-

0.00

VIP

Figure 9 VIP plot produced by strongest GA-PLS chromosome consisting of all non-QCT descriptors plus QCT descriptors at level C

descriptor seems to capture the overall positive binding charge profile of the phenyl ring that would favour the docking of the DHP.2 QCT descriptors corresponding to bonds C16 C17 C13 C14 and C4 X9 appear third, fourth and fifth respectively in the VIP plot. The two former bonds are on the dihydropyridine ring (Fig. 3) and do not convey any immediate importance. However, the para-position on the phenyl ring is indirectly involved in controlling the activity. Bulk at this position is known to displace the molecule from an ideal docking position, thus weakening drug–receptor interactions and thus decreasing the biological activity.

314


3.3. PCA-ANN The network was able to provide very good fit r 2 = 0972, but the predictive ability was only slightly better than chance, q 2 = 051. Figure 10 compares the observed and predicted log1/EC50 values. The resulting model is consistent in its predictive ability with that of Viswanadhan et al.6 However, it seems that although a better r 2 value was obtained here, overfitting by Viswanadhan et al. was not detrimental to the q 2 of their model. The low q 2 value obtained in the present work could be attributed to the small dataset. Ideally, there should have been at least four times as many observations as weights. To this end, applying a neural network to 41 observations was somewhat optimistic. Nonetheless, this was a first attempt at implementing a nonlinear method into the QTMS tool and shows that it is possible to assimilate new methods into our existing QSAR procedure.

3.4. SOM (Kohonen map) The three subsets of compounds (ortho-, meta- and para-substituted DHPs) were subject to a separate SOM analysis. The most pronounced clustering pattern appeared for the para-substituted class, which is the only one we report on here. The U -matrix is shown in Figure 11. Two clusters emerge, one in dark red hues at the top right corner and one in dark blue extending over the bottom half of the map. The dark red cluster corresponds to neurons associated with the PCs of bonds C4 X9 and C1 C6 . The SOM tells us that this set is very different from the other PCs, without explaining exactly why. In spite of several attempts to link this result to the (supervised) PLS analysis, we cannot present a rigorous and clearcut case at this stage and more work is needed. It is just very interesting to note that in terms an unsupervised analysis is splitting off these electronic QCT descriptors from the others.

9 8

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7 6 5 4 3 2

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9

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Figure 10 Observed versus predicted log1/EC50 values from PCA-NN study


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Figure 11 U -matrix of trained Kohonen map for the PCs of all para-substituted 1,4dihydropyridines

4. Conclusion The QTMS technique, which we have been developing over the years, is tested by applying it to a well-known and most challenging QSAR, the calcium channel blockers DHPS. The action of these compounds is complex because substituents (attached to the phenyl ring) act in different ways depending on their position (ortho, meta and para). This complexity is reflected by the type of descriptor that is able to account for the observed activity. We show that even in this convoluted activity pattern, the quantum topological descriptors are able to capture the electronic effect and hence are able to replace the classic Hammett parameters. If the latter type of electronic descriptor is not known for a given substituent, then QTMS can still proceed by taking advantage of currently available computer power and mature ab initio algorithms. Considerable improvements in statistical coefficients and a substantial reduction in the number of descriptors were realised when combining a GA with the standard PLS technique. Taking a stepwise approach in building models at various regions in the molecules and introducing different combination of descriptors provided a picture consistent with previous studies of other researchers. Application of a neural network showed promise in fitting the data, confident that any overfitting had not occurred. However, due to the modest size of the dataset, the network’s ability could not be properly tested. Also, the network did not offer the same level of insight into the activity of DHPs as that gained by the GA-PLS analysis. Nevertheless, quantum topological descriptors have been shown to be sound substitutes of well-established classical empirical parameters such as the Hammett constant and to work well with nonelectronic parameters.

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Index

Acetylene, 77–80, 157, 169 Activation energy, 88, 96, 150, 153, 155, 177, 263, 264, 290 Adamo, C., 287 Aizman, A., 139 Alikhani, M. E., 47 Aromaticity, 33, 42, 63, 69, 77, 79, 112–13, 203, 204–210, 211, 213, 214–15 Atoms in molecules (AIM), 14, 59, 62–3, 67, 206 Ayers, P. W., 1, 3, 5, 33

De Proft, F., 1, 2, 8 Density functional theory (DFT), 1, 19, 31, 102, 120, 122, 140, 203, 269, 272, 287, 288 Diels–Alder reactions (DA), 101, 109, 110, 142, 144, 147, 150, 153, 156, 158 Domingo, L. R., 139 Duran, M., 31 Dyson orbitals, 87, 88, 89, 92, 98 Electron affinity, 12, 23, 32, 123, 125, 189, 206 Electron density, 1, 3, 4, 5, 6, 8, 10, 11, 12, 15, 20, 22, 28, 32, 47, 48, 50, 59, 61, 62, 63, 64, 67, 72, 73–5, 78, 90, 91, 93, 95, 102, 103, 148, 167, 169, 177, 191, 196, 223, 225, 228, 258, 274, 277, 288, 289, 291, 297, 305 Electron-donating group (EDG), 105, 189 Electron localization function (ELF), 50, 51, 57, 82, 87, 88, 99, 121, 205, 258, 288 Electron withdrawing group (EWG), 106, 108–11, 112 Electronegativity, 2, 7, 8, 19, 23–7, 32, 102, 107, 112, 122, 123–4, 133, 141, 142, 177, 183, 188, 189, 191, 216, 221, 269, 270, 274, 283 Electrophile, 22, 23, 25, 102, 105, 106, 107, 108, 112, 113, 114, 115, 124, 125, 126, 128, 129, 133, 139, 140, 141, 142, 143, 144, 145, 146, 147, 150, 151, 152, 155, 156, 157, 161, 162, 164, 165, 166, 167, 168, 169, 170, 176, 177, 178, 189, 190, 193, 196 Electrophilicity index, 19, 25, 27, 139, 140, 141, 143, 148, 155, 157, 159, 168, 170, 171, 172, 174, 175, 176, 177, 178, 179, 180, 181, 183, 185, 186, 195–7, 270 Ensemble, 8, 9, 10, 273, 289 Ethane, 72, 73, 261

Bader, R. F. W., 5, 47, 48, 58, 59, 62, 63, 67, 124, 206, 287, 288, 298, 305 Benzene, 35, 36, 39, 40, 77, 79, 80, 112, 125, 128, 129, 176, 205 Benzocyclobutadiene, 41, 43 Bifurcation, 48, 49, 52, 67–8, 69, 70, 73, 78, 79, 80, 205 Blancafort, L., 31 Bond length alternation (BLA), 34–7, 38, 39, 40, 41, 42, 43 Carbenes, 139, 187–9, 190, 196 Catastrophe (theory), 47–8, 49–50, 53, 54, 73 Cedillo, A., 2, 8, 19 Chamorro, E., 57 Chattaraj, P. K., 31, 33, 34, 41, 269 Chaudry, U. A., 301 Chemical potential, 2, 8, 10, 20, 22, 23, 24, 25, 27, 32, 34, 48, 50, 102, 103, 123, 140, 141, 146, 147, 148, 164, 167, 188, 191, 270, 273, 274, 275, 277, 279 Ciofini, I., 287 Conceptual DFT, 2, 32, 50, 102, 103, 141 Contreras, R., 139 Cyclic hydrocarbons, 204, 206, 208, 209, 261, 263, 264, 265 Cyclobutane, 36, 39, 129 319

320 Ethylene, 128, 130, 142, 143, 146, 147, 149–53, 157, 178, 183, 185, 186 Excited state, 33, 36–8, 271, 272, 273, 275, 276 External potential, 1, 3–7, 8, 20, 23, 24, 25, 27, 28, 31, 32, 33, 34, 41, 102, 103, 197, 269, 271, 272, 273, 276 Fourré, I., 47 Friedel–Crafts, 139, 176, 196 Frontier molecular orbital (FMO), 101, 143 Fuentealba, P., 57, 274 Fukui function, 8, 10, 11, 12, 19, 20, 21, 22, 25, 26, 27, 50, 51, 102–104, 105, 114, 121, 141–2, 156, 157, 159, 166, 167, 168, 170, 190, 193–4, 270, 274 Geerlings, P., 1, 2 Global properties, 22, 25, 48, 50, 144, 153, 154, 162, 163, 164, 169, 178, 179, 192 Gold clusters, 219–22, 225, 226, 228, 230, 233, 234, 235, 238, 239, 240, 241, 243–4 González, P., 203 Grand, A., 101 Ground state, 3, 5, 7, 9, 13, 20, 21, 22, 32, 37, 38, 81, 82, 119, 122, 123, 140, 144, 147, 150, 153, 154, 158, 162, 164, 166, 167, 169, 171, 176, 177, 179, 184, 187, 189, 190, 192, 219, 271, 273, 275, 276, 282 Gutiérrez-Oliva, S., 101 Hammond (postulate), 48, 102 Hardness (MHP, PMH), 32, 33, 34, 35, 37, 102, 103, 105, 115 Hartree–Fock (HF), 55, 60, 89, 119, 274 Heine, T., 203 Hohenberg–Kohn (HK), 5, 26, 103, 273, 274 Holographic (electron density theorem), 1, 4, 5, 6 HOMO (highest occupied molecular orbital), 26, 32, 35, 36, 37, 38, 91, 101, 105, 106, 141, 143, 152, 159, 160, 168, 169, 204, 208, 211–13, 221, 226–8, 230, 252, 257, 264, 265, 270 HSAB (hard–soft acid–base) principle, 32, 102, 103, 115, 155, 269, 282, 283 Hydrogen bonding, 140, 193, 221, 222, 223, 224, 229, 234, 238, 239, 243, 244 Information (theory), 13, 14, 270 Ionization energy, 3, 12, 97, 119, 120, 122, 123, 124, 131, 133

Index Jahn–Teller, 32, 38, 44 Joubert, L., 287 Kernel, 21, 22, 25, 28, 29, 88, 274 Koopmans, 32, 38, 96, 97, 119, 120, 206 Kryachko, E. S., 219 Lewis (acid, base), 22, 23–5, 27, 28, 57, 58, 81, 98, 140, 146, 152, 155, 189, 213, 287 Local properties, 22, 23, 25, 29, 50, 102, 124, 150, 158, 167 Luis, J. M., 31 LUMO (lowest unoccupied molecular orbital), 26, 32, 35, 36, 37, 38, 102, 141, 143, 152, 159, 160, 212, 213, 221, 226, 227, 243, 244, 257, 270 Melin, J., 87 Méndez-Rojas, M. A., 251 Merino, G., 203, 251 Metal clusters, 203, 207, 208, 209, 216 Methane, 72, 252 Morell, C., 101 Murray, J. S., 119, 128 Naphthalene, 36, 41 Neural network, 301, 302, 307, 308, 309, 314, 315 Nucleophile, 22, 23, 28, 101, 105, 106, 107, 108, 112, 113, 115, 140, 141, 143, 145, 146, 147, 150, 151, 152, 153, 155, 156, 157, 161, 162, 164, 165, 167, 168, 170, 178, 180, 187, 189, 196 Ortiz, J. V., 87 Parr, R. G., 2, 8, 14, 31, 32, 33, 34, 41, 102, 122, 139, 140, 189 Partial least square (PLS), 301, 305, 306, 307, 308, 310, 311, 312, 313, 314, 315 Pearson, R. G., 25, 33, 37, 102, 103, 146, 147, 269 Pérez, P., 139 Pericyclic, 72, 73, 74, 146, 150, 153, 155 Phenanthrene, 36, 42 Poater, J., 203 Polarizability (MPP), 31–2, 33, 34, 35, 36–7, 39, 41, 43, 44, 130–3, 183, 193, 197, 221, 270, 273, 274, 275, 276, 282, 283

Index Politzer, P., 119, 120, 125, 274 Popelier, P. L. A., 48, 289, 301 Propene, 72, 105, 106 Proton (acceptor, donor, transfer), 221, 222, 223, 230, 233, 234, 235, 237, 238, 240, 243 Pyrazine, pyridine, pyrimidine, 35, 36, 38, 41 Quantitative structure–activity relationship (QSAR), 301, 302, 303, 305, 306, 308, 314, 315 Quantum fluid, 271, 272, 283 Rate coefficient, 170–82, 196, 197 Reactivity–selectivity (descriptors), 101–103, 104, 106 Regioselectivity, 108, 109, 141, 143, 152, 155, 156, 157, 159, 161, 166, 167, 168, 169, 170, 181, 182, 196, 271, 282, 283 Remacle, F., 219 Robles, J., 203 Rydberg, 87, 88, 91, 92, 93, 94, 95, 97, 98, 277, 283

321 Santos, J. C., 57, 205 Sarkar, U., 269 Seabra, G., 87 Selectivity, 101, 108, 112–13 Shape function, 1–3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15 Silvi, B., 47, 57, 64, 66, 67, 68, 82 Similarity (index), 13 Singh, N., 301 Softness, 2, 10, 11, 21, 22, 25, 27, 33, 36, 37, 44, 51, 102, 103, 130, 142, 164, 271, 273 Solá, M., 31 Superelectrophilicity, 139, 189–95, 196 Topology (topological analysis), 48, 57, 59, 64, 68, 69, 73, 88, 90, 91, 93, 95, 97, 102, 104, 258, 287, 288, 301, 305 Toro-Labbé, A., 101, 245, 298 Torrent-Sucarrat, M., 31 Transition state, 29, 72, 73, 74, 75, 88, 96, 98, 99, 102, 145, 146, 148, 149, 150, 152, 155, 156, 159, 164, 166, 168, 170, 176, 196, 254, 259, 288, 289, 289–93, 296 Vela, A., 251


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