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CHEMISTRY RESEARCH AND APPLICATIONS

QUANTUM FRONTIERS OF ATOMS AND MOLECULES No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

CHEMISTRY RESEARCH AND APPLICATIONS Additional books in this series can be found on Nova’s website under the Series tab. Additional E-books in this series can be found on Nova’s website under the E-book tab.

CHEMISTRY RESEARCH AND APPLICATIONS

QUANTUM FRONTIERS OF ATOMS AND MOLECULES

MIHAI V. PUTZ EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Quantum frontiers of atoms and molecules / editor, Mihai V. Putz. p. cm. Includes index. ISBN 978-1-61324-867-6 (eBook) 1. Chemical bonds--Mathematical models. 2. Dirac equation. 3. Quantum chemistry. I. Putz, Mihai V. QD461.Q36 2009 541'.28--dc22 2010001746

Published by Nova Science Publishers, Inc. † New York

CONTENTS Foreword

ix

Chapter 1

Fulfilling Dirac’s Promise on Quantum Chemical Bond Mihai V. Putz

Chapter 2

Duality within the Structure of Complementarity: Right Where It Has No Place to Be Constantin Antonopoulos

21

Chapter 3

Complementarity Out of Context: Essay on the Rationality of Bohr’s Thought Constantin Antonopoulos

41

Chapter 4

Molecular Integrals over Slater-Type Orbitals. From Pioneers to Recent Developments P.E. Hoggan, M.B. Ruiz and T. Özdoğan

61

Chapter 5

Tunneling Dynamics and Its Signatures in Coupled Systems S. Ghosh and S.P. Bhattacharyya

91

Chapter 6

Theoretical Calculation of the Low Laying Electronic States of the Molecular Ion CsH+ with Spin-Orbit Effects M. Korek and H. Jawhari

111

Chapter 7

Theoretical Explanation of Light Amplifying by Polyethylene Foil Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski and Bratislav Tosic

141

Chapter 8

Anharmonic Effects in Normal Mode Vibrations: Their Role in Biological Systems Attila Bende

157

Chapter 9

Emergent Properties in Bohmian Chemistry Jan C.A. Boeyens

191

1

vi

Contents

Chapter 10

The Algebraic Chemistry of Molecules and Reactions Cynthia Kolb Whitney

217

Chapter 11

Quantum and Electrodynamic Versatility of Electronegativity and Chemical Hardness Mihai V. Putz

251

Chapter 12

Physics and Chemistry of Carbon in the Light of Shell-Nodal Atomic Model G.P. Shpenkov

277

Chapter 13

Molecular Modeling of the Peanut Lectin - Carbohydrate Interaction by Means of the Hybrid QM/MM Method Alexei N. Pankratov, Nikolay A. Bychkov and Olga M. Tsivileva

325

Chapter 14

Electron Density Distributions of Heterocycles: A Shortcoming of the Resonance Model Ricardo A. Mosquera, Marcos Mandado, Laura Estévez and Nicolás Otero

343

Chapter 15

Electromerism in Small Molecule Activation by Metal Centers of Biological Relevance Radu Silaghi-Dumitrescu

367

Chapter 16

Structural Modelling of Nano-Carbons and Composites Mihai Popescu and Florinel Sava

387

Chapter 17

Nanostructure Design—between Science and Art Mircea V. Diudea

425

Chapter 18

Quantifying Structural Complexity of Graphs: Information Measures in Mathematical Chemistry Matthias Dehmer, Frank Emmert-Streib, Yury Robertovich Tsoy and Kurt Varmuza

479

Chapter 19

Topological Indices of Nanostructures Ali Reza Ashrafi

499

Chapter 20

On Uniform Representation of Proteins by Distance Matrix M. Randić, M. Vračko, M. Novič and D. Plavšić

521

Chapter 21

Timisoara Spectral – Structure Activity Relationship (SpectralSAR) Algorithm: From Statistical and Algebraic Fundamentals to Quantum Consequences Mihai V. Putz and Ana-Maria Putz

539

Chapter 22

On Plots in QSAR/QSPR Methodologies Emili Besalú, Jesus Vicente de Julián Ortiz and Lionello Pogliani

589

Contents

vii

Chapter 23

Application of the Fuzzy Logic Theory to QSPR-QSAR Studies Pablo R. Duchowicz and Eduardo A. Castro

607

Chapter 24

Modeling the Toxicity of Alcohols. Topological Indices versus Van Der Waals Molecular Descriptors Dan Ciubotariu, Vicentiu Vlaia, Ciprian Ciubotariu, Tudor Olariu and Mihai Medeleanu

629

Index

669

FOREWORD ATOMS AND MOLECULES AS THE QUANTUM FRONTIERS OF LIFE Elements of Nature: elements of Life: atoms and molecules. From the earlier time of rational modeling of the Universe, the atomistic vision (Leucippus & Democritus) gave the mechanistic vision upon which all objects are created by unitary elementary entities; they eventually later found the autonomic conceptualization by Leibniz’s monads — the veritable frontiers, two-sided coins, of Creation. On one side they realize the first step from the nonorganized vacuum towards the matter structure, while on the other hand preserve the entangled connection and unity between all the created/observed (and beyond) world. From the epistemological point of view, the Atom at the base – frontier of micro-to-macro Universe marks a drastic unification of the ancient beliefs and myths of the Elements of Nature: water, earth, fire, air – all of them being in one way or another fluidization, sublimation, exaltation or aeration of atomic systems in various conjugations or composites; moreover, Plato’s geometric elements attributing the octahedrons to air, cubes to earth, tetrahedrons to fire, icosahedrons to water, while dodecahedrons build the constellation and paradise – appears as a further atomic combination, yet in a more organized paradigm. Atoms therefore — the first frontier of microcosm, the most pre-eminent physical system in which the fermions (electrons) live in a boson environment (atoms as a whole) — so simple in principle to describe and speculate upon, so difficult to model and quantify. The reason is because of being material and also immaterial, since it spans in principle the whole Universe by its wavefunction. Moreover, all living Nature behaves like its most simple benchmark: Hydrogen. We put under the Hydrogenic wave-function: life and death follows its localized-delocalized curvatures, all events encountered as the Hydrogen wave-function shape; acceleration of particles and bodies, the maximum metastable-equilibrium — the climax of life, followed by the smooth descent toward the zero ground of being, leaving nevertheless space for believers regarding the eventual unification of minus and plus infinites, such that all of Nature works in cycles, with the satisfaction of energy conservation. This is Nature — this is Physics; but maybe it is not enough. Just when more than one electron is stabilized in an atomic system — the atomic world diversifies in such manner that the multiple can spring out of Monads, the Elements become now the Periodic Elements, the Chemistry arises. As such the Chem listens by the words of his father Noah in bringing new life in the Universe by dissemination, and cross-fertilization of the earth’s seeds, as such the cheos gives the fluids of Life, and archeos

x

Mihai V. Putz

the archetypes of living, the paradigms of Nature. Living with more than one electron in an orbital, the appearance of duality, of spinning, that’s Chemistry. As a name enough obscure, partly mythical from the dark side of Egypt, partly theological from the sacral combination it provides (El-Chemia: Alchemy) towards the molecular world. With this we arrive at the bonding mystery; it is called in general the chemical bond, but stays as the frontier of the bigger world to come: the Bios. The biological component of Nature may be approached through the generalization of the chemical bonding idea, or the resonance between the ionic and covalent bindings, to the ligand-receptor one. Such picture is fully feasible after rooting in the hard and soft acids and bases theory in which terms any chemical reaction can in principle be formulated. On the other hand, for the bio-, eco-, and pharmaco-logical involved systems the chemical bond appears unstable respecting the isolated systems, due to the environment damping; in other words, a toxicological action (meaning a chemical binding with a toxic effect) is diminishing in time as a consequence of metabolic actions (intern factors) and of the environment (extern factors). This way, it raises the challenge in formulating a theory of the chemical bond variable in time at the biomolecular level, specialized on the in vitro enzymic case, and then extended to the in vivo situations in order to better cover the record of the (non)toxic or therapeutic actions in organisms and the environment. Recent attention was jointly focused on eco-, bio-, as well as on pharmacosciences. In this stage new pharmacophoric reactivity indices may be formulated with the help of quantum molecular topology combined with the graph theory (due to its versatility to be iteratively computed from atoms to molecules) producing a direct quantification of the toxic or therapeutic effect for a given chemical. The Hydrogenic atoms, the many-electron atoms, and the (bio-)molecules: the Physics, Chemistry, and Biology — and the wholeness they cover in a rheological manner with the help of the quanta, fields and particles, in reciprocal transformation, combination, excitation — the thin red line of the Universe. Life’s frontiers —the quantum-verse, the true nature of the uni-verse! With the actual ever-expanding developments of the edge technology with direct impact on life and environment, a lucid review of the main foreground conceptual realms of electronic matter at the level of atoms and molecules is by this volume unfolded aiming to offer a unitary perspective of the quantum principles as applied to many-electronic states, either in isolate and interacting context, to chemical bond and bonding as well to the relation between the physical-chemical structure and chemical-biological reactivity and activity manifestations of nature. The volume widely gives larger and deeper coverage of the electronic matters through the physical, chemical and biological ordered systems, with their increasing complexity through gradually presenting the matter structure from the physical to chemical to biological manifestations in an inter-disciplinary cross-fertilization manner. With equilibrated contents provided by important scientists worldwide with a valuable impact on quantum fundaments and applications, the book presents and reviews the avant-garde contributions for the XXI century. In fact, the present volume steps aside to serve for the unification of the physicalchemical-biological manifestation of atoms in molecules and in nanostructures by means of expanding the quantum frontiers by conjunction with either relativity or topological or information or graph theories as well. The book successfully balances among the physical, chemical and biological sides of the quantum theory and of its applications emphasizing both conceptual and computational sides while experiment is addressed only for reference; in this regard the book is more focused on why rather than how quantum effects are produced with

Foreword

xi

the inner belief that this way the second issue is self-contained in the first one. The book is addressed to a large audience as well as to advanced research wisely combining the epistemological, heuristic and philosophic aspects of quantum manifestation of matter in atoms, molecules and of their combination in complex nano- and bio- structures. On the other hand, the book likes to show how far the quantum theory furnishes the background and the framework in which the simple as well as most complex electronic structures may unfold and evolve in an interacting environment. Overall, searching the unity of the manifestation forms of the chemical bonding at various levels of matter organization had become a very active interdisciplinary field in the last years, being one of the main goals in the frame of the nanosciences. The quantum paradigm of bonding unification through the formulation of a minimal set of concepts and quantities having as much universal multi-electronic relevance as possible represents a real challenge for the conceptualization and prescription of the viable applicative directions of the nanosystems, from atoms to biomolecules. For that we are certain that the present edited volume will have a special role for making further advancement in improving the scientific knowledge in this priority domain of research. And last, but not least, Editor and Authors like to sincerely thank all the NOVA team involved in the present challenging editorial venture, and to NOVA Vice-president Nadya Columbus especially, for kind assistance and patience throughout all stages of publication. Mihai V. PUTZ (Volume Editor) Chemistry Department West University of Timişoara Romania

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 1-19

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 1

FULFILLING DIRAC’S PROMISE ON QUANTUM CHEMICAL BOND Mihai V. Putz* Laboratory of Computational and Structural Physical Chemistry, Chemistry Department, West University of Timişoara, Str. Pestalozzi No.16, Timisoara, RO-300115, Romania

Abstract One of the most fundamental issues of quantum chemistry, the forming and electronic description of bonding, is here unfolded at the level of Dirac’s theory, in conjunction with the density functional concept of chemical action or, equivalently, with the beloved electronegativity for practical applications of encountering atoms in molecules. The resulting chemical bonding equation allows for the first time the geometrical identification of molecular region along bonding in which the parallel and anti-parallel spin-electrons coexist in antibonding and bonding states, respectively.

1. Introduction After the Nobel Prize 1933 jointly awarded to the “productive” contribution on atomic theory for Schrödinger and Dirac’s theories of electrons, there was created the impression that at least the chemistry world of phenomena will be entirely explained [1-17]. This is not minor, because of the huge importance the Chemistry of atoms and molecules plays in life and in metabolism [18]. Yet, soon it became clearer that the chemical systems are not as simple as the physical models intend to imply; for instance, no spherical molecule exists to transfer upon the principles of special orthogonal groups; or even more complicated the chemical systems consist of many—but numerable—electrons, therefore not amendable with the thermodynamic physical limit ( N → ∞ ); or more subtle, the chemical bond is supporting *

E-mail addresses: [email protected], [email protected]. Tel: +40-256-592633; Fax. +40-256-592620,

2

Mihai V. Putz

both the parallel and anti-parallel spins in different states, inversely known as anti-bonding and bonding states, emphasizing that the first instance is firstly involved in the chemical reactivity through belonging to the so-called valence state [3-5]. Although the molecular orbital theory attempted to explain this intriguing behavior by the anti-symmetry rule of a given state’s spin-electron wave function, i.e. the total Pauli antisymmetry wave function may be obtained either by convoluting symmetrical spin with antisymmetrical coordinate wave functions—for anti-bonding or anti-symmetrical spin with symmetrical coordinate wave function—for bonding states, this remains only as a qualitative description with no practical effect on geometrical description of anti-bonding parallel spins’ coexisting along the bonding [11-13]. One can say that since electrons are waves they virtually occupy all space of bonding with an anti-symmetrical coordinate wave that eventually does not interfere with that symmetrical one coming from the bonding wave; very true, but also this seems a qualitative argument [8]. Therefore, one should be next interested in describing the geometrical locus along the chemical bond where the bonding and anti-bonding waves behave as within resonators (since they are stationary in the formed molecule), and eventually explaining why they do not interfere (i.e., they are orthogonal states belonging to separate Hilbert spaces); and all these with the help of Dirac’s theory directly since the spin problem is aiming to be responded, and not by employing the Schrödinger indirect argumentum of coordinate symmetry [9]. To this end, the present work likes to step forward in elucidating the mystery of “the chemical bond” through combining the fundamental Dirac equation with spinorial solution, generalized from the Schrödinger equation, while being combined with the recently advanced chemical action description of bonding [19,20].

2. Dirac-Schrödinger Equivalence There is a known fact that the basic Dirac equation unfolds as the temporal generalized operatorial form [1,21-23]

i=∂ t [Ψ ] = Hˆ Dir [Ψ ]

(1)

in a very similar shape with the Schrödinger one, however with the Dirac Hamiltonian specialization: 0 Hˆ Dir = Hˆ Dir + vˆ( x)

(2a)

with the free particle and applied potential components:

G Gˆ 0 Hˆ Dir = −i=cαˆ ⋅ ∇ + mc 2 βˆ ,

(2b)

vˆ( x) = V ( x) βˆ ,

(2c)

Fulfilling Dirac’s Promise on Quantum Chemical Bond

3

respectively, with m - the particle mass, c - the light velocity, = - the Planck constant, while

G the introduced special operators αˆ , βˆ assume the Dirac 4D representation:

⎡0

αˆ k = ⎢ ⎣σˆ k

σˆ k ⎤

, k = 1,2,3 , 0 ⎥⎦

⎡1ˆ

0⎤ ˆ⎥ ⎣0 − 1⎦

βˆ = ⎢

(3a)

(3b)

in terms of bi-dimensional Pauli and unitary matrices (operators)

⎡1 0⎤ ⎡0 1 ⎤ ⎡0 − i ⎤ ⎡1 0 ⎤ , σˆ 1 = ⎢ , σˆ 2 = ⎢ , σˆ 3 = ⎢ ⎥ ⎥ ⎥ ⎥ ⎣0 1 ⎦ ⎣1 0 ⎦ ⎣i 0 ⎦ ⎣0 − 1⎦

σˆ 0 = 1ˆ = ⎢

(4)

and with the wave function featuring the so called spinorial (bi-dimensional) equivalent formulation

⎡ϕ ⎤ − =i E⋅t [Ψ ] = ⎢ ⎥ e ⎣φ ⎦ ⎧⎡ϕ ⎤ − =i E ⋅t , E > 0 anti − bonding states ⎪⎢ ⎥ e ⎪⎣ 0 ⎦ =⎨ i ⎪⎡ 0 ⎤ e + = E ⋅t , E < 0 bonding states ⎪⎢φ ⎥ ⎩⎣ ⎦

⎡ − =i E ⋅t ⎤ ϕe = ⎢ i ⎥ (5) ⎢ + = E ⋅t ⎥ ⎢⎣φe ⎥⎦ However, there also arises the question whether the general Dirac equation (1) may be reduced or transformed so that to represent the eigen-equation for the electronic states for a given quantum system. For this, through closely analyzing the form of Eq. (1) with all its contribution, one may resume the free motion Dirac operator to the working form [8]

∂ ∂ ∂ ∂ Dˆ = −i=Aˆ σˆ 0 + i=σˆ 1 + i=σˆ 2 + i=σˆ 3 + Cˆ ∂t ∂x1 ∂x 2 ∂x3 and employing it to the stationary operatorial equation:

(6a)

4

Mihai V. Putz

⎡ Ψ e i (kx−ωt ) ⎤ ⎡ζ ⎤ 0 = Dˆ [Ψ ] = Dˆ ⎢ 1 −i (kx−ωt ) ⎥ = Dˆ ⎢ ⎥ ⎣ξ ⎦ ⎣Ψ2 e ⎦

(6b)

with the oscillatory phase written so that to be in accordance with the prescription of eq. (5) for the Planck energy-frequency identification:

E = =ω

(7)

In these conditions, one notes that for the time and coordinate derivatives yields:

−ζ ∂ [Ψ ] = iω ⎡⎢ ⎤⎥ , ∂t ⎣ξ ⎦

(8a)

−ζ ∂ [Ψ ] = −ik k ⎡⎢ ⎤⎥ ∂xk ⎣ξ ⎦

(8b)

which reduce the above stationary condition (6) to the form

(=ωAˆ1ˆ + =k σˆ 1ˆ)⎡⎢−ξζ ⎤⎥ + Cˆ ⎡⎢ζξ ⎤⎥ = 0 k

k

⎣

⎦

(9)

⎣ ⎦

Choosing now appropriately (that stands for the optimization procedure) the matrices

⎡ 0 0 ⎤ ˆ ⎡0 2 m ⎤ Aˆ = ⎢ ⎥ , C = ⎢0 0 ⎥ ⎣1 0⎦ ⎣ ⎦

(10)

the last form (9) further rearranges as

⎡=k k σˆ k ⎢ ˆ ⎣ =ω1

2m1ˆ ⎤ ⎡− ζ ⎤ ⎥⎢ ⎥ = 0 =k k σˆ k ⎦ ⎣ ξ ⎦

(11a)

leaving with the system:

⎧⎪− =k k σˆ k ζ + 2m1ˆξ = 0 ⎨ ⎪⎩− =ω1ˆζ + =k k σˆ k ξ = 0 Now, since solving the first equation of the system (11b) in one variable, say

(11b)


ξ=

=k k σˆ k 1ˆ p σˆ 1ˆ ζ = k k ζ 2m 2m

5

(12a)

and substituting into the second one, there is obtained:

( p σˆ )( p σˆ ) p 2σˆ 2 p2 ˆ 1ζ E1ˆζ = k k k k ζ = k k ζ = 2m 2m 2m

(12b)

where the above Planck relationship was supplemented by the companion de Broglie one for the momentum,

=k k = p k

(13)

σˆ k2 = 1ˆ

(14)

while the Pauli matrices basic property

applies, as may be immediately verified from their realization of eq. (4). Nevertheless, the eq. (12a) represents in fact the eigen-equation for the free motion, supporting the latent generalization to the bounded state, either in anti-bonding or bonding existence

Eζ = εζ , Eξ = εξ

(15)

This is an interesting result because abolished many odd perception about Dirac equation and its meaning; actually, there follows that: •

•

•

Dirac equation is formally related with the temporal Schrödinger one, while producing the same eigen-problems, thus describing in essence the same nature of the electronic motion; The spin information modeled by the bi-dimensional spinors is not necessarily a relativistic effect (beside completing the 2+2=4 relativistic framework dimension of the Dirac equation) but merely a quantum one since fulfilling the eigen-value problems, each separately; The two spinors of the Dirac equation may be associated with the bonding (for negative energies) and anti-bonding (for positive energies) of a system, being thus suited for physically modeling of the chemical bond, beside the common interpretation of negative/positive spectrum of free positronic/electronic energies in the Dirac Sea.

However, before effectively pursuit to the chemical bonding description based on Dirac equation one needs some background of the recent non-orbitalic quantum modeling of the chemical bond.

6

Mihai V. Putz

3. Quantum Chemical Bond 3.1. Binding Functions and the Chemical Bond Employing the dimensional quantum-relativity relationship

< energy > ⋅ < distance >~ Joule ⋅ meter ~ = ⋅ c

(16)

recently, there were introduced the chemical binding functions [19]:

⎧1, λ → 0 f α (λ , C A ) = 1 − ΩλC A = ⎨ ⎩− ∞ , λ → ∞

(17a)

⎧1, λ → 0 f β (λ , C A ) = exp(− ΩλC A ) = ⎨ ⎩0, λ → ∞

(17b)

called as the anti-bonding and bonding functions, for the reason grounded on their asymptotical behavior, respectively; the introduced Ω-factor accounts for assumed dimensionless nature of functions (17), being adequately settled as:

Ω= and where

1 = 0.506773 ⋅ 10 −3 J −1 m −1 =c

(18)

λ stands for the localization distance, while C A stays for the chemical action

[19,20]

G G G C A = ∫ ρ (x )V ( x )dx ≅ χ [ρ ]

(19)

o

expressed in A (ångstrom) and eV (electron-volts), respectively. Note that in eq. (19) the equivalence between chemical action and electronegativity χ [ρ ] , as electronic density functionals, was assumed based on their similar nature in convoluting the applied potential with the concerned electronic density ρ [24], although electronegativity is currently unfolded as a generalization of the chemical action, being dependent on it, showing a more complex density functional expression at various localization levels [25, 26]. Nevertheless, the bonding functions (17) reciprocally combine within a paradigmatic AB molecule, with a coordinate system centered in A, to provide the electronic pair-localization region within the bond length RAB by means of the binding equations (see Figure 1) [19],

(

)

(

)

(I) : f αA λI , χ A = f βB RAB − λI , χ B ,

(20a)


(

)

(

(II) : f αB RAB − λII , χ B = f βA λII , χ A as the interval

)

7 (20b)

λII − λI or as the single point λII = λI for the hetero- and homo- bonding

systems, i.e. having different or identical isolated electronegativities

χ A and χ B ,

respectively.

Figure 1. Geometrical loci of the sigma-bonding (σ-B in blue), anti-bonding (⎤-B in red), nobonding (∅-B in orange), and pi-bonding (π-B in green) for chemical binding from equal electronegativity influences of two systems A and B throughout equations (20) with binding functions (17) through constants and parametric settings as = = c = 1 , χ A = χ B = 1 , RAB = 1.

In each of above equations (20) the binding “points” I and II appear as the informational crossing (transfer) between the anti-bonding and bonding functions of both A and B systems driven by their electronegativities; however, they fix the all types of involved bonding regions (see Figures 1 and 2) as follows [19]: •

the sigma-bonding region (σ-B in Figures 1 and 2) is uniquely defined and has no “nodes” or discontinuities; it is delimited by the area under bonding crossing functions inside of the pairing interval bordered by the projection of the points I and II along the bond; it corresponds to the consecrated bonding obtained by the composed wave-function density Ψ A + ΨB

•

2

in the conventional molecular orbital

(MO) theory [4]; the anti-bonding region (⎤-B in Figures 1 and 2) is represented by the two parts spanning the space from the systems A and B until the sigma-bonding limit, being defined by the area under bonding functions f βA and f βB but outside of the interval

8

Mihai V. Putz fixed by the projection of points I and II on the bond length; it corresponds to the 2

anti-bonding state density ΨA − ΨB with separated parallel electronic spins in MO •

•

theory [4]; the no-bonding region (∅-B in Figures 1 and 2) is composed of two parts, one in each binding side respecting sigma-bonding, being formed by the area delimited by all the binding functions of (17) around the binding points I and II, outside of their projected interval on the bond length, while not intersecting between them and with the bond length; the pi-bonding region (π-B in Figures 1 and 2) spans the bond length entirely without crossing it, thus having nodes on it, being resulted from the area defined by all the binding functions of (17) around the binding points I and II, partially outside and partially inside (with a common point inside the projected interval of the points I and II on bond; therefore, this region is compatible with the consecrated pi-bond type of the MO theory.

Figure 2. Geometrical loci of the bonding regions as in Figure 1 for chemical binding from different electronegativity influences of two systems A and B throughout equations (20) with binding functions (17) through constants and parametric settings as = = c = 1 , χ A = 2 χ B = 2 , RAB = 1 .

Note that the difference between the equal and different electronegativity influences on bonding in Figures 1 and 2 is reflected in sigma-bonding shift towards the more electronegative bonding component, while slightly enlarging the spanning interval of projection of points I and II on bond in Figure 2; moreover, the bond of Figure 2 is accompanied by a slightly decreasing of the sigma bonding apex value on binding probability, being now about 0.5 respecting the recorded 0.6 value in Figure 1. This may lead with the


9

meaningful consequence in describing the covalency (in Figure 1) and ionicity (in Figure 2) characters of chemical bonding, in terms of quantum tunneling of the sigma bonding region: • •

covalent binding is characterized by a symmetric higher and thinner well of electrons, being those more localized on middle of bond; ionic binding features a dissymmetric taller and thicker well of electrons with more delocalized pairing electrons towards the more electronegative component.

The bonding regions may appear as the consequence of equilibrium between binding functions (17) that caries the density functional information either as chemical action or as electronegativity. As a consequence, all above identified binding regions are defined within positive (0, 1) realm of binding functions (17) allowing the natural probabilistic interpretation for their inside. Nevertheless, there remains to explore their influence on bonding within the Dirac equation framework, and how the Dirac equation in generals influences the chemical bonding phenomenology when the spin (or spinors) are involved. This will be addressed in the sequel.

3.2. Chemical Bond by Dirac Equation The above spinorial identification as bonding and anti-bonding, see eq. (5), may be now combined with the introduced bonding and anti-bonding functions (17) so that to create the actual working binding spinor:

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ ⎢ ⎜1 − =c ⎟ exp⎜ − = E ⋅ t ⎟ ⎥ [Ψ ] = ⎢ ⎝ λχ⎠ ⎝ i ⎠ ⎥ ⎞ ⎛ ⎞ ⎢exp⎛⎜ − exp⎜ + E ⋅ t ⎟⎥ ⎟ ⎢⎣ ⎝ =c ⎠ ⎝ = ⎠⎥⎦

(21)

where the previous chemical action dependence was here reconsidered as the more generalized (density functional) electronegativity. Next, we impose the condition the spinor (21) fulfilling the Dirac equation (1); for this we separately express the involved terms, while self-understanding the presence of the (bidimensional) unitary and other Dirac operators on both spinorial upper and down components so that the implicit total dimension of the wave-function to be completed to four-dimensional space: •

the time derivative Dirac term is directly computed as:

⎡ ⎛ λχ ⎞⎛ i ⎞ ⎛ i ⎞⎤ ⎢i=⎜1 − =c ⎟⎜ − = E ⎟ exp⎜ − = E ⋅ t ⎟⎥ ⎠⎝ ⎠ ⎝ ⎠⎥ i=∂ t [Ψ ] = ⎢ ⎝ λχ i i ⎛ ⎞ ⎛ ⎞ ⎢ i= E exp⎜ − ⎟ exp⎜ + E ⋅ t ⎟ ⎥⎥ ⎢⎣ = ⎝ =c ⎠ ⎝ = ⎠⎦

10

Mihai V. Putz

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ ⎢ E ⎜1 − =c ⎟ exp⎜ − = E ⋅ t ⎟ ⎥ ⎝ ⎠ ⎝ ⎠ ⎥ =⎢ λχ i ⎛ ⎞ ⎛ ⎞ ⎢− E exp⎜ − exp⎜ + E ⋅ t ⎟⎥ ⎟ ⎢⎣ ⎝ = ⎠⎥⎦ ⎝ =c ⎠ •

(22)

the space coordinate Dirac derivative needs the pre-requisite of simple derivative

∂ k λ = ∂ k λk λk =

λk λ

(23a)

providing the yield:

⎤ ⎡ ⎛ λ χ⎞ ⎛ i ⎞ − i=cαˆ k ⎜ − k ⎟ exp⎜ − E ⋅ t ⎟ ⎥ ⎢ ⎝ = ⎠ ⎝ =cλ ⎠ ⎥ − i=cαˆ k ∂ k [Ψ ] = ⎢ ⎢− i=cαˆ k ⎛ − λk χ ⎞ exp⎛ − λχ ⎞ exp⎛ + i E ⋅ t ⎞⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎥ ⎢⎣ ⎝ =c ⎠ ⎝ = ⎠⎦ ⎝ =cλ ⎠ ⎤ ⎡ χ ⎛ i ⎞ i λk exp⎜ − E ⋅ t ⎟ ⎥ ⎢ ˆ σk ⎤ λ ⎝ = ⎠ ⎥ ⎢ ⎛ λχ ⎞ ⎛ i ⎞⎥ 0 ⎥⎦ ⎢ χ i λ exp⎜ − ⎟ exp⎜ + E ⋅ t ⎟⎥ ⎢⎣ λ k ⎝ =c ⎠ ⎝ = ⎠⎦

⎡0 =⎢ ⎣σˆ k

⎡ χ ⎛ λχ ⎞ ⎛ i ⎞⎤ ⎢i λ σˆ k λk exp⎜ − =c ⎟ exp⎜ + = E ⋅ t ⎟⎥ ⎝ ⎠ ⎝ ⎠⎥ =⎢ χ i ⎛ ⎞ ⎥ ⎢ i σˆ k λk exp⎜ − E ⋅ t ⎟ ⎥⎦ ⎢⎣ λ ⎝ = ⎠ •

(mc

(23b)

the free + potential term:

2

(

⎡ mc + V ( x) + V ( x) βˆ [Ψ ] = ⎢ 0 ⎣

)

2

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ 1− exp⎜ − E ⋅ t ⎟ ⎥ ⎜ ⎟ ⎢ ⎤ ⎝ 0 =c ⎠ ⎝ = ⎠ ⎥ ⎥⎢ 2 − mc + V ( x) ⎦ ⎢exp⎛ − λχ ⎞ exp⎛ + i E ⋅ t ⎞⎥ ⎜ ⎟⎥ ⎢⎣ ⎜⎝ =c ⎟⎠ ⎝ = ⎠⎦

)

(

)

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ 2 ⎢ mc + V ( x ) ⎜1 − =c ⎟ exp⎜ − = E ⋅ t ⎟ ⎥ ⎝ ⎠ ⎝ ⎠ ⎥ =⎢ λχ i ⎛ ⎞ ⎛ ⎞ ⎢− mc 2 + V ( x ) exp⎜ − exp⎜ + E ⋅ t ⎟⎥ ⎟ ⎢⎣ ⎝ =c ⎠ ⎝ = ⎠⎥⎦

(

(

)

)

With these, the Dirac equation (1) now provides the system of equations:

(24)

Fulfilling Dirac’s Promise on Quantum Chemical Bond λχ

11

− E ⋅t χ ⎛ λχ ⎞ − = E ⋅t ⎛ λχ ⎞ − = E ⋅t = i σˆ k λk e =c e = + 1ˆ(mc 2 + V ( x) )⎜1 − 1ˆ E ⎜1 − , ⎟e ⎟e λ =c ⎠ =c ⎠ ⎝ ⎝

i

− 1ˆ E e

−

λχ =c

e

i E ⋅t =

i

i

(25a)

λχ

i i − − E ⋅t E ⋅t χ 2 c = = = ˆ = i σˆ k λk e − 1(mc + V ( x) )e e λ

(25b)

Next, through getting out from the second equation (25b) the term containing the covariant product: λχ

i

− χ σˆ k λk = 1ˆ(mc 2 + V ( x) − E )e =c λ

(26a)

it is then replaced in the first equation (25a) to obtain:

(mc

2

λχ −2 ⎛ λχ + V ( x) − E )⎜⎜1 − + e =c =c ⎝

⎞ ⎟=0 ⎟ ⎠

(26b)

whose solutions expresses the energy conservation

E = mc 2 + V ( x)

(27)

and the Dirac adapted bonding equation

−e

−2

λχ =c

= 1−

λχ =c

(28a)

Yet, due to the negative sign of the left hand side of eq. (28) one may infer that it is just one solution of a quadratic equation, say

e

−4

λχ =c

≅ 1− 2

λχ =c

(28b)

providing the second accompanied solution

e

−2

λχ =c

≅ 1−

λχ =c

(28c)

However, there is noted the formal difference between eqs. (28a) and (28c) not only because of sign but also due to the approximate nature of the second, coming from the form (28b) in short range of binding distance regime λ ≅ 0 . Nevertheless, the appearance of two

12

Mihai V. Putz

(±) forms of Dirac chemical bonding equation is in accordance with the manifestation of the Dirac positive/negative manifestation of energies respecting the electronic/positronic motions within the Dirac Seas, respectively. Still, for the chemical bond description the difference in sign allows for further mixing of the bonding equations for a paradigmatic AB molecule generating more bonding points so modeling in more detail the bonding with spins in bonding and anti-bonding states. Therefore, the actual working Dirac binding functions are: •

The Dirac anti-bonding function remains the same as given within density kernel approach by eq. (17a):

f αDir (λ , χ ) = 1 − •

λχ

(29)

=c

The Dirac bonding function is modified respecting the previous one given by eq. (17b) while being two-folded:

λχ ⎞ ⎛ f βDir ⎟, ( + ) (λ , χ ) = exp⎜ − 2 =c ⎠ ⎝

(30a)

λχ ⎞ ⎛ f βDir ⎟ ( − ) (λ , χ ) = − exp⎜ − 2 =c ⎠ ⎝

(30b)

Now, the bonding geometric loci are determined, for the molecule AB, by the system of equations: −B B (I) : f αA (λI , χ A ) = f βDir ( + ) (RAB − λI , χ ) ,

(31a)

−A A (II) : f αB (RAB − λII , χ B ) = f βDir ( + ) (λ II , χ ) ,

(31b)

−A A (III) : f αA (λIII , χ A ) = f βDir ( − ) (λIII , χ ) ,

(31c)

(

)

(

−B (IV) : f αB RAB − λIV , χ B = f βDir RAB − λIV , χ B ( −)

)

(31d)

which is regarded as Dirac generalization of the previous one of eqs. (20) by means of the last two equations which quantifies the “interference” effect of the anti-bonding and the negative bonding functions belonging to the same atom in the to be transferred towards a virtual bonding partner. The representations of Figures 3-4 show how the Dirac binding functions and equations (29)-(31) provides more insight in modeling of chemical bonding respecting the previous density functional ones of Figures 1 and 2.


13

The differences comes from two basic facts: the bonding function (17b) takes through Dirac equation two forms, i.e. it degenerates into one positive and other negative, see eqs. (30a) and (30b), respectively, while having also the modified argumentum. Instead, the antibonding equation (17b) is Dirac preserved either as in form and multiplicity. Due to this fact, depending on the electronegativity differences between the bonding partners the anti-bonding spin state may be located in various locations between the mixed (positive) bonding-antibonding crossing points I and II, eqs. (30a) and (30b), and the self (negative) bonding – antibonding crossing points III and IV, eqs. (30b) and (30c). Even more, for equal electronegativity three types of parallel spin (antibonding) separation may arise as illustrated by the Figures 3a, 3b and 3d, i.e. as being delocalized outside, precisely localized at the edge and delocalized inside of the sigma-bonding region, respectively, while the Figure 3c illustrates the case when the bonding pairing is precisely localized on bond. Actually, following the bonding points delivered by the system (31) one has, for the equal electronegativity cases of Figures 3a-3d, the following configurations respecting the electronegativity equal values, respectively: Figure 3a:

χ A = χ B = 1 : IV II < I < III N N< ↑

(32a)

↑

↑↓

providing the anti-bonding (parallel) spins delocalized outside of the region with delocalized anti-parallel pairing electrons; Figure 3b:

χ A = χ B = 2 : IV =N I
0 = Δt, since, in accordance with it, the very fact that E cannot be defined in an instant, t→0, but only over an interval t>0, is the raison d’etre of an uncertainty in time. Now it is known that, in accordance with Error Theory, the expression Δ(1/t) is equal with Δt/t2. Therefore, by substituting for Δt/t2 in ΔE=hΔ(1/t), we obtain: ΔE=hΔt/t2. Since, however, in my initial derivation, t>0=Δt, the denominator of the fraction becomes (Δt)2. Hence, ΔE=hΔt/(Δt)2. This, in turn, yields ΔE=h1/Δt and this, finally, ΔEΔt=h. The point of importance is this: Unless it is assumed in this line of reasoning that t>0=Δt, which is the exact same assumption in my initial derivation, ΔEΔt=h will never follow from ΔE=hΔν, itself derived tautologically from E=hν, thus contradicting the theory. In short, if E=hν implicitly entails ΔE=Δhν, then ΔE=Δhν can only be turned into ΔEΔt=h –and indeed how can it not be? – iff t>0=Δt, in other words, iff the t of the primary relation, E=hν, is interpreted as a limiting time latitude for the definition of the energy, exactly as postulated in my first deduction. This second proof of ΔEΔt=h, not only reconfirms the first but, in addition, it furnishes in the process a by and large self-contained deduction of ΔEΔt=h, fully crosschecking with the first.

58

Constantin Antonopoulos

Notes 1. Mainly by Dr. V. Karakostas. See his “Nature of Physical Reality in the Light of Quantum NonSeparability” [Oviedo, Spain, 7–13 August 2003] or his “Nonseparability, Potentiality and the Context-Dependence of Quantum Objects”. Dialectica, 2005 (to be published). 2. Beyond Reason: Essays on the Philosophy of Paul Feyerabend. Gonzalo Munevar ed., Kluwer, Dordrecht, 1991. 3. Plus, that is, eight other people signing the text. John Earman and Wesley Salmon are two of them. (See 2d paragraph from the end, Section 3.) 4. The general idea is that “when language-games change, then there is change in concepts, and with the concepts the meanings of words change” [Wittgenstein, 1977, 66.]. 5. The very expression “joint applicability” of the classical concepts, eo ipso excludes waves and particles as its possible recipients, which are –trivially– not jointly applicable in classical mechanics. Only the variables of the products Et and pq are. 6. The correct thing to say, of course, is that the two quantum relations, when submitted to a Fourier treatment, will alternatively represent the system as either an extended, plane wave or as a localizable wave packet. The former allows the determinations of E and p, since a unique wavetrain is employed, the latter, the determinations of t and q, since a superposition of many waves is employed. And thus the classical concepts, E and p, on the one side, t and q, on the other, will through this process assume an incompatibility analogous to that between “one” and “many”. The sole duality, therefore, derivable from this reasoning is that between waves and wave packets. That is to say, of waves alone, though in antithetic wave profile configurations. Particles are absent even here. 7. This is what Hooker has earlier referred to as “the factual absence” for classical compatibility. For a thorough analysis of logical vs factual incompatibility see my essays [2, 1994, pp.187-9 and 4, 223-5, 2005]. 8. I can hardly overemphasize the importance of this word, soon to be abandoned by Bohr for the sake of the dominant term, CTY, [1934, 19], although the former shows exactly how complementary counter happenings at either end of the indivisible cluster result from uneven balance. The two notions, I should think, go hand in hand. 9. “More strikingly, Zeno shows that no body can ever start its journey: it can never take the first step, since there is no first step to take” [Barnes, 1982, 262]. This, of course, is Kant’s problem in the Second Antinomy, that between Composite and Simple and the frustration thereby of constructing the former by repetitive accumulations of the latter. It is anticipated, in a uniquely important degree, by Zeno’s paradox of extension: “How can a line of finite length be divided into infinitely many parts of finite length? And how can a line which is made up of lengthless parts add up to a line which has length?” [Harrison, 1996, 273]. 10. Dimly, Duality is discerned in the distant horizon. What can only manifest undivided aspects of itself, will reasonably manifest such aspects as may conflict with one another, when what should otherwise be only a part of the entity, must

Complementarity Out of Context

59

now extend over its whole profile instead; and may thus conflict with its alternative, indivisibly manifested profile. Duality is not a primitive assumption in this syllogism. It is only a side product of Indivisibility. 11. The aim is, of course, an equalitarian ideology, offered in abundance through ‘humanitarian’ incommensurability. Incommensurable scientific beliefs let alone moral, social, political or aesthetic ones, are impossible to bring to conflict, therefore impossible to select from. They are all just as good (or just as bad, if it came to that). For the essentially ideological character of context-dependence see [Katz, 1978, 364] and, it goes without saying, A. Sokal’s pseudo-paper [Longino, 1977, 119]. But Feyrabend himself has beaten them all to it, openly confessing that his scientific relativism is preferable because more humane [1971, 33; and 1978 passim]. Scientific truth must be silenced, if it conflicts with politics.

References [1] Antonopoulos C. “Innate Ideas, Categories and Objectivity”. Philosophia Naturalis, 26, 2, 1987. [2] Antonopoulos C. “Indivisibility and Duality; A Contrast”. Physics Essays, 7, 2, 1994. [3] Antonopoulos C. “Investigating Incompatibility: How to Reconcile Complementarity With EPR”. Annales de la Fondation Louis de Broglie, 30, 1, 2005. [4] Antonopoulos C. “Making the Quantum of Relevance”. Journal for General Philosophy of Science, 36, 2, 2005. [5] Barnes, J. The Presocratic Philosophers. Routledge and Kegan Paul, London, 1982. [6] Bohr, N. Atomic Theory and the Description of Nature. Cambridge University Press, 1934. [7] Bohr, N. Atomic Physics and Human Knowledge. New York, 1958. [8] Bohr, N. Essays 1958-62 on Atomic Physics and Human Knowledge. Suffolk, 1963. [9] Darrigol, O. “Strangeness and Soundness in de Broglie’s Early Works”. Physis, 30, 1993. [10] Feyerabend, P.K. “Complementarity”. Proceedings of the Aristotelian Society, supplementary volume, xxxii, 1958. [11] Feyerabend, P.K. “Problems in Microphysics”.Frontiers of Science and Philosophy, R.Colodny, ed., University of Pittsburgh Press, 1962. [12] Feyrabend, P.K. “How to Be a Good Empiricist”. The Philosophy of Science, P.H. Nidditch ed., Oxford University Press, 1971. [13] Feyerabend, P.K. Science in a Free Society, NLB 1978. [14] Faye, J., Folse, H. eds. Niels Bohr and Contemporary Philosophy. Dordrecht 1994. [15] Folse, H. The Philosophy of Niels Bohr. Amsterdam, 1985. [16] Hanson, N.R. Patterns of Discovery. Cambridge University Press 1961. [17] Harrison, C. “The Three Arrows of Zeno”. Synthese, 107, 1996. [18] Hooker, C.A. “The Nature of Quantum Mechanical Reality”. Paradigms and Paradoxes, The University of Pittsburgh Press, 1972. [19] Kant, Imm. The Critique of Pure Reason. Transl. N.K.Smith, Macmillan, 1973. [20] Karakostas, V. “The Nature of Physical Reality in the Light of Quantum Nonseparability”. Oviedo, Spain, 7–13 August 2003.

60

Constantin Antonopoulos

[21] Karakostas V. “Nonseparability, Potentiality and the Context-Dependence of Quantum Objects”. Dialectica, 2005 (to be published). [22] Katz, J. “Semantics and Conceptual Change”. The Philosophical Review, 88, 1979. [23] Kuhn, Th. The Structure of Scientific Revolutions. The University of Chicago Press, 1970. [24] Leibniz, W. Monadology. The European Philosophers from Descartes to Nietzsche. M.C.Beardsley ed., New York, 1960. [25] Longino, H.E. “Alan Sokal’s ‘Transgressing Boundaries’”. International Studies in the Philosophy of Science, 11, 2, 1997. [26] Mackay, D.M. “Complementarity”. Proceedings of the Aristotelian Society, suppl. volume xxxii, 1958. [27] Munevar, G. “Bohr and Evolutionary Relativism”. Explorations in Knowledge, xii, 2, 1995. [28] Munevar, G., Ed., Beyond Reason: Essays on the Philosophy of Paul Feyerabend. Dordrecht, 1991. [29] Petersen, A. “On the Philosophical Significance of the Correspondence Argument”. Boston Studies in the Philosophy of Science, Volume 1966-68. [30] Rosenfeld, L. “Foundations of Quantum Theory and Complementarity”. Nature, 190, 1961. [31] Plotnitsky, A. Complementarity. London 1994. [32] Salmon Marilee H., Earman, J., Glymour C., Lennox J., Machamer P., McGuire, J.E., Norton J.D., Salmon Wesley C., Schaffner, K.N. Introduction to the Philosophy of Science, Prentice Hall, New Jersey 1992. [33] Wittgenstein, L. On Certainty, transl. by G.E.Anscombe, B.Blackwell, Oxford 1977.

In: Electrostatics: Theory and Applications Editor: Camille L. Bertrand, pp. 61-89

ISBN 978-1-61668-549-2 c 2010 Nova Science Publishers, Inc.

Chapter 4

M OLECULAR I NTEGRALS OVER S LATER -T YPE O RBITALS . F ROM P IONEERS TO R ECENT D EVELOPMENTS ¨ Philip E. Hoggan1,∗, Mar´ıa Belén Ruiz2,† and Telhat Ozdo˘ gan3 1 LASMEA, UMR 6602 CNRS, University Blaise Pascal, 24 avenue des Landais, BP 80026, 63171 AUBIERE Cedex, France 2 Department of Theoretical Chemistry of the Friedrich-Alexander-University Erlangen-Nürnberg, Egerlandstraße 3, D-91058 Erlangen, Germany 3 Department of Physics, Faculty of Arts and Sciences, Rize University, 53100 Rize, Turkey

Abstract It can readily be demonstrated that atomic and molecular orbitals must decay exponentially at long-range. They should also possess cusps when two particles approach each other. Therefore, Slater orbitals are the natural basis functions in quantum molecular calculations. Their use was hindered over the last four decades by integration problems. Consequently, Slater orbitals were replaced by Gaussian expansions in molecular calculations (in spite of their more rapid decay and absent cusps). From the 90s until today considerable effort has been made by several groups to develop efficient algorithms which have fructified in new computer programs for polyatomic molecules. The key ideas of the different methods of integration: one-center expansion, Gauss transform, Fourier transform, use of Sturmians and elliptical co-ordinate methods are reviewed here, together with their advantages and disadvantages, and the latest developments within the field. A recent approximation separating the variables of the Coulomb operator will be described, as well as its usefulness in molecular calculations. Recently, due to the developments of the computer technology and the accuracy of the experiments, there is a renewed interest in the use of Slater orbitals as basis functions for Configuration Interaction (CI) and Hylleraas-CI atomic and molecular calculations, and in density functional and density matrix theories. ∗ E-mail † E-mail

addresses: [email protected] addresses: [email protected]

¨ P.E. Hoggan, M.B. Ruiz and T. Ozdo˘ gan

62

Keywords: Slater orbitals, computer programs, Kato conditions, accurate molecular wavefunctions

1.

Introduction

Slater-type orbitals (STO) [1] are the natural basis functions in quantum molecular calculations. Nevertheless, their use has been rather restricted, mostly due to mathematical integration difficulties. Even today there are no simple general algorithms to solve all the integrals appearing in a Hartree-Fock (HF) or Configuration Interaction (CI) molecular calculation, where integrals involving up to four atomic centers may appear. In spite of these difficulties the research on Slater orbitals has always continued. The reason is the requirement for large basis sets of Gaussian orbitals (GTO) and large wave function expansions to perform more accurate calculations of energy and properties of ever larger interesting systems. As a consequence those calculations need enormous computational times. In 1981, in a Congress in Tallahassee about Slater type orbitals, Milan Randic described the situation: ”Gaussian functions are not the first choice in theoretical chemistry. They are used (...) primarily because molecular integrals can be evaluated, not because they posses desirable properties. Today this may be a valid reason for their use, but tomorrow they may be thought of as bastard surrogates, which served their purpose in the transition period, have no longer viable merits and will fall into oblivion” [2]. The use of an expansion of GTOs instead of an STO was then a pragmatic solution and originally intended for solving the problems in the calculation of the first molecules on early mainframe computers. The GTO expansion together with the popular distribution of computer programs like GAUSSIAN have encouraged the use of GTOs for accurate calculations of large systems. The limits are receding with respect size of the systems and dimension of the wave function, i.e. HF calculations of clusters of hundreds of atoms, CI calculations including hundreds of thousands of Slater determinants. In spite of the rapid development of the computer technology and the availability of supercomputers, the computational times are unreasonably long, so that the computational chemist is restricted i.e. to perform numerous test calculations. This motivates the search for basis functions, where fewer would give a good CI, in particular. The possibility of using Slater orbitals, where a minimal basis would consist in one function per atom would provide a forward impulse to theoretical and computational chemistry. Since the difficulties are of a purely mathematical nature, e.g. definite integrations, it would be worthwhile pursuing investigations of Slater orbitals. The purpose of this paper is to explain the key ideas about Slater orbitals for readers outside the field. It is beyond our scope to review the whole work of the all authors in this field, what would deserve a longer treatment. The history of Slater orbitals and the first computer programs using them is exposed and the currently used computer programs are listed. The STO and GTO are defined and compared. The methods used in the literature are explained recalling in the key ideas in which these methods are based. The last developments within the field are reported.

Molecular Integrals over Slater-Type Orbitals

2.

63

Early History of the Slater Orbitals

The history of STOs is the history of theoretical chemistry. In 1928 Slater [1] simplified the hydrogen-like orbitals (which are eigenfunctions of the Hamiltonian for a one-electron atom) obtaining the orbitals which bear his name. Curiously Slater called these orbitals at that time Hartree orbitals. Slater orbitals are a simplification of the hydrogen-like orbitals, which are eigenfunctions of the atomic one-electron Schrödinger equation. Brief time-line of events in molecular work over Slater type orbitals to date: 1928 Slater and London. 1929 Hylleraas: He atom. 1933 James and Coodlidge: Hylleraas calculations on H2 . 1949 Roothaan LCAO paper. 1950 Boys: first Gaussian expansion of STO published. 1951 Two-center Coulomb Integrals. Roothaan formulae. 1954 Boys and Shavitt ’Automated calculations’. 1958 Tauber: Work on analytic two-center Exchange integrals: Poisson equation. 1962 Scrocco: first publishes STO work, (in Italian) but with a programme. This follows early molecular work in 1951-53. [3, 4]. 1963 Clementi produces tables of optimised single zeta basis sets for atoms. Shavitt B-Functions described. 1970 The Journal of Chemical Physics published work on STO codes by E. Scrocco and R. Stevens. Gaussian 70 prepared for QCPE by J. Pople and R. Ditchfield. 1973 E. J. Baerends: numerical integration over STO used for ADF DFT code. 1978 Filter and Steinborn: Fourier transform work. B-functions and plane-wave expansion of Coulomb operator. 1981 ETO conference in Tallahassee. Weatherford and Jones. 1994 First STOP (Slater Type Orbital Package, QCPE 667 1996) code. Bouferguene and Hoggan. 2001 First SMILES (Slater Molecular Integrals for Large Electronic Systems) code. Fernández Rico, López et al. 2008 Gill: Coulomb resolution.

64


Very soon with Slater at MIT, researchers broached the problem of evaluating the twoelectron integrals in this basis. During the 1950s the Chicago group led by Mulliken took on the task of evaluating all the molecular integrals. Roothaan treated the Coulomb and Hybrid two-center integrals [5,6], Rüdenberg the exchange integrals [7]1 . Among the many authors who were working around the world on the solution of the necessary integrals one may mention Masao Kotani in Japan [8], who wrote the famous tables of integrals which bear his name and that were widely used. Coulson in Oxford (England) proposed a method to evaluate the three- and four-center integrals [9], Löwdin in Uppsala [10], and young American scientist called Harris [11] were involved. Work in the early 50s mostly focused on integrals over STO. The interest was to make the first theoretical calculations of some molecules starting with the diatomic systems H2 , N2 . For three-center molecules the problem of integration was encountered (orbital translation). Mulliken and Roothaan called this ”The bottleneck of Quantum Chemistry” [12], Mulliken mentioning it in his Nobel Lecture in 1966, on the molecular orbital method. Boys in Cambridge published his landmark paper [13] containing the evaluation of three- and four-center integrals using Gaussian functions, for which he derived the so-called product theorem: the product of two Gaussian functions located on different centers is a new Gaussian function located on a new center. Thus four-center electron distributions could be reduced to single-center distributions and evaluation was analytically facilitated. Boys regarded his work as an existence theorem. It was to change the course of molecular computations. Note that the product theorem for Slater orbitals leads to complicated infinite sums, making evaluation awkward compared with the simple closed forms for Gaussians. In 1954 Boys, Shavitt et al [14] expanded Slater orbitals into Gaussians to perform quantum mechanical calculations. In 1963 Clementi presented the so-called basis set using Slater orbitals [15]. Later Pople would base his programs on Boys’ pragmatism.

3.

History of the STO Computer Programs

The first (and surely the last) manual calculation of a molecule, the N2 molecule, was done by Scherr in 1956. It was necessary the work of 2 (sometimes it appears 3 ) men for 2 years. Afterwards this calculation was reproduced by the first digital computer calculation [16, 17], taking 35 minutes. In 1962 Shull initiated the Quantum Chemistry Program Exchange (QCPE) at Indiana University. The first automatic computer program was POLYATOM [18] which used nevertheless GTOs with SCF-LCAO. The program was developed at MIT in 1963 when Slater was there. In 1963 the program IBMOL [19] was developed by Clementi and others when he visited the Chicago group. In 1968 a STO code was developed by Scrocco and Tomasi from Pisa. Preliminary work by Scrocco is reported in Italian as early as 1962 [4]. 1 The two-center two-electron integrals are classified according to the centers a, b. Writing them according the charge distributions [Ω(1)|Ω(2)] the Coulomb integrals are [aa|bb], the hybrids [aa|ab] and the exchange integrals [ab|ab]. The most difficult are the exchange integrals because the charge distribution of every electron is located over two centers.


65

This program was also used by Berthier in France. The program ALCHEMY in 1968 was originally developed using Slater orbitals by Clementi and the staff of the IBM laboratory in San Jose [20], afterwards, the new ALCHEMY 2 by Bagus and others used GTOs. The program DERIC [21] by Hagstrom in 1972 perform STO calculations of two-center molecules. In the 80s, the advent of GAUSSIAN [22] saw development in the STO field hibernate somewhat. By the 90s several groups around the world developed new STO computer programs which are now distributed. The program STOP, by Bouferguène and Hoggan [23] was published first in 1996. It is based on the single center strategy and was first presented in 1994 at the 8th ICQC in Prague. New versions appeared, the latest (parallel) in 2009. Then in 1998 a program was written using B-functions by Steinborn, Weniger, Homeier et al, in Regensburg [24]. The program SMILES by Fernández Rico, López, Ema, and Ram´ırez in Madrid appeared in 1998 and new versions have appeared, the latest in 2004 for the HF and CI calculations of molecules [25]. The program CADPAC [26] in Cambridge uses techniques like density fitting, involving auxiliary Slater type orbital basis sets to perform Hartree-Fock and Density Functional Theory (DFT) calculations with a reduced number of indices in requisite integrals. They aimed to obtain better Nuclear Magnetic Resonance (NMR) chemical shifts on the basis involving nuclear cusps. In the density functional theory field in 2001 the program ADF (Amsterdam Density Functional) [27] begun in 1973 by Baerends et al uses Slater orbitals for their calculations. This much-used package offers a very extensive series of atomic basis sets for input, including most elements. It is a numerical grid strategy and this review will not detail it. The program ATMOL of Bunge et al performs large highly accurate CI calculations on atoms using Slater orbitals [28]. In the first century of the third millennium much interest is concentrated in generating more efficient calculation algorithms, use of non-integer Slater orbitals, numerical solution of integrals when using B-functions and in the electron correlation when using Hylleraas wave functions.

4.

Slater Orbitals & Gaussian Orbitals

It is well known that hydrogen-like orbitals are the solution of the Schödinger equation for a one-electron atom. For helium and atoms with more electrons the Schödinger equation has no analytical solution due to the potential term 1/ri j which correlates the (otherwise) independent electrons. It is assumed that for systems with N ≥ 2 this form of the exponential e−αr will be the asymptote of the formal solution. The hydrogen-like orbitals have nodes, i.e. the 2s orbital is of the form (1 − br) e−αr , and higher quantum number orbitals are similar but STOs are node-less. A related problem appears for Gaussians. In 1928 Slater [1] regarded the hydrogen-like orbitals as polynomials in r which make the calculations messy and proposed the use of single powers of r i.e. linear combinations of hydrogen-like terms. A picture which helps to visualize the differences between Slater and Gaussian orbitals is the representation of the 1s orbital function of both types, see Figure 1.

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Figure 1. Comparison of the shape of a STO and GTO functions. STOs represent well the electron density near the nucleus (cusp) and far from the nucleus (correct asymptotic decay). STOs thus resemble the true orbitals. Conversely, the GTOs have erroneous shape near and far from the nucleus (no cusp). One can observe that far from the nucleus the GTOs tend to zero much faster than STOs. To reproduce a 1s STO using 3 GTOs (the so-called minimal GTO basis) an orbital is obtained with the shape of a Gauss curve, no cusp, see Figure 2. To reproduce a single STO many GTOs are necessary, but the electron cusp at the nucleus is missing. This is one of the reasons of the slow convergence of the wave function solutions to the exact (HF or CI) result. In general, if the basis function is not a formal solution of the Schrödinger equation its convergence is slower. That means that more Slater determinants are required to obtain the same result. Thus Slater orbitals show faster convergence when increasing their number. Another advantage of Slater orbitals is the size of the basis, one orbital per electron is of reasonable quality and multiple-zeta basis sets converge fast to the Hartree-Fock limit. therefore, the number of integrals to be evaluated is dramatically smaller. CI is spectacularly more efficient. Finally, conceptually the Slater orbitals give a more intuitive description of the atomic orbitals and of the molecular orbitals (MO) formed with them. The disadvantages of Slater orbitals have been already mentioned: the three- and fourcenter two-electron integrals are the bottleneck. There is no general analytical solution for them, which would be the most effective and fastest way of calculation. Instead there are a number of approximate methods of calculation, involving infinite series, or truncated approximations to the Coulomb operator itself. They will be treated in the next Sections. The radial Slater functions do not represent the bonding region adequately, it being then necessary to add higher angular momentum functions. It is nevertheless possible to use linear combinations restoring radial nodes. This approach is advocated particularly for ADF, where the hydrogen-like basis is obtained by fixing the coefficients for combining Slater functions. Another disadvantage is that some of the two-center integrals since the times of Roothaan and Rüdenberg have been solved for a co-axial conformation of the atomic coordinate systems (the z-axes point to each other) that is not the molecular frame. Therefore rotations and reflections are necessary. These problems have been solved, but it requires


67

Figure 2. Construction of a STO with 3 GTOs. additional calculations [29]. Nowadays, Slater orbitals are used in atomic calculations, especially in highly accurate calculations of atoms using Hylleraas wave functions (with explicit ri j dependence, and also in diatomics. They are used in DFT and in Density matrix theories. Traditionally they have been used in semi-empirical calculations where of course the three- and four-center integrals were neglected. The Gaussian orbitals are generally used in standard quantum mechanical calculations. As explained they are not shaped like analytical orbitals, with no cusp at the nucleus, for that reason they are not good for the calculation of properties where the density at the nucleus has to be well described. Also the radial dependence is not well represented and the number of integrals increases with the dimension of the basis dramatically. The major advantage of GTOs is the existence of a product theorem. Over many years, workers have improved the calculation of the necessary integrals, having achieved a considerable speed-up. For example the Coulomb operator with a Laplace transform enables to calculate three- and four-center integrals like two center integrals. Concluding, the main defect of GTO expansions is the absent cusp which slows the convergence and the large number of integrals to be computed.

5.

Types of Exponentially Decaying Orbital, Based on Eigenfunctions for One-Electron Atoms

In general one calls Slater-type orbitals those with an exponential radial factor of the form rn e−αr , for n a positive integer (or 0). The atom-centered Slater orbitals are defined as: ϕnlm (r) = rn−1 e−αrYlm (θ, φ),

(1)


68

where n, l, m are the quantum numbers. Ylm (θ, φ) are the spherical harmonics defined using the Condon-Shortley phase: Ylm (θ, φ) =

m

(−1)

2l + 1 (l − m)! 4π (l + m)!

1/2

Plm (cos θ)eimφ ,

(2)

Plm (cos θ) are the associated Legendre functions. The spherical harmonics are eigenfunctions of the angular momentum operator Lˆ 2 and its z-projection Lˆ z . The complex spherical harmonics are used mainly in atoms and in developing theories because it is easier to work out general formulae and derivations with them. The real spherical harmonics are linear combinations of the complex ones. These are used mainly in molecules. Note that they are written using polar coordinates. They can be also straightforwardly converted into Cartesian Slater orbitals by the exchange: x = r sin θ cos φ,

(3)

y = r sin θ sin φ,

(4)

z = r cos θ,

(5)

obtaining in general: χnlm (r) = xnx yny znz rn−1 e−αr .

(6)

Cartesian Slater type orbitals are very seldom used compared with Cartesian Gaussians, that are an almost systematic choice. When the principal quantum number n in Eq. (1) is a non-integer we have the NISTOs (Non Integer Slater Orbitals). The main difficulty when working with these orbitals is during the derivations a binomial has to be used with an non-integer power what leads to an infinite expansion. These orbitals are widely investigated in the present [30]. The additional flexibility of using non-integer quantum numbers brings a lowering in the energy results. There is the possibility to transform also the polar coordinates to elliptical coordinates. Traditionally the Elliptical Slater orbitals have been used as basis functions for two-center molecules [31]- [33]. These orbitals are known to lead to lower energy results, see Ref. [34]. Using ξ = λ1 = ra + rb and η = µ1 = ra − rb : ϕnlm (r) = ξn ηl (ξ2 − 1)m/2 (1 − η2 )m/2 e−αξ eimφ ,

(7)

where ξ, η, φ are the elliptical coordinates. Now we go to orbitals which are linear combinations of Slater orbitals: B-functions [35], hydrogen-like, Sturmians [36]. The B-functions are Bessel functions. The orbitals have some helpful properties like a compact Fourier transform. Written in the form n

Bnlm (r) =

(2n − j − 1)!

∑ 22n+l−1 (n + l)!(n − j)!( j − 1)! (ζr)l+ j−1 e−ζrYlm (θ, φ),

j=1

(8)


69

one can see that they are a linear combination of Slater orbitals. The angular parts are the spherical harmonics. The hydrogen-like orbitals which are solutions of the Schrödinger equation for the hydrogen atom have a radial part which is a Laguerre polynomial. The polynomial and the exponent coefficient depend on the atomic number Z and the principal quantum number n: 2Zr l − Zr m 2l+2 χnlm (r) = Nnl Ln−l−1 (9) r e n Yl (θ, φ). n Due to that fact, the hydrogen-like orbitals do not form a complete set (for finite n), they need orbitals of the continuum to be complete. This would be important for the convergence of the solutions. Shull and Löwdin [37] realized that this was due to the dependence of Z with n that dilates the orbitals and they proposed the following orbitals where these were substituted by adjustable parameters, i.e. usual orbital exponents: 2l+2 χnlm (r) = Nnl Ln−l−1 (2αr) rl e−αrYlm (θ, φ),

(10)

so these orbitals form a complete set. These orbitals were subsequently called Coulomb Sturmians because they fulfill the so-called Sturm-Liouville theorem for eigenfunctions of such differential equations, with central Coulomb attraction. In the Section 7 methods of the literature we will see how these kinds of orbitals have been used.

6.

Types of Integral over Slater Orbitals

Due to the form of the Hamiltonian and of its expectation value we find the following kinds of integrals. First the integrals which appear when using Hartree-Fock and CI wave functions, in general ab initio methods. The integrals are classified according the number of electrons and centers which are linked. We present them in order of difficulty.

6.1.

One-Electron Integrals

These are the one- and two-center overlap integrals ha|bi, kinetic energy integrals ha|bi and two-center nuclear attraction ones ha|1/rb |bi. Other case of one-electron integral is the three-center nuclear attraction, originated from the nuclear attraction operators in the Hamiltonian: ha|1/rc |bi.

6.2.

Two-Electron Integrals

They can be up to four-centers because of the determinant giving the wave-function and thus the four orbitals which form the integral. According to the number of centers: The two-center integrals have been traditionally the most investigated, they have the following nomenclature: The Coulomb integrals where the charge distribution of every electron is located at a center: [aa|bb]. Hybrid integrals, one charge distribution is located at one center and the other over two centers [aa|ab] and their equivalents [bb|ab]. The exchange integral is more difficult, it leads in case of different exponents to an infinite

70


sum. Every electron is located in two centers: [ab|ab]. To solve these integrals a change to elliptical co-ordinates is useful. The Coulomb operator in elliptical co-ordinates contains associated Legendre functions of the first and second kind, for which integration is very difficult. In the case of slightly different exponents there are some singularities. In actual calculations, the Coulomb and Hybrid integrals are calculated exactly, numerous methods exist. The exchange integrals are calculated with great accuracy. The three-center integrals are of several types [aa|bc], [ab|ac]. For different exponents there is no general solution. The four-electron integrals are of the type [ab|cd].

6.3.

Three- and Four-Electron Integrals

They appear in the Hylleraas-CI method [38] when using one inter-electronic distance ri j per configuration. For the two-center case they have been solved generally by Budzinski [39]. Three- and higher number of centers have not been solved yet. These can be many-center integrals, as every electron from right and left in the expectation value operator may be in a different center. These integrals are of the type, i.e. the easier [aa|r12 r13 |ab|bb], to the most difficult [ab|r12 r13 |ab|ab]. Four-electron ones [aa|r12 r13 /r14 |bb|ab|bb], and so on. For three- and higher number of centers one would find three- and four electron integrals with as many centers as the molecule has up to 8. These integrals are still not solved. Interest nowadays focusses on the solution of two and three center molecules using explicitly correlated methods.

7.

Methods in the Literature

In this section the main methods of evaluation of the three- and four-center integrals over Slater orbitals from the literature will be explained. The methods are approximate because they consist in transformations, expansions or include numerical integrations. Therefore they are not as effective as analytical integration would be. Nevertheless, by these methods the evaluation of these integrals is possible and the programs are even as competitive as those using Gaussians.

7.1.

Single-Center Expansion

The single-center expansion method requires expanding the Slater orbitals located at different centers at only one of them and then as for atoms to perform the integrations. The translation method consists in selecting an atom as origin then the translation of other orbitals from their atom to the origin. Therefore both methods are essentially the same. To expand one function centered in A at another point B the following expansion: ∞ Z ϕAi = ∑ ϕAi χB j dτ χB j . (11) j=1

This formula is due to Smeyers [40]. In brackets, the requisite coefficients. The different methods of single-center expansion differ in the way to calculate these coefficients.


71

This method was first proposed by Barnett and Coulson [9] in 1956 using radial orbitals (s-orbitals) and was called the zeta function method because of expansions in terms of successive derivatives with respect to exponents. The method has similarities with the alpha function method of Löwdin [10]. Harris and Michels [41] extended the method to angular general orbitals in 1965. This method has been used by Smeyers, Jones, Guseinov, Fernández Rico et al, and others. The idea is the translation of an orbital from one point to the other. The translation of a spherical harmonic is a limited expansion, the translation of the radial part is nevertheless an infinite expansion. This situation can be best explained with formula of Guseinov [42]: χn,l,m (ζ, rA ) =

∞ n′ −1

l′

∑ ∑ ∑

n′ =1 l ′ =0 m′ =−l ′

Vnlm,n′ l ′ m′ (ζ, RAB ) χn′ ,l ′ ,m′ (ζ, rB ),

(12)

where V are the coefficients of the expansion. The method is very stable but it requires computation of a lot of terms to obtain sufficient correct decimal digits, therefore this method needs very long computational times.

7.2.

Gaussian Expansion

This is the Boys-Shavitt method [43], which consist in solving some integrals over Slater orbitals expanding them into a finite series of Gaussians: NG

e−αr = ∑ ci e−αi r , 2

(13)

i=1

ci and αi are obtained by minimizing the least squares. This method and some improvements of this method are used at present in the program SMILES [25]. As NG is usually larger than the number of the primitives when using only Gaussian basis sets, the number of integrals to calculate is large. The method is very stable and robust. It requires lengthy computational times to get accurate integral values.

7.3.

Gaussian Transform Method

The Gaussian transform method by Shavitt and Karplus 1965 [44] has been probably the most used method. It consists in the Laplace transform of the exponential function, here exemplified by the simplest one i.e. a 1s orbital: α e−αr = √ 2 π

Z ∞ 0

s−3/2 e−α

2 /(4s)

2 ds e−sr .

(14)

Every Slater exponential within the integral is transformed into a Gaussian one, for that one has to solve the integrals over s which have a special form. This integral has to be solved numerically. This is the disadvantage of the method.

7.4.

Fourier-Transform Method

The B-functions Eq. (8) proposed by Filter and Steinborn in 1978 [35] have a highly compact Fourier transform. The group of Steinborn has developed this method [24]. The

72


Figure 3. Transformation from polar to elliptical coordinates. evaluation of integrals using B-functions leads to some integrals including a Bessel function of first kind which is oscillatory: Z ∞ 0

rn e−αr Jl+1/2 (rx)dr.

(15)

To evaluate these Safouhi [45,46] used the SD-transform, due to Sidi [47], which consists in substituting this integral by a sine integral which has the same behavior. It needs numerical integration.

7.5.

Use of Sturmians

The Sturmians were proposed by Shull and Löwdin in 1956 [37]. Smeyers used the Sturmians to evaluate three-center nuclear attraction integrals using the one-center expansion [40]. Guseinov 2001 used also them [48]. The Sturmians Eq. (10) satisfy the SturmLiouville theorem: 2αn m 2 m 2 ∇ Sn,l = α − Sn,l . (16) r The so-called Coulomb Sturmians orthogonalise the Coulomb potential in their argument. This generally applies to the attraction term, at least for one-electron functions. Geminals useful for explicit correlation have also been used. A seminal text by Avery gives more details to the interested reader on this subject [36].

7.6.

Elliptic Coordinate Method

The elliptic coordinate method is the transformation of the polar orbital coordinates into elliptical ones λ, µ according to Figure 3. The two coordinate systems pointed to each other so that the elliptical angle φ coincides with polar angle φ. This transformation is: r1a =

R (λ1 + µ1 ), 2

r1b =

R (λ1 − µ1 ), 2

(17)

Molecular Integrals over Slater-Type Orbitals 1 + λ1 µ1 , λ1 + µ1

cos θ1b =

1 − λ1 µ1 , λ1 − µ1

(18)

[(λ21 − 1)(1 − µ21 )]1/2 , λ1 + µ1

sin θ1a =

[(λ21 − 1)(1 − µ21 )]1/2 , λ1 − µ1

(19)

cos θ1a = sin θ1a =

The volume element and the domain change are: Z ∞ 0

r2 dr

73

Z π 0

sin θdθ

Z 2π 0

dφ →

R3 8

Z ∞ 1

dλ1

Z +1 −1

dµ1 (λ21 − µ21 )

Z 2π 0

dφ1 .

(20)

The method has been used by numerous authors: Mulliken, Rieke, Orloff, Rüdenberg, Roothaan, Eyring, Randic, Saika, Yoshimine, Maslen and Trefry, Guseinov, Bosanac, ¨ Randic, Harris, Fernandez Rico, Lopez, Ozdogan and many others. Some types of three¨ electron integrals have been recently solved by Ozdogan and Ruiz using this method [49].

8. 8.1.

General Two-electron Exponential Type Orbital Integrals in Poly-Atomics without Orbital Translations Introduction

Now, the Coulomb resolution will be presented. This is a readily controlled approximation to separating the variables in the 1/r12 which, in recent work by Gill and by Hoggan is shown to spell the end of exponential orbital translations and ensuing integral bottlenecks. This section advocates the use of atomic orbitals which have direct physical interpretation, i.e. hydrogen-like orbitals. They are Exponential Type Orbitals (ETOs). Until 2008, such orbital products on different atoms were difficult to manipulate for the evaluation of two-electron integrals. The difficulty was mostly due to cumbersome orbital translations involving slowly convergent infinite sums. These are completely eliminated using Coulomb resolutions. They provide an excellent approximation that reduces these integrals to a sum of one-electron overlap-like integral products that each involve orbitals on at most two centers. Such two-center integrals are separable in prolate spheroidal coordinates. They are thus readily evaluated. Only these integrals need to be re-evaluated to change basis functions. The above is still valid for three-center integrals. In four- center integrals, the resolutions require translating one potential term per product. This is outlined here and detailed elsewhere. Numerical results are reported for the H2 dimer and CH3 F molecule. The choice between Gaussian and exponential basis sets for molecules is usually made for reasons of convenience at present. In fact, it appears to be constructive to regard them as being complementary, depending on the specific physical property required from molecular electronic structure calculations. As regards exponential type orbitals (ETOs) such as Slater functions, it seems to be difficult to evaluate two-electron integrals because the general three- and four-center integrals evaluated by the usual methods require orbital translations. Some workers avoid the problem using large Gaussian expansions, as in SMILES [50, 51].


74

It would be helpful to devise a separation of the variables of integration. This would eliminate orbital translations, although some other translations remain involving a simple analytic potential. The present work describes a breakthrough in two-electron integral calculations, as a result of Coulomb operator resolutions. This separates the independent variables of the operator and gives rise to simple analytic potentials. The two-center integrals are replaced by sums of overlap-like one-electron integral products. One potential term in these products requires translation in four-center terms, which is significantly simpler to carry out than that of the orbitals. This implies a speed-up for all basis sets, including Gaussians. The improvement is most spectacular for exponential type orbitals. A change of basis set is also facilitated as only these one-electron integrals need to be changed. The Gaussian and exponential type orbital basis sets are, therefore interchangeable in a given program. The timings of exponential type orbital calculations are no longer significantly greater than for a Gaussian basis, when a given accuracy is sought for molecular electronic properties. Numerical values for all two-electron integrals evaluated using Coulomb resolutions as well as total energies will be tabulated for the H2 dimer and CH3 F molecule.

8.2.

Basis Sets

Although the majority of electronic quantum chemistry uses Gaussian expansions of atomic orbitals [13, 43], the present work uses exponential type orbital (ETO) basis sets which satisfy Kato’s conditions for atomic orbitals: they possess a cusp at the nucleus and decay exponentially at long distances from it [52]- [54]. It updates a ‘real chemistry’ interest beginning around 1970 and detailed elsewhere [3, 4, 15, 27, 44, 55, 56]. Slater type orbitals (STOs) [57, 58] are considered here. STOs allow us to use routines from the STOP package [23, 59] directly. The integrals may be evaluated after Gaussian expansion or expressed as overlaps to obtain speed up [60]. Exponents may be optimized solving a secular determinant as in [61].

8.3.

Programming Strategy

Firstly, the ideal ab initio code would rapidly switch from one type of basis function to another. Secondly, the chemistry of molecular electronic structure must be used to the very fullest extent. This implies using atoms in molecules (AIM) and diatomics in molecules (DIM) from the outset, following Bader (in an implementation due to Rico et al [50]) and Tully [62] implemented in our previous work [59, 63], respectively. The natural choice of atomic orbitals, i.e. the Sturmians or hydrogen-like orbitals lend themselves to the AIM approach. To a good approximation, core eigenfunctions for the atomic hamiltonian remain unchanged in the molecule. Otherwise, atom pairs are the natural choice, particularly if the Coulomb resolution recently advocated by Gill is used. This leads us to products of auxiliary overlaps which are either literally one- or two- centered, or have one factor of the product where a simple potential function is translated to one atomic center. The Slater basis set nightmare of the Gegenbauer addition theorem is completely avoided. Naturally, the series of products required for, say a four-center two-electron integral may require 10 or even 20 terms to converge to chemical accuracy, when at least one


75

atom pair is bound but the auxiliaries are easy to evaluate recursively and re-use. Unbound pairs may be treated using approximate methods. Now, the proposed switch in basis set may also be accomplished just by re-evaluating the auxiliary overlaps. Furthermore, the exchange integrals are greatly simplified in that the products of overlaps just involve a two-orbital product instead of a homogeneous density. The resulting cpu-time growth of the calculation is n2 for SCF, rather than n4 . Further gains may be obtained by extending the procedure to post-Hartree-Fock techniques involving explicit correlation, since the r12 −1 integrals involving more than two electrons, that previously soon led to bottlenecks, are also just products of overlaps. This Coulomb resolution is diagonal in Fourier space in some cases.

8.4.

Avoiding ETO Translations for Two-Electron Integrals over Three and Four Centers

Previous work on separation of integration variables is difficult to apply, in contrast to the case for Gaussians [64, 65]. Recent work by Gill et al [66] proposes a resolution of the Coulomb operator, in terms of potential functions φi , which are characterized by examining Poisson’s equation. In addition, they must ensure rapid convergence of the implied sum in the resulting expression for Coulomb integrals J12 as products of ”auxiliaries” i.e. overlap integrals, as detailed in [66]: J12 = hρ(r1 ) φi (r1 )i hφi (r2 ) ρ(r2 )i, with implied summation over i.

(21)

This technique can be readily generalized to exchange and multi-center two-electron integrals. For two-center terms it is helpful to define structure harmonics by Fourier transforms, limiting evaluation to non-zero terms [67]. Note, however, that in four-center integrals, the origin of one of the potential functions only may be chosen to coincide with an atomic (nuclear) position. Define the potential functions [67]: φi = 23/2 φn l (r)Ylm (θ, φ) . Omitting the spherical harmonic term gives radial factors: φn l (r) =

Z +∞ 0

hn (x) jl (rx)dx, with jl (x) denoting the spherical Bessel function.

(22)

Here, hn (x) is the nth member of any set of functions that are complete and orthonormal on the interval [0, +∞), such as the nth order polynomial function (i.e. polynomial factor of an exponential). The choice made in [66] is to use parabolic cylinder functions (see also another application [51]), i.e. functions with the even order Hermite polynomials as a factor. This is not the only possibility and a more natural and convenient choice is based on the Laguerre polynomials Ln (x): Define: √ hn (x) = 2 Ln (2 x)e−x . (23)


76

These polynomial functions are easy to use and lead to the following analytical expressions for the first two terms in the potential defined in (22): V00 (r) =

V10 (r) =

√

2[

√ tan−1 (r) , 2 r

tan−1 (r) 2 − ], r (1 + r2 )

(24)

(25)

Furthermore, higher n expressions of Vn0 (r) all resemble (25) (see [67] eq (23)): Vn0 (r) =

n √ 1 sin(2 k tan−1 (r)) 2 (tan−1 (r) + ∑(−1)k ), r k 1

(26)

and analytical expressions of Vnl (r) with non-zero l are also readily obtained by recurrence. These radial potentials can generally be expressed in terms of hypergeometric functions, whether the choice of polynomial is the present one, i.e. Laguerre, or Hermite polynomials, as in [66]. This structure has been used to confirm the results of [67] using a rapid code in C [68]. Spherical harmonics are translated using Talman’s approach [69]. The displaced potential in one factor of the product of ’auxiliaries’, from four-center integrals is readily expanded in two-center overlaps, after applying Euler’s hypergeometric transformation. [70, 71]. The auxiliary overlap integrals hρ(r1 ) φi (r1 )i and hφi (r2 ) ρ(r2 )i will involve densities obtained from atomic orbitals centered on two different atoms in exchange multi-center two-electron integrals. The overlap integrals required in an ETO basis are thus of the type: hψa (r1 ) ψb (r1 ) φi (r1 )i =

µmax

∑ Nµ (n1 , n2 , ni , li , |mi | αβ) s(n1, l1 , m, n2, l2 , αβ),

(27)

µ=0

with: α = ζ1 R and β = ζ2 R. Slater exponents. In three-center overlaps, Nµ is a normalised Racah coefficient [71]. In two-center cases the sum reduces to a single normalisation term, N0 . A Fourier transform approach is also being investigated, extending [67]. The real space core overlaps then take the form: λ 1 1 (α + β) B j (α − β) , (28) s(n1 , l1 , m, n2 , l2 , α, β) = Dl1 ,l2 ,m ∑ Yiλj Ai 2 2 ij Yiλj is a matrix with integer elements uniquely determined from n, l and m. Dl1 ,l2 ,m is a coefficient that is independent of the principal quantum number. It is obtained upon expanding the product of two Legendre functions in this co-ordinate system. Symmetry conditions imply that only m1 = m2 = m lead to non-zero coefficients: Z ∞ 1 1 Ai exp − (α + β)µ µi dµ, (α + β) = (29) 2 2 1 Bj

Z 1 1 1 (α − β) = exp − (α − β)ν ν j dν. 2 2 −1

(30)


77

Here, recurrence relations on the auxiliary integrals A and B lead to those for the requisite core integrals [72, 73]. These integrals may be pre-calculated and stored. Such integrals appear for two-center exchange integrals and three- and four-center integrals (although just in one factor for three-center Coulomb terms). Note that exchange integrals require distinct orbitals ψa and ψb . In the atomic case, they must have different values for at least one of n, l, m or ζ. In the two-center case, the functions centered at a and b may be the same. The product does not correspond to a single-center density: it is two-centered. Equation (27) then illustrates the relationship to the one-electron two-center overlap integral, although it clearly includes the extra potential term from the Coulomb operator resolution. This assumes tacitly that the potential obtained from the Coulomb operator resolution be centered on one of the atoms. Whilst this choice can be made for one pair in a four-center product, it cannot for the second. There remains a single translation for this potential in one auxiliary of the two in a product representing a four-center integral and none otherwise. This method obviates the need to evaluate infinite series that arise from the orbital translations efficiently. They have been eliminated in the Coulomb operator resolution approach, since only orbitals on two centers remain in the one-electron overlap-like auxiliaries. These can be evaluated with no orbital translation, in prolate spheroidal co-ordinates, or by Fourier transformation [67, 71].

8.5.

Numerical Results of Coulomb Resolutions: Efficiency

First a test system is studied, built up of four hydrogen atoms. The second example is the full RHF calculation of CH3 F using the Coulomb resolutions. Consider the H2 molecule and its dimer/agregates. In an s-orbital basis, all two-center integrals are known analytically, because they can be integrated by separating the variables in prolate spheroidal co-ordinates. A modest s-orbital basis is therefore chosen, simply for accuracy demonstration on a rapid calculation, for which some experimental data could be corroborated. The purpose of this section is to compare evaluations using the Coulomb resolution to the exact values, obtained analytically. The IBM Fortran compiler used is assumed to be reliable to 14 decimals in double precision. The worst values in the Coulomb resolution approximation have 10 correct decimals for two-center integrals with a 25-term sum. Timings are then compared for translation of a Slater type orbital basis to a single center (STOP) [59] with the Poisson equation solution using a DIM (Diatomics in molecules or atom pair) strategy and finally to show that the overlap auxiliary method is by far the fastest approach, for a given accuracy (the choice adopted is a sufficient six decimals, for convenient, reliable output). H2 molecule with interatomic distance of 1.402 atomic units (a.u.). One and two-center Coulomb integrals may be obtained analytically and Coulomb resolution values compare well with them [66]. The two-center exchange integrals are dominated by an exponential of the interatomic distance and thus all have values close to 0.3. The table is not the full set. All index ‘15‘ terms, involving 1sa1 (1) 1sb1 (2) are given, to illustrate symmetry relations. Note that this is by no means the best possible basis set for H2 , since it is limited to

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Table 1. Atomic exchange integrals (6 distinct single center values between pairs of different AOs). AOs (zeta) 1sa1 1.042 999 1sa2 1.599 999 2sa1 1.615 000 2sa2 1.784 059 1sb1 1.042 999 1sb2 1.599 999 2sb1 1.615 000 2sb2 1.784 059

Label 1 2 3 4 5 6 7 8

[a(1)b(2)a′ (2)b′ (1)] 1212 1313 1414 2323 2424 3434 2121 3232

Value 0.720 716 0.585 172 0.610 192 0.557 878 0.607 927 0.602 141 0.720 716 0.557 878

Table 2. Two-center exchange integrals. All pair permutations possible. Some are identical by symmetry. Labels 1515 1516 1517 1518 1527 1528 1538 2525 2516 2517 2518

Value 0.319 902 0.285 009 0.325 644 0.324 917 0.291 743 0.293 736 0.329 543 0.260 034 0.254 814 0.290 533 0.290 149

l = 0 functions (simply to ensure that even the two-center exchange integral has an analytic closed form). The total energy obtained for the isolated H2 molecule is -1.1284436 a.u. as compared to a Hartree-Fock limit estimate of -1.1336296 a.u. Nevertheless, the Van der Waals well, observed at 6.4 au with a depth of 0.057 kcal/mol (from Raman studies) is quite reasonably reproduced [74]. Dimer geometry: rectangular and planar. Distance between two hydrogen atoms of neighboring molecules: 6 a.u. Note that this alone justifies the expression dimer, the geometry corresponds to two almost completely separate molecules. However, the method is applicable in any geometry (for 3 a.u. all three- and four-center integrals evaluated by Coulomb resolution agree with those of STOP to at least 6 decimals). Timings on an IBM RS6000 Power 6 workstation, for the dimer (all four-center integrals in msec): STOP: 12 POISSON: 10 OVERLAP: 2. Total dimer energy: -2.256998 a.u. This corresponds to a well-depth of 0.069 Kcal/mol, which may be considered reasonable in


79

Table 3a. Orbital exponents. AO No. 01 02 3-5 06 07 8-10 H

n 1 2 2 1 2 2 1

l 0 0 1 0 0 1 0

m 0 0 m 0 0 m 0

zeta 5.6727 1.6083 1.5679 8.5600 2.5600 2.5200 1.2400

Table 3b. Selected examples of three-center exchange integrals. Integral h2sC 2sF |2sC 1sHa i h2sC 2sF |2sC 1sHa i h2sC 1sF |1sC 1sHa i h2sC 1sF |2sC 1sHa i h1sC 2pzF |2pzC 1sHa i h2sC 2pzF |2pzC 1sHa i h2pzC 1sF |1sC 1sHa i h2pzC 1sF |2sC 1sHa i h1sC 1sF |1sC 1sHa i h1sC 1sF |2sC 1sHa i h1sC 2sF |1sC 1sHa i h1sC 2sF |2sC 1sHa i h2pzC 2sF |1sC 1sHa i h2pzC 2sF |1sC 1sHa i

Value 0.4970 48510 ×10−1 0.8420 56635 ×10−2 0.5737 90540 ×10−3 0.3789 18525 ×10−2 0.1587 58344 ×10−2 0.5258 34208 ×10−2 0.1025 32536 ×10−2 0.6772 76818 ×10−2 0.1099 00118 ×10−6 0.6794 54131 ×10−6 0.1446 31297 ×10−2 0.4235 59085 ×10−2 0.1112 10955 ×10−1 0.6738 14908 ×10−1

Integral h2sF 1sHa |1sF 2sC i h2sF 1sHa |2sF 2sC i h2sF 1sHa |2pzF 2sC i h2sF 1sHa |1sF 2pzC i h2sF 1sHa |2sF 2pzC i h2sF 1sHa |2pzF 2pzC i h1sHa 2sF |1sHa 2sC i h1sHa 2sF |1sHa 2pzC i h1sHa 2pzF |1sHa 2pzC i h1sF 1sHb |2sF 1sC i h1sHb 2sF |1sHb 1sC i h2sC 1sHa |1sC 1sHb i h1sC 1sHa |1sC 1sHb i h2sC 1sHa |2sC 1sHb i

Value 0.1014 05594 ×10−2 0.9341 35949 ×10−2 -0.8442 95091 ×10−2 0.1813 23479 ×10−2 0.1379 64387 ×10−1 -0.1135 01125 ×10−1 0.1252 319411 ×10−1 -0.1591 49899 ×10−2 0.1772 90873 ×10−2 0.2287 77210 ×10−4 0.1939 63837 ×10−2 0.2034 841982 ×10−1 0.7154 932331 ×10−2 0.1137 390852

view of the basis set.

8.6.

Selected Exchange Integrals for the CH3 F Molecule (Evaluated Using the Coulomb Resolution)

Geometry and exponents are those of previous work [75]: Tetrahedral angles, with C-H 2.067 and C-F 2.618 a.u. No symmetry is assumed but geometric relationships are observed, as well as those due to m values, at least to the nano-Hartree accuracy chosen. For illustrative purposes, three-center exchange integrals are tabulated in a real basis. Timings on IBM RS6000 Power 6 workstation for all two-electron integrals: STOP: 1.21 s, OVERLAP: 0.17 s. All the two-electron integrals are identical to better than six significant figures with those obtained using the STOP software package [59]. The factor limiting precision in this study is the accuracy of input. The values of Slater exponents and geometric parameters are required to at least the accuracy demanded of the


80

integrals and the fundamental constants are needed to greater precision.

8.7.

Conclusions

A remarkable gain in simplicity is provided by Coulomb operator resolutions [66], that now enables the exponential type orbital translations to be completely avoided in ab initio molecular electronic structure calculations. This breakthrough that Coulomb resolutions represent (in particular with the convenient choice of Laguerre polynomials) in the ETO algorithm strategy stems from a wellcontrolled approximation, analogous to the resolution of the identity. The convergence has been shown to be rapid in all cases [67]. The applications to H2 dimer Van der Waals complexes and CH3 F uses a general code within the STOP package [59]. They show the Coulomb resolution can be used to give fast and accurate results for basis sets of s and p Slater type orbitals. Generalisation is in progress. Numerical vales for the H2 dimer geometry and interaction energy agree well with complete ab initio potential energy surfaces obtained using very large Gaussian basis sets and data from vibrational spectroscopy [74].

9.

Explicitly Correlated Methods for Molecules

The application and development of such methods to determine accurately the ground and excited states, and properties of diatomic and triatomic molecules is very promising and more interesting for the Computational Chemist than the atomic case. There is nowadays a growing interest in this field. Subroutines and programs which perform these calculations are often requested in the community. The investigation of these integrals should be approached within the Molecular Orbital method (MO) [76], because the MO wave function is the simplest wave function for a molecular system. As Coulson [77] discussed, the MO method permits the visualization of electrons and nuclei and interpretation of individual electrons and their orbital exponents better than the wave functions written in elliptical coordinates. The wave functions constructed with elliptical orbitals are of two types, the so-called James-Coodlige [78] wave functions (one-alpha exponent), recently extended to the twoalpha case [33], and Kolos-Wolniewick [79,80] wave functions (with both orbital exponents alpha, and beta ). Both have been applied to the H2 molecule. The elliptical wave functions are the natural representation of a two-center problem but for three-center and larger molecules the use of the MO method becomes necessary. Frost [81] used the MO method and the Correlated Molecular Method (CMO) in H2 calculations. About the extension of the method he wrote: ”The extension of CMO-type wave functions to more complex molecules does not seem feasible at the present time. The new integrals which will be introduced would involve more than two centers if more nuclei were involved and higher atomic orbitals than 1s if more electrons were considered, and their evaluation would be extremely difficult”.


81

Recently, impressive calculations using Hylleraas wave functions have been done for H2 , see Table 4. Hylleraas [33], the Iterative Complement Iteration method (ICI) [82], and explicitly correlated Gaussian (ECG) [83] calculations of the hydrogen molecule, Hylleraas calculations on HeH+ and some other species [84] leading to −2.9710784698 a.u. using 9576 configurations and calculations of He2 using 4800 optimized ECG configurations with energy −5.80748359014 a.u. [83] achieved the highest known accuracy in molecules (picohartree accuracy is more than that of chemical measurements, e.g. a micro cm−1 , a nano eV or micro cal/mol. Although one must recall that in the calculation of properties according to Drake [85], only half of the digits of the energy are kept). Note also that input exponents, distances and some fundamental constants may limit accuracy of calculations compared with measurements and that molecules may not be rigid. Eventually, dynamics and the effect of the Born-Oppenheimer approximation should be included. Hylleraas-CI (Hy-CI) was applied in 1976 to LiH molecule by Clary [32] using elliptical STOs. For two-center molecules the three-electron and four-electron integrals occurring in the Hy-CI have been developed by Budzinski [39]. Another type of explicitly correlated wave functions are the ones that use Gaussian orbitals. Clementi et al extended the Hy-CI to molecules using Gaussian orbitals [86], and applied it to the calculation of H3 . The ECG wave function is appropriate also for molecules [83, 87], as the inter-electronic distance r12 is a Gaussian exponent. This leads to results, which are comparable with Hylleraas calculations [83]. The R12 -wave function proposed by Kutzelnigg and Klopper [88, 89] has the merits to fulfill the cusp condition, to use Gaussian functions avoiding the three- and four-center integration problems, and to include precisely r12 , involving electrons 1 and 2, close to the nucleus, where the probability that r12 = 0 is larger, also these electrons are present at any system starting from helium atom. The r12 variation influences energy. The R12 wave function, developed for molecular calculations is nowadays widely used and combined with all kinds of methods. The occurring three- and four-electron integrals are calculated in terms of two-electron ones. Due to the use of a single r12 value, the accuracy achieved for atomic calculations is lower than the accuracy of Hy and Hy-CI calculations. Recent improvements of the method [90]- [92] can achieve microhartree accurate energy results for chemically interesting systems. Table 4. Highly accurate calculations on the H2 molecule with different types of wave functions at R=1.4011 a.u. Authors 1933 James and Coolidge 1960 Kolos and Roothaan 1968 Kolos and Wolniewicz 1995 Wolniewicz 2006 Sims and Hagstrom 2007 Nakatsuji 2008 Cencek and Szalewicz

type w. f. JC KR KW KW JC ICI ECG,opt

Confs. 5

833 7034 6776 4800

Energy (a.u.) -1.1735 -1.17214 -1.174475 -1.17447467 -1.17447593139984 -1.17447571400027 -1.17447571400135


82

Short wave function expansions lead to very good results. When a large number of configurations are used (up to 10000) the energy results are beyond pico-hartree accuracy, while the CI wave function would need in the order of millions of configurations.

10.

Highly Accurate Calculations Using STOs

Another problem appearing in these calculations is the digital erosion. For many operations and subtraction numbers of similar value some digits can be lost leading to erroneous results. Quadruple precision avoids this, about 30 decimal digits are correct on our computer. Other possibility is high precision arithmetic software. Some programs are available like Bailey’s MPFUN [93], the Brent and Miller program packages [94, 95]. One example of the use of Slater orbitals in the present are the highly accurate calculations of small molecules using explicitly correlated wave functions i.e. wave functions where the inter-electronic coordinate ri j is included explicitly in the wave function. These are the Hylleraas and Hylleraas-CI wave functions, ICI method, compared with the explicit correlated Gaussians ECG and the R12 method.

11.

Closing Remarks

We conclude with the words of G. Berthier: GTOs are like medicine, you have to use them as long as they are healing, but once they don’t work any more, you much change them, Gaston Berthier, Interview, Paris, 2nd June 1997. Recently, a whole book ”Recent Advances in Computational Chemistry: Molecular Integrals over Slater Orbitals ” was dedicated to a mathematical review of methods of integration over Slater orbitals and Hylleraas wave functions [96].

Acknowledgements The authors would like to thank very much Profs. Milan Randic, Ante Graovac, Roberto Todeschini and Peter Otto for their interest in Slater orbitals.

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[81] Frost A.A. and J. Braunstein J. Hydrogen molecule energy calculation by correlated molecular orbitals. J. Chem. Phys. 1951 19, 1133-1138. [82] Kurokawa Y., Nakashima H. and Nakatsuji H. Free iterative-complement-interaction calculations of the hydrogen molecule. Phys. Rev. A, 2005 72, 062502. [83] Cencek W. and Szalewicz K. Ultra-high accuracy calculations for hydrogen molecule and helium dimer. Int. J. Quantum Chem. 2008 108, 2191-2198. [84] Zhou B.L., Zhu J.M., and Yan Z.C. Ground state energy of HeH+ . Phys. Rev. A 2006 73 064503. [85] Drake G.W.F. High precision theory of atomic helium. Phys. Scr. 1999 T83, 83-92. [86] Frey D., Preiskorn A., Lie G.C., and Clementi E. HYCOIN: Hylleraas Configuration Interaction method using Gaussian functions, in Modern Techniques in Computational Chemistry: MOTECC-90, Clementi E. Ed., ESCOM Science Publ.: Leiden 1990, Chapter 5, pp. 57-97. [87] Rychlewski J. and Komasa J. Explicitly correlated functions in variational calculations, in Explicitly Correlated Wave Functions in Chemistry and Physics, Rychlewski J. Ed., Kluwer Academic Publishers: Netherlands 2004, pp. 91-147. [88] Kutzelnigg W. r12 -dependent terms in the wave function as closed sums of partial wave amplitudes for large l. Theor. Chim. Acta 1985 68, 445-469. [89] Klopper W. and Kutzelnigg W. Møller-Plesset calculations taking care of the correlation cusp. Chem. Phys. Lett. 1987 134, 17-22. [90] Klopper W. and Noga J. Linear R12 terms in Coupled Cluster theory, in Explicitly Correlated Wave Functions in Chemistry and Physics, Rychlewski J. Ed., Kluwer Academic Publishers: Netherlands 2004, pp. 149-183. [91] Cardoen W., Gdanitz R.J., and Simons J. Transition-state energy and geometry, exothermicity, and van der Waals wells on the F + H2 → FH + H ground-state surface calculated at the r12 -ACPF-2 level. J. Phys. Chem. A 2006 110, 564-571. [92] Explicitly Correlated Wave Functions in Chemistry and Physics, Rychlewski J. Ed., Kluwer Academic Publishers: Netherlands, 2004.


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[93] Bailey D.H. High-precision software directory. Available from: http://crd.lbl.gov/˜dhbailey/mpdist/mpdist.html [94] Brent R.P. A Fortran multiple-precision arithmetic package, ACM Transactions on Mathematical Software (TOMS), v.4 n.1, pp.57-70, 1978. [95] Miller A.J. Alan Miller’s Fortran software. Available from: http://users.bigpond.net.au/amiller/ [96] Recent Advances in Computational Chemistry: Molecular Integrals over Slater Or¨ bitals, Ozdogan T. and Ruiz M.B. Eds., Transworld Research Network: Kerala, India, 2008.



Chapter 5

T UNNELING DYNAMICS AND I TS S IGNATURES IN C OUPLED S YSTEMS S. Ghosh and S.P. Bhattacharyya∗ Department of Physical Chemistry Indian Association for the Cultivation of Science Jadavpur, Calcutta 700 032, INDIA

Abstract It has been demonstrated through numerical experiments that tunneling in a symmetric double well may be either suppressed or enhanced by a Morse oscillator coupled to it depending on the form of coupling. An external time varying electric field that causes 0+ − 0− transition in the double well affects the tunneling rate. A well defined minimum in the rate is observed for λ = λc for which maximum energy transfer from the double well to the Morse mode takes place. If the field is chosen to cause 0 → 1 transition in the Morse mode, the dissociation probability is influenced by the tunneling mode, being maximized for a particular value λ = λ′c . In the uncoupled system, field with similar intensity practically fails to cause any dissociation. The tunneling dynamics is analyzed in a situation when the particle has a coordinate dependent mass as is often assumed in the charge transport in hetero-structures. The tunneling time or rate are seen to be very significantly affected by the nature of the coordinate dependence of the tunneling mass.

1.

Introduction

As a purely quantum phenomenon, tunneling is ubiquitous in microscopic systems. It pervades all areas in physics, chemistry and biology and ever since its use in nuclear physics in explaining the decay of α particles from atomic nuclei, the importance of tunneling has been increasingly recognized. The study of electron tunneling in condensed matter physics has led to the Josephson effect and the tunneling diode [1] . The tunneling of proton or hydrogen atom has often been invoked in chemistry to explain unusual features of chemical ∗ E-mail

address: [email protected]

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S. Ghosh and S.P. Bhattacharyya

reaction rates or mechanisms. In fact, the atom tunneling phenomenon has been recognized to be important in science of various types of materials and biology. Starting from atom tunneling reactions in quantum solid hydrogen or in solid or liquid Helium, there have been studies relating to rather unusual aspects of tunneling reactions of organic substances, tunneling insertion reaction of carbenes and heavy particle tunneling. The role of atom tunneling reaction in vitamin-C in the suppression of mutation or of vitamin-E in its antioxidant, pro-oxidant and regeneration reactions have attracted serious attention from biologists. The possible occurrence of spontaneous tunneling elimination of hydrogen molecules from hydrocarbon cations has led to serious questions about our conventional idea of a stable chemical structure [2]. The relevance of tunneling in the interpretation of molecular and crystal structure in very low temperature regimes can hardly be overestimated. It would be appropriate therefore to review briefly how the idea of tunneling was developed and exploited in different fields of science.

2.

Historical Development

A. Tunneling in Physics The theory of α radioactivity proposed by Gamow [3] explained the law of exponential −Γt by solving the Schrodinger decay P(t) = N(t) ¨ equation for the α particle inside the N0 = e nucleus where the attractive nuclear force and coulomb repulsion were assumed to provide an effective barrier confining the α particle. Gamow imposed the ’outgoing wave’ boundary condition at large distances from the center of the nucleus and found that the Schrodinger ¨ equation does not have solutions for the real energies while for complex energies, it had solutions. Gamow interpreted the imaginary part of the energy as the decay width Γ2 and obtained a relation between Γ2 and the energy of the emitted α particle. The use of complex energy meant the use of a non-hermitian hamiltonian and the idea was criticized as quantum mechanics worked with hermitian operators. The same result was later obtained by Bohr by considering states with real energies and working with a hermitian hamiltonian [4]. The idea of resonant tunneling was introduced by Gurney who realized that particles with low energies that match with quasi-stationary energies of the nucleus could easily penetrate the barrier [5]. The importance of the idea in artificial disintegration can hardly be overestimated. The idea of tunneling was soon exploited in other areas of physics. Notably, many attempts were made to relate the dynamics of electron current in the metal semi-conductor systems to the tunneling of electrons in solids. The discovery of transistors in 1947 rekindled interest in the tunneling of electrons in solids the occurrence of which was conclusively proved by L. Esaki (1957) who discovered the tunneling diode [6]. Close on the heels of the discovery of tunneling diode, Giaever found if one or both the metals are superconducting, the voltage-current plots could lead to measurement of energy gaps in superconductors [7]. Josephson discovered that the superconductors separated by a thin layer of insulating oxide provide a system in which a second current (over and the above the Giaever’s current), the so-called supercurrent, exists and that it is caused by tunneling of electrons in pairs [8]

Dynamics and Its Signature

93

B. Tunneling in Chemistry Tunneling has a long history in chemical kinetics. Traditionally, curvature in Arrhenius plots of rate constants has been interpreted as a signature of tunneling. It is generally very difficult to observe such curvatures in Arrhenius plots for gas phase chemical reactions while such curvatures are a common-place occurrence in chemical reactions in condensed phases below 100 K. Accurate theoretical calculations, however, indicate that tunneling can contributes significantly to the reaction rates even at room temperature where the Arrhenius plots are very nearly linear [8]. In fact, ’tunneling’ can be taken to be synonymous with a chemical reaction occurring at energies less than the barrier energy (Ea ). The barrier arises on the 3N − 6 dimensional potential energy surface in the Born-Oppenheimer approximation as the nuclei of the N atom reactive system move breaking and making bonds. Traditionally one identifies the one dimensional minimum energy path (x) as the reaction path along which an effective potential energy Ve f f (x) is defined by adding the vibrational energies ε(x) along the reaction path associated with the nuclear motion perpendicular to it [9]. This vibrationally adiabatic approximation reduces the multidimensional problem into a one dimensional problem, for which tunneling has a unique definition. The tunneling probabilities calculated using the idea described above are generally a bit too small due to the neglect of the reaction path curvature. Several alternative approaches have been explored for better representation of the tunneling path, for example, the least action ground state method [10] and the ’tunneling tube method’ [11]. The barrier energy on the reaction −Ea

path can be identified with the activation energy (Ea ) in the Arrhenius rate law k = Ae Kb T which has been recognized as the central law of chemical kinetics. Many attempts have been made to derive the central law of chemical kinetics. The transition state theory (TST) of Eyring [12] was the first attempt in this direction. The TST was derived assuming complete thermodynamic equilibrium wherein the possibility of reverse transition was neglected, which was later taken care of in an ad hoc manner by introducing a transmission coefficient in the pre-exponential factor. Kramers’ [13] treated elementary rate process in the presence of a medium as a generalized Brownian motion in a potential field and showed how the rate constant would be influenced by the viscosity of the medium. Kramers’ results have been derived from the TST applied to an extended system of the reactants and reservoir of oscillators which provide a frictional force and a random force along the reactive mode [14]. Kim and Hynes [15], Truhler etal. [16] introduced additional coordinates describing the dynamics of the solvent mode along with the reagent mode within the framework of TST and derived results equivalent to those obtained by Kramers. Chemical reactions at sufficiently low temperatures are marked by remarkably special features. Goldanski [17], showed that the rate constant of a chemical process has a low temperature limit (non-zero) as T → 0 and introduced the concept of crossover temperature (Tc )which divides the whole temperature interval into over and under barrier regimes. Goldanskii assumed that the tunneling particle moves through the saddle point for obtaining TC . A tunneling particle may not, however move in the traditional classical manner along the adiabatic reaction coordinate through the first order saddle point, but take a short cut wherein it experiences a higher barrier, but a shorter tunneling length. In the absence of highly precise data for the reaction systems, numerical calculations for obtaining the tun-

94


neling rate constant become difficult. One is then left with the alternative of relating the special features of tunneling chemical reactions to the changes in the reaction barrier due to the vibrations (intermolecular) [18]. Atom transfer reactions are the most thoroughly investigated reactions [19]. Here the transferring atom moves between the two potential wells. A reversible transfers leads to tunneling level splitting (spectral signature of tunneling) and an irreversible transfer precipitates chemical reactions (signature of tunneling dynamics). If the period of inter-well quantum oscillation is smaller than the relaxation time, coherent transfer takes place while the process is incoherent if the opposite situation is encountered. If the two time scales are comparable, the process is better described by density matrix methods. Ivanov and Kozhushner [20] showed that the time period of oscillation (Ω−1 ) and the particle transfer time (the tunneling time τ) play an important role. If Ωτ > 1, the transfer takes place at average positions of the reagents and the surrounding molecules. If Ωτ = 1, the tunneling particle adjusts its positions corresponding to those of the reagents and the surroundings. The synchronous motion of the reagent and the surroundings may either hasten or delay the particle transfer rate [22]. In a tunneling reaction we have three subsystems taking part of which the electronic and the intramolecular subsystem constitute the fast variables and the inter subsystem variables constitute the slow variable. Using the Fermi Golden rule, and a modified theory of radiationles transition, an expression for the tunneling rate constant has been obtained [21] ktunneling =

2π ~

Aνi ∑hφν f |V (R)|φνi i2 δ(~ν f + ∆E − ~νi ). νf

The transition matrix element delicately depends on the distance between the reagents and that means intermolecular vibrations play a dominant role in shaping the tunneling rate constant. Such vibrations cause variations in the tunneling distance. Vibrations that bring the reagent closer are called promotive modes which are taken into account by invoking either the Einstein or the Debye model of promotive modes [22-24], depending upon the situation. It has been shown that in atom tunneling reactions in the solid phase the low temperature limit of ktunneling exists for nonendothermic processes only. Along with inter-molecular vibrations (which can change inter-reagent separation), medium reorganization and under barrier friction play significant roles in determining temperature and pressure dependence of tunneling rate constants. The effect of reorganization of the medium alone leads to a temperature dependence that is independent of the form of the barrier. We note that the tunneling particle is assumed to interact with phonons only in the initial and final states, but not during the course of tunneling. The interaction of the tunneling particle with phonons in the underbarrier regimes lead to the appearance of frictional effects that depend rather sensitively on the form of the barrier and its modulations, if any.

C. Tunneling in Coupled Systems With the preceeding background in view, our purpose in this chapter has been to investigate typical signatures of tunneling dynamics in several coupled systems. In one of these, the tunneling mode is described by a symmetric double well potential which is coupled to,


95

let us say, a bond stretching mode. The latter is represented by a typical Morse potential and the form and strength of the coupling are assumed to vary. We propose to investigate how the mode of coupling with bond stretching mode influences the tunneling dynamics in the double well and vice versa. If the bond stretching mode is externally driven, how does it affect the tunneling dynamics? These questions are addressed in subsequent sections. Normally, the tunneling particle is assumed to have a fixed mass- fixed in space and time. However, in charge transfer and tunneling of electrons through heterostructures, the effective mass of the electron is often assumed to be coordinate dependent. Could the coordinate dependence of the mass of the tunneling particle have a typical signature on the tunneling dynamics? We have investigated the question through numerical experiments. Let us now focus on the general methodology used in our explorations.

3.

The Method

For the calculations, we have used time dependent Fourier grid hamiltonian method [25-19]. The basic framework of the approach adopted in the present series of calculations is described in this section.

A. Dynamics of the Coupled System in the Absence of Driving Let H0 (x, y) be the hamiltonian of a particle of mass m moving on the x − y plane in a potential V0 (x, y), where V0 (x, y) is assumed to be additively separable into a double well potential V0 (x) and a Morse potential V0 (y) . Thus H0 (x, y) = T0 (x) + T0 (y) +V0 (x, y) = T0 (x) +V1 (x) + T0 (y) +V2 (y) = H0 (x) + H0 (y).

(1)

Let us assume that x represents the tunneling coordinate and y stands for a bond stretching coordinate. To be more specific, we may take H0 (x) is taken to represent the Hamiltonian describing the motion of a quantum particle in a asymmetric double well potential V1 (x) while H0 (y) is assumed to represent the motion along a bond stretching coordinate described by an appropriate Morse potential V2 (y) . We may now introduce a coupling between the two modes by using the interaction potential Vint (x, y) so that the total hamiltonian H(x, y) that describes the coupled system can be written as H(x, y) = H0 (x, y) + λVint (x, y).

(2)

λ being the strength of the coupling between the tunneling and bond stretching coordinates. The purpose of the present study has been to investigate how the tunneling dynamics gets affected by the coupling with the bond stretching mode. Let us assume that the eigen functions φi (x) of H0 (x) and χi (y) of H0 (y) are known to start with. The states of the coupled system can be described by superposition of the products of the eigenstates of the uncoupled system [H0 (x) and H0 (y)]. Thus we may write ni n j

|ψ(x, y,t)i = ∑ ∑ ci j (t)|φi (x)χ j (y)i. i

j

(3)

96


The product functions generated by the eigenfunction of the uncoupled system described by the hamiltonians.H0 (x) and H0 (y) serve as the basis of the coupled system. The introduction of the coupling term Vint (x, y) would cause the superposition in equation 3 to evolve in time. The combination coefficients ci j ’s are therefore taken to be time dependent. Time evolution equations for the combining coefficients are obtained from the time dependent Schrodinger equation by following the standard procedures and are given by i~

dci j = dt

mx ,my

∑ k,l

ckl hφxk χyl |H0 (x, y)|φxi χyj i

(4)

for i = 1, mx , j = 1, ny We may call it a time dependent configuration interaction type of formalism. To find out the time independent basis functions |φi (x)i and |χ j (y)i we make use of the Fourier grid hamiltonian (FGH) technique [29] and compute the eigenfunctions and eigenvalues of the uncoupled model systems as follows H0 (x)|φi (x) = εxi |φi (x)i,

(5)

where nx

∑ wxpi |x p i∆x,

(6)

H0 (y)χi (y) = εyi |χi (y)i,

(7)

|φi (x)i =

p=1

and

where |χyi i =

ny

∑ wyqi |yq i∆y.

(8)

q=1

wxp,i and wyq,i are the corresponding grid point amplitudes along x and y coordinates, respectively. We note that the coordinates are uniformly discretized, ∆x and ∆y being the uniform grid spacing along the x and y coordinate, respectively. Using equations (6) and (8) the time evolution equations of the combining coefficients ci j (i = 1, nx ; j = 1, ny ) of equation 4 reduce to nx ,ny dci j i~ = ∑ ckl ∑ wxpi wyq j H(x p , yq )wxpk wyql (∆x∆y)2 = ∑ ckl Hi j,kl . dt p,q k,l k,l

(9)

In matrix form equation 9 can be written as ˙ = HC(t). i~C(t)

(10)

The time integration may be done by employing sixth order Runge-Kutta method. When a quantum mechanical particle moves in a double well there is a nonzero probability that the particle tunnels from one well to the other well whatever be its energy relative to the barrier


97

top. Let us suppose the particle was initially localized in the left well of the uncoupled symmetric double well. The lowest energy localized states (φL and φR ) can be described by linear combinations of the two lowest energy eigen-states of even (ψ+ 0 ) and odd parity − (ψ0 ), respectively 1 − φL = √ (ψ+ (11) 0 + ψ0 ), 2 1 − φR = √ (ψ+ (12) 0 − ψ0 ), 2 where φL and φR represent the states localized in the left and the right well, respectively. In the absence of any coupling between the tunneling mode and the bond stretching mode the particle tunnels coherently from the left well to the right well. If one takes φL to be the initial state, tunneling probability is obtained by calculating the probability of finding the particle in the right well at energies less than the barrier energy. The corresponding probability PR (t) at any particular instance is given by (the barrier is located at x = 0) PR (t) =

Z ∞ 0

|ψ(x,t)|2 dx.

(13)

In the two dimensional case, the corresponding probability can be computed by using PR (t) =

Z +∞ Z +∞ y=−∞ x=0

|ψ(x, y,t)|2 dxdy.

(14)

In the CI formalism used by us in the FGH basis, equation 14 can be written as ni ,n j

PR (t) =

∑ i, j

|ci j |2

nx

ny

nx −1 2

q=1

∑ ∑ wxpi wyq j ∆x∆y.

(15)

The rate of tunneling can then be obtained from the average slope of the PR (t) - t plot. Alternatively, the tunneling rate can be calculated from the average rate of the change of the hx(t)i with time plot, where hx(t)i is given by hx(t)i = hψ(x, y,t)|x|ψ(x, y,t)i.

(16)

Substituting the expression for hψ(x, y,t)| from equation (4) we get hx(t)i =

∑ c∗i j (t)ck j (t)hφxi |x|φxk i

(17)

i, j,k

nx

=

∑ c∗i j ck j ∑ wxpi x p wxpk .

i, j,k

(18)

p

The computed rate of tunneling is related to the tunneling splitting in the double well [30]. When the double well gets coupled to a bond stretching mode described by a Morse oscillator, we may envisage three probabilities: (a) The bond stretching mode enhances the tunneling rate. We may call it a promoting mode. (b) The bond stretching mode reduces the tunneling rate. We call it a suppressing mode (c) The bond stretching mode does not affect the tunneling rate at all. It acts as a passive mode We anticipate that the type of the effect could depend on the nature of the coupling (see later).

98


B. Dynamics of the Coupled System in the Presence of External Driving When a spatially homogeneous external electric field is applied along the x direction the perturbed hamiltonian H(x,t) can be represented as H(x,t) = T (x) +V1 (x) +V (x,t) = H0 (x) +V (x,t),

(19) (20)

where V (x,t) = xeεx sin(ωt).

(21)

ε is the field intensity and e represents the charge carried by an electron. Similarly, with an electric field applied along y direction H(y,t) = H0 (y) +V (y,t),

(22)

V (y,t) = yeεy sin(ωt).

(23)

where

The time evolution equations now turn out to be (with field along x direction) i~

nx ,ny dci j = ∑ ckl ∑ wxpi wyq j (H(x p ,t) + H0 (yq ) + λVint (x p , yq ))wxpk wyql (∆x∆y)2 dt p,q k,l

(24)

for i = 1, nx ; j = 1, ny . These equations can be integrated numerically over ckl ’s at t = 0 are provided.

4.

Results and Discussion

A. Tunneling in the Coupled System in Absence of External Driving We take the model potentials V1 (x) = ax4 − bx2 + c,

(25)

(−β(y−ye )) 2

V(2) (y) = D(1 − e

) .

(26)

The system parameters are listed in table-1. FGH calculations have been done with 151 grid points in each coordinate. The grid lengths along x axis and y axis are 6 a.u and 10 a.u, respectively. When V1 (x) and V2 (y) are coupled through an interaction potential Vint (x, y) and the coupling strength λ. the shape of the two dimensional potential energy surface (PES) gets modified depending on the functional form of V (x, y) and strength λ. Figure 1a shows the two dimensional potential energy surface (PES) when V1 (x) and V2 (y) are uncoupled, i.e. λ = 0. Figure 1b shows the two dimensional PES for Vint (x, y) = λxy, (λ = 0.001). Evidently, for the particular functional form the symmetry of the double well potential gradually gets distorted (along the y axis). The left well becomes deeper than the right well. By increasing the coupling strength the asymmetry between the two wells


99

Figure 1. Potential Energy Surfaces for different forms of interaction potentials Vint (x, y). (a) Vint (x, y) = 0; (b) Vint (x, y) = λxy; (c) interaction potential Vint (x, y) = λx2 y2 ; (d) Vint (x, y) = λ(x2 y + xy2 ).

increases and the coherence in the tunneling process is progressively disturbed. Figure 2a shows the plot of the computed tunneling rate versus the coupling strength. The initial wave function ψ(x, y,t = 0) corresponds to the state in which the Morse mode and the tunneling mode are both in the ground state. From figure 2a we can clearly conclude that the tunneling rate decreases almost linearly with increasing coupling strength for the particular functional form of interaction potential Vint (x, y) = λxy. Figures 1c and 1d show the two dimensional PES for the functional forms Vint (x, y) = λx2 y2 and Vint (x, y) = λ(x2 y + y2 x), respectively (with λ = 0.001). In figure 1c the symmetry of the double well potential remains intact but the barrier height of the double well decreases (along y axis). As we go on increasing λ with the functional form Vint (x, y) = λx2 y2 the barrier height gradually decreases and as a consequence the tunneling rate increases almost linearly which is very clearly reflected in figure 2b. If Vint (x, y) is taken to have the form Vint (x, y) = λ(x2 y+y2 x), the PES (figure 1d) is obtained in which the barrier height of the double well potential increases as well as the symmetry of the double well potential is lost. The tunneling rate decreases (figure 2c) as λ increases. These results indicate that the form (Vint (x, y)) and strength (λ) of the interaction potential between the symmetric double well (tunneling mode) and the Morse oscillator (bond stretching mode)are very important in determining the nature and the extent of the influence that the coupling could

100


Figure 2. Tunneling rate in the coupled systen in absence of external driving when (a) Vint (x, y) = λxy; (b) Vint (x, y) = λx2 y2 ; (c) V (x, y) = λ(x2 y + xy2 ). have on the tunneling dynamics in the double well.

B. Tunneling Dynamics in the Coupled System in the Presence of an External Electric Field Coupled to the Tunneling Coordinate We apply an external time varying electric field along the tunneling coordinate with an intensity of 0.01 a.u and frequency of 0.00227 a.u. The specific frequency matches with the 0 → 1 transition frequency (ω0→1 )of the double well. We take the product of the lowest even parity state the double well and the ground state of the Morse oscillator, as the initial state; ψ(x, y,t = 0) = φ+ 0 χ1 .

(27)

For the uncoupled composite system, λ = 0 and we see the particle execute a to and fro movement between the two wells. Figure 3a shows the plot of < x > vs t for the uncoupled double well Morse oscillator system. The variation of < x > seems to be very regular and periodic in nature. The corresponding quantum phase space diagram (figure 3b) shows that


101

Figure 3. Tunneling dynamics in the presence of an external time varying field coupled to the tunneling coordinate x with Vint (x, y) = λxy (a) < x > versus t profile at λ = 0; (b) ’Quantum phase space’ diagram along λ = 0 for the tunneling mode; (c) < x > versus t profile at λ = 0.0001; (d) ’Quantum phase space’ diagram along λ = 0.0001 for the tunneling coordinate; (e) ’Quantum phase space’ diagram for λ = 0.0001 for the bond stretching mode.

102


the particle is symmetrically distributed in the two wells ( the symmetry about < x >= 0 line is maintained). Tunneling rate is calculated from the average slope of < x > versus t plots. When the symmetrical double well system gets coupled to the Morse oscillator with a coupling term Vint (x, y) = λxy the dynamics gets modified. We have shown the variation of < x > with t for λ = 0.0001 for such a system in figure 3c which is substantially different from the corresponding picture of the uncoupled system (figure 3a). However, figure 3c also exhibits regularity and periodicity. The ’quantum phase space’ picture (figure 3d) is still symmetric about the < x >= 0 line, but the area occupied in the phase space diagram has decreased substantially. It indicates transfer of some energy from the tunneling to the bond stretching mode. Figure 3e displays the ’quantum phase space’ structure along y direction (the bond stretching mode). The occupied region of the phase space in figure 3e is rather small which indicates that the energy transfer from the x mode to the y mode (i.e. from the tunneling to the bond stretching or the Morse mode) is also small for λ = 0.0001. However, because of this small energy transfer the tunneling rate gets reduced. We have investigated the dynamics at different values of λ and we have found that the energy transfer from the x mode to the y mode becomes maximum at λ = 0.003. The tunneling rate in turn passes through a minima for λ = 0.003 (figure 4a). Figure 4b shows the variation of < x > with time for λ = 0.003. At this λ value the ’quantum phase space’ diagram for the Morse mode occupies a larger area which again indicates that the energy transfer (figure 4d) from the tunneling to the Morse mode for the given value of λ is higher. The population of the first excited state of the Morse oscillator grows (figure 4d) with oscillations which is reflected in the increase of the amplitude of the oscillation of the average bond length < y > (figure 4e). It is clear therefore that direct excitations by external field coupled to the tunneling mode can influence the bond stretching mode coupled to it.

C. Tunneling Dynamics in the Presence of External Field Coupled to the Bond Stretching Mode If we locally excite the Morse mode with an external time varying electricfield of intensity 0.01 a.u, and frequency ω0→1 (y) (frequency of the 0 → 1 transition in the Morse mode), we do not observe any significant probability of bond dissociation in the uncoupled system. But if we introduce an interaction potential Vint (x, y) = λxy, the dissociation probability slowly rises as λ increases. Table-3 shows the values of the computed dissociation probability and tunneling rate for various values of coupling strength (λ). At λ = 0.004 we observe a dissociation probability of 0.37 which is the maximum for the given intensity. From table-3 it is clear that tunneling is strongly impeded when the tunneling mode gets coupled to a stretching mode that is locally excited. Quantum phase space picture along the tunneling coordinate at λ = 0.004 is depicted in figure 5a. Figure 5b exhibits the corresponding picture for the stretching mode at λ = 0.004. The growth of dissociation probability attains a maximum at λ = 0.004 (Table-3). Figures 5c and 5d show the ’quantum phase space’ diagrams for the stretching mode for λ = 0.001 and λ = 0.006, respectively. In both the cases the area occupied in the phase space is smaller compared to what is observed for λ = 0.004 (figure 5b). It indicates that the maximum energy transfer from the symmetric double well to the Morse mode takes place at λ = 0.004 which is


103

Figure 4. Tunneling dynamics in the presence of an external time varying field coupled to the tunneling coordinate with Vint (x, y) = λxy (a) Tunneling rate versus coupling strength λ; (b) < x > versus t profile for λ = 0.003 along the tunneling coordinate; (c) ’Quantum phase space’ diagram for λ = 0.003; (d) Population of the ground and the first excited states of the bond stretching mode for λ = 0.003; (e) < x > versus t profile at λ = 0.003 for the bond stretching mode.

104


Figure 5. Tunneling dynamics in the presence of an external time varying field coupled to the bond stretching coordinate with Vint (x, y) = λxy (a) Quantum phase space diagram for λ = 0.004 along the tunneling coordinate; (b) “Quantum phase space” diagram for λ = 0.004 along the bond stretching coordinate; (c) “Quantum phase space” diagram for λ = 0.001 along the bond stretching coordinate; (d) “Quantum phase space” diagram for λ = 0.006 along the bond stretching coordinate; (e) Dissociation probability versus time for λ = 0.004.


105

reflected in the relatively high dissociation probability achieved at the given intensity of the external field . Figure 5e shows the plot of the computed dissociation probability against time at λ = 0.004 a.u. The dissociation probability grows fast and attains a relatively high value. The tunneling, on the other hand gets quenched completely. Thus, excitations in the non-tunneling mode in the coupled system can be exploited for controlling tunneling.

D. Tunneling Dynamics of a Particle with Coordinate Dependent Mass Tunneling of electron through heterostructures is complicated by the interaction of the tunneling particles with many centres of scattering. Instead of taking these complexities directly into account in the calculation, it is often expedient to replace the real system by a model one in which the tunneling potential remains unaffected, but the tunneling particle is assumed to have coordinate dependent mass. The question that arises now concerns the signature of the coupling of the tunneling particle with the lattice or equivalently of the coordinate dependent mass [31], on the dynamics of the tunneling. We have carried out a series of experiments within the framework of the basic methodology described in SectionII. The numerical experiments are done with a symmetric double well potential and are subdivided into two classes. 1. The mass of the tunneling particle varies symmetrically along the tunneling coordinate. 2. The mass variation is asymmetric

Figure 6. Pattern of coordinate dependent mass variation (a) mass distribution is gaussian; (b)mass distribution has a minima at the barrier top. Under category 1 three possibilities have been investigated: (a) the tunneling mass is constant m0 everywhere, (b) the tunneling particle has mass m0 at x = 0 where the barrier height is maximum while away from the barrier top, the mass decreases. More specifically, 2 the mass has a gaussian profile along the tunneling coordinate (figure 6a), m(x) = m0 e−βx , and (c) the tunneling mass is m0 at the bottom of the left or right well and it decreases as 2 2 it approaches the barrier top at x = 0 (figure 6b) i.e m(x) = m0 e−β(x −a ) . x = ±a being the location of the well minima. Figure 7a shows how the dynamics appears to be when the tunneling mass (m0 ) is fixed along x. It is perfectly coherent and the particle oscillates back and forth between the right and the left wells. When the tunneling mass is higher in the

106


Figure 7. The pattern of tunneling dynamics displayed when (a) tunneling mass is independent of x i.e. m(x) = m0 ; (b) tunneling mass has a maxima at the bottom of the wells; (c) tunneling mass has a maxima on the barrier top.

Figure 8. The mass distribution plot when it is sharply peaked at x = −α. wells and lower in the barrier, the tunneling becomes slower, still remaining coherent (figure 7b). However, if the tunneling mass is lower in the well and increases as it approaches the barrier, the tunneling rate is increased very significantly and the coherent oscillation frequency becomes much larger (figure 7c). In such a situation very significant increase in tunneling rate through heterostructure should be seen. It is clear that mass concentration in the well lowers the tunneling rate while mass concentration in the barrier region enhances tunneling. Under category-2, we have considered an asymmetric distribution of the tunneling mass along the tunneling coordinate by assuming m(x) =

m0 . |x + α|

(28)

The function is peaked at x = −α (figure 8). The larger the value of α the larger is the shift of the peak towords the region x < 0. Figure-8 shows the mass variation pattern along the tunneling coordinate for different values of α. The tunneling time [32] has been computed for each value of α with m0 being assumed to be the proton mass and reported in table-IV. It is clearly seen that tunneling time increases as α increases meaning thereby that the rate of


107

tunneling decreases. The asymmetric mass variation of the type displayed in figure 8 along the tunneling coordinate could lead to quenching tunneling. The coupling with the lattice (’environment’) can therefore modulate tunneling current in hetero structures. Table 1. Morse and Symmetric double well potential parameters used in model calculations Parameters a b c D β xe µ

Values (a.u) 0.1 0.12 0.04 0.135 0.731 2.23 250.0

Table 2. Energies of the Morse and the double well oscillators

1st state energy(a.u) 2nd state energy (a.u)

double well 0.02270 0.02496

Morse Oscillator 0.00439 0.01267

Table 3. Computed dissociation probability and tunneling rate in a symmetric double well and Morse oscillator system for various strength of coupling parameters λ 0 0.001 0.002 0.003 0.004 0.005 0.006

Dissociation probability 0.04 0.05 0.1 0.18 0.37 0.05 0.05

Tunneling rate(au−1 ) 0.000769 0 0 0 0 0 0

Table 4. Computed tunneling time for various α Tunneling time 5.0 ∗ 105 1.0 ∗ 106 3.3 ∗ 106

α 0.5 1.0 1.5

108

5.


Conclusion

In a tunneling system where the tunneling motion of the particle along the coordinate gets coupled to non-tunneling motion along another coordinate, the coupling leaves its signature on the dynamics in different ways. It can quench tunneling, depending upon the strength and form of coupling. Similarly the tunneling motion can also affect the dynamics along the non-tunneling (e.g. bond stretching) coordinate. Experimentally, little seems to be known about the possible impact that excitation in the tunneling coordinate could have on the bond stretching and the dissociation dynamics of a mode coupled to the tunneling coordinate and vice-versa. Work along these lines could enrich our knowledge about the dynamics of coupled quantum system.

References [1] M. Razavi, Quantum Theory of Tunneling, World Scientific, 2003. [2] R. P. Bell, The Tunneling Effect in Chemistry, Chapman & Hall, London, 1980. [3] G. Gamow, The quantum theory of nuclear disintegrationm, Nature 1928, 122, 805806; Zur Quantentheorie des Atomkevnes, Z. fur. Phys. 1928, 51, 204–212. [4] M. Born, Zur Theorie des Kernzerfalls, Z. fur. Phys. 1928, 58, 306–321. [5] R. W. Gurney, Nuclear levels and artificial disintegration, Nature 1929, 123, 565–566. [6] L. Esaki, Long journey into tunneling, Proc. IEEE 1974, 62, 825–831. [7] I. Giaever, Electron tunneling and superconductivity, Science 1974, 183, 1253–1258. [8] B. D. Josephson, The discovery of tunneling supercurrents, Science 1974, 184, 527– 530. [9] G. C. Schatz,Tunnelling in bimolecular collisions, Chem. Rev. 1987, 87, 81–89; Quantum effects in gas phase bimolecular chemical reactions, Ann. Rev. Phys. Chem. 1988, 39, 317–340. [10] D. G. Truhler, A. D. Isaacson, B. C. Garrett, Theory of chemical reaction dynamics, D. C. clary, Dordrecht Ed.; NATO ASI series. Series C, Mathematical and physical sciences; vol. 170, NATO ASI series., no. 170. CRC Press, Boca Raton, FL, 1985, Vol. 4, pp. 65–137. [11] B. C. Garrett and D. G. Truhlar, A least action variational method for calculating multidimensional tunneling probabilities for chemical reactions, J. Chem. Phys. 1983, 79, 4931–4938. [12] H. Ushiyama and K. Taktsuka, Semiclassical study on multidimensuional effects in tnneling chemical reactions: tunneling paths and tunneling tubes, J. chem. Phys. 1997, 106, 7023–7035.


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[13] H. Eyring, The activated complex in chemical reactions, J. Chem. Phys. 1935, 3, 107– 115. [14] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 1940, 7, 284–304. [15] P. Hanggi, P. Talkner, M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys. 1990, 62, 251–270. [16] H. J. Kim, J. T. Hynes, A theoretical model for SN1 ionic dissociation in solution. 1. Activation free energetics and transition-state structure, J. Am. Chem. Soc. 1992, 114, 10508–10528. [17] D. G. Truhlar, G. K. Schenter, B. C. Garrett, Inclusion of nonequilibrium continuum solvation effects in variational transition state theory, J. Chem. Phys. 1993, 98, 5756– 5770. [18] V. I. Goldanskii, Role of the tunneling effect in the kinetics of chemical reactions at low temperatures Dokl. Akad. Nauk. Phys. Chem. 1959, 124, 1261–1264. [19] V. A. Benderskii, V. I. Goldanskii, A. A. Ovchinnikov, Effect of molecular motion on low temperature and other anomalously fast chemical reactions in solid phase, Chem. Phys. Lett. 1980, 73, 492–495. [20] T. Miyazaki; Ed. Springer, Atom tunneling phenomena in Physics, Chemistry and Biology, Berlin, 2004, 36. [21] G. K. Ivanov and M. A. Kozhushner, Sov. J. Chem. Phys. 1983, 2, 1299. [22] G. K. Ivanov, M. A. Kozhushner, L. I. Trakhtenberg, Temperature dependence of cryochemical H-tunneling reactions, J. Chem. Phys. 2000, 113, 1992–2002. [23] V. I. Goldanskii, L. I. Trakhtenberg and V. N. Fleurov, Tunneling Phenomena in chemical Physics, Gordon and Breach Science Publishers, New York, 1989. [24] L. I. Trakhtenberg, V. L. Klochikhin, S. Ya. Pshezhetskii, Theory of tunnel transitions of atoms in solids, Chem. Phys. 1982, 69, 121–134. [25] L. I. Trakhtenberg in ref-18, 46–47. [26] C. C. Marston, G. B. Baliant Kurti, The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions, J. Chem. Phys. 1989, 91, 3571–3576. [27] S. Adhikari, P. Dutta and S. P. Bhattacharyya, A time-dependent Fourier grid Hamiltonian method. Formulation and application to the multiphoton dissociation of a diatomic molecule in intense laser field, Chem. Phys. Lett. 1992, 91, 574–579. [28] P. Dutta, S. Adhikari and S. P. Bhattacharyya, Fourier grid Hamiltonian method for bound states of multidimensional systems. Formulation and preliminary applications to model systems, Chem. Phys. Lett. 1993, 212, 677–684.

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[29] S. Adhikari, P. Dutta and S. P. Bhattacharyya, Properties, dynamics, and electronic structure of atoms and molecules applications of a local grid method for modeling chemical dynamics at a mean-field level, Int. J. Quant. Chem. 1996, 59, 109–117. [30] G. B. Balint Kurti, C. L. Ward, R. N. Dixon, A. J. Mulholland, The calculation of product quantum state distributions and partial crsssections in time-dependent molecular collision and photodissocistion theory, Comput. phys. Commun. 1991, 63, 126–134. [31] A. V. Kolesnikov, A. P. Silin, Quantum mechanics with coordinate-dependent mass, Phys. Rev. B 1999, 59, 7596–7599. [32] Kaushik Maji, C. K. Mondal, S. P. Bhattacharyya, Tunneling time and tunneling dynamics, International Reviews in Phys. Chem. 2007, 26, 647–670.



Chapter 6

THEORETICAL CALCULATION OF THE LOW LAYING ELECTRONIC STATES OF THE MOLECULAR ION CSH+ WITH SPIN-ORBIT EFFECTS M. Korek* and H. Jawhari Faculty of Science, Physics Department, Beirut Arab University, P.O.Box 11-5020 Riad El Solh, Beirut 1107 2809, Lebanon

Abstract Research studies on ultracold molecules are a current and great challenge in the spectroscopic study of alkali dimers, because of their importance in the cooling and trapping of atoms and molecules, their role in high precision spectroscopy and Bose-Einstein condensation (BEC). Using the high reliability of the ab initio technique combined with the easily amenable phenomenology core polarization concept, the theoretical calculation of the electronic structure of the molecular ion CsH+ has been performed. This ion is treated as a one electron system where the interaction between the outer electron and the atomic core of Cs+ is modulated through non empirical relativistic effective one-electron core potential. The lowest 2 71 electronic states for the ion CsH+ have been calculated for the molecular states Λ(+) and Ω and dissociating into the 16 asymptotes considered, i.e. up to 16 states 2Σ+, 9 states 2Π, 4 states 2Δ, 25 states Ω=1/2, 13 states Ω=3/2 and 4 states Ω=5/2. Some avoided crossings are pointed out for the symmetries 2Σ+, 2Π, Ω=1/2 and Ω=3/2, their positions rAC and the energy difference ΔEAC at these positions have been determined. For 19 bound states, the harmonic vibrational constant ωe, the internuclear distance re and the electronic transition energy with respect to the ground Te have been calculated. Using the canonical functions approach, we calculate in the present work Ev, Bv and Dv of the molecular ion CsH+ up to the vibrational levels v = 19 for 17 electronic states. From the calculated values of Ev for a given vibrational level v and by using a cubic spline interpolation between each 2 consecutive points of the potential energy curves, the rmin and rmax of the turning points have been investigated for a these bound states. Permanent dipole moments M a (r) as well as all non-zero transition dipole moments *

M ab (r)

(a≠b ) have been calculated for each electronic state (a,b) under

E-mail address: [email protected]

112

M. Korek and H. Jawhari consideration and in the whole range of r investigated here. The comparison of the present results with those available in literature shows a very good agreement.

1. Introduction Beginning in the late 1980s, methods to use laser light to cool and confine atoms at unprecedented temperatures have made a dramatic impact on atomic physics. The cooling and manipulation of cold molecules is likely opening up new branches of research. As a gas of molecules is cooled, their average velocity is decreased and the spread of their molecular velocities narrowed. This is important not only for studying molecular physics, but also for studying fundamental physics. The internal structure of certain molecules provides an ideal “laboratory” for sensitive measurements of fundamental physical quantities. Ultracold molecules are of a current and great challenge in the spectroscopic study of alkali dimers, because of their importance in the cooling and trapping of atoms [1,2] and molecules [3], their role in high precision spectroscopy [4], Bose-Einstein condensation (BEC) [5], atomic clocks, ultrasensitive isotope detection, quantum information, and processing ultracold collisions. At ultracold temperatures, the collisions of atoms, which may be characterized by s-wave scattering lengths, have received considerable attention because of their importance in cooling and trapping of atoms and molecules [6] and their role in high precision spectroscopy [7]. Collisions of ions involve higher-order partial waves because of the long-range attractive polarization forces and because of the possibility of charge transfer. Ion-atom charge transfer collisions are of great interest both theoretically and experimentally. Another especially promising area will be the study of collisions between ultracold molecules, in a regime where they behave like waves, perhaps giving rise to a new chemistry [8]. They may also allow for the study of collective quantum effects in molecular systems, including BEC [9]. There are proposed experiments to study polar molecular systems in order to measure the electron’s permanent electric dipole moment (EDM), the lifetime of long-lived energy levels, and the effects of the dipole-dipole interactions on the molecular samples properties [10]. Ultracold polar molecules interact with each other via highly anisotropic electric dipoledipole forces providing access to qualitatively new regimes previously unavailable by ultracold homonuclear_nonpolar systems [11-12]. Novel phenomena are expected from ultracold polar molecules as new features in phase diagrams of degenerate states [13] and anisotropic collisions caused by anisotropic dipole-dipole interaction [14]. The sympathetic cooling of these mixtures has led to the achievement of simultaneous quantum degeneracy of bosonic and fermionic species, producing BEC and Fermi-Bose mixtures [15-16]. Theory groups developed the methods to map problems, such as the BEC-BCS crossover, superfluid phases, Bose glasses and spin-charge separation from solid state quantum systems to the pure world of degenerate quantum gases [17-19]. Meanwhile, a growing community developed the technique of atom chips. From this dataset and the experience of three generations of atom chip experiments on hand, a truly next-generation experiment was designed. Furthermore, the interaction of bosons and fermions with attractive and repulsive interactions can be studied without the need of a Feshbach resonance, as the interaction between Rb and Li is repulsive and the interaction between Rb and K is attractive [20]. Recent advances in precision control of the optical spectrum emitted by a femtosecond laser have made a revolutionary impact on the fields of optical frequency metrology (via self-referenced, ultra-broad bandwidth

Theoretical Calculation of the Low Laying Electronic States…

113

frequency combs) and ultrafast optical science (via carrier-envelope phase stabilized pulse trains). These advances have, in effect, provided an entirely new class of optical sources available for experimental investigations which is the Femtosecond lasers+cold atoms/molecules. In the three past decades a limited number of theoretical studies have been performed for the molecular ions AH+(where A is an alkaline atom). The potential energy curves of these molecular ions are well needed to understand ion-atom collision [21] and to serve as input for diatomics-in-diatomics studies of the potential energy surfaces for A+H2 collisions [22]. These systems are interesting for the theory of chemical bonding since each involves a single valence electron. Interest in alkali dimers is closely related to developments in the ultra-cold alkali dimers atom trapping, which are at the root of photoassociation spectroscopy. The charge transfer in collision of neutral alkaline atoms with protons affects the ionization balance in the atmospheres of planets, dwarf stars, and the interstellar medium [23-27]. At low temperature, such as in ultracold experiments, collision energies are much les than 1eV, and radiative charge transfer may become dominant over nonradiative charge transfer. Neutralization of H+ is of interest in the area of plasma fusion as a method of energetic neutral beam injection in fusion reactors. At lower energies, this charge exchange process is used to make metastable [H(2s)] hydrogen used for atomic experiments [28,29] and for the creation of spin polarized proton beam for injection into large accelerators [30]. Due to these results, the lack of the theoretical calculation on a certain number of alkali dimers, and because of its use particularly in the domain of quantum computer to create the qubite [31], we investigated recently theoretical calculation of these molecules and their ions [32-38]. Using an improvement on the ab initio pseudo-potential method [39-45], we investigate in this chapter the lowest 29 electronic states of Λ-representation (neglect spin-orbit effect), the lowest 42 electronic states of Ω-representation (including spin-orbit effect), and the spectroscopic constants of the regularly bound states. Based on the canonical functions approach [46-48], a rovibrational study has been done to calculate the eigenvalue energies Ev, the rotational constants Bv, the centrifugal distortion constants Dv, and the abscissas of the turning points (rmin, rmax) for 17 electronic states. Moreover, the dipole moment functions and the transition dipole moment are calculated for many of the states in the Ω-representation.

2. The Theory A. Ab initio Calculation For the molecular ion CsH+ the energies for the molecular states including the spin-orbit effect Ω=1/2, 3/2, and 5/2 have been obtained from the treatment of the total Hamiltonian Ht=He+WSO where He is the Hamiltonian in the Born-Openheimer approximation for the calculation of the energies for the molecular states labelled 2S+1Λ(+/-) and WSO is the spin-orbit pseudo-potential. The Spin-orbit (SO) effects are considered for Cs while they are neglected for H. The CsH+ ion is treated as a one electron system where the interaction between the outer electron and the atomic core of Cs+ is modulated through non empirical relativistic effective one-electron core potential of the Durand and Barthelat type [42-44]. The electroncore interaction is represented by the effective potential

114

M. Korek and H. Jawhari 2

V[r ] = ∑ U [r ]P =0

where ℓ is the orbital angular momentum and Pℓ corresponds to the projection operator on the subspace defined by the Ym spherical harmonies with a given ℓ. U[r] is written as: 2

U [ r ] = ∑ ci r ni e −α r

2

i

i =1

with c, n and α adjusted to fit the energy and wave functions of the valence Hartree-Fock orbitals. Core valence effects including core-polarization and core valence correlation are taken into account by using an ℓ-dependent core-polarization potential of the Foucrault et al. type [49]

1 ∑α k f k . f k 2 k

Vcpp = −

where the index k labels the ionic cores, αk is the static dipole polarizability of the ionic core, fk is the electric field action on the ionic core k due to the valence electrons and the other core. The ℓ-dependent form proposed by Foucrault et al [49] ∞

fk = ∑

m=+

∑ F (r

=0 m = −

ik

, rk )

m > kk < m

with ⏐ℓm>k are spherical harmonic centered on the core k and rkℓ are cut-off parameters. For l

the one-valence-electron atom Cs the parameters rk have been determined in order to reproduce the experimental values of the ionization potential IP, as well as the transition energies for the atomic. In this way rl has been obtained for l = 0, 1, 2 and we have chosen

rk2 = rk3 . [50] The parameters defining the core-polarization potentials and the comparison of the calculated IPs and atomic transition energies with the experimental values of the atom Cs are given in Ref. [51] The core-core interaction is evaluated as the ground state energy for the molecular ion RbH2+ instead of the approximation 1/r which not accurate enough for this species, at least for small values of the internuclear distance. In the present calculation including the spin-orbit effect the total Hamiltonian Ht is diagonalized in the basis of the SΛΣ states yielding the relativistic Ω adiabatic states. The symmetry used in this calculation is being C∞v with a common set of molecular orbitals for all symmetries. Semi-empirical spinorbit pseudo-potentials have been designed for Cs atom [52-53]. The present investigation of the electronic structure including the spin-orbit effect for the molecular ion CsH+ has been performed by using the package CIPSO (Configuration Interaction by Perturbation of a multiconfiguration wave function with Spin-Orbit interaction) of the Laboratoire de Physique


115

Quantique Toulouse, France, which allows a full CI calculation as well as perturbative CI calculations with SO effects.

B. Vibration – Rotation Calculation In the Rayleigh-Schrödinger perturbation theory the eigenvalue EvJ and the eigenfunction ΨvJ are given respectively by

E vJ = ∑ e n λ n

(1)

ΨvJ (r ) = ∑ Φ n (r )λ n

(2)

n =0

n =0

where r is the internuclear distance, v and J are respectively the vibrational and rotational quantum numbers, λ = J (J + 1) , and e0 = Ev, e1=Bv, e2 =- Dv , …, φ0 is the pure vibrational wave function and φn its rotational corrections. By replacing Eqs.(1) and (2) into the radial Schrödinger equation [54-58]

⎡ d 2 2μ λ⎤ ⎢ 2 + 2 (E vJ − U(r )) − 2 ⎥ ΨvJ (r ) = 0 r ⎦ ⎣ dr

(3)

φ '0' (r ) + [e 0 − U (r )]φ 0 (r ) = 0

(4)

one can write [36]

φ1'' (r ) + [e 0 − U(r )] φ1 (r ) = −[e1 − R (r )] φ 0 (r )

(5-1)

φ2'' (r ) + [e0 − U (r )]φ2 (r ) = −[e1 − R (r )]φ1 (r ) − e2 φ0 (r )

(5-2)

n

φ 'n' (r ) + [e 0 − U(r )] φ n (r ) = R (r ) φ n −1 − ∑ e m φ n −m (r )

(5-n)

m =1

where R(r)=1/r2, the first equation is the pure vibrational Schrödinger equation and the remaining equations are called the rotational Schrödinger equations. One may project Eqs.(5) onto φ0 and find [47]

< φ 0 | φ 0 > e1 =< φ 0 |

1 | φ0 > r2

(6-1)

116

M. Korek and H. Jawhari

1 | φ1 > −e1 < φ 0 | φ1 > r2

(6-2)

n −1 1 | φ n −1 > − ∑ e m < φ n − m | φ 0 > 2 m =1 r

(6-n)

< φ 0 | φ 0 > e 2 =< φ 0 | < φ 0 | φ 0 > e n =< φ 0 |

Once e0 is calculated from Eq.(4), e1, e2, e3 … can be obtained by using alternatively Eqs.(5) and (6).

3. The Results Atomic Calculation The electronic structure of the molecular ion CsH+ is studied with and without the spinorbit (SO) coupling. The spin-orbit effects has been studied by using the package CIPSO (Configuration Interaction by Perturbation including Spin-Orbit coupling) of the "laboratoire de physique Quantique Toulouse-France". The values of the cut-off parameters involved in the polarization potentials are given in Table 1 [38,59]. Table 1. Parameters for the polarization potential of the Cs and H atoms α(ao3) 15.117 1.000

Atom Cs H

rko(ao) 2.6915 1.000

rk1(ao) 1.8505 1.000

rk2(ao) 2.8070 1.000

Table 2. Energies of the lowest lying levels of the H-atom Configuration 1s

Term 2 S

2p

2 0

2s

2

3p

2 0

3s

2

3d

2

4p

2 0

4s

2

4d

2

P

S

P

S

D

P

S

D

J Etheoretical(cm-1) 1/2 0.00 1/2 82312.7386 3/2 82324.9802 1/2 82324.9802 1/2 97568.0092 3/2 97568.0092 1/2 97568.0092 3/2 97568.0092 5/2 97568.0092 1/2 102948.0576 3/2 102948.0576 1/2 102948.0576 3/2 102948.0576 5/2 102948.0576 Average relative error

Eexperimental(cm-1) 0.00 82258.9206 82259.2865 82258.9559 97492.2130 97492.3214 97492.2235 97492.3212 97492.3574 102823.8505 102823.8962 102823.8549 102823.8961 102823.9114

δE/E% 0.00 0.07 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.12 0.12 0.12 0.12 0.12 0.09


117

By using the Gaussian basis set given in Ref. [32] for the cesium atom and the modified basis cc-pv6z [60] for the hydrogen atom we calculate the energy levels up to 4d and 8s respectively for H and Cs-atoms. The comparison of these values, in Tables 2 and 3, to those obtained experimentally [61-63] shows an excellent agreement with an average relative error δE/E=0.09% for the H-atom and 0.01% for the Cs-atom. This agreement confirms the validity and the accuracy of the chosen basis sets in this calculation. Table 3. Energies of the lowest lying levels of the Cs-atom Configuration

Term

6s

2

6p

2 0

5d

2

S

P

D

7s

2

7p

2 0

6d

2

8s

2

S

P

D S

J

Etheoretical(cm-1)

Eexperimental(cm-1)

δE/E%

1/2 1/2 3/2 3/2 5/2

0.00 11172.7488 11728.5025 14480.8502 14579.0300

0.00 11178.2686 11732.3079 14499.2584 14596.8423

0.00 0.04 0.03 0.12 0.12

1/2

18537.3276

18535.529

~0.00

1/2 21837.0533 3/2 22019.7714 3/2 22776.7197 5/2 22818.5012 1/2 24354.4958 Average relative error

21765.35 21946.396 22588.8210 22631.6863 24317.0185

0.32 0.33 0.83 0.82 0.01 0.23

3.3. Spin-Orbit Effects Neglected For the molecular ion CsH+ the potential energy curves (PECs) of the 2Λ(+) states have been investigated in the range 3.0a0≤r≤60a0 of the internuclear distance and dissociating into the 16 asymptotes considered up to the dissociation limit H+(4d 2D3/2,5/2)+ Cs(1S0); i.e. up to 16 states 2Σ+, 9 states 2Π, 4 states 2Δ, as displayed in Table 4. This table also reports the calculated energies at the dissociation limits; the comparison of these values to those obtained experimentally [62-63] shows a very good agreement with an average relative error of 0.06 %. The PECs versus the internuclear distance for the 2Λ(+)-states are plotted respectively in the Figs1-3. Among the calculated 29 PECs 22 states are proved to be attractive. For each bound state the harmonic vibrational constant ωe, the rotational constant Be, the internuclear distance at equilibrium re and the electronic transition energy with respect to the ground state Te are calculated by fitting the energy data around the equilibrium position to a polynomial in terms of the internuclear distance r, these values are given in Table 5. The comparison of our calculated values of re and Be of the ground state with those available in literature shows a good agreement with relative errors 3.6% and 7.5% respectively. Double minima potentials are obtained for the states (14)2Σ+, (16)2Σ+, and (9)2Π, the minima of these potentials are given in Table 5. No comparison for the other results since they are given here for the first time.

118


Table 4. Numbering of the various Λ-states of the CsH+ correlated adiabatically to the 16 lowest dissociation limits Cs

H

2

1

2

1 0 2 78268.9598 3 1 82318.8594 4 82324.9802 5 2 89719.5854 6 3 1 92798.8999 7 96806.2874 8 97568.0092 9 4 97568.0092 10 5 2 97568.0092 11 6 100197.3722 12 7 3 101066.5703 13 102623.4556 14 102948.0576 15 8 102948.0576 16 9 4 102948.0576 Average relative error

S0 6s 2S1/2 1 S0 1 S0 2 6p P1/2,3/2 5d 2D3/2,5/2 7s 2S1/2 1 S0 1 S0 1 S0 7p 2P1/2,3/2 6d 2D3/2,5/2 8s 2S1/2 1 S0 1 S0 1 S0

1s S1/2 1 S0 2 2p P1/2,3/2 2s 2S1/2 1 S0 1 S0 1 S0 3s 2S1/2 3p 2P1/2,3/2 3d 2D3/2,5/2 1 S0 1 S0 1 S0 4s 2S1/2 4p 2P1/2,3/2 4d 2D3/2,5/2

Σ+

2

Π

2

Δ

Etheoretical(cm-1)

Eexperimental(cm-1)

δE/E%

0 78268.9598 82259.1035 82258.9559 89724.2480 92817.0101 96804.4888 97492.2235 97492.2672 97492.3393 100124.8328 100879.2135 102585.9783 102823.8549 102823.8733 102823.9037

0.00 0.00 0.07 0.08 ~0.00 0.02 ~0.00 0.08 0.08 0.08 0.07 0.18 0.03 0.12 0.12 0.12 0.06

460000

E(cm-1)

440000

420000

400000 0

10

20

30

40

R(Bohr)

Figure 1. Continued on next page.

50

60


119

400000

E(cm-1)

300000

200000

100000

0 0

10

20

30

40

50

60

R(Bohr)

Figure 1. Potential energy curves of the states 2Σ+ of the molecule CsH+. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

40

50

R(Bohr)

Figure 2. Potential energy curves of the states 2Π of the molecule CsH+.

60

120

M. Korek and H. Jawhari Table 5. Calculated spectroscopic constants for the various [(k)2Λ] states of CsH+ [(k)2s+1Λ]

Te(cm-1)

[(1)2Σ+]

0.00

[(2)2Σ+]

re(Å)

204220.06

3.372 3.25a 3.467

2 +

401701.74

2 +

[(4) Σ ] [(5)2Σ+]

ωe(cm-1)

Be(cm-1)

409.50

1.479 1.59a 1.402

4.611

263.99

0.792

414566.28

5.178

129.26

0.628

418211.74

7.854

115.45

0.273

2 +

423296.11

9.699

96.250

0.179

2 +

428459.83

11.65

81.54

0.124

2 +

432489.77

15.185

41.21

0.073

2 +

[(9) Σ ]

433523.86

24.742

6.017

0.029

2 +

435366.04

16.978

46.20

0.058

2 +

[(11) Σ ]

438110.28

19.817

40.05

0.042

[(12)2Σ+]

441472.00 444554.93 Max 444583.4 443451.88

11.932 13.921 At r=14.4 23.834

66.71 62.13

0.117 0.086

32.93

0.029

[(15) Σ ]

444571.45

17.765

72.78

0.053

447113.83 Max 447239.8 447111.94

15.030 At r=16.6 25.870

78.56

0.074

[(16)2Σ+]

7.20

0.025

2

204796.07

3.519

344.80

1.359

2

[(2) Π]

414065.64

3.425

457.16

1.437

[(3)2Π]

424463.77

6.946

53.227

0.344

[(4)2Π]

429363.48

9.583

59.41

0.183

[(6) Π]

438641.40

14.171

36.11

0.083

[(8)2Π]

444031.66

18.80

17.15

0.047

447229.32 Max 447249.3 447118.05

13.270 At r=14 23.338

46.83

0.095

[(9)2Π]

7.14

0.030

[(3) Σ ]

[(6) Σ ] [(7) Σ ] [(8) Σ ] [(10) Σ ]

[(14)2Σ+] 2 +

[(1) Π]

2

460.350

(a) Ref. [64].

These PECs present avoided crossings in quite complex forms (humps and wells) at short and large value of the internuclear distance which are due to either crossings or avoided crossings. The internuclear distance at the avoided crossing RAC with the energy difference ΔEAC between two corresponding states at these points for the different states are given in Table 6. We show avoided crossing between the states (9)2Σ+ and (8)2Σ+ in Figs 4 as illustration.


121

460000

-1

E(cm )

440000

420000

400000 0

10

20

30

40

50

60

R(Bohr)

Figure 3. Potential energy curves of the states 2Δ of the molecule CsH+.

Table 6. Some avoided crossing between 2Λ+ states of the molecular ion CsH+ (n+1)State / (n)State

rAC(Bohr)

ΔEAC(cm-1)

(8)2Σ+/(7)2Σ+

12.4

1573.97

(9)2Σ+/(8)2Σ+

10.4

1461.58

(9)2Σ+/(8)2Σ+

18.4

33.14

(12)2Σ+/(11)2Σ+

8.3

543.19

(12)2Σ+/(11)2Σ+

11.5

547.72

(12)2Σ+/(11)2Σ+

20.6

504.07

2 +

2 +

6.4

357.90

2 +

2 +

17.8

473.14

2 +

2 +

32

118.38

2 +

2 +

(16) Σ /(15) Σ

10.6

154.25

(5) 2Π/(4) 2Π

8.2

1886.88

(9) 2Π/(8) 2Π

8.3

1630.83

(15) Σ /(14) Σ (15) Σ /(14) Σ (15) Σ /(14) Σ

122

M. Korek and H. Jawhari 434400 (8)Σ (9)Σ

E(cm-1)

434300

434200

434100

434000 18.1

18.3

18.5

18.7

R(Bohr)

Figure 4. Avoided crossing between (9)2Σ+ and (8)2Σ+states of the molecule CsH+.

3.4. Spin-Orbit Effects Included Energy calculation for the Ω-representation is performed for the states corresponding to the 16 lowest dissociation limits, i.e. up to H+(4d 2D3/2,5/2)+ Cs+(1S0) is the range of internuclear distance r from 3ao to 60ao. Consequently 25 states of Ω=1/2, 13 states of Ω=3/2 and 4 states of Ω=5/2 are correlated adiabatically as shown in Table 7. The PECs of these states are displayed in Figs 7, 8 and 9 respectively. Among these calculated PECs 24 states are proved to be attractive and the other are repulsive. Table 7. Numbering of the various Ω-states of CsH+ correlated adiabatically to the 25 lowest dissociation limits Cs 1

S0 6s 2S1/2 1 S0 1 S0 2 6p P1/2,3/2 5d 2D3/2,5/2 7s 2S1/2 1 S0 1 S0

H+

Ω=1/2

2

1 2 3,4 5 6,7 8,9 10 11 12,13

1s S1/2 1 S0 2 2p P1/2,3/2 2s 2S1/2 1 S0 1 S0 1 S0 2 3s S1/2 3p 2P1/2,3/2

Ω=3/2

Ω=5/2

1 2 3,4

5

1


123

Table 7. Continued Cs 1

S0 2 7p P1/2,3/2 6d 2D3/2,5/2 8s 2S1/2 1 S0 1 S0 1 S0

H+

Ω=1/2

Ω=3/2

Ω=5/2

3d 2D3/2,5/2 1 S0 1 S0 1 S0 4s 2S1/2 4p 2P1/2,3/2 4d 2D3/2,5/2

14,15 16,17 18,19 20 21 22,23 24,25

6,7 8 9,10

2

11 12,13

3

4

For each bound state the harmonic vibrational constant ωe, the rotational constant Be, the internuclear distance at equilibrium re and the electronic transition energy with respect to the ground state Te are calculated by fitting the energy data around the equilibrium position to a polynomial in terms of the internuclear distance r, these values are given in Tables 8 and 9 respectively along with the main parents 2Λ(+) of the states Ω=1/2 and Ω=3/2 near the equilibrium positions. It should be noticed that, such identification was not possible for other states since their minima are situated close to the crossings between 2Λ(+) states. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

R(Bohr)


40

50

60

124


E(cm-1)

300000

200000

100000

0 0

10

20

30

40

50

60

50

60

R(Bohr)

Figure 7. Potential energy curves of the states Ω=1/2 of CsH+. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

R(Bohr)


40


125

400000

E(cm-1)

300000

200000

100000

0 0

10

20

30

40

50

60

R(Bohr)

Figure 8. Potential energy curves of the states Ω=3/2 of the molecule CsH+. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

40

50

60

R(Bohr)

Figure 9. Potential energy curves of the states Ω=5/2 of CsH+.

Since the spin orbit coupling has no effect on the Σ−state we compare these results with those available in literature without spin orbit. This comparison shows a very good agreement with a relative errors 3.6% and 7.5% respectively for re and Be for the ground state. No comparison for the other results since they are given here for the first time.

126


Table 8. Calculated spectroscopic constants for the various (n)Ω=1/2 states of CsH+ n[(k)2s+1Λ]

Te(cm-1)

re(Å)

ωe(cm-1)

Be(cm-1)

(1) [(1)2Σ+]

0.00

3.372 3.25a

460.356

1.479 1.59a

(2) [(2)2Σ+]

204220.06

3.467

409.473

1.402

(3) [(1)2Π]

204795.65

3.519

345.114

1.359

(4) [(3)2Σ+]

401700.63

4.611

263.463

0.792

3.426 At r=3.9 5.177

456.004

1.437

(5) [(4)2Σ+]

414063.89 Max 414752.7 414565.97

129.213

0.628

(6)

414792.88

4.169

480.602

0.974

418211.24

7.855

115.627

0.272

(8) [(6) Σ ]

423222.19

9.696

96.473

0.179

(9) [(3)2Π]

424450.23

7.574

1105.84

0.541

(10) [(7)2Σ+]

428383.70

11.694

78.041

0.123

(11) [(4) Π]

429284.02

10.256

58.061

0.159

(12) [(8)2Σ+]

432489.53

15.182

41.239

0.073

(15) [(10)2Σ+]

435366.45

16.977

46.185

0.058

(16)

438617.06 Max 438026.4 438110.06

12.252 At r=13.7 19.815

26.045

0.082

40.064

0.042

(17) [(6) Π]

438666.78

15.846

95.567

0.066

(18) [(12)2Σ+]

441471.64

11.934

66.424

0.117

(21) [(14) Σ ]

443451.88

23.834

32.939

0.029

(22)

444555.10 Max 444583.4 444026.53 444571.46 Max 444632.2 444080.58

12.624 At r=13.8 17.430 17.765 At r=18.5 30.278

92.079

0.085

58.353 72.777

0.044 0.053

15.702

0.017

(24) [(16) Σ ]

447113.96

15.019

79.332

0.075

(25)

447237.12 Max 447249.2 447225.70 Max 447239.8 447118.11

12.224 At r=13.5 15.187 At r=15.9 23.973

67.025

0.081

225.477

0.042

102.062

0.028

2

(5) [(2) Π]

(7) [(5)2Σ+] 2 +

2

(16) [(11)2Σ+] 2

2 +

(22) [(8)2Π] (23) [(15)2Σ+] (23) 2 +

(25) [(9)2Π] (25) [(9)2Π] (a) Ref.[64].


127

Double minima potentials are obtained for the states (5)Ω=1/2, (16)Ω=1/2, (22)Ω=1/2, (23)Ω=1/2 and (13)Ω=3/2, while the (25)Ω=1/2 is a triple well potential state. Table 9. Calculated spectroscopic constants for the various (n) Ω=3/2 states of CsH+ n[(k)2s+1Λ]

Te(cm-1)

re(Å)

ωe(cm-1)

Be(cm-1)

(1) [(1)2Π]

204796.49

3.519

344.896

1.359

2

414067.25

3.424

458.206

1.438

2

(4) [(3) Π]

424505.55

7.170

14.270

0.295

(5)

414565.97

5.177

129.213

0.628

(2) [(2) Π]

(8) [(6)2Π]

438666.91

14.144

36.317

0.084

2

444033.05

18.801

17.144

0.047

2

447229.54 Max 447249.4 447118.05

13.272 At r=13.4 23.337

46.757

0.095

7.246

0.030

(11) [(8) Π] (13) [(9) Π] (13) [(9)2Π]

At the internuclear distance at equilibrium re, the SO splitting for the states (1, 2, 3, 6, 8, 9)2Π have been identified and evaluated, the difference between the lowest and highest energy are 0.84cm-1, 3.36cm-1, 0.13cm-1, 6.52cm-1, 3.84cm-1 and 0.6cm-1 respectively. In the Ω-representation the PECs present avoided crossings in quite complex forms (humps and wells) at short and large value of the internuclear distance which are due to either crossings or avoided crossings of the Λ-states. The internuclear distance at the avoided crossing rAC with the energy difference ΔEAC between two corresponding states at these points are given in Tables 10 and 11. These avoided crossings are drawn, as illustration, in figures 10 and 11. Table 10. Some avoided crossings between Ω=1/2 states of CsH+ . rAC and ΔEAC are respectively the internuclear distance and the energy difference at the avoided crossing between the two corresponding states (n+1)Ω =1/2/(n)Ω=1/2

rAC(Bohr)

ΔEAC(cm-1)

(3)Ω=1/2/(2) Ω=1/2 (5)Ω=1/2/(4) Ω=1/2 (6)Ω=1/2/(5) Ω=1/2 (8)Ω=1/2/(7) Ω=1/2 (9)Ω=1/2/(8) Ω=1/2 (11)Ω=1/2/(10) Ω=1/2

5.1 4.7 7.7 5.2 13.9 17.7 6.1

25.18 169.97 46.92 137.64 109.66 378.17 8.87

Crossing between (n)state/(m)state (2)2Σ+/(1)2Π (3)2Σ+/(2)2Π (4)2Σ+/(2)2Π (5)2Σ+/(3)2Π (6)2Σ+/(3)2Π (7)2Σ+/(4)2Π (8)2Σ+/(5)2Π

(13)Ω=1/2/(12) Ω=1/2

12.1 18.8 18.2

81.59 18.14 52.56

(8)2Σ+/(5)2Π (9)2Σ+/(5)2Π (9)2Σ+/(5)2Π

(14)Ω=1/2/(13) Ω=1/2

Aviod crossing (n)state/(m)state

128

M. Korek and H. Jawhari Table 10. Continued

(n+1)Ω =1/2/(n)Ω=1/2

rAC(Bohr)

ΔEAC(cm-1)

Crossing between (n)state/(m)state

(16)Ω=1/2/(15) Ω=1/2

11.4

65.71

(10)2Σ+/(6)2Π

(17)Ω=1/2/(16) Ω=1/2

29.4

66.70

(11)2Σ+/(6)2Π

8.8

3.26

(11)2Σ+/(7)2Π

11.5

519.22

(12)2Σ+/(11)2Σ+

20.6

502.78

(12)2Σ+/(11)2Σ+

8.3

545.70

(12)2Σ+/(11)2Σ+

11.1

121.37

(12)2Σ+/(7)2Π

12

88.07

(12)2Σ+/(7)2Π

17.7

4.62

(12)2Σ+/(7)2Π

34.5

62.83

(12)2Σ+/(7)2Π

16

44.80

(13)2Σ+/(8)2Π

21.7

18.20

(14)2Σ+/(8)2Π

25.8

6.20

(14)2Σ+/(8)2Π

35

73.10

(14)2Σ+/(8)2Π

32.2

163

56.5

5.5

(15)2Σ+/(8)2Π

26.4

8.5

(16)2Σ+/(9)2Π

31

13.5

(16)2Σ+/(9)2Π

(18)Ω=1/2/(17) Ω=1/2

(19)Ω=1/2/(18) Ω=1/2

(21)Ω=1/2/(20) Ω=1/2 (22)Ω=1/2/(21) Ω=1/2

(23)Ω=1/2/(22) Ω=1/2 (25)Ω=1/2/(24) Ω=1/2

Aviod crossing (n)state/(m)state

(15)2Σ+/(14)2Σ+

Table 11. Some avoided crossings between Ω=3/2 states of CsH+ (n+1)Ω =3/2/(n)Ω=3/2

rAC(Bohr)

ΔEAC(cm-1)

Crossing between (n)state/(m)state

(6)Ω =3/2/(5)Ω=3/2

9

63.07

(4)2Π/(2)2Δ

(7)Ω =3/2/(6)Ω=3/2

8.2

1848.77

(8)Ω =3/2/(7)Ω=3/2

4.6

14.75

(5)2Π/(3)2Δ

(9)Ω =3/2/(8)Ω=3/2

7.4

20.47

(6)2Π/(3)2Δ

(12)Ω =3/2/(11)Ω=3/2

11.8

26.63

(8)2Π/(4)2Δ

Avoid crossing (n)state/(m)state (5)2Π/(4)2Π

Each Ω-state, except for few ones, has more than one main parent Λ-state. This is proved in Tables 12 and 13 by showing the percentage of parent Λ-states over a certain range of the internuclear distance of Ω-states.


129

425000 (8)Ω=1/2 (9)Ω=1/2

424800

(6)Σ+ (3)Π

E(cm-1)

424600

424400

424200

424000 13.2

13.4

13.6

13.8

14

14.2

14.4

14.6

R(Bohr)

Figure 10. Avoided crossing between (8)Ω=1/2 and (9)Ω=1/2 is due to crossing between (6)2Σ+ and (3)2Π. 445200 (22)Ω=1/2 (23)Ω=1/2 (14)Σ+ (15)Σ+

E(cm-1)

444800

444400

444000 30

31

32

33

34

35

R(Bohr)

Figure 11. Avoided crossing between (22) Ω=1/2 and (23) Ω=1/2 is due to avoided crossing between (14) 2Σ+ and (15) 2Σ+.

Table 12. Parent states for the potential energy curves of (n)Ω=1/2 (n)

From

to

(1)

3

60.5

(2)

3

5

(3)

3

5

(4)

3

4.7

(5)

3

4.7

(6)

3

7.8

(7)

3

5

(8)

3

5.1

(9)

3

4.7

(10)

3

4.7

% State 100% (1)2Σ+ 3.4% (1)2Π 3.4% (2)2Σ+ 2.9% (2)2Π 2.9% (3)2Σ+ 8.3% (4)2Σ+ 3.4% (3)2Π 3.6% (5)2Σ+ 2.9% (4)2Π 2.9% (6)2Σ+

From

to

5

60.5

5

60.5

4.7

60.5

4.7

7.9

7.8

60.5

5

60.5

5.1

14

4.7

13.9

4.7

17.6

% State

96.5% (2)2Σ+ 96.5% (1)2Π 97% (3)2Σ+ 5.5% (2)2Π 91.6% (2)2Π 96.5% (5)2Σ+ 15.4% (3)2Π 16% (6)2Σ+ 22.4% (4)2Π

From

to

% State

7.9

60.5

91.5% (4)2Σ+

14

60.5

13.9

60.5

17.6

60.5

80.9% (6)2Σ+ 81.1% (3)2Π 74.6% (7)2Σ+

Table 12. Continued (n)

From

to

(11)

3

17.6

(12)

3

6

(13)

3

6

(14)

3

60.5

(15)

3

11.5

(16)

3

4.7

(17)

3

5

(18)

3

8.8

(19)

3

60.5

3

6.6

16

60.5

(20)

% State 25.3% (7)2Σ+ 5.2% (8)2Σ+ 5.2% (5)2Π 100% (9)2Σ+ 14.7% (6)2Π 2.9% (7)2Π 3.4% (10)2Σ+ 10% (11)2Σ+ 100% (12)2Σ+ 5.7% (8)2Π 77.9% (13)2Σ+

From

to

17.6

60.5

6

12

6

12.4

11.5

60.5

4.7

11.3

5

8.7

8.8

60.5

6.6

11.8

% State 74.6% (4)2Π 10.4% (5)2Π 11.1% (8)2Σ+

85.2% (10)2Σ+ 11.4% (10)2Σ+ 6.4% (7)2Π 89.9% (7)2Π

9% (13)2Σ+

From

to

12

60.5

12.4

60.5

11.3

60.5

8.7

60.5

11.8

16

% State

84.3% (8)2Σ+ 83.6% (5)2Π

85.6% (6)2Π 90.1% (11)2Σ+

7.3% (8)2Π

Table 12. Continued (n)

From

to

3

6.7

16

60.5

3

3.7

21.6

60.5

3

5.2

3

13.3

32

44

3

26.4

44

60.5

(21)

(22)

(23)

(24)

(25)

% State 6.4% (13)2Σ+ 77.4% (8)2Π 1.2% (13)2Σ+ 67.6% (8)2Π 3.8% (14)2Σ+ 17.9% (15)2Σ+ 20.8% (9)2Π 41% (15)2Σ+ 28.3% (9)2Π

From

to

6.7

11.5

3.7

5.2

5.2

13.3

13.3

26.4

44

60.5

26.4

31

% State 8.3% (8)2Π

2.6% (9)2Π

14% (9)2Π 22.7% (9)2Π 28.8% (16)2Σ+ 8% (9)2Π

From

to

11.5

16

5.2

21.6

13.3

60.5

26.4

32

31

44

% State 7.8% (13)2Σ+

28.5% (14)2Σ+

82.1% (15)2Σ+ 9.7% (16)2Σ+

22.6% (15)2Σ+

Table 13. Parent states for the potential energy curves of (n)Ω=3/2 (n)

From

to

(1)

3

60.5

(2)

3

60.5

(3)

3

60.5

(4)

3

60.5

(5)

3

9

(6)

3

9

(7)

3

21.8

(8)

3

4.5

(9)

3

7.2

(10)

3

60.5

(11)

3

11.7

(12)

3

11.7

(13)

3

60.5

% State 100% (1)2Π 100% (2)2Π 100% (1)2Δ 100% (3)2Π 10.4% (2)2Δ 10.4% (4)2Π 32.6% (5)2Π 2.6% (5)2Π 7.3% (6)2Π 100% (7)2Π 15.1% (4)2Δ 15.1% (8)2Π 100% (9)2Π

From

to

9

60.5

9

60.5

21.8

60.5

4.5

7.2

7.2

60.5

11.7

60.5

11.7

60.5

% State

89.5% (4)2Π 89.5% (2)2Δ 67.3% (2)2Δ 4.6% (3)2Δ 92.6% (3)2Δ

84.8% (8)2Π 84.8% (4)2Δ

From

to

% State

7.2

60.5

92.7% (6)2Π

134


3.5. The Vibration-Rotation Calculation The canonical functions approach [1, 2, 3] enables us to calculate the eigenvalue energy Ev, the rotational constant Bv, the centrifugal distortion constant Dv at any vibrational level even near dissociation. However, this approach fails if avoided crossings between states occur because of the break down of the Born-Oppenheimer approximation at these points. Here for the alkali dimmer CsH+, these constants have been calculated for 8 states in Λ-representation and 9 states in the Ω-representation up to vibrational level v=19. From the cubic spline interpolation between each two consecutive energy values of the PECs, and by using eigenvalue energies Ev, the abscissas of the turning points (rmin, rmax) of the above mentioned states have been calculated. These constants for the states (1, 2, 3)2Σ+ are reported in the Tables 14 to 16, and the other are given in Ref. [59]. No comparison of these values with other results since they are given here for the first time. Table 14. Values for the eigenvalues (Ev), the abscissas of the turning points (rmin, rmax), the rotational constant (Bv), and the centrifugal distortion constant (Dv) for the state (1)2Σ+ of the molecular ion CsH+ v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

0 1 2 3

214.728 587.566 873.605 1089.894

3.156 3.049 2.997 2.966

3.716 4.131 4.552 5.035

1.423 1.276 1.326 0.971

7.036 9.568 11.934 14.5043

Table 15. Values for the eigenvalues (Ev), the abscissas of the turning points (rmin, rmax), the rotational constant (Bv), and the centrifugal distortion constant (Dv) for the state (2) 2 + Σ of the molecular ion CsH+ v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

0 1 2 3

190.687 501.900 737.346 921.226

3.240 3.134 3.083 3.051

3.849 4.319 4.795 5.287

1.3260 1.1693 1.0219 0.8949

7.9254 10.5019 11.8595 12.8388

Table 16. Values for the eigenvalues (Ev), the abscissas of the turning points (rmin, rmax), the rotational constant (Bv), and the centrifugal distortion constant (Dv) for the state (3) 2 + Σ of the molecular ion CsH+ v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

0 1 2 3 4

129.638 377.385 608.215 820.972 1014.498

4.290 4.096 3.981 3.899 3.835

5.014 5.384 5.696 5.999 6.314

0.7785 0.7490 0.7169 0.6819 0.6435

2.9326 3.0982 3.3159 3.5967 3.9675


135

Table 16. Continued v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

5 6 7 8 9 10 11 12

1187.593 1339.148 1468.346 1574.907 1659.412 1723.533 1769.979 1802.086

3.785 3.745 3.714 3.690 3.761 3.658 3.648 3.641

6.655 7.038 7.484 8.018 8.676 9.504 10.564 11.938

0.6011 0.5544 0.5031 0.4475 0.3887 0.3287 0.2700 0.2149

4.4562 5.0841 5.8789 6.8411 7.9306 9.0650 10.1569 11.1810

3.6. Dipole Moment Knowledge of the permanent or transition dipole moment is essential. For the molecular ion CsH+ we calculate the transition electric dipole moment and the permanent electric dipole moment values of the different bound Ω-states as functions of the internuclear distance r. This data provides us information about the most efficient scheme of forming the CsH+ molecular ion. Moreover, our ab initio potentials for the excited states can be used to identify the complex behavior of the transition dipole moment as function of a

r. Permanent dipole moments M a (r) as well as all non-zero transition dipole moments

M ab (r)

M ab =≺ Ψea | μ e (r ) | Ψeb have been calculated for each electronic states (a, b) under consideration and in the whole range of r investigated here. Ψe and Ψe are respectively the electronic wave functions of two a

b

different electronic states and μe(r) is the permanent electronic dipole moment. This dipole moment function has been calculated for most of the Ω-states [59]. In Fig 14 we show, as illustration, the transition dipole moment between the states (22)Ω=1/2 → (21)Ω=1/2. The three peaks at 21.7, 25.8 and 35.0 Bohr respectively, correspond to crossing or avoid crossing between the states (14)2Σ+ and (8)2Π as shown in these figure15, 16 and 17and given in Table 10.

136


2

1

0 20

22

24

26

28

30

32

34

36

38

40

R(Bohr)

Figure 15. Variation dipole moment between (22) Ω=1/2 → (21) Ω=1/2 states. 445600 (21)Ω=1/2 (22)Ω=1/2 (14)Σ+

445400

E(cm-1)

Re(r)

3

(8)Π

445200

445000

444800 21

21.4

21.8

22.2

R(Bohr)

Figure 16. Crossing between (14)2Σ+ and (8)2Π states.

22.6


137

444800 (21)Ω=1/2 (22)Ω=1/2 444700

(14)Σ+ (8)Π

E(cm-1)

444600

444500

444400

444300 24.8

25.2

25.6

26

26.4

26.8

R(Bohr)

Figure 17. Crossing between (14)2Σ+ and (8)2Π states. 444600

(21)Ω=1/2 (22)Ω=1/2

444400

(14)Σ+ (8)Π

E(cm-1)

444200

444000

443800

443600 33

34

35

36

37

R(Bohr)

Figure 18. Crossing between (14)2Σ+ and (8)2Π states.

38

138


4. Conclusion Using an ab initio approach the potential energy has been calculated for 16 states 2Σ+, 9 states 2Π, 4 states 2Δ, 25 states Ω=1/2, 13 states Ω=3/2 and 4 states Ω=5/2 of the molecular ion CsH+. For 19 bound states the harmonic vibrational constant ωe, the internuclear distance re and the electronic transition energy with respect to the ground Te have been calculated. Based on the canonical functions methods Ev, Bv and Dv have been calculated up to the vibrational levels v = 19 for 17 electronic states. From the calculated values of Ev for a given vibrational level v and by using a cubic spline interpolation between each two consecutive points of the potential energy curves the rmin and rmax of the turning points have been a

investigated for these bound states. Permanent dipole moments M a (r) as well as all nonb

zero transition dipole moments M a (r) (a≠b ) have been calculated for each electronic states (a,b) under consideration and in the whole range of r investigated here. The comparison of the present results with those available in literature shows a very good agreement.

References [1] [2] [3]

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[50] S. Magnier and Ph. Millié, Phys. Rev. A. 53, 204 (1996). [51] A. R. Allouche, M. Korek, K. Fakherddin, A. Chaalan, M. Dagher, F. Taher and M. Aubert-Frécon, J. Phys. B: At. Mol. Opt. Phys. 33, 2307 (2000). [52] A.R.Allouche, G. Nicolas G, J.C. Barthelat and F. Spiegelmann, J. Chem. Phys. 96, 7646 (1992). [53] S. Rousseau S, A.R. Allouche, and M. Aubert-Frécon, J. Mol. Spectrosc. 203, 235 (2000). [54] M. Korek and H. Kobeissi. Can. J. Phys. 73, 559 (1995). [55] M. Korek. Can J. Phys. 75, 795 (1997). [56] M. Korek and K.Fakhreddine. can. J. Phys. 78, 969 (2000). [57] M. Korek, B. Hamdan and K.Fakhreddine. Physica scripta. 61, 66 (2000). [58] G. Herzberg, Spectra of diatomic molecule, Van Nostrand, Toronto, 1950. [59] H. Jawhari, Master Thesis, Beirut Arab University (2008). [60] See EPAPS Document No. E-JCPSA6-129-605840 for supplementary tables. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html. [61] S. A. Hammoud, Master Thesis, Beirut Arab University (2007) [62] National Institute of standards and Technology (NIST): Tables of Spectra of Hydrogen, Carbon, Nitrogen, and Oxygen Atoms and Ions, C.E. Moore, edited by J.W. Gallagher. CRC Handbook of Chemistry and Physics, Edition 76 (CRC Press, Boca Raton, FL), 336 pp. (1993) [63] National Institute of Standards and Technology, http://physics.nist.gov/ PhysRefData/ASD/ [64] L.Von szentpaly, P. Fuentealba, H. Preuss and H. Stoll, Chem. Phys. Letters, 93, 8, (1982).



Chapter 7

THEORETICAL EXPLANATION OF LIGHT AMPLIFYING BY POLYETHYLENE FOIL Vjekoslav Sajfert1a,, Dušan Popov2,b, Stevo Jacimovski3,c and Bratislav Tosic4,d 1

Technical Faculty “Mihajlo Pupin” Zrenjanin, Serbia 2 Universitatea “Politehnica”, Timisoara, Romania 3 Police Academy Belgrade, Serbia 4 Vojvodina Academy of Science and Arts Novi Sad, Serbia

Abstract In connection with the experimental result which stated that polyethylene foil amplifies about three times the penetrated light, we propose two theoretical explanations of this phenomenon. One of them is that several amplified peaks are the consequence of the forming of solitons in a polyethylene chain whose velocities are close to the velocity of sound. Forming of solitons, together with boundary conditions in a polyethylene macromolecules chain, which contain about thirty monomers, lead to the amplification of light. The second explanation requires introduction of homeopolar excitons in polymer macromolecules. Both energy gap of homeopolar excitons and width of homeopolar exciton zone are of the same order of magnitude. It means that transitions in a very wide zone give light quanta which are able to amplify the initial light. In order to avoid some confusion and misunderstandings, we wish to point out the following. Atoms and molecules as the whole are treated classically (transition through potential barriers, for example, etc.). The exception to this rule are phonon theories of crystals where the phonon is considered as a quanta of boson field, i.e., it means that, in the theory of mechanical oscillations, molecules and atoms as the whole are treated quantum mechanically. On the other hand, elementary excitations in crystals such as excitons, vibrons, spin waves, and ferroelectric excitations, etc., which arise from changes of some parts of atoms or molecules are treated quantum mechanically exclusively.

a

E-mail address: [email protected]. E-mail address: [email protected]. Correspondent author. c E-mail address: [email protected]. d E-mail address: [email protected]. b

142

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al. In the analyses of this work, the excitations of an individual molecule subsystem (i.e. the quantum objects) would serve as an explanation of the light amplification by a polymer chain.

1. Introduction In some previous works [1,2] it was experimentally found that the polyethylene foil noticeably amplifies intensity of light, and that amplification is proportional to the foil thickness. It was separated seven lines from mercury lamp and each of them was amplified. The experimental procedure is described in detail in [1], and therefore it will not be repeated here. We shall quote only the results of the experiment. The seven lines of mercury lamp were amplified 3–4 times after passing the light through a polyethylene foil. The wavelengths of amplified peaks were 254, 315, 366, 436, 506, 577 and 624 nanometers. Seven initial lines which are separated, are the more probable consequence of impurities of Hg, since in the foil of absolutely pure Hg exists only one level at λ = 253.7 nm. We shall try to explain the amplification of incident peaks in the frames of two models. One model is based on the idea that solitons, whose velocity is close to the velocity of sound appearing in polyethylene molecular chains, create the conditions for light amplification. The second model requires introducing homeopolar excitons in the system of possible excitations in the polyethylene chain. In this case, amplification of light by polyethylene arises as a consequence of the fact that the width of homeopolar exciton zone is of the same order of amplitude as the energy of excitation of individual polyethylene monomer. The amplifications of peaks were produced by one polyethylene foil whose thickness was 2 mm. The peaks were amplified 3–4 times by this foil. The amplifications produced by two foils (thickness: 4 mm) could not be registered since the devices were not able to register highness of peaks. It means that amplifications sharply increase with respect to the thickness of polyethylene foil. The theoretical explanation of this amplification of light intensity, given in [1], was based on inverse population of electrons. Namely, it was assumed that in the polyethylene chain exist metastable energy levels where electrons gathered and that they coherently transit to ground state. This coherent transition, as in a laser, leads to the energy amplification. In the work [2], the amplification was explained by exciton and soliton transitions. The excitons were not of dipole-dipole type, but of exchange (homeopolar) type, since polyethylene monomers are forming linear polyethylene macromolecules (polyethylene chains) by homeopolar forces whose potentials are of the order of magnitude of excitation energy of an electron in isolated monomer. The behavior of such excitons will be analyzed in this work with the goal to explain amplification of the light.

2. Excitons and Solitons in Infinite Polyethylene Chains Frenkel excitons [3] more often appear in a molecular crystal where dipole-dipole interactions propagate excitations of an isolated molecule produced by visible light. Excitation energy of an isolated molecule lies between 2.5 eV and 5 eV, while energies of

Theoretical Explanation of Light Amplifying by Polyethylene Foil

143

dipole-dipole interactions are 50–100 times lower. In such situation, energy of exciton and energy of visible photons are practically identical. For forming polyethylene linear chain made of monomers C2H4 are responsible exchange forces (covalent connection of monomers) [4,5]. Since potential energies between two electrons are reflexive, the potential energy is of positive sign. On the other hand if electron spins are parallel, the configurationally parts of wave function must be antisymmetric and this leads to negative exchange integrals which are binding monomers in chain. Consequently, if one isolated monomer of chain is excited by quanta of visible light this excitation propagates along the chain in covalent forces field. The matrix elements of covalent potential are of the order of five electronvolts. It means that they are of approximately of same magnitude as excitations of an isolated molecule. It is essential difference with respect to excitons in molecular crystals. In polyethylene chains wideness of exciton zone is practically same as the energy of excitation of an isolated monomer. It means that in polyethylene chains can appear excitons whose energy is two or three times higher than energy of visible light which produces excitons. This amplifying is registered in experiment described in the previous section. The mechanical oscillations interacting with excitons lead to forming of new quasiparticles - solitons [6], which can have higher energy than exciton. Besides, the solitons are qusiparticles of stable form moving superfluidely and having higher luminescence time than excitons. Luminescence time for singlet excitons is about 10

−8

s, for triplet excitons

−3

−2

−1

(corresponding to parallel electron-spins) 10 s [7] and for solitons about 10 s - 10 s [8]. The long times of luminescence means that metastable state are lasting sufficiently long to cause coherent illumination. The upper ideas will be demonstrated on the simple case of two level monomer excitation. The electronic Hamiltonian of monomer chain can be written in the following way:

H = ∑ ε s ans+ ans + n,s

1 2

∑W (s , s , s , s )a nm

1

2

3

n, m s1 , s 2 , s 3 , s 4

4

+ ns 1

+ ams a ans 2 3 ms 4

(1)

where a are Fermi operators of electrons localized in monomer and ε (which are of the order 2.5 -5 eV) are energies of excitations of an isolated monomer. The matrix element W include Coulomb and exchange forces and they are given by

( ) ( ) n e− m Ψ (ξ ) Ψ (ξ )

Wnm = ∫ d 3ξ n d 3ξ m Ψs*1 ξ n Ψs*3 ξ m

2

s4

m

s2

n

(2)

where ξ are internal coordinates of monomers. Because we consider two-level scheme, indices s1, s2 , s3 , s4 must have only two values: 0 - corresponding to ground state and f corresponding to excited state.

144

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al. Here and below, for reasons of simplifying the writing of equations, we will use in the

denominator the following notation: n − m ≡

ξn − ξm . +

+

+

Introducing the operators of monomers excitations P = a f a0 and P = a0 a f we

(

easily concluded that they are closed in electron subspace 10 ,0 f operators P

+

)

(

)

and 00 ,1 f . The

and P are Pauli operators which in the lowest ASQ approximation can be +

substituted by Bose operators B and B . +

Extracting from the Hamiltonian P only quadratic terms in P and substituting P with Bose operators B , we obtain the following excitonic Hamiltonian in ASQ approximation +

+

(ground state terms as well as the terms proportional to P P and PP are omitted):

V ( f , f ,0,0) + V (0,0, f , f ) ⎤ + ⎡ H = ∑ ⎢ε f − ε 0 − V (0,0,0,0) + ⎥ Bn Bn 2 n ⎣ ⎦ +

1 ∑ 2 [Wnm ( f ,0,0, f ) + Wnm (0, f , f ,0)]Bn+ Bm

(3)

n, m

where V =

∑ Wn − m . m

It will be now shown that W ( f , f ,0,0 ) and W (0,0, f , f

while W ( f ,0,0, f

) are positive Coulomb terms

) and W (0, f , f ,0) are negative exchange terms.

The antisymmetric electron pair function is given by:

Ψnm =

[ ( ) ( )

( ) ( )]

1 Ψ f ξ n Ψ0 ξ m − Ψ f ξ m Ψ0 ξ n 2

(4)

and, as a consequence of this fact, the integral of the electron-electron interaction

∫

3

d ξn d

3

* ξm Ψnm

e2 Ψnm n−m

consists from four integrals:

∫

d 3ξ n d 3ξ mΨ nm*

( ) ( )

( ) ( )

e2 1 e2 Ψ nm = { ∫ d 3ξ n d 3ξ mΨ f* ξ n Ψ 0* ξ m Ψ0 ξm Ψ f ξn n−m 2 n−m

( ) ( ) n e− m Ψ (ξ )Ψ (ξ )

+ ∫ d 3 ξ n d 3ξ mΨ 0* ξ n Ψ f* ξ m

2

f

m

0

n


145

( ) ( ) n e− m Ψ (ξ )Ψ (ξ ) }

− ∫ d 3 ξ n d 3ξ mΨ 0* ξ n Ψ f* ξ m =

2

0

m

f

n

1 [Wnm ( f , f ,0,0) + Wnm (0,0, f , f ) + Wnm ( f ,0,0, f ) + Wnm (0, f , f ,0)] 2

(5)

Introducing notations

Δ = ε f − ε0 D=

V ( f , f ,0,0 ) + V (0,0, f , f ) − V (0,0,0,0) 2 1 − Wnm = [Wnm ( f ,0,0, f ) + Wnm (0, f , f ,0)] 2

(6)

and taking the nearest neighboring approximation ( Wn, n ±1 = W ), the Hamiltonian (3) can be written as (see [5,6]):

H exc = (Δ + D )∑ Bn+ Bn − W ∑ Bn+ (Bn +1 + Bn −1 ) ; W > 0 n

(7)

n

By means of Fourier transformations

Bn =

1 N

∑

eikan Bk

(8)

k

the Hamiltonian (7) goes over to

H exc = ∑ Ek Bk+ Bk

(9)

Eexc (k ) = Δ + D − 2W cos ak

(10)

k

where

Now we can estimate exciton energy in polymer chain with exchange interaction. We will take Δ = 3 eV , D = 1 eV (matrix elements W ( f , f ,0,0 ) are bigger than matrix

elements W (0,0,0,0 ) and W ~2.5 eV. The maximal energy of excitons for ak = π is 9 eV

and it is 3 times higher than energy excitations (energy of photons). This corresponds to the results of quoted experiment. In the case of dipole-dipole interaction we can take D − 2W cos ak = 0.06 eV , so that Eexc (k )max = 3.06 eV . This example points out a

146

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al.

sharp difference between exciton energies in systems with dipole-dipole interactions with respect to those energies in systems with exchange interactions. At the end, we shortly expose the soliton characteristics of this exciton system. To the excitonic Hamiltonian we add the phonon Hamiltonian:

H ph =

1 2M

∑

pn2 +

n

C (un − un −1 )2 ∑ 2 n

(11)

where M is the mass of monomer, C is the Hook's constant of monomer chain, un are displacements of monomers and pn = u are corresponding momenta. The exciton phonon Hamiltonian is given as follows:

∂W + ⎡ ∂D + ⎤ ∂u H ep = ∑ ⎢2 Bn Bn − Bn (Bn +1 + Bn −1 )⎥ a ∂a ⎦ ∂ (na ) n ⎣ ∂a

(12)

Using complete Hamiltonian of the system

H = H exc + H ph + H ep

(13)

and equations of motion for B we obtain

∂W ⎡ ∂D (Bn +1 + Bn −1 )⎤⎥ a ∂u (14) EBn = (Δ + D )Bn − W (Bn +1 + Bn −1 ) + ⎢2 Bn − ∂a ⎣ ∂a ⎦ ∂ (na ) and

∂ 2u n C (un +1 + un −1 − 2un ) = M dt 2

(15)

The equations can be averaged by coherent states

< Ψep eβ n (Bn − Bn )Bn e −β n (Bn − Bn ) Ψep >=< Ψep eβ n (Bn − Bn )Bn+ e −β n (Bn − Bn ) Ψep >= βn +

+

+

Ψep >= nexc > n ph > ;

+

β*n = βn

(16)

and

< Ψep

γn pn ei u

ne

−

γn pn i

Ψep >= − γ n ; γ* = γ

(17)

After that we go over to continual variable na → x and to common variable ξ = x − vt , where v is velocity of soliton. After quoted operations, the equation (15) goes over to:


(v

2

− s2

147

)ddξγ = 0

(18)

C M

(19)

2

2

where

s2 = a2

is the square velocity of sound in the chain. The boundary condition for this equation is

γ (0 ) = 0 ,

dγ dξ

ξ =0

= M −1 p0

(20)

where p0 is the beginning momentum which produce mechanical oscillations. By the way, one of explanations of appearance of tsunami is based on such beginning conditions. By means of conditions (20) we obtain

dγ M −2 p02 = dξ v 2 − s 2

(21)

The equation (14), written in continuum, after the substitution (21), becomes:

⎡ 2aM −2 p 2 ⎛ ∂V ∂W E − ⎢Δ + D − 2W + 2 2 0 ⎜ − v − s ⎝ ∂a ∂a d 2β ⎢⎣ = ∂W ⎞ ⎛ dξ 2 a 2 ⎜W + a ⎟ ∂a ⎠ ⎝

⎞⎤ ⎟⎥ ⎠⎥⎦

β=0

(22)

If the coefficient in second term of (22) is positive the solution of (22) is

β = c1 cosh Qξ = c1 cosh Q( x − vt )

(23)

where

Q=

⎡ 2aM − 2 p 2 ⎛ ∂V ∂W ⎞⎤ E − ⎢Δ + D − 2W + 2 2 0 ⎜ − ⎟⎥ v − s ⎝ ∂a ∂a ⎠⎦⎥ ⎣⎢ ∂W ⎞ ⎛ a 2 ⎜W + a ⎟ ∂a ⎠ ⎝

(24)

The details of further calculations one can find in ref. [6, pp. 27-31]. Here they will not be quoted. For our aims it is important to say that the wave function of soliton is proportional to reciprocal value of β , i.e.

148


Ψsol =

c2 1 = β cosh Q( x − vt )

The function Ψsol is taken as Ψsol = cosh

−1

(25)

[Q(x − vt )]

since Q contains arbitrary

parameter soliton energy E. Soliton energy Esol can be determined from normalization condition: +∞

1 2 dξ Ψsol = 1 ∫ a −∞

(26)

2 =1 aQ

(27)

which reduces to

wherefrom we obtain

E = Δ + D + 2W +

2aM −2 p02 ⎛ ∂V ∂W − ⎜ v 2 − s 2 ⎝ ∂a ∂a

∂W ⎞ ⎟ + 4a ∂a ⎠

(28)

Soliton wave function (25) is of the stable form and propagates practically superfluidly. Comparing (26) with (10) we see that soliton energy is some higher than the exciton one, but when s → v , i.e. when soliton velocity becomes close to sound velocity then soliton energy becomes noticeably higher than exciton energy. It means that in the soliton model of excitations in polyethylene, amplification can be considered as a consequence of sharp increase of soliton energies when velocity of solitons becomes very close to the velocity of sound.

3. Green’s Function of Homeopolar Excitons The general form of electron Hamiltonian, where overlapping of electron wave functions of neighbor molecules is weak, is given by N

H = ∑ ∑ ε s ans+ ans + n=0 s

+

8 1 N Wn − m ( s1 , s2 , s3 , s4 )ans+ 1 ans 2 ans+ 3 ans 4 ∑ ∑ 2 n, m = 0 s1 , s 2 , s3 , s 4 = 0

Here a and are a electron operators,

(29)

ε s is energy of electron in an isolated monomer

and W are matrix elements of reflexive electron potential

e2 taken over products of n−m


149

configuration electron functions and spin electron functions. For two electrons forming bonds between two monomers spin functions is even (it corresponds to parallel spins) while the configurationally function must be odd. It gives attractive transition potentials between monomers whose order of magnitude is 5 eV, i.e. of the same order of magnitude as excitation energies of electrons in an isolated monomer. It should be noted that in the case of dipole-dipole interactions (Frenkel excitons) attractive potentials between molecules are of the order of 10-2 eV. Using the well known procedure, taken from the theory of Frenkel excitons for two level schemes of electron excitations [3, 4] and going over to approximate second quantization approach, we can write the Hamiltonian (29) in the nearest-neighboring approximation as: N

N

N

n=0

n=0

n=0

H = Δ ∑ Bn+ Bn + ∑ (Dn , n +1 + Dn , n −1 )Bn+ Bn −∑ Bn+ (J n , n +1Bn +1 + J n , n −1 Bn −1 ) (30) where

Δ = ε f − ε 0 ; Dn, n ±1 =

J n, n ±1 =

Here

Wn , n ±1 ( f , f ,0,0) + Wn , n ±1 (0,0, f , f ) − Wn , n ±1 (0,0,0,0) (31) 2

Wn , n ±1 ( f ,0,0, f ) + Wn , n ±1 (0, f , f ,0) ; D > 0, J > 0 2

ε f denotes energy of excited electron, while ε 0 denotes electron energy in ground

state. Assuming that polyethylene linear chain has N monomers (N is 30-50 monomers) we must introduce the boundary conditions

D0, −1 = DN , N +1 = 0 ; J 0, −1 = J N , N +1 = 0

(32)

The exciton system with Hamiltonian (2) will be analyzed by means of the “real space” Green’s function method. The double-time Green’s functions are

⎧1, n = m (33) Gnm (t ) =>= θ (t ) < Bn (t ), Bm+ (0) > ; θ (t ) = ⎨ ⎩0 , n ≠ m

[

where

[, ]

is the commutator,
is the thermal average over the grand canonical

ensemble and θ (t ) is the Heaviside function. Using the equations of motion for Bose operators B and taking into account the boundary conditions (32) we will obtain, after Fourier transformations time-frequency, the system of three difference equations defining Green’s functions,

150


J [Gn +1, m (ω ) + Gn −1, m (ω )] + ρGn , m (ω ) = JG1, m (ω ) + ( ρ + D)G0, m (ω ) =

i δ n, m ; 1 ≤ n ≤ N − 1 2π i δ 0, m ; n = 0 2π

JGN −1, m (ω ) + ( ρ + D)GN , m (ω ) = where

i δ N ,m ; n = N 2π

ρ = E − Δ − 2D

(34)

(35)

(36)

(37)

It can be shown that the system of equations (34) - (36) reduces into unique equation (39) by the substitution: N +1

Gn,m (ω ) = ∑ Aν (m, ω ) Fν (n)

(38)

ν =1

and N

i

∑ (2 J cosϕν + ρ )Aν (m,ω ) Fν (n) = 2π δ ν =0

n,m

; n = 0, 1, 2,…, N

(39)

where

Fν (n) = sin(n + 1)ϕν −

D sin nϕν J

(40)

Parameters φν are the solutions of the transcendental equation 2

D ⎛D⎞ sin( N + 2)ϕν − 2 sin( N + 1)ϕν + ⎜ ⎟ sin Nϕν = 0 J ⎝J⎠

(41)

The Kronecker symbol will be represented as follows N +1

δ n, m = ∑ Fν (n) Φν (m) ; n, m ∈ {0, 1, 2,…, N }

(42)

ν =1

Putting equation (42) into (39) we obtain

Aν ( n , ω ) = Φ ν ( n ) gν ( ω )

(43)


151

where

gν ( ω ) =

1 i 2π ω − Ων

(44)

After substituting the equations (42) - (44) into equation (39), we get

Ων =

Eν

; Eν = Δ + 2 D − 2 J cos ϕν

(45)

The spectral intensity of Green’s functions:

Gn , m (ω ) =

i N Fν (n) Φν (m) ∑ 2π ν = 0 ω − Ων

(46)

is given by [5, 6] N +1

I n, m (ω ) = ∑ ν =1

Fν (n) Φν (m) ω

e

k BT

δ (ω − Ων )

(47)

−1

By means of the formula (3.19 ) we can determine the correlation function ∞

N +1

Cn , m (t ) =< Bm+ (0) Bn (t ) >= ∫ dω e − iω t I n , m (ω ) = ∑ ν =1

−∞

Fν (n) Φν (m) e

Eν k BT

e

i − Eν t

(48)

−1

and the concentration of excitons N +1

Cn , m (0) =< Bm+ (0) Bn (0) >= ∑ ν =1

Fν (n) Φν (m) e

Eν k BT

(49)

−1

This formula gives the exciton energies in analysed polyethylene chain.

4. The Exciton Wave Function of Polyethylene Chain The equation (45) gives the energies of excitons in polymer chain. Using this formula we can find the transitions whose wavelenghts correspond to experimentally obtained wavelenghts , which are quoted in Introduction. Besides these wavelenghts, we shall determine the probabilities of finding of exciton at given energy level, as well as the probabilities of exciton transition between two energy levels. These probabilities can be find out by means of exciton one particle wave function

152

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al. N +1

| Ψν >= ∑ Aν (n) Bn+ | 0 >

(50)

n =1

On the basis of equations motion and boundary conditions, it is obtained the system of homogeneous diference equations for coefficients Aν(n):

An +1 + An −1 + ρAn = 0 ; 1 ≤ n ≤ N − 1 D⎞ ⎛ A1 + ⎜ ρ + ⎟ A0 = 0 ; n=0 W⎠ ⎝ D⎞ ⎛ AN −1 + ⎜ ρ + ⎟ AN = 0 ; n=N W⎠ ⎝

(51)

which, by the substitution N +1 D ⎡ ⎤ N +1 An = ∑ Aν ⎢sin( n + 1)ϕν − sin nϕν ⎥ = ∑ Aν Fν ( n) J ⎣ ⎦ ν =1 ν =1

(52)

goes over to the unique equation N +1

∑ (2 J cosϕν + ρ )Fν (n) = 0 ; ν

n ∈ (0,1, 2,…, N )

(53)

=1

where Fν (n) is given by (40) and parameters φν are real solutions of the equation (41). Consequently, finally we have N

Bν+ | 0 >= ∑ Fν (n)Bn+ | 0 >

(54)

n=0

Multiplying equation (54) with Φν(m), summing both sides of obtained equality over ν and taking into account (42), we obtain: N +1

Bn+ | 0 >= ∑ Φν (n)Bν+ | 0 >

(55)

ν =1

+

The normalised function Bn 0 is given by

Bν+ | 0 >=

N

1 N

∑ Fν (n) n =0

2

∑ Fν (n)B n =0

+ n

|0>

(56)


153

+

while normalised function Bν 0 is:

Bn+ | 0 >=

N +1

1 N +1

∑ Φν (n) ν 2

∑ Φν (n)Bν ν

+

|0>

(57)

=1

=1

The dependence of exciton energies with respect to

ϕν , which are obtained from (17) for

values Δ = 3 eV; D = 2.5 eV; J = 2.7 eV, is given in Figure 1.

Figure 1. The energy levels of excitons in polyethilen chain.

Table 1. Experimental and theoretical wavelengths of transitions leading to amplifications

λexp (nm) λtheor (nm) Transition Amplification factor ν →ν ' 254 315 366 436 506 577 624

256 314 363 439 503 574 625

32 → 18 27 → 18 24 → 17 32 → 22 23 → 18 30 → 23 29 → 23

4.46 4.16 3.81 4.46 3.67 4.39 4.33

154


Experimental and theoretical wavelenghts of peaks as well as the theoretical amplifying of peaks are given in the Table 1. The theoretical values λ are determined as

hc (Eν − Eμ ) ; ν > μ . −1

On the other hand, the probabilities of finding excitons on levels before luminiscence are given Table 2. Table 2. The probabilities of existence at stable levels where Φν =

Pν

Pν =

Φν2

N +1

∑ Φν2

1 31 ∑ Φν (n ) 32 n = 0

Pν (%)

ν =1

P23 P22 P18 P17

0.299281 0.072849 0.265614 0.152059

8.956 0.531 7.055 2.312

Table 3. The transition probabilities of exciton to stable levels

Pν , μ =

Φν Φ μ N +1

∑ Φν ν

2

Pν , μ (%)

=1

P32;18 P30;23 P32;23 P27;18 P32;22 P24;17 P23;18

3.454 7.794 6.289 0,796 0.947 3.652 7.949

Similarly, the transitions probabilities from higher level Eν to the stable levels Eμ quoted in the previous Table 2. are presented in the Table 3.

Conclusion In this work are analyzed two possible mechanisms of light amplification by polyethylene foil, registered in experiments with a mercury lamp. In these experiments the light, after transition through polyethylene foil was about 3 times more intensive than in the cases when the foil is absent. In the first model it was assumed that in the polymer chain solitons are forming whose velocities were close to the velocity of sound and this caused the light amplification.


155

In the second model was introduced the concept of homeopolar excitons having approximately equal gap and zone width. The analyses by means of Green's function as well as the wave of homeopolar excitons wave function were determined metastable levels of homeopolar excitons and probabilities of transition levels to ground state. Calculated theoretical values of mentioned probabilities were in good agreement with experimental data.

Acknowledgements This work was supported by the Serbian Ministry of Science and Technology: Grant No 141044, by Vojvodina Academy of Sciences and Arts, and by the Provincial Secretariat for Science and Technological Development of the Autonomous Province of Vojvodina (Project 114-451-00615/2007-06).

References [1] [2]

[3] [4] [5] [6] [7] [8]

Janjić J. D., Tošić B. S., Scientific Bulletin of the „Politehnica“ University of Timisoara, ROMANIA, Transactions on Mathematics and Physics 2006, 51, 80. Janjić J. D., Tošić B. S., Sajfert V., "Luminescence Mechanism of Light Amplyfing by Polyethylene, Bulletin of the “Politehnica” University of Timisoara, ROMANIA, Transactions on Mathematics and Physics 2008, 53, 91. Agranovich V.M., JETP 1959, 37, 430. Agranovich V. M., Theory of Excitons; Nauka: Moscow, RUSSIA, 1978. Тyablikov S.V., Methods in the Quantum Theory in Magnetism, Plenum Press: New York, USA, 1967. Tošić B. S., Statistical Physics, Faculty of Natural Sciences, Novi Sad, SERBIA, 1978. (in Serbian) Masterson W. W., Hurley C. N., Chemical Principles and Reactions, Saunders College Publishing, Phyladelphia, USA, 1993; pp. 579. Brown, W.H., Introduction in Organic Chemistry, Saunders College Publishing Phyladelphia, USA, 1997; pp. 428.



Chapter 8

ANHARMONIC EFFECTS IN NORMAL MODE VIBRATIONS: THEIR ROLE IN BIOLOGICAL SYSTEMS Attila Bende* Molecular and Biomolecular Physics Department, National Institute for Research and Development of Isotopic and Molecular Technologies, Donath street, No: 65 – 103, Ro-400293, Cluj-Napoca, ROMANIA

Abstract The intra- and intermolecular hydrogen-bond dynamics plays an important role in thermal stability and molecular functionality of biomolecules like DNA base pairs and proteins. The phenomena of thermal relaxation, conformational changes as well as the ultrafast nonradiative decays in molecules, especially in biomolecules are realized among other physical effects also with the help of molecular vibrations. Intermolecular hydrogen- or weak chemical bonds (covalent bond including H atoms) usually present large anharmonic normal mode vibrations. The anharmonic effects of normal mode vibrations for some small molecular model systems (formamide and urea) as well as for guanine-cytosine and adenine-thymine DNA base pairs are presented considering DFT and ab initio second order Møller-Plesset (MP2) theoretical methods. The role of basis set superposition errors in harmonic and anharmonic frequency calculations is briefly discussed. It was observed that anharmonic effects can significantly change the blue- or red-shifted harmonic frequency values. Large anharmonic effects were found in the case of protonated molecular structures, especial for Hoogsteen conformation of guanine-cytosine base pairs.

1. Introduction Hydrogen bonding is ubiquitous in nature and governs a wide array of chemical and biological processes ranging from local structure in molecular liquids to the structure and folding dynamics of proteins [1, 2]. Although the hydrogen bond is well studied, its lowfrequency vibrations—the large-amplitude motions involving stretching and bending along the actual hydrogen-bond coordinates—have been rarely investigated [3]. Information about *

E-mail addresses: [email protected], [email protected]

158

Attila Bende

these vibrations offers exceptional insight into the potential energy surface of the interaction and so further enhances our understanding of the hydrogen bond and its impact on molecular structure and dynamics. The “C=O···H-N” type hydrogen-bond (H-bond) is one of the most frequently occurring Van der Waals (VdW) bonds in the biological systems. They can be found as a main component of the DNA bases pair’s interaction systems or in protein α-helix and β-sheets. In this sense, not only is the cognition of molecular structures important, but also their dynamics, which in essence represent their biological functionality. Vibrational spectroscopy is a powerful tool for getting information about the complex dynamics of atoms in H-bonds. In particular, such calculations provide sensitive information on the potential energy surfaces of these systems [4, 5]. In the case of infrared (IR) spectroscopic studies, the most evident effect of hydrogen bonding are the red shift of the high-frequency X–H···Y stretching mode, its intensity increase and band broadening; the latter is often accompanied by the development of peculiar band-shapes. The large increase of the bandwidth, the band asymmetry, the appearance of subsidiary absorption maxima and minima, such as Evans windows, peculiar isotope and temperature effects and the similarity of features of the line shape when passing from the condensed to the gas phase [6, 7] are the challenge for theories of H-bonds. Computational studies are essential in this context since comparison of calculations to the measured spectroscopic data provides a test of the potential surface [8–11]. While calculations at the level of the harmonic approximation are very useful, they are often not of sufficient accuracy [8, 9, 12–14]. One approach is to apply empirical scaling factors in order to represent the anharmonic effects [15, 16]. However, such an approach, while useful in many cases, has no fundamental (quantum mechanical) basis. Moreover, it does not provide any insights into the nature of the anharmonic part of the potential, which itself is of great interest. Calculations of the spectra of large molecules beyond the harmonic approximation remain a major challenge. The goal of this Chapter is to give an accurate description of intermolecular normal modes and to present different intermolecular interaction effects which could influence the monomer type vibrations, considering the formamide and urea dimer cases and the guaninecytosine DNA base pair molecular system.

2. Theoretical Background and Computational Methods Regarding the theoretical methods, it is well established that for an accurate description of HBs, ab initio techniques which include the electronic correlation level are needed, i.e. with an error bar of less than 0.04 eV (1 kcal/mol) for predicting the HB strength. Thus the observed underestimation of the HB strength by the Hartree-Fock (HF) calculations (where the electron correlation effects are missing) is overcome using correlated methods like second order Møller-Plesset perturbation theory (MP2) or coupled cluster (CC) methods. But it is necessary to use very large high quality basis sets to expand the wave function and to get reliable HB properties. This fact and the necessity of correlated methods to accurately describe HBs make such studies computationally too expensive and only applicable to molecular complexes of at most a few tens of atoms. Therefore strategies to study HBs with similar accuracy to MP2 or higher levels of theory but computationally less expensive are needed. In this vein, density functional theory (DFT) is a method that includes electronic correlation. Unfortunately the accuracy of DFT to describe the HB interaction relies on the

Anharmonic Effects in Normal Mode Vibrations

159

applied functional to approximate the electronic exchange - correlation (XC) contribution. To overcome this problem, first we try to find those XC functionals which could describe HBs with an approximate accuracy as MP2. In the last few years a number of significant works could be found in the literature, which compared the efficiency of different DFT functionals with those obtained using MP2 method. Besides of the accurate electron correlation methods, based on the perturbational, coupled-cluster or Kohn-Sham (KS) scheme, the basis set superposition errors (BSSE) present a very important source of discrepancies. Many papers have indicated that the impact of BSSE on the geometries of weakly bound systems is smaller for DFT methods than those for ab initio methods such as MP2, but their influence even in case of DFT could not be neglected [17–19]. The reason why BSSE is similar in HF and KS methods is that the KS method does not require the high oscillatory basis functions which are needed to describe electron correlation in traditional electronic structure methods, and which are virtually impossible to include at saturation.

2.1. The BSSE Problem The BSSE is a pure “mathematical effect” and it is an important problem to solve when we study a weakly bonded molecular complex. This effect appears only as a result of the use of finite basis sets, because the description of the monomer is actually better within the supermolecule than that which one has for the free monomers by applying the same basis set, so thus leads to incomplete description in the individual monomers. Due to BSSE, the calculated interaction energies show too deep minima and the computed potential energy surface (PES) is distorted. The most important and straightforward a posteriori correction scheme, the so-called “function counterpoise” (CP), or simply the Boys–Bernardi method, was introduced by Jansen and Ross [20] and, independently, by Boys and Bernardi [21] in 1969/1970. In this CP scheme, the monomer energies are recalculated by using the whole supermolecular basis set and these corrected monomers are used in the molecular interaction energy calculations. A conceptually different way to handle the BSSE problem is to apply the “chemical Hamiltonian approach” (CHA) for the case of intermolecular complexes proposed by Mayer [22] in 1983. (For a detailed review on CHA, see Ref. [23].) By using the a priori CHA method one can eliminate the nonphysical terms of the Hamiltonian, which leads to wave functions free from the nonphysical delocalizations caused by BSSE. Using this CHA scheme, several approaches have been developed both at the HF [24–37] and correlated [38– 45] levels of theory to study the structures and interaction energies for different van der Waals and H-bonded systems.

The CP Scheme The simplest definition of the uncorrected interaction energy between two molecules is the difference of the supermolecular energy and the sum of the free monomer energies, each calculated in its own basis set: unc ΔE AB = E AB ( AB ) − E A ( A) − EB (B ) ,

(1)

160

Attila Bende

where EAB(AB), EA(A), and EB(B) denote the total energy of the AB “supermolecule” and the energy of the A and B monomers, respectively. The notations in parentheses indicate that basis sets corresponding to (sub)system A, B, and AB, respectively, were used. To compute unc , we need to use (nearly) complete basis sets the correct value of the interaction energy ΔE AB on the supermolecule and on each monomer, which is usually impossible in practice. CP In the CP scheme, the interaction energy ΔE AB is defined as the difference of the supermolecule and monomer energies, all computed in the same supermolecule basis set: CP ΔE AB = E AB ( AB ) − E A ( AB ) − EB ( AB )

(2)

Using Eqs. (1) and (2), one can define the BSSE content in the interaction energy as unc CP δEBSSE = ΔE AB − ΔE AB

= E A ( AB ) − E A ( A) + EB ( AB ) − EB (B )

(3)

According to Eq. (3), the CP-corrected potential energy surface (PES) of the dimer becomes unc E CP ( AB ) = ΔE AB − δE BSSE

unc = ΔE AB − E A ( AB ) + E A ( A) − E B ( AB ) + E B (B )

(4)

Equation (4) shows that by considering only the intermolecular internal coordinates as optimized parameters one has to calculate three different total energies to determine the CPcorrected PES.

The CHA Scheme In the alternative a priori CHA scheme introduced by Mayer [22, 23] one can omit the BSSE caused terms of the Hamiltonian, which is a conceptually different way of handling the BSSE problem. The CHA procedure permits the supermolecule calculations to remain consistent with those for the free monomer performed in their original basis sets. The basic idea of Mayer’s scheme is that one can divide the Born–Oppenheimer Hamiltonian into two parts:

Hˆ BO = Hˆ CHA + Hˆ BSSE

(5)

where Hˆ CHA is the BSSE-free part of the Hamiltonian and Hˆ BSSE is the “unphysical” part of the Hamiltonian that is responsible for the BSSE. The only difficulty of this scheme is that the resulting “physical” Hamiltonian Hˆ CHA is not Hermitical, so one cannot expect the BSSE-


161

free Hamiltonian Hˆ CHA to be Hermitical either. Based on this CHA Hamiltonian, Mayer and Vibók developed different SCF-type equations [28]:

Hˆ CHA ΨCHA = ΛΨCHA ,

ECHA / CE =

ΨCHA Hˆ BO ΨCHA ΨCHA ΨCHA

(6)

.

(7)

In the CHA framework [32–35] described by Eqs. (6)–(7) the non-Hermitical CHA Hamiltonian Hˆ CHA is used only to provide the BSSE-free wave function (or a perturbative approximation to it) but the energy should be calculated by using the conventional (Hermitical) Born– Oppenheimer Hamiltonian Hˆ BO , making not trivial the question of how one should calculate the second-order energy correction. The 0th-order Hamiltonian is defined in terms of the CHA Fockian [45]:

Hˆ 0 =

∑ ε ϕˆ ϕ~ˆ p

+ p

− p

,

(8)

p

+ − where ϕˆ p and ϕ~ˆ p are the creation and annihilation operators.

To obtain the first-order wave function in the perturbation theory, the non-Hermitical CHA Hamiltonian could be partitioned as

Hˆ CHA = Hˆ 0 + VˆCHA ,

(9)

where Hˆ defined by Eq. (8), is the Møller–Plesset type unperturbed Hamiltonian, which is 0

also non-Hermitical, and VˆCHA represents the perturbation. At the same time, the perturbation energy could be obtained considering the following partition of the Born–Oppenheimer Hamiltonian Hˆ BO :

Hˆ BO = Hˆ 0 + Vˆ

(10)

Then the energy up to second order could be presented as

E

(2 )

=

Ψ0 Hˆ BO Ψ0 Ψ0 Ψ0

where J2 is the generalized Hylleraas functional:

+ J2 ,

(11)

162

Attila Bende

J2 =

[ (

) (

)]

1 2 Re Qˆ Ψ1 Vˆ Ψ0 − Re Ψ1 Hˆ 0 − E0 Ψ1 , Ψ0 Ψ0

(12)

Here Ψ0 is the unperturbed wave function, E0 is the zero order energy ( Hˆ Ψ0 = 0

= E0 Ψ0 ), Ψ1 is the first order wave function of the perturbation, and Qˆ is the projection operator on to the orthogonal complement to Ψ0 . These equations define our working formula at the second-order perturbation level. This formalism is called “CHA/MP2” theory [40].

2.2. Anharmonic Approach of Normal Mode Vibrations Computation of harmonic force field and implicit the harmonic vibrational frequencies many times give results which are very far from the experimentally measured values. The explanation seems to be quite simple. Beyond of harmonic approximation, many other contributions like solvent effects, anharmonic corrections or cluster effects, are not taken into account. Usually, all of these corrections were considered by multiplying the harmonic frequency values with a scaling factor. Unfortunately, the collective contribution of different approximations could not always be fitted into the previous scaling scheme. So, it is very important to study them separately considering different theoretical models which describe these effects. The linear relationship between normal coordinates Q and Cartesian displacement X coordinates are:

Q = L+ M 1 2 X ,

(13)

where, by convention, all the components of Q and X vanish at the reference geometry, M is the diagonal matrix of atomic masses, and L is the matrix of (columwise) eigenvectors of the mass weighted Cartesian force constant matrix M-1/2FM-1/2. The second-derivative matrix over normal coordinates, Φ is

Φ = L+ M −1 2 FM −1 2 L

(14)

and is diagonal when evaluated at the equilibrium geometry with eigenvalues λ proportional to the squares of harmonic vibrational frequencies ω. In this way we can evaluate the third and fourth order energy derivatives with respect to normal coordinates by numerical differentiations of analytical Hessian matrices at geometries displaced by small increments δQ from the reference geometry [46, 47]: 1 ⎛ Φ (δQi ) − Φ jk (− δQi ) Φ ki (δQ j ) − Φ ki (− δQ j ) Φ ij (δQk ) − Φ ij (− δQk ) ⎞⎟ (15) Φ ijk = ⎜ jk + + ⎟ 3 ⎜⎝ 2δQi 2δQ j 2δQk ⎠


Φ ijkk =

163

Φ ij (δQk ) + Φ ij (− δQk ) − 2Φ ij (0 )

(16)

δQk2

1 ⎛ Φ (δQk ) + Φ ii (− δQk ) − 2Φ ii (0 ) Φ kk (δQi ) + Φ kk (− δQi ) − 2Φkk (0) ⎞ ⎟⎟ (17) + Φ iikk = ⎜⎜ ii δQk2 δQi2 2⎝ ⎠ In case of nonlinear molecules these computations require at most the Hessian matrices at 6N-11 different points, N being the number of atoms in molecules. Expanding the vibrational Hamiltonian Hvib using the vibrational wavefunctions in the framework of the second-order perturbation theory, one can obtain the following expression:

vi H vib vi = ξ 0 +

⎛

1⎞

1 ⎞⎛

⎛

∑ ω ⎜⎝ n + 2 ⎟⎠ + ∑ ξ ⎜⎝ n + 2 ⎟⎠⎜⎝ n i

i

ij

i

i≤ j

j

1⎞ + ⎟ 2⎠

(18)

where the ξ constants are simple functions of the cubic and quartic force constants.

2.3. Density-Fitting Local Perturbation Methods Hyper-accurate quantum chemical techniques, suffer from high-degree polynomial scaling. For example, CCSD scales as O(N6) and MP2 scales as O(N5) where N is some measure of system size. In this sense, to develop new techniques which reduce their highdegree polynomial scaling are very important in order to describe larger molecular systems. In the last decade were developed two important methods (density fitting and local perturbational or coupled cluster approximation) which all together can change the highdegree scaling and reduce them close to the linear dependency. In this way, medium sized molecular systems (60 – 80 atoms) can be treated at triple-zeta basis set quality level.

Density Fitting In the HF theory, calculations of electron repulsion integrals (ERI)

G G ψ p (1)ψ q (1)ψ r (2)ψ s (2 ) pr r qs = pq rs = dr1 dr2 r12 −1 12

∫ ∫

*

*

(19)

are the most expensive computational procedures. The computational effort for ERI valuation and transformation can be reduced by 1–2 orders of magnitude using density fitting (DF) methods [48 – 51]. In this approach the one-electron charge densities in the ERIs, which are binary products of orbitals, are approximated by linear expansions in an auxiliary basis set

| pq ) ≈ | ~ pq ) = DApq | A)

(20)

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Attila Bende

This leads to a decomposition of the 4-index ERIs in terms of 2- and 3-index ERIs, and the O(N4) dependence of the computational cost is reduced to O(N3)

( pq | rs ) ≈ ( ~pq | ~r s ) = DApq J AB DBrs ,

(21)

G G A(1)B(2) J AB = dr1 dr1 r12

(22)

where

∫ ∫

Local Approximation In the LMP2 method the occupied space is spanned by localized molecular orbitals (LMOs), which can be obtained from the canonical orbitals by standard localization procedures as proposed by Boys [52] or Pipek and Mezey[53]. The virtual space is spanned by a basis of non-orthogonal projected atomic orbitals (PAOs),

1 = Tabij Ψijab

(23)

which are obtained from the AO basis functions by projecting out the occupied orbital space

1 =

∑ ∑[ T]

ij ab

Ψijab

(24)

i , j∈P ab∈ i , j

In the following, PAOs will be labeled a,b. Since these functions are inherently local, one can introduce two approximations: First, excitations from a pair of occupied LMOs can be restricted to subsets of PAOs that are spatially close to the two LMOs. The number of functions Nij in each of these subsets (pair domains) is independent of the molecular size, and the number of excitations for each electron pair reduces from Nvirt2 to Nij2. Second, the integrals (ai|bj) for distant orbitals i and j can be approximated by multipole expansions [54] or neglected. The remaining number of non-distant orbital pairs (ij), and therefore the total number of excitations, scales linearly with molecular size.

3. Results and Discussions 3.1. Formamide Dimers Formamide (FA) is the simplest molecule that contains a peptide linkage built by the carbonyl and amino groups; therefore, we can consider the formamide dimer (FA–FA) as the simplest model of the pairing of nucleic acids and the formamide–water (FA–WA) complex as a hydration of proteins, respectively. Geometry structures [3, 55–64] and vibrational spectra [3, 56, 60, 65–68] of the FA–FA and FA–WA dimers have been the subject of many studies using different ab initio [HF and MP2] methods, which give valuable information about the structure and dynamics of the H-bonds in molecular systems.


165

The standard HF, MP2, and CP-corrected HF/MP2 calculations were performed using the Gaussian 03 computer code [69]. The CHA/CE- and CHA/MP2-type calculations were done by generating the input data (integrals and RHF orbitals) with a slightly modified version of HONDO-8 [70]. In these calculations the CHA/SCF code [25–27] and the CHA/MP2 program of Mayer and Valiron [40, 42] were used. For the frequency calculations based on Wilson’s G-F method, the program written by Beu [71] was applied. We considered six different basis sets: 6-31G, 6-31G**, 6-31G**++, D95V, D95V**, and D95V++**. 6-31G to 6-31G**++ are standard Pople basis sets; D95V to D95V++** are Dunning/Huzinaga valence basis sets. The conventional supermolecule geometries were optimized at both the HF and MP2 levels, applying the analytical gradient method included in the Gaussian 03; the CHA- and CP-corrected geometries were calculated by using a numerical gradient method [inverse parabolic interpolation (IPI) [72]] in internal coordinates including only internal coordinates with intermolecular character (one bond, two angles, and three torsion angles). The reason for this choice is that the CPU time of MP2-CHA program is fairly big. To test the applicability of our numerical gradient method we performed several sample calculations using both the IPI method and the analytical gradient built into Gaussian 03. There is practically no difference between them. For conventional uncorrected cases we also performed similar calculations to check the values of the force constants and harmonic vibrational frequencies. The uncorrected HF and MP2 results for the force constants (in internal coordinates) and for the harmonic vibrational frequencies were obtained by using the standard routines of the Gaussian 03 program. As for the CHA- and CP-corrected calculations, at first the numerical second derivatives of the energies were calculated to obtain the CHA and CP force constants and then the NOMAD program [71] was applied to obtain the appropriate CHA and CP harmonic vibrational frequencies. As we are interested in the BSSE content in the molecular interaction energies, only those components of the force constant matrix were recalculated that correspond to intermolecular internal coordinates. The anharmonic frequencies were obtained using the standard full-CP method implemented in Gaussian 03.

The Geometry Structure Table I shows the optimized geometry parameters for the FA–FA complex (Figure 1), using the conventional (Uncorr.), CHA, and CP schemes at both the HF and MP2 levels. The FA–FA dimer has planar geometry configuration: the global minimum for the dimer is a cyclic structure of C2h symmetry involving two equivalents N-H···O=C intermolecular hydrogen bonds. Two other planar minima has been identified [56, 62, 73] that establish a single N-H·O=C HB building up the linear and zig-zag configurations. In the work we consider just the planar cyclic dimer configuration, for which the BSSE-corrected geometrical parameters are presented. Once this assumption is made, the only variables left are the rHO bond length and two angles, αNHO and αHOC, which could be associated with the in-plane vibration normal modes, while all three torsion angles having intermolecular character are kept constant, their normal modes representing out-of-plane vibrations. The results show that only the rHO bond length has an important BSSE correction (0.06 Å for MP2-CHA/6-31++G**, 0.08 Å for MP2-CHA/D95V++**, and 0.06 Å for MP2-CP/6-31++G** and MP2-CP/D95V++**), which increase the bond size. Furthermore the change in the αNHO and αHOC angle values is

166

Attila Bende

insignificant but their corresponding force constants also include BSSE effects. Unfortunately, the experimental value for rHO bond length presented in Ref. [63] does not have the desired precision, so we could not compare with high precision the uncorrected values and the given BSSE corrected bond lengths. Both the corrected and the uncorrected values are close to the experimental value, 1.9 Å (1.87 Å for MP2/D95V++** and 1.9 Å for MP2/6-31++G** in the uncorrected case; 1.95 Å for the CHA- and CP-corrected cases). However, we can compare our calculated result for the rNO intermolecular distance with the experimental values obtained by Itoh and Shimanouchi [66]. Table 1. Intermolecular coordinate for the FA–FA dimer computed at the HF and second-order Møller–Plesset perturbation theory (Uncorr., CHA, CP) level, using D95V, D95V**, D95V++**, 6-31G, 6-31G**, and 6-31++G** basis sets rHO (Å)a Basis

ΑNHO (Deg.)

RHF

MP2

RHF

MP2

RHF

MP2

Uncorr.

1.904

1.872

129.7

125.9

165.6

169.3

(66)

CHA

1.950

1.936

128.9

125.1

166.3

170.1

CP

1.943

1.960

129.4

125.7

165.8

169.5

D95V**

Uncorr.

2.002

1.862

125.7

122.1

168.9

172.6

(120)

CHA

2.029

1.912

125.0

122.1

169.6

172.6

CP

2.019

1.924

124.9

121.8

169.7

172.8

D95V++**

Uncorr.

2.021

1.874

125.2

120.9

169.4

173.7

(150)

CHA

2.039

1.915

125.2

122.5

169.4

172.1

CP

2.034

1.936

125.2

122.4

169.4

172.2

D95V b

Method

αHOC (Deg.)

6-31G

Uncorr.

1.919

1.912

126.8

122.0

168.0

172.4

(66)

CHA

1.934

1.923

126.8

123.0

168.0

171.3

CP

1.939

1.965

127.6

123.7

167.3

170.7

6-31G**

Uncorr.

1.998

1.887

122.7

119.0

171.5

175.0

(120)

CHA

2.007

1.898

123.6

120.9

170.5

173.0

CP

2.025

1.947

123.6

120.4

170.6

173.6

Uncorr.

2.017

1.899

125.4

121.7

169.2

172.8

CHA

2.034

1.954

125.1

122.6

169.4

171.9

CP

2.038

1.954

125.2

122.5

169.3

172.1

6-31++G** (150) a

Experimental value ≈ 1.9 Å, taken from Ref. [63]. The number of basis functions are given in parenthesis.

b

The calculated lengths are 2.895 Å for the uncorrected case, 2.970 Å for CHA-type, and 2.955 Å for CP-type BSSE-corrected cases, while the X-ray data for the formamide crystal [66] gives 2.935 Å for the rNO intermolecular distance. If we suppose that the intermolecular bond length in the crystal phase is a bit shorter than in the gas phase, it can be considered that the corrected values are very close to the experimental.


167

Figure 1. The formamide dimer.

Table 2. Interaction energiesa (in kcal/mol) for different FA–FA dimer geometry computed at the HF and second-order Møller–Plesset perturbation theory (Uncorr., CHA, CP) level, using D95V, D95V**, D95V++**, 6-31G, 6-31G**, and 6-31++G** basis sets Basis Method Uncorr.

D95V

D95V**

D96V++**

RHF

MP2

RHF

MP2

RHF

MP2

-17.018

-18.370

-12.912

-16.921

-12.043

-15.903

CHA

-15.279

-14.511

-12.086

-13.971

-11.567

-13.140

CP

-15.185

-14.046

-11.886

-13.552

-11.547

-13.201

Basis Method

6-31G RHF

6-31G** MP2

RHF

MP2

6-31++G** RHF

MP2

Uncorr.

-17.387

-18.799

-14.212

-18.330

-12.385

-15.365

CHA

-15.580

-14.496

-12.576

-13.851

-11.965

-13.228

CP

-14.956

-13.705

-12.032

-13.492

-11.837

-13.399

a

Experimental value ≈ -13.967 kcal/mol; taken from Ref. [63].

In Table 2 we present the calculated intermolecular binding energies, considering the optimized geometry in the given basis and using the given methods (uncorrected, CHA, and CP) and levels of theory (RHF and MP2). The experimental value was obtained using Rydberg electron transfer technique between laser-excited atoms; the molecular system [63] is 606 meV, which corresponds to 13.967 kcal/mol. Our BSSE-corrected results (13.288 kcal/mol for MP2-CHA/6-31++G**, 13.399 kcal/mol for MP2-CP/6-31++G**, 13.140

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Attila Bende

kcal/mol for MP2-CHA/D95V++**, and 13.201 for MP2-CP/D95V++**) are very close to experimental values, whereas the uncorrected results show more than 1.4 kcal/mol difference. Moreover, we can obtain reasonable binding energy value even if we use the 6-31G and D95V bases without diffuse or polarization functions, applying BSSE correction for uncorrected geometry at the same time.

Harmonic and Anharmonic Frequencies The FA–FA dimer has 30 vibrational normal modes from among which 24 (12 for each monomer) vibrations are characteristic to the monomer-type vibrational motion, while six modes have a pure intermolecular character. Table 3. The uncorrected and CP-corrected harmonic (ν) and diagonal anharmonic (x) frequency of FA–FA dimer computed at MP2 level of theory, using 6-31++G** basis set Nr.

νdim

νCP

νmon

xdim

xCP

xmon

Assign.

intramolecular 1

3770.0 3769.9

3770.3 3770.1

3814.1

-32.5 -32.9

-32.4 -32.8

-40.5

N-N a.

2

3465.8 3424.8

3467.5 3426.4

3660.9

-49.9 -58.9

-49.2 -57.8

-36.1

N-N s.

3

3103.9 3101.6

3104.2 3101.9

3083.9

-33.0 -33.1

-33.0 -33.1

-67.7

C-H

4

1795.5 1773.9

1796.4 1774.4

1794.9

-24.0 -22.3

-24.4 ???

-6.6

C=O

5

1677.0 1666.6

1677.2 1667.9

1654.2

-9.5 -3.9

-9.4 -4.0

-8.5

H-N-H

6

1444.0 1444.8

1446.7 1445.8

1444.0

-3.8 -3.9

-3.9 -3.9

-9.1

O=C-H

7

1367.7 1353.1

1367.4 1352.4

1295.9

-2.9 -2.6

-2.8 -2.6

-5.0

C-N

8

1112.1 1106.0

1111.6 1105.6

1075.2

-0.8 -0.7

-0.8 -0.7

-1.3

C-N-H

9

1060.2 1046.2

1067.7 1056.1

1041.0

0.5 1.1

-1.1 -0.7

-1.3

OofP

10

824.3 785.1

827.8 789.7

628.1

-15.5 -6.5

-20.7 -8.3

-15.9

Torsion

11

622.8 605.7

621.2 604.8

565.6

-0.6 -0.2

-0.6 0.3

1.2

O=C-N

12

413.9 394.5

425.5 404.4

276.1

17.8 28.2

12.9 24.8

-462.7

Torsion


169

Table 3. Continued Nr.

νdim

νCP

νmon

xdim

xCP

xmon

Assign.

intermolecular 1

47.5

118.3

451.2

47.6

OofP

2

29.6

99.7

2101.0

11.0

OofP

3

9.2

49.0

16800.0

1.5

OofP

4

171.8

176.2

-3.1

-2.3

H…O

5

212.6

217.3

-1.8

-1.2

H…H

6

136.4

136.2

-0.2

-0.1

O…O

Usually the vibration frequencies of molecular complex are evaluated at the harmonic approximation applying the Wilson F-G analysis and using the BSSE-uncorrected Hessians. On the other hand, the vibrational frequencies, in particular there with an intermolecular character, contain considerable anharmonic effects and therefore it is difficult to follow these two important corrections in a distinct way. Considering the monomer-type vibrations (Table 3) it can be found that two different dimer frequencies correspond to the similar monomer vibrations, but their values are usually shifted due to the intermolecular interaction. Taking in to account the full CP-corrected values in the dimer calculations, we found another frequency shift, but in this case due to the BSSE effects. Moreover, the frequency values show an important basis size effect at the MP2 level, which implies the shifts in dimer frequency values will change. The results of the anharmonic frequency corrections show a more complex picture. Because of the large numbers of the anharmonic frequencies in Table 3 only the diagonal elements of anharmonic frequency matrix are presented. The most important effect in the anharmonic values is given by the influence of the adjacent molecule, which generates substantial shifts in the anharmonicity of different monomer normal modes within the dimer system. Although the above-mentioned “cluster” effect is quite uniform, the basis size effects become much more complicated. In the case of hydrogen bond stretching (N–H, C–H) and angle-bending vibrations, changes in the anharmonic frequency are not so important, but the torsion angle and C=O stretching modes show very dissimilar results. Considering the full CP-corrected BSSE-free anharmonic frequency calculations, no major corrections can be found for the monomer-type vibrations, which in practice mean that their effects could be generally neglected for the FA–FA dimer. Regarding the intermolecular normal modes, in addition to the “cluster” and basis size effects, the BSSE corrections become very important, especially for “out of plane” normal modes (see MP2/6-31++G**). After these, considering the collective effects of basis size and BSSE corrections on the intermolecular vibration frequencies, it can be concluded that major corrections are obtained in the cases of the harmonic approximation given by the quality (applying polarization and diffuse basis sets) and the BSSE of the applied basis sets, which is followed by the similar correction of the anharmonic approximation. With respect to the intermolecular normal modes, the out-of-plane vibration with low frequencies usually shows a very dissimilar and unrealistic anharmonic correction, especially for the MP2/6-31++G** CP-corrected case. This phenomenon may be related to many facts: i) in the CP method the PES is very flat, the intermolecular force constants are very small [74] and the numerical

170

Attila Bende

calculations could give significant errors; ii) the role of the well-balanced basis set is very important, therefore we consider that the 6-31++G** does not give us adequate results. For example, the 6-31++G(2d, 2p) basis set could be a more suitable choice, but the available computer capacity does not allow us to perform such full-CP anharmonic calculations.

3.2. Urea Dimer The urea dimer presents to some extent a good similarity with the formamide-formamide system. The difference between these two systems is the existence of the second cyclic system in the urea dimer. In the case of the urea dimer one of the HBs is the weak C-N···H-N hydrogen bond. The planarity (or nonplanarity) of the urea dimer system has been the focus of a number of studies. Masunov and Dannenberg [75, 76] considered different levels of theory (HF, MP2 and DFT with and without BSSE corrections), and the most stable conformation was found to be the non-planar structure (using the MP2 method with the D95++** basis set). However, they accentuate that inclusion of vibrational and thermal corrections in the calculations of the molecular structure might give an effectively planar structure. We consider the papers of Rousseau and Keuleers [77, 78] as very important work in elucidating the urea structure, where detailed descriptions of vibrational spectra of urea both in gas and crystal phase are presented. Their concisely conclusion was that the vibrational analysis of solid urea and of the gas phase of urea are not comparable, which is mostly due to the different planar or non-planar conformation of urea in different states. The goal of this study is to give an accurate description of intermolecular normal modes and to present different intermolecular interaction effects which could influence the monomer type vibrations, considering the cyclic urea dimer case. Accordingly, several DFT functionals were tested by comparing them with the corresponding MP2 results. The BSSE was corrected using the counterpoise (CP) method [20, 21] as implemented in the Gaussian03 package suite [69]. The uncorrected and BSSE-corrected energies, geometries, harmonic frequencies and their anharmonic corrections [79, 80] were calculated for MP2 and DFT levels of theory using D95V, D95V** and D95V**++ basis sets [81]. The combined local and density fitting approximations for MP2 and Coupled-Cluster methods (DF-LMP2 and DF-LCCSD(T)) were taken into account considering Molpro program package suite [82].

The Geometry Structure In Table 4 we list the interaction energies (given in kcal/mol) and intermolecular HB distances (in Å) obtained for the cyclic urea dimer structure with MP2 method and six different DFT XC functionals (both in uncorrected and CP-corrected cases) using D95V**++ basis set. In the last five rows of Table 4 the same interaction energy and intermolecular HB distance obtained with DF-LMP2 method using D95V**++ and cc-pVQZ basis sets are also shown, as well as the interaction energy obtained using DF-LCCSD(T) method. The cyclic urea dimer is bonded by two equivalent C=O···H-N HBs presented in Figure 2.


171

Table 4. The he ε interaction energies (in kcal/mol) and R intermolecular distances (in Ǻ) in case of cyclic urea dimer structure, optimized at different levels of theory and considering the D95V**++ and cc-pVQZ basis sets Method

Cyclic

MP2

NoCP

a

b

CP BLYP

ε

RO…H

-16.53

1.86

-13.18

1.93

NoCP

-14.15

1.84

CP

-13.52

1.86

NoCP

-15.04

1.84

CP

-14.43

1.85

NoCP

-16.81

1.79

CP

-16.12

1.80

NoCP

-12.33

1.96

CP

-11.76

1.98

NoCP

-20.74

1.75

CP

-20.08

1.75

NoCP

-15.59

1.81

CP

-15.01

1.86

-

-13.26

1.93

-

-2.21

-

DF-LMP2/vqz

-

-14.83

1.86

LMP 2 E disp

/vqz

-

-3.56

-

DF-LCCSD(T)e/vqz

-

-14.29

-

B3LYP PBE HTCH407 KMLYP c

BHLYP

DF-LMP2 LMP 2 E disp d

a

Without counterpoise correction With counterpoise correction c BHandHLYP d cc-pVQZ e The LMP2 optimized geometry was used. b

Figure 2. The urea cyclic dimer.

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Attila Bende

Table 5. The harmonic (ν) and anharmonic (a) frequencies (in cm-1) of intramolecular normal modes (H – N: ν1 – ν4 and O=C: ν5,ν6 bond stretching) of cyclic urea dimer obtained at MP2, B3LYP and BHLYP levels of theory with and without BSSE correction and using the D95V++** basis set No

ν1

Meth

MP2 B3LYP BHLYP

ν2

MP2 B3LYP BHLYP

ν3

MP2 B3LYP BHLYP

ν4

MP2 B3LYP BHLYP

ν5

MP2 B3LYP BHLYP

ν6

MP2 B3LYP BHLYP

dim ν NCP

dim ν CP

3782.5 3782.4 3712.1 3712.1 3848.3 3848.3 3752.3 3751.7 3657.4 3675.7 3813.7 3813.4 3649.7 3649.7 3589.9 3589.9 3723.5 3723.5 3457.8 3419.6 3340.0 3297.2 3510.8 3478.8 1798.3 1777.9 1768.6 1743.7 1855.9 1825.3 1693.2 1684.7 1656.9 1651.0 1722.7 1721.6

3784.1 3784.0 3712.6 3712.6 3848.8 3848.8 3759.4 3758.6 3675.9 3675.3 3814.7 3814.4 3651.1 3651.0 3590.2 3590.1 3723.8 3723.7 3509.0 3480.0 3348.9 3307.3 3518.2 3483.2 1801.2 1781.0 1769.4 1744.3 1856.6 1825.6 1693.1 1687.4 1659.1 1653.5 1724.3 1723.3

ν mon 3770.1 3706.8 3841.3 37701 3706.6 3841.1 3641.5 3578.8 3720.1 3638.9 3582.5 3715.1 1853.2 1797.0 1887.4 1823.2 1629.0 1696.2

dim a NCP

dim aCP

3608.7 3608.7 3546.4 3546.4 3688.1 3688.0 3584.7 3584.3 3512.3 3511.7 3652.1 3652.1 3493.6 3493.7 3441.9 3441.8 3580.2 3580.0 3229.3 3149.0 3144.0 3008.5 3322.0 3305.0 1764.9 1729.1 1722.6 1769.3 1813.0 1781.8 1648.3 1650.1 1619.9 1602.4 1681.6 1682.4

3610.5 3610.4 3546.7 3546.7 3688.6 3688.6 3592.6 3592.2 3514.3 3513.3 3653.8 3653.8 3495.3 3495.2 3442.1 3442.0 3580.6 3580.3 3388.8 3370.4 3148.1 2984.4 3307.5 3325.0 1765.6 1740.9 1723.9 1700.5 1800.9 1782.6 1647.3 1646.6 1622.3 1610.6 1683.7 1685.4

a mon

νexp

3596.1 3544.5

3450a 3348b

3658.4 3596.3 3543.9

3444 3435

3684.8 3484.3 3438.9

3349 3345

3580.3 3482.3 3435.3

3331 3330

3575.3 1782.1 1751.5

1683 1683

1841.4 1728.1 1589.4

1625 1627

1659.0

a

At T = 20 ºC At T = -196 ºC

b

The interaction energy and HB distance values show that a good fitting of DFT values with MP2 results are not obvious for any of the selected XC functionals. In the case of cyclic structure one can observe that the best agreement for the intermolecular interaction energies and intermolecular RO...H distances are given by the BHLYP and B3LYP XC functionals. The


173

interaction energy values are: -15.04 kcal/mol for B3LYP, -15.59 kcal/mol for BHLYP and 16.53 kcal/mol for MP2, while the intermolecular distance for RO...H is: 1.84 Å (B3LYP), 1.85 Å (BHLYP) and 1.86 Å (MP2). At the same time, reasonable values could be also obtained applying the BLYP functional, while in the case of KMLYP, PBE and HTCH407 the energy and geometry parameter results are quite different from the MP2 values. If we compare the same ε and R values at DF-LMP2 level of theory, but obtained with different basis sets (D95V**++ versus cc-pVQZ), one could see that the energy results increase with 1.58 and 2.39 kcal/mol, respectively, while for the R distance we get smaller values with 0.06 and 0.07 Å, respectively. Focusing on the dispersion part of the intermolecular interaction energy, significant dispersion energy growth can be found (1.35 kcal/mol for cyclic dimer and 1.71 kcal/mol for asymmetric case), when the larger cc-pVQZ basis set is used against the D95V**+ one. These represent the major contribution to the total interaction energy increase. In order to consider higher correlation effects, other than those included in the MP2, the interaction energies were computed applying the DF-LCCSD(T) method, using the same cc-pVQZ basis set and taking into account the DF-LMP2 optimized geometry.

Harmonic and Anharmonic Frequencies The cyclic urea dimer has 42 vibrational normal modes from which a number of 36 (18 for each monomer) vibrations are characteristic to monomer-type vibrational motion. The last 6 normal modes with the lowest frequency values have purely intermolecular character. These 18 monomer frequencies of dimer structure are split in a form of frequency pairs (doublets). For the whole theoretical IR spectra, see Figure 3. Compared to the individual monomer lines they present different frequency shifts and their magnitudes depend very much on the dimer molecular symmetry.

Figure 3. The theoretical IR spectra of urea cyclic dimer obtained at MP2-CP, DF-LMP2 and B3LYPCP levels of theory, using D95V**++ basis set.

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In Table 5 were collected the harmonic frequencies (doublets) and their anharmonic corrections for those monomer type normal modes of which vibrational motions are substantially perturbed by the adjoining molecule. They were determined in advance by identifying the frequency value which corresponds to each normal mode vibration and by visualization of their vibrational characters [83]. It was found that, the ν1, ν2 and ν3 intramolecular normal modes are N-H bond stretching vibrations where the H atoms do not take part in intermolecular HB formations. The ν4, ν5 and ν6 are one N-H and two C=O bond stretching vibrations located at the N-H···O=C intermolecular HBs. It can be observed that the doublet frequency splits for ν1, ν2 and ν3 normal modes are almost irrelevant. More precisely, the frequency shifts, which are induced by each monomer on the adjoining molecules, have the same magnitude. Only those normal modes presents different frequency shifts and implicitly larger doublet splits where the perturbation of HB vibration is present (ν4, ν5 and ν6). As it can be seen in Table 5 the dimer frequency shift could be attributed to several effects like: anharmonic corrections, BSSE effects, or intermolecular effects. It should be mentioned that BSSE is not a real physical effect and normally it must be considered together with the intermolecular effects. But, in order to see how important frequency shifts could the BSSE error induce we consider as a separate effects.

Table 6. The harmonic (ν) and anharmonic (a) frequencies (cm-1) of intermolecular normal modes of cyclic urea dimer obtained at MP2, B3LYP and BHLYP levels of theory with and without BSSE correction and using the D95V++** basis set No. νI

νII

νIII

νIV

νV

νVI

Meth. MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP

Cyclic dim ν NCP

dim ν CP

dim a NCP

dim aCP

159.3 157.8 158.4 154.3 152.0 154.7 126.8 134.6 133.4 94.4 93.7 93.7 65.1 67.9 68.6 35.8 45.4 45.8

141.5 150.6 156.0 135.2 137.9 150.7 121.9 127.2 133.4 89.9 81.5 94.1 62.0 50.5 67.6 38.0 46.8 45.0

151.9 155.2 151.5 144.7 147.4 143.9 121.6 135.0 127.0 80.5 93.8 78.7 55.6 66.9 53.8 29.5 44.1 43.9

132.4 150.1 153.1 123.7 137.4 141.3 117.6 132.6 132.6 75.3 80.7 80.7 54.4 53.0 55.9 33.8 45.7 54.4


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In the case of N-H bond stretching vibrations, the magnitudes of anharmonic corrections (frequency red-shift) are quite large (≈ 150 - 170 cm-1 for ν1, ν2 and ν3, and >200 cm-1 for ν4, respectively) while in the case of a1, a2 and a3 anharmonic corrections, for both of doublet frequency values, the magnitude of frequency shifts are the same. For a4 correction the size of doublet frequency shifts is different (ex. at MP2 level ν′4-a'4=3457.8 - 3229.3 cm-1 = 228.5 cm-1 and ν"4 - a"4 = 3419.6 - 3149.0 cm-1 = 270.6 cm -1). Regarding to ν5 and ν6 C=O stretching modes one can see that the anharmonic corrections are much smaller than in the HN case, their shifts are about 30-40 cm-1 and the behavior of their frequency split is similar to ν4 mode. The correction scheme for ν 4 and ν 5 is as follows:

ν 4' : ν mon = 3638 .9 cm −1 ⎯dimer ⎯⎯→ν dim ⎯anh. ⎯→ ⎯ ν anh ⎯BSSE ⎯⎯→ν BSSE = 3388 .8 cm −1 -

ν : ν

mon

ν : ν

mon

ν : ν

mon

" 4

' 5

" 5

= 3638 .9 cm

−1

= 1853 .2 cm

−1

= 1853 .2 cm

−1

-

⎯ ⎯⎯→ν

dim

⎯ ⎯⎯→ν

dim

⎯ ⎯⎯→ν

dim

dimer

-

dimer

-

⎯ ⎯→ ⎯ ν

⎯ ⎯⎯→ν BSSE = 3370 .4 cm −1

⎯ ⎯→ ⎯ ν

anh

⎯ ⎯⎯→ν BSSE = 1765 .6 cm −1

⎯ ⎯→ ⎯ ν

anh

⎯ ⎯⎯→ν BSSE = 1740 .9 cm −1

-

-

dimer

+1

anh

anh.

anh.

+2

-

anh.

-

BSSE

BSSE

+0.

BSSE

+1

Considering the scheme of frequency corrections by dimer, anharmonic and BSSE effects presented above, one could be observe that they bring different contributions in the case of the ν 4 monomer frequency value (3638.9 cm-1), where finally we got a double-split frequency pair (ν 4' and ν 4" ) of 3370.4 cm-1 and 3388.8 cm-1 values. Similar situation can be found for the ν 5 monomer frequency value where we have two frequency values with 24.7 cm-1 distance between them. Beside of the monomer-type frequencies, the molecular association induces a group of another six normal mode vibrations which can be called intermolecular normal modes. They can be found in the very-far region (10–250 cm-1) of the molecular IR spectra and show the relative vibrations of two “rigid” urea monomers according to the six degree of freedom which derive from the intermolecular coordinates. The frequency values of these intermolecular normal mode vibrations are shown in Table 6 obtained both at MP2 and DFT (B3LYP and BHLYP) levels of theory. They have strictly intermolecular character and show only anharmonic and BSSE corrections. Scrutinizing the results of normal mode vibrations for the cyclic dimer one can observe that there are three frequencies of which motion take place in the supermolecular plane (νI – νIII) – let calls them “in-plane” vibrations, while the another three intermolecular normal modes (νIV – νVI) show “out-of-plane” vibrations. If we consider together the amount of anharmonic and BSSE corrections one can see that the “inplane” vibrations are more affected by these errors than “out-of-plane” normal modes. Yet analyzing separately for only one mode, it can be seen that they have almost similar magnitude, becoming equally relevant corrections for the intermolecular normal mode vibrations. At the same time, if we consider the off-diagonal vibration couplings of these normal modes, one can find that frequencies which belong to the group of “in-plane” or “outof-plane” their vibrations are strongly coupled inside the group, but much less coupled between the groups. In the case of asymmetric urea dimer the separation of the above mentioned “in-plane” and “out-of-plane” group of vibrations is not so obvious, but similarly

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to the cyclic dimer the BSSE and the anharmonic effects are equally relevant. In both cases of dimer structures the collective effect of BSSE and anharmonic corrections presents about 10 – 15% from the frequency values. The intra – intermolecular normal mode couplings could give us more detailed information about how strongly the intra- and intermolecular normal modes are coupled between them. Since the intra- and intermolecular normal modes have very different vibrational frequency values one should obtain a strong coupling only in some special cases. Analyzing the anharmonic coupling matrix we found that in case of cyclic dimer the νI – νIII intermolecular normal modes are coupled only with ν 4' and ν 4" (ex. x(ν 4" -ν I ) = +6.73 cm-1) intramolecular vibrations, while couplings between ν 5' and ν 5" intra-normal modes and νI – νIII inter-normal modes are almost missing. This could be explained with the fact that molecular vibrations are performed in the same molecular plane and along the same vibrational direction. In the case of asymmetric urea dimer neither for ν 4' and ν 4" , nor for ν 5' and ν 5" so significant anharmonic coupling with the intermolecular normal modes could be found than for the cyclic structure. In this case, the only relevant coupling is x(ν 5" -ν II ) = -3.86

cm-1. If one compares the selected theoretical normal mode frequencies with the experimental data (Table 5), the more appropriate values are obtained for the CP-corrected anharmonic frequencies, both for cyclic and asymmetric structures. At the same time, it can be seen that the dimer approximation is not satisfactory in order to reproduce these experimental data with the desired high accuracy. Thus, other extended theoretical calculation where also solvent effects and larger cluster sizes are taken into account would be absolutely necessary. In addition, other spectroscopic data, like vibrational absorption intensities calculated via atomic polar tensor [84] and comparing them with the measured vibrational intensities would be also useful in order to explain the unusual behavior of urea in gas or solid phase.

3.3. Guanine-Cytosine DNA Base Pair The guanine-cytosine (GC) base pair's intermolecular interaction can be considered as a contribution of two very important HBs: a number of two C=O···H-N and one PN···H-N bonds. Due to the biological manifoldness of DNA base pair conformations (dry-DNA, wetDNA, double helix, super-double helix, etc.) their theoretical characterization is much diversified. A given theoretical model should take into account the backbone and environment effects as well as the influence of the DNA-protein interactions. In virtue of this fact, choosing the correct theoretical method is essential in the investigation of DNA base pairs. In the last five years the substantial computer advance gave for molecular modeling scientists the opportunity to include accurate electron correlation effects in their calculations. Šponer et al. [85] and Podolyan et al. [86] used the well-known MP2 method with different basis sets (mostly including the polarization effects) and they give a very detailed description of the interaction energies and geometry structures of the adenine-thymine and guaninecytosine base pairs. At the same time the normal mode analysis of molecular vibration could give us supplementary information about the efficiency of different methods. In connection with this, several vibrational frequency calculations and experimental measurements were


177

performed [9, 86, 87, 88, 89] in order to study different interaction effects in normal mode vibrations of base pairs (mostly for guanine-cytosine dimer). For instance, the intermolecular interactions could significantly influence the intramolecular normal mode vibrations such as red-shift or improper blue-shifting [90, 91] of the vibrational frequencies. Furthermore, the intermolecular normal mode vibrations depend very much on the applied method or basis sets, and last but not least on the BSSE effects [3, 74, 92, 93]. The different Watson-Crick and Hoogsteen G-C geometries, harmonic frequencies and their anharmonic corrections [79, 80] were calculated at DFT level of theory considering the PBE exchange-correlation functional implemented in Gaussian03 [69] program package and using the 6-31G basis set. The harmonic frequency values were scaled using the 0.986 factor corresponding to PBE/6-31G type calculations.

The Geometry Structure In Figures 4 and 5 two different conformation (4: Watson-Crick [94] and 5: Hoogsteen [95]) of G-C base pair are presented. Beside of the well-known Watson-Crick base pair configuration the Hoogsteen base pairs have been known for more than 40 years. In Hoogsteen base pairs the N face of guanine is hydrogen bonded to cytosine N side by an H+ proton. Such interactions were postulated in U(A·U) triple helices [96] and it was also found in chemically modified nucleic acids [97, 98]. Isolated Hoogsteen base pairs have been reported in some protein/DNA complexes [99] and occasionally in RNA [100]. Its crystal structure is presented in [101].

Figure 4. The Watson-Crick conformation of Guanine-Cytosine base pair.

Figure 5. The Hoogsteen conformation of Guanine-Cytosine base pair.

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The geometry of the DNA base pair was the subject of many scientific works, among which the theoretical investigation performed by Guerra at al. [102] and van der Wijst [103] are the most detailed ones. They present a systematic comparison of different DFT exchangecorrelation functionals and basis sets as well as they also compare the intermolecular interaction energy values and H-bond distances with the experimental results. The best agreement was obtained for the BP86 and PW91 exchange-correlation functionals, while the widely used B3LYP functional consistently underestimates hydrogen-bond strengths and overestimates hydrogen-bond distances. Nearly same good results can be obtained also with the PBE functional as it found in case of BP86 and PW91 ones. Considering the above mentioned method and basis set, for the intermolecular distances were obtained the following values: a) Watson-Crick: d1(O···H-N) = 2.80 Å, d2(N···H-N) = 2.93 Å and d3(O···H-N) = 2.92 Å, b) Hoogsteen: d1(N···H+-N) = 2.66 Å, d2(O···H-N) = 3.08 Å.

Harmonic and Anharmonic Frequencies The guanine-cytosine binary system in its Watson-Crick configuration has a number of 29 atoms and presents a number of 81 normal mode vibrations from which 6 have intermolecular character. In Figure 6 the IR absorption spectra for two characteristic spectral regions of 500 – 2000 cm-1 and 2750 – 3750 cm-1, respectively, are presented.

Figure 6. The IR absorption spectra for Watson-Crick configuration of guanine-cytosine DNA base pair at harmonic and anharmonic approximation.

The black vertical line shows the normal mode vibrational frequencies in harmonic approximation, while the red lines are frequencies where the anharmonic approximation was also taken into account. The values show a considerable frequency shift due to the anharmonic approximation. These shifts are very pronounced in case of 2750 – 3750 cm-1 spectral region, where the C-H and N-H covalent stretching vibration are located.


179

Figure 7. The IR absorption spectra for Hoogsteen configuration of guanine-cytosine DNA base pair at harmonic and anharmonic approximation.

Figure 8. The IR absorption spectra for Watson-Crick and Hoogsteen configuration of guanine-cytosine DNA base pair at harmonic approximation.

The guanine-cytosine binary system in its Hoogsteen configuration has a number of 30 atoms and presents a number of 84 normal mode vibrations from which 6 have intermolecular character. In Figure 5 can be observed that the N···H+-N intermolecular bond is stabilized with the help of the H+ proton. The H+-N is not a real covalent bond and therefore its stretching vibration is also different from the usual H-N stretching vibrations. In case of Watson-Crick configuration there are three characteristic stretching vibrations: the first one is a pure H4-N4 vibration (ν5 = 3292.9 cm-1) of O6···H4-N4 intermolecular donor-acceptor complex, while the second and third ones are a combined N1-H1 and N2-H2 symmetric (ν9 = 3085.8 cm-1) and asymmetric (ν10 = 3027.8 cm-1) stretching vibrations of N1-H1···N3 and N2H2···O2 intermolecular donor-acceptor complexes. In case of Hoogsteen configuration there are only two characteristic vibrations which belong to the intermolecular interaction region:

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the first is an N-H stretching vibration (ν10 = 3014.4 cm-1) of N-H···O donor-acceptor complex, while the second one is also an N-H stretching vibration (ν11 = 2053.2 cm-1) but as a component of the N···H+-N unusual intermolecular donor-acceptor complex. If one considers the anharmonic corrections, the similar behavior can be observed as it was found in case of formamide and urea systems. There are large anharmonic shifts for all three ν5, ν9 and ν10 stretching vibrations of Watson-Crick configuration: a5 = 263.1 cm-1, a9 = 294.8 cm-1 and a10 = 362.4 cm-1. In case of Hoogsteen conformation the anharmonic shifts are: a10 = 187.6 cm-1 and a11 = 958.0 cm-1. The a11 anharmonic shift is very large compared with the harmonic frequency values. This can be explained with the unusual nature of N···H+-N intermolecular donor-acceptor complex. Two different type intermolecular normal mode vibrations can be found both in case of Watson-Crick and Hoogsteen configurations. In one of them the vibration motion occurs in the plane defined by the monomer molecules (here we have a number of three normal modes), while in the second case the intermolecular vibrational motions are out-of-plane vibrations. Since the characterization of the out-of-plane vibrations is quite difficult, this will not constitute the subject of our further investigation. Accordingly, the three intermolecular normal mode (in-plane) frequencies are: νI = 101.2 cm-1, νII = 125.5 cm-1 and νIII = 131.0 cm-1 for Watson-Crick conformation and νI = 88.9 cm-1, νII = 109.4 cm-1 and νIII = 170.6 cm-1, for Hoogsteen conformation, respectively. Similar to the previous urea case presented in section 3.2, there are also significant interand intramolecular normal mode couplings. In case of Watson-Crick system the intramolecular ν9 and ν10 stretching vibrations are relatively strong coupled (about 5-6 cm-1) with all three νI, νII and νIII intermolecular normal modes. For Hoogsteen configuration these couplings are stronger: 17.0-21.0 cm-1.

3.4. Adenine-Thymine DNA Base Pair The adenine-thymine binary system (Watson-Crick configuration) has a number of 30 atoms and presents a number of 84 normal mode vibrations from which 6 have intermolecular character. For its geometry structure see Figure 9. Since we are interested in the stretching vibrations (ν4 and ν11) of those intramolecular covalent bonds where H atoms are involved, as well as in those intermolecular vibrations of which motion is situated mostly in the molecular plane (ν78, ν80 and ν82), we present only these specific normal mode vibrations. At the same time, one can identify another group of seven normal modes (ν41, ν42, ν54, ν60, ν61, ν64, and ν69) specific for purine and pyrimidine ring vibrational deformation which can significantly disturb the H-bond vibrations. All these normal mode frequency values in harmonic, BSSE corrected harmonic, and anharmonic approximations are presented in Table 7. Considering the BSSE correction, the most affected normal modes are the ν4 and ν11 N-H stretching vibrations. In these two cases we found 76.6 cm-1 and 429.5 cm-1 frequency increasing, respectively. Similar findings can be obtained in case of νI and νII intermolecular normal modes, where in spite of the fact that BSSE effects not show large frequency shifts, they could represent significant corrections compared with the uncorrected frequency values. Regard to anharmonic corrections, one can observe that large frequency shifts are obtained in the case of N-H covalent-bond stretching vibrations (281.1 cm-1 for ν4 and -160.4 cm-1 for ν11). Considering the ring deformation normal mode


181

vibrations (ν41, ν42, ν54, ν60, ν61, ν64, and ν69), the anharmonic correction is more important than BSSE effects, but even so, their frequency shifts are less than 20 cm-1. As it was concluded in the previous case of urea dimers, the anharmonic and BSSE corrections could be considered in a very good approximation as additive effects. According to this fact, we present in the fifth column of Table 7 the integral correction (νint) of the anharmonic and BSSE effects. The results show that in some frequency cases we have an opposite contribution of frequency shifts, while in some other cases the anharmonic and BSSE collective corrections increase the magnitude of the frequency shift. Analyzing the anharmonic coupling matrix one observes strong vibrational coupling between ν11 intra- and ν78 intermolecular normal modes (x11,I = 8.2 cm-1) as well as between ν11 intra- and νII intermolecular normal modes (x11,II = 10.4 cm-1). The case of this strong coupling could be explained by the same fact that in the case of cyclic urea dimer namely, by the presence of the same molecular plane for both normal modes and the same vibrational direction for ν11 and νI modes. In other cases there is no significant coupling between the intra- and intermolecular normal modes.

Table 7. The harmonic (ν) and anharmonic (a) frequencies (in cm-1) of some selected intramolecular and intermolecular normal modes in adenine-thymine DNA base pair, obtained at B3LYP level of theory and using D95V basis set (The ν int is the predicted frequency value, considering, together, the BSSE and anharmonic corrections) Nr.

dim ν NCP

dim ν CP

dim a NCP

νint

ν4

3329.3

3405.9

3048.2

3124.8

ν11

2546.7

2976.2

2386.3

2815.8

ν41

1035.2

1035.7

1018.5

1019.0

ν42

1031.4

1024.2

1014.2

1007.0

ν54

742.4

745.9

726.9

730.4

ν60

637.7

630.3

632.8

625.4

ν61

607.4

611.1

601.2

604.9

ν64

550.2

550.4

542.0

542.2

ν69

397.2

404.1

388.8

395.7

νI

119.9

108.2

115.4

103.7

νII

114.6

101.6

110.3

97.3

νIII

66.8

60.3

58.3

51.8

Krishnan et al [104] show that to compare the experimentally observed IR spectra of adenine-thymine base pair with the calculated frequencies is not a simple task. First of all because the Watson-Crick configuration of A-T base pair is not the most stable isomer conformation [105]. Making a detailed theoretical anharmonic frequency analysis they were able to assign the IR-UV double resonance spectra [106] also to a particular isomer which is not the Watson-Crick structure. A direct experimental assignment of N-H stretching vibrations in A-T oligomers in condense phase is very difficult because the N-H stretching

182

Attila Bende

vibration spectral region overlap with the water’s O-H stretching spectral region. Reducing the water content of the A-T oligomers does not solve the problem because they do not adopt a well define structure at extremely low water concentration. In order to identify and to characterize the N-H stretching vibrations of A-T base pair oligomers in DNA Heyne et al [107] use the two-color IR pump-probe technique to overcome the above mentioned experimental problem. They found that the adenine ν(NH2) absorbs at 3215 cm-1 and has pronounced anharmonic couplings to the ν(C=O) mode of the thymine and δ(NH2) mode of the adenine.

Figure 9. The adenine-thymine DNA base pair.

4. Conclusion In this Chapter a detailed investigation, including geometry structure, harmonic and anharmonic frequency calculations, on cyclic conformation of formamide and urea dimers, as well as on the guanine-cytosine and adenine-thymine DNA base pairs were presented. Considering the results some general conclusions can be drawn. First of all, in order to describe accurately the geometry structures of the molecules one needs to consider proper ab initio methods, which are able to describe correctly the different intermolecular interaction effects (electrostatic, polarization, induction, dispersion and three- or four-body terms). Including all this effects in our calculation, high level electron correlation methods like perturbation or coupled-cluster methods as well as basis sets which contain successively larger shells of polarization (correlating) functions (d, f, g, etc.) are required. For these calculations one need a huge amount of computer capacity and therefore we can only limit to a small size molecular systems (up to 15-20 atoms). Recently introduced local and densityfitting approximations are very promising tools, they can reduce very much the calculation efforts and in this way the molecular size limit could be extended. These methods are also free from the mathematical artifact called basis set superposition error or BSSE. In order to obtain accurate description for the infrared absorption spectra, the harmonic approximation is not good enough and therefore the anharmonic approximation is also should be included. Including these effects, some normal mode vibrations, like X-H (X=C or N), C=O and N-H+ stretching vibrations, need more detailed investigation. For the N-H and C=O covalent-bond stretching vibrations the anharmonic frequency correction is significant. In case of Hoogsteen conformation of guanine-cytosine base pair the anharmonic approximation is comparable with the harmonic frequency value. For the same N-H and C=O covalent-bond


183

stretching vibrations the BSSE corrections are also significant, but only in the case when the normal mode vibrations are in the intermolecular region. Normal mode vibrations which are located in the intermolecular region, the influence of the adjoining molecule (dimer effect) on the vibrational frequency is comparable with the magnitude of the BSSE and anharmonic corrections. Analyzing the magnitude of the anharmonic and BSSE corrections, it was found that their contributions in the harmonic frequency shift could be considered in a very good approximation as an additive effect. The anharmonic coupling between intra- and intermolecular normal modes is significant only when the motion of normal mode vibrations occur in the same molecular plane (the plane defined by those atoms which move during the vibration) and along of the same vibrational direction. Using the classical DFT functionals and applying double-zeta quality basis sets, like 6-31G or D95V, one could obtain a good qualitative description of the anharmonic corrections, but further investigations considering the recently developed DFT or perturbation methods and triple-zeta quality basis sets are needed in order to obtain also a good quantitative description for them. The strong coupling between different intra- and intermolecular normal modes can be considered as an important and biologically relevant effect. In this way the vibrational relaxation processes can take place very easily and some strongly excited localized vibrations can wither away as thermal vibrations.

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Chapter 9

E MERGENT P ROPERTIES IN B OHMIAN C HEMISTRY Jan C.A. Boeyens Unit for Advanced Study, University of Pretoria, South Africa

Abstract Bohmian mechanics developed from the hydrodynamic interpretation of quantum events. By this interpretation all dynamic variables retain their classical meaning in quantum systems. It is of special significance in chemistry as a discipline which is traditionally based on the point electrons of quantum field theory. It could be more informative to assume a non-dispersive electronic spinor, or wave packet, with divergent and convergent spherical wave components, and with many properties resembling those of a point particle. Complex chemical matter is endowed with three attributes: cohesion, conformation and affinity, which can be reduced to the three fundamental electronic properties of charge, angular momentum and quantum potential, known from the wave structure. The chemical effects of these respective scalar, vector and temporal principles, all manifest as extremum phenomena. The optimal distribution of electronic charge in space appears as Pauli’s exclusion principle. Minimization of orbital angular momentum becomes the generator of molecular conformation. Equilization of electronegativity, the quantum potential of the valence state, dictates chemical affinity. The chemical environment is said to generate three emergent properties: the exclusion principle, molecular structure and the second law of thermodynamics. These concepts cannot be predicted from first fundamental principles. Only by recognition of the emergent properties of chemistry is it possible to simulate chemical behaviour. The exclusion principle controls all forms of chemical cohesion, atomic structure and periodicity; molecular structure underpins vector properties such as conformational rigidity, optical activity, photochemistry and other stereochemical phenomena; while transport properties and chemical reactivity depend on the second law. Simulation of these chemical concepts by constructionist procedures, starting from basic physics, is impossible. The ultimate reason is that complex chemical properties are not represented by quantum-mechanical operators in the same sense as energy and momentum. The Bohmian interpretation, which enables the introduction of simplifying emergent parameters, in analogy with classical procedures, allows the calculation of molecular properties by generalized Heitler-London methods, point-charge simulation and molecular mechanics.

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Introduction

The editorial challenge to address the quantum frontiers of atoms and molecules in chemistry cuts far deeper than a survey of current activity in quantum chemistry, which for all practical purposes means ab-initio computational chemistry. There is no quantum theory of chemistry: Quantum mechanics originated as a theory to understand radiation and its interaction with sub-atomic matter. It gave birth to the modern science of spectroscopy, in which form it stimulated the development of sophisticated observational techniques that revolutionized physics, chemistry and biology. However, the early promise of a quantum theory of matter in general has not come to fruition. A theory that fails to elucidate the nature of electrons, atoms and molecules can never lead to an understanding of chemistry. It may enable computations of unrivalled complexity, but without a conceptual framework the results have no meaning. As a theory of spectroscopy, quantum mechanics serves to relate measured quantities, such as frequency and wavelength to the dynamic concepts of energy and angular momentum, by means of differential operators e.g.: −i~∂/∂t → E,

i~∂/∂ϕ → L.

Although chemically more useful information on molecular structure and electronegativity is also embedded in the state functions, there are no known operators whereby to extract this information directly. The current computational alternative fails for the same reason. Minimization of the scalar quantity, energy, can never generate three-dimensional structure, which is a vector concept. The failure to rationalize chemical behaviour is not a failure of quantum theory, but rather, a failure of the traditional interpretation of the theory. The ruling interpretation of quantum theory, known as the Copenhagen interpretation, incorporates a number of features totally at variance with chemistry. It defines quantum objects, including photons, electrons, atoms and molecules, as structureless point particles without extension. It reduces a continuously varying density, such as the electronic charge distribution in atoms, molecules and crystals to a probabilistic function. It offers no rationale for the occurrence of stationary states, apart from a postulate. Non-local effects are forbidden and the doctrine teaches more about measurement problems and quantum uncertainty than about chemical interaction. The unfortunate reality is that Schrödinger’s description of elementary matter as wave structures, which could not be reconciled with the Copenhagen orthodoxy, was ruled unacceptable and re-interpreted in terms of the quantum jumps and probability density of the particle model. This compromise resulted in the awkward concept of wave-particle duality, familiar to all modern chemists, but understood by none. The challenge that we are facing here is to retrace our steps to the point where the time-honoured concepts of classical chemistry merge in a natural way with the ideas of wave mechanics and start rebuilding a theory that ′′ stimulate(s) the mutual understanding of the various branches of chemistry and its neighbouring sciences′′ , realizing that ′′ the main stumbling block for the development of a theory of large and complex molecular systems is not computational but conceptual′′ [1].

Emergent Properties in Bohmian Chemistry

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I propose to start with an outline of the essential concepts of cohesion, conformation and affinity; show how these relate to the more fundamental concepts of space, matter and number; and the derived concepts of interaction, waves and periodicity. Next we examine the Schrödinger formalism, the Bohmian interpretation and the application of wave mechanics, supported by higher-level emergent properties. The photoelectric effect, probably the most effective argument to have established the particle nature of photons is shown to be explained more convincingly by the transactional-wave model of electromagnetic interaction.

2.

The Fundamental Concepts

Any object or event observed in Nature can always be considered as the product of more primitive events or the interaction between more primitive entities. It is quite natural, in this reductionist spirit, to ascribe the actions and features of a living organism to some lower-level activity of biological cells, which in turn is driven by intracellular chemical interactions. The molecular building blocks of biological cells are assumed to consist of atoms, held together by electromagnetic forces, while the sub-atomic nucleons interact with strong interaction, and so on, ad infinitum. Not really. The reduction has to stop somewhere. As progressively smaller entities are implicated at the more primitive levels, it is reasonable to conclude that the cascade ends in a void, traditionally assumed to be the same as space, or the vacuum. At this point it is easy to get distracted by the interminable philosophical dispute about the possible existence or non-existence of a void. As a practical alternative I prefer to define space in geometrical terms and the vacuum as a physical entity.

2.1.

Space

The simplest way of looking at space is as a coordinate system that serves to describe the relative positions of observable objects. The intuitively most obvious is a cartesian system on three orthogonal axes. However, this may not necessarily be the most convenient coordinate system. Although it works well in a laboratory environment, it is well known to be inappropriate in geographical context, which is simplified by using spherical trigonometry. By the same argument cartesian coordinates may not be the most convenient to map the relative positions of astronomical objects. To first approximation the planets and most of their moons have been assumed to move in a single plane, called the ecliptic, around the sun. Individual elliptic orbits of planets and moons have a simple description in terms of Kepler’s laws in a plane, but most of these planes are known to make non-zero angles with the ecliptic. Moving into interstellar, or even intergalactic, space the situation becomes more complicated when dealing with cosmological distances, times and velocities, which demands relativistic rather than Galilean kinematics. Special relativity is conveniently formulated in four-dimensional Minkowski space and general relativity requires non-Euclidean geometry, i.e. curved space. In general relativity the geometry of space is described by a curvature tensor, which is linearly related to a stress tensor that describes the distribution of matter in the cosmos.

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This reciprocal relationship shows that an empty universe has zero curvature and that curved space generates matter. The mechanism whereby curvature generates matter is visualized in the process of covering a curved surface with an inflexible sheet. The higher the curvature, the poorer the fit and the more the wrinkles in the cover that cannot be smoothed away. Such wrinkles in space are interpreted as matter and energy – the content of the stress tensor. 2.1.1.

Number

Figure 1. Spiral structure of a fossilized nautilus shell. Not only the topology of space-time, but also the physical content of the universe, resembles the natural number system in remarkable detail [2]. This explains the unreasonable effectiveness of mathematics as a scientific tool and the success of number theory to predict natural phenomena as a manifestation of cosmic symmetry [2, 3]. The physical world, as an image of the natural numbers, can never be known in more detail than the number system. Concepts such as infinity and singularity, poorly understood mathematically, therefore make no physical sense. On the other hand, the concept of high-dimensional space, readily manipulated mathematically, and, although physically difficult to visualize, has a legitimate place in scientific discourse. To deal with the ubiquitous, but bothersome, infinities of physics, the infinity concept of projective geometry can be used to define both the number system and the physical cosmos as closed. Realizing that any closed system is periodic, by definition, a wave structure of the vacuum and periodicity of matter are inferred. The implied cosmic symmetry is referred to as self-similarity. The chambered structure of a nautilus shell, shown in Figure 1, is one of the best-known examples of this symme-


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try type. All the chambers have the same shape and only differ in size, which increases regularly along a golden logarithmic spiral. The same pattern occurs in the arrangement of growing seed buds in a sunflower head and in the image of a spiral galaxy. Recognition of the same growth features [2, 4] in atomic nuclei, atoms, covalent molecules and the solar system, reveals self-similarity on a much wider scale. The number theory of self-similarity shows that all of these structures are based on the Fibonacci number sequence, which converges to the golden ratio.

2.2.

Vacuum

The fabric of space is a matter of conjecture. However, if tangible matter occurs on curving flat space, flat space is not void. A useful analogy is to picture the vacuum as a regular undulating expanse filled by waves of constant wavelength. When curved, interference of the primary waves produces persistent wave packets, earlier identified as wrinkles. In local space such a wave packet is conveniently described as the superposition of threedimensional spherical waves, converging to and diverging from a centre of mass. These waves are the retarded and advanced solutions of the general wave equation, which implies motion, either forward or backward in time. 2.2.1.

Wave Packets

A typical wave packet generated by such a superposition of waves is shown in Figure 2. The tangent curve follows the amplitude of the 1/r Coulomb potential, which reflects the actual charge density, except when r → 0. The secondary waves propagate with the group velocity vg of the system and the primary waves have phase velocity vφ , such that √ vg vφ = c2 , where c = 1/ ǫ0 µ0 , is the velocity of light in the vacuum. Such a wave packet [4]: sin kr iωt cos kr (1) or Φ = Ae kr kr has been shown [5] to describe elementary waves, equivalent to the postulated elementary distortions of space, i.e. the elementary particles of atomic physics. Interpreted as an electron, the distance between nodal points represents λdB = h/me vg , the de Broglie wavelength of a free electron and λC = 2π/k = h/me c, the Compton wavelength. The amplitude of the standing wave is proportional to the electronic charge. Φ0 = A, in eqn.(1), represents a wave packet with charge proportional to 0 or ±A. Electrons and protons, despite their difference in mass have charges of ±e. The neutron is neutral. The field intensity ΦΦ∗ = A2 (sin kr/kr)2 = C/r2 defines the force between charges, in line with Coulomb’s law, except when r → 0. The breakdown of Coulomb’s law, which occurs naturally for charged wave structures is equivalent to the special renormalization postulate in quantum field theory. 2.2.2.

Electron and Atom

A particle image, as shown in Figure 3, is obtained by rotating the diagram of Figure 2 about two axes perpendicular to x. The charge density can either contract or expand,

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Figure 2. One-dimensional section through a spherical wave packet with components converging on and diverging from x0 . depending on environmental pressure. The minimum radius that can be reached on compression depends on the rest mass of the object. For an electron r0 = e2 /m0 c2 .

Figure 3. Wave structure of a free electron with de Broglie wavelength λdB = h/me vg . Given the inferred flexible structure of an electron as a continuous indivisible charge, the self-similarity of atoms and the solar system is not obvious. The pioneering work on the planetary model of atomic structure was firmly based on Kepler’s model of elliptic orbits. Two crucial parameters that define a Kepler ellipse are the semimajor axis and the eccentricity, which characterize the size and shape of an orbit. By Newton’s laws these respective parameters are related to the energy and angular momentum of the orbital motion. In retrospect Kepler’s laws are seen to embody the general conservation principles for energy and angular momentum in celestial mechanics. The efforts of Bohr and Sommerfeld to explain electronic motion in atoms by the same model were spectacularly successful, despite a few subtle, but fatal, defects. Whereas Kepler’s model is valid in a gravitational field, it needs modification in an electromagnetic field as an accelerated charge radiates energy and an accelerated point charge therefore cannot maintain a stable orbit. Conservation


197

of angular momentum in a central electrostatic field should rather be interpreted as conservation of the spherical shape of a continuous charge. Polar deformation under an external influence is described by the three-dimensional surface harmonics, or eigenfunctions, of the circulation Laplacian, with discrete eigenvalues: L2 Ylml = l(l + 1)k 2 Ylml ,

Lz Ylml = ml kYlml .

This classical result acquires quantum-mechanical meaning by equating the arbitrary constant k with ~, the elementary unit of angular momentum [6].

Figure 4. Phase-locked cavity with perfectly reflecting walls, filled with radiation in the form of standing waves. The difference between atomic and planetary systems goes a long way towards understanding of cosmic self-similarity. Although electrons in an atom are spread in three dimensions and planets orbit the sun in an approximately two-dimensional plane, both arrangements depend parametrically on the golden ratio. In the same way sunflower seeds of varying size are closely packed in a plane, compared to the three-dimensional stacking of nucleons; both styles conditioned by the golden ratio. The only common factor in all cases is the general curvature of space. Evidently, the curvature of cosmic space must be a function of the golden ratio, from the sub-atomic to supergalactic scales. 2.2.3.

Mass

Not only the charge, but also the characteristic mass and spin of sub-atomic species are accounted for by their wave structure. Jennison and Drinkwater [7] demonstrated that microwave radiation trapped in a phase-locked cavity, as in Figure 4, generates an interaction pattern which is mathematically equivalent to a system with inertial mass. Disturbing the equilibrium by a pulse that moves a cavity wall at velocity δv for a period δt, which is matched to the wave propagation across the cavity, modifies the internal pressure by Doppler shifting of the waves and sets the entire system into motion with velocity 2δv. The radiation pressure on the walls is balanced by an electromagnetic field, which keeps the system in static equilibrium. In the real vacuum the analogue of the phase-locked cavity is a standing wave, filled with radiation of Compton wavelength, internal energy E, in equilibrium with the external radiation (wave) field. Simple calculation [7] shows that

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the inertial mass of the wave packet obeys Newton’s law, F = ma, on identifying E/c2 with the rest mass.

2.2.4.

Spin

The standing-wave description of an electron defines it as an integral part of the vacuum, not obviously free to move without impediment. Linear motion of an electron must then clearly lead to continuous drag and deformation of both electron and its immediate environment, culminating in rupture of the vacuum and creation of a turbulent state. An electron that rotates in the vacuum, although more symmetrical, winds up the connecting medium until it shears and develops a discontinuity along a cylindrical surface. The only motion that occurs without distortion of the spherical wave packet or mechanical entanglement of the environment is rotation around a point. Unlike axial rotation this mode is more like a continuous wobble that returns to the original situation after two complete revolutions. The three dimensions of space participate equally in the motion without the transfer of rotational energy from the spinning object to the connecting medium. The strain that builds up during the first part of the rotational cycle relaxes during the second part. Apart from half-frequency cyclic disturbance in the connecting medium, the electron is free to move through the vacuum without permanent entanglement. An object, which performs this type of spherical rotation, is described mathematically by a spinor, or a quantity that reverses sign on rotation through an odd multiple of 2π radians.

3.

Quantum Theory

Quantum theory started with the discovery of line spectra and Balmer’s observation that the spectral lines of atomic hydrogen obey a digital formula, later generalized to: ν = Rc

1 1 − 2 2 n1 n2

,

n2 > n1 = 1, 2, 3 . . .

(2)

The first sensible explanation of the formula was proposed in 1904 by Nagaoka who used the planet Saturn with its system of rings as the basis of an atomic model, with electrons at energy levels (rings) in simple numerical order, orbiting a heavy positively charged nucleus. Experimental confirmation of such an arrangement was found by Rutherford in 1910 and a dynamic model, based on Planck’s quantum condition, E = hν, was proposed in 1914 by Bohr. Where Nagaoka argued that electrodynamically stable orbits required a standing electron wave of length λ = 2πr/n at an average distance r from the nucleus, Bohr postulated quantum stability for an orbiting electron with angular momentum p = nh/2π ≡ n~. With electrostatic and mechanical forces in balance, (using electrostatic units, 4πǫ0 = 1): p2 e2 = , r2 mr

E =T +V =

e2 e2 e2 − =− , 2r r 2r


199

the postulate leads to the Balmer formula 2π 2 me4 hν = ∆E = h2 En =

2π 2 me4 , n2 h2

1 1 − , n21 n22 n2 h2 rn = 2 2 . 4π me

(3) (4)

The rest is history. When de Broglie rediscovered the Nagaoka condition in 1924 by postulating that all matter has an associated wavelength of λ = h/mv, only the mathematical framework for defining a general wave formalism of electronic behaviour, was lacking.

3.1.

Wave Mechanics

The mechanical behaviour of a Newtonian particle is described correctly by three quantities – energy, momentum and angular momentum, which describe the motion as a function of either time, displacement or rotation. In wave formalism each of these parameters is specified as a periodic function [8] with respect to time (τ ), translation (λ) or rotation (ϕ): ω = 2π/τ, E = ~ω,

k = 2π/λ, p = ~k,

ml = 2π/ϕ, Lz = ~ml .

The carrier of the electromagnetic field is described by the differential wave equation: ∇2 Ψ = µǫ

1 ∂2Ψ ∂2Ψ = 2 2. 2 ∂t c ∂t

(5)

To remain consistent with the previous relationships the dynamic variables need to be specified as differential operators: E → −~i∂/∂t,

p → −~i∇,

Lz → ~i∂/∂ϕ,

which can be checked by direct substitution. To allow for the first-order temporal dependence of the energy, the equation for matter waves is restricted to processes which only depend on time through a factor exp(2πiνt), leading to the final form: 2 ~ ∂Ψ 2 ∇ + V Ψ = ±~i (6) 2m ∂t which formally resembles the classical Hamiltonian definition of total energy, as H =T +V =

p2 + V = E. 2m

(7)

By defining a density function ρ = ΨΨ∗ and a current density j=

~ (Ψ∗ ∇Ψ − ψ∇Ψ∗ ) 2mi

(8)

there follows a continuity equation as in classical hydrodynamics ∂ρ + divj = 0. ∂t

(9)

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A general expression for a one-electron wave function over all available states X ck ψk e2πiνt Ψ=

(10)

k

may be used to calculate the current density over two states k and l: j=

~e X ck cl (ψl ∇ψk − ψk ∇ψl ) e2πi(νk −νl )t . mi

(11)

k,l

If only a single eigenvibration is excited, the current disappears and the distribution of electron density remains constant. Otherwise an electron flows from one state to another in an exchange that involves a photon to keep the energy in balance. This flow of electricity can hardly be described as a quantum jump. More realistically the vibrations of the two affected states (emitter and acceptor) are seen to interact and generate a beat (wave packet) that moves to the state of lower energy. The virtual photon that links two equilibrium states turn into a real photon that carries the excess energy, either into or away from the system. In chemical applications Schrödinger’s equation is best known in its amplitude form, which is obtained by substituting Ψ = ψ exp(2πiνt), followed by elimination of the time parameter to give: 2m ∇2 ψ + 2 (E − V )ψ = 0. (12) ~ In spherical polar coordinates this equation, for the hydrogen problem, separates into independent radial and angular equations: l(l + 1) 2m d2 R 2 dR R = 0, + + (13) E − V (r) − dr2 r dr ~2 r2 1 ∂ ∂Y 1 ∂Y sin θ + + l(l + 1)Y = 0, (14) sin θ ∂θ ∂θ sin2 θ ∂ϕ2 with separation constant λ = l(l + 1), integer l. The angular part is further separable into: d2 Φ + m2l Φ = 0 (ml = −l . . . l), dϕ2 m2l dΘ 1 d sin θ + l(l + 1) − Θ = 0. sin θ dθ dθ sin2 θ

(15) (16)

An electron associated with a stationary proton (V = e2 /r) defines the only problem of some chemical significance for which the radial equation has been solved. Since the proton is here regarded as a point particle, the system does not represent a wave-mechanical model of a hydrogen atom, despite contrary claims in all chemistry texts. Like the Bohr model, it defines a set of quantized energy levels to match most spectroscopic measurements, apart from the Lamb shift, fairly well. The angular equations are valid for central-field problems and produce quantized values of the orbital angular momentum. These eigenvalues should not be confused with the angular momenta of an orbiting particle. They are, more appropriately, considered as symmetry


201

parameters, such that ml = 0 defines a spherically symmetrical charge distribution. For given l there is always an odd number of 2l + 1 sub-levels with different quantum numbers P ml , which, for many-electron systems, can be chosen in such a way that l ml = 0, in all cases. The choice reflects the electrostatic property of the charge distribution to assume spherical symmetry. A hydrogen atom, by this model, has ml 6= 0 only for excited states, which spontaneously relax to the ml = 0 spherically symmetrical ground state. 3.1.1.

Electron Spin

Schrödinger’s equation appears incomplete in the sense of lacking an operator for spin, only because its eigenfunction solutions are traditionally considered complex variables. The wave function, interpreted as a column vector, operated on by square matrices, such that abbreviated to

ei(ωt−kx) 0 0 e−i(ωt−kx)

φ1 e+ φ2 e−

φ1 φ2

=

φ1 ei(ωt−kx) φ2 e−i(ωt−kx)

.

, represents a spinor that moves in the x-direction. By forming the

derivatives: ∂φ = iω ∂t

φ1 e+ φ2 e−

∂2φ = k2 ∂x2

φ1 e+ φ2 e−

,

,

it follows that (in three dimensions): −i

∂φ ω = 2 ∇2 φ. ∂t k

This is Schrödinger’s equation, providing (~k)2 = 2m~ω, i.e. −i

~ 2 ∂φ = ∇ φ, ∂t 2m

as in (6):V = 0

which shows ~ω = E = p2 /2m, k = 2π/λ, p = h/λ. This result is interpreted [4] to show that a region of the continuum, which rotates in spherical mode, interacts with its environment by generating a wave-like disturbance at half the angular frequency of the core. The angular momentum on the surface of a unit sphere is L = mω. At λ = 2π, k = 1, the spin angular momentum follows as L = ~/2, with intrinsic magnetic moment µ = ~e/2mc.

3.2.

Bohmian Mechanics

The connection between wave mechanics and hydrodynamics, expressed by equations (7) and (8), was developed in more detail by Madelung, writing the time dependence of Ψ

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as an action function, Ψ = ψe2πiνt → ReiS/~, which seperates (5) into a coupled pair that resembles the field equations of hydrodynamics: ∂S (∇S)2 ~2 ∇2 R + − + V = 0, ∂t 2m 2mR ∂R2 R2 ∇S = 0, +∇· ∂t m

(17) (18)

which describe the irrotational flow of a compressible fluid, assuming R2 to represent the density ρ(x) of a continuous fluid with stream velocity v = ∇S/m. It was shown that both density and flux vary periodically with the same periodicity as νik = (Ei − Ek )/h, that results from superposition of states i and k. This means that radiation is not due to quantum jumps, but rather happens by slow transition in a non-stationary state. An attractive feature of the hydrodynamic model is that it obviates the statistical interpretation of quantum theory, by eliminating the need of a point particle. It is worth noting that the assumption of a point electron derives from the observation that it responds as a unit to an electromagnetic signal, which must therefore propagate instantaneously through the interior of the electron, at variance with the theory of relativity. However, by now it is known from experiment that non-local (instantaneous) response is possible in quantum systems and the initial reservation against Madelung’s proposal and Lorentz’s definition of an electron as a flexible sphere should fall away. On reinterpretation it was pointed out by David Bohm that equation (17) differed from the classical Hamilton-Jacobi equation only in the term Vq = −

~ 2 ∇2 R . 2mR

(19)

The quantity Vq , called quantum potential vanishes for classical systems as h/m → 0. A gradual transition from classical to quantum behaviour is inferred to occur for systems of low mass, such as sub-atomic species. All dynamic properties of classical systems should therefore be defined equally well for quantum systems, although the relevant parameters are hidden [10]. 3.2.1.

Quantum Potential

As for the classical potential, the gradient of quantum potential energy defines a quan.. tum force. A quantum object therefore has an equation of motion, m x= −∇V − ∇Vq . For an object in uniform motion (constant potential) the quantum force must vanish, which requires Vq = 0 or a constant, −k say. Vq = 0 defines a classical particle; alternatively1 −(V + Vq ) = T , the kinetic energy of the system. Hence ~2 ∇2 R/2mR = −E, which rearranges into 2mE ∇2 R + 2 R = 0 ~ Schrödinger’s equation for a free particle. 1

It is a common misconception that Vq = T for a free electron – compare [11]. Stationary states do not occur for Vq = 0, but when Vq = −V .


203

The quantum potential concept is vitally important for understanding the structure of an electron and of quantum systems in general. The fact that the amplitude function (R) appears in both the numerator and denominator of Vq implies that the effect of the wave field does not necessarily decay with distance and that remote features of the environment can affect the behaviour of a quantum object. The quantum potential for a many-body system: Vq =

n X i=1

~2 − 2mR

∇2i R mi

depends on the quantum state of the entire system. The potential energy between a pair of entities, Vq (xi , xj ) is not uniquely defined by the coordinates, but depends on the wave function of the entire system, Ψ. This condition defines a holistic system in that the whole is more than a sum of the parts. The instantaneous motion of one part depends on the coordinates of all other parts at the same time. That defines a non-local interaction of the type assumed to exist within an indivisible electron, and now inferred to occur in all quantum systems, including molecules. If the system is distorted locally, the entire system responds instantaneously. As the quantum potential is not a function of distance, the behaviour of a composite system depends non-locally on the configuration of all constituents, no matter how far apart. In a chemical context the properties, structure and rearrangement of molecules must depend intimately on the quantum potential. It is necessary to give up the notion that molecular rearrangement involves the breaking and making of bonds and rather consider it as a modification of the intramolecular electronic wave interference pattern. However, all systems are not correlated equally well. Whenever a wave function can be written as a product Ψ(r1 , r2 , t) = ΦA (r1 , t)ΦB (r2 , t) the quantum potential becomes the sum of two terms: Vq (r1 , r2 , t) = VqA (r1 , t) + VqB (r2 , t). The two sub-systems evidently behave largely independently. That is a good description of a molecular crystal, or liquid, with relatively weak interaction between molecular units. Systems like these are better described as partially holistic. The contentious issue of quantum-particle trajectories is put into perspective by the Bohmian model. One interpretation is that the quantum electron has an unspecified diffuse structure, which contracts into a classical point-like object when confined under external influences. The observed trajectory, as in a cloud chamber, may be considered to follow the centre of gravity. In a two-slit experiment an electron wave passes through both slits to recombine, with interference, but without rupture. The interference pattern disappears on closure of one slit or when the slits are too far apart, compared to the de Broglie wavelength. It now behaves exactly like a classical particle, when forced through a single slit2 [12]. 2

The de-Broglie – Bohm formulation of particle plus pilot wave is considered an unnecessary complication by this author. Instead, Ψ may be thought of as a state of vibration of empty space.

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3.2.2.

Stationary States

Writing the wave equation in two equivalent forms: Ψ(x, t) = Ψ0 e−iEt/~, Ψ(x, t) = R(x, t)eiS(x,t)/~, and noting that R(x, 0) = R0 (x); S(x, 0) = S0 (x); Ψ0 = R0 eiS0 /~, it follows that: S(x, t) = S0 (x) − Et,

(20)

R(x, t) = R0 .

The unexpected conclusion is that a real wave function, Ψ0 = ψ, implies S0 (x) = 0 and hence the momentum ∇S = p = 0 and E = V + Vq . Those states with ml = 0 all have real wave functions, which therefore means that such electrons have zero kinetic energy and are therefore at rest. The classical (electrostatic) and quantum forces on electrons in such stationary states are therefore balanced and so stabilize the position of the electron with respect to the nucleus. For the hydrogen atom in the ground state, R(r) = N e−r/a0 and hence, d2 R N = 2 e−r/a0 , 2 dr a0 such that, from (19), Vq = ~2 /2ma20 . In general Vq =

~2 , 2mr2

(21)

and the quantum force on the electron: Fq =

∂Vq ~2 =− 3 ∂r mr

whereas the electrostatic force F = e2 /r2 . These forces are in balance when ~2 e2 = ; mr3 r

r=

~2 = a0 , me2

the Bohr radius. This means that V = Vq at r = a0 /2, halfway between proton and electron. 3.2.3.

Orbital Angular Momentum

Orbital angular momentum is perhaps the most awkward concept to visualize as the property of a quantum-mechanical point electron, but is readily understood in hydrodynamic analogy. Like tidal motion, atomic orbital motion in a continuous spherical charge cloud consists of the propagation of a wave disturbance, without matter circulation, as first proposed by Nagaoka and described by the quantized spherical surface harmonics, Ylml = N Plml eiml ϕ , in terms of Legendre polynomials, P .


205

In Bohmian formalism angular momentum is described by rotation of the phase function: S(x, t) = S0 (x) − Et

= ml ~ϕ − Et.

The wavefronts S=constant are planes parallel to and rotating about the z-axis, with angular velocity ∂ϕ/∂t = E/ml ~. Single-valuedness of Ψ = R exp(iS/~) requires that Ψ(S) = Ψ(S + 2πn~) = ψ(S + nh). This is interpreted to mean that n = |ml | wave crests occur during each cycle. Positive and negative values of ml represent anticlockwise and clockwise rotations respectively. This interpretation of orbital angular momentum has a formal resemblance to the semiclassical model of Bohr and Sommerfeld, but there is no physical rotation of charge. Two electrons with magnetic quantum numbers of ±ml have wave structures that rotate, in phase, in opposite directions, with resultant distortion of zero. Quenching of orbital angular momentum during chemical interaction between neighbouring atoms happens by the same principle. The wave pattern in the case where l 6= 0 and ml = 0 is to be interpreted as the three-dimensional analogue of the circular modes of a vibrating drumhead. There is no axial component to the disturbance. The wave motion is more like spherical vibration, compared to spherical rotation that causes electron spin and which can be oriented in the polar direction of a magnetic field.

4.

Chemical Change

In the same sense that biological activity is more than chemical change, chemical effects depend on a number of emergent properties unknown to physics. The concepts of chemical affinity, cohesion and structure were discovered experimentally and not anticipated from first principles. Although chemical events can therefore not be inferred from the laws of physics, the Bohmian interpretation of quantum mechanics provides an attractive framework for their understanding. The fundamental reason for this emergence is the chemical environment. The interaction between chemical species, partially characterized in isolation, is as hard to predict as the behaviour of an individual in a crowd. Not being acquainted with the concepts molecule, phase transition and free energy, there is no possibility of deriving the laws of chemical affinity, reactivity and composition from the quantum numbers that quantify the energy and angular momentum of electrons in isolated atoms. The problem is approached here by examining the possible modes of interaction between charges and the response of atoms to close confinement.

4.1.

Interaction Theory

Interaction at a distance is interpreted in modern theories as a field phenomenon. The electromagnetic field, described by Maxwell’s equations as waves, propagate through the vacuum, with a constant velocity that depends on the permittivity and permeability of free √ space, c = 1/ µ0 ǫ0 . The wave equation (4) has solutions Ψ(t) and Ψ(−t), known as retarded and advanced waves, respectively. The transmission of electromagnetic energy

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Jan C.A. Boeyens

between an emitter and a distant receptor is assumed to be negotiated by a pair of retarded and advanced waves. As a spherical wave signal from the emitter reaches an acceptor, it responds with an advanced return signal that reaches the emitter at the exact moment of first emission, to establish a standing wave, known as a photon. Further interaction depends on the potential energy difference between emitter and receptor. Transfer of excess energy occurs by relaxation of the standing wave, which is experimentally observed as photon emission. Alternatively the standing wave, known as a virtual photon, that exists between interacting sites, becomes balanced against external factors, at a distance that defines the electrostatic force of interaction between the charges as: F =

q1 q2 . 4πǫ0 r2

All chemical interactions are of this type [4]. In Bohmian formalism the theory predicts the stability of atomic matter as a function of the fine-structure constant. Sommerfeld [13] – (p.107) introduced the fine-structure constant as α = v1 /c = e2 /4πǫ0 c~ (= 2πe2 /ch, in esu), where v1 is the velocity of an electron in the first Bohr orbit. More generally, the parameter α′ = v/c for a freely moving electron with de Broglie wavength λdB = h/mv and Compton wavelength λC = h/mc is defined, more appropriately as α′ = λC /λdB . An electron in a hydrogenic stationary state has nλdB = 2πn2 a0 , hence: e2 . αn = n~c In the Bohmian interpretation an atomic stationary state occurs when the potential energy of the electron, at rest, is balanced by the quantum potential. The relativistic mass of an electron at the position of the nucleus, with respect to the rest mass mo in the ns state, would be

i.e.,

mo mo =√ m= p 2 2 1 − α2 1 − v /c α2 =

m2 − m2o 4π 2 e4 En = = , m2 n2 h2 c2 mc2

Hence En = ∆m′ c2 ,

where ∆m′ = m − m2o /m ≃ m − mo .

This is interpreted here to show that an electron in a stationary state has its mass reduced, with respect to the nucleus, by an amount ∆m′ , which reappears as the binding energy −En . The same argument explains nuclear binding energy as a mass defect. Transition of an electron with n > 1 to a lower unoccupied energy level by emission of a photon with energy hν and spin ~, is anticipated. However, in the 1s state with quantum number l = 0, there is no orbital angular momentum to transfer in promoting photon emission and the ground state remains stable. The calculation does not imply different velocities for the electron at different energy levels – only a quantized change in de Broglie wavelength. The mass-energy difference amounts to exchange of a (virtual) photon in the form of a standing wave between the charge centres.


207

With the classical radius of the electron defined as r0 = e2 /mc2 it is noted that r0 me4 = = a0 m~2 c2

e2 ~c

2

= α2 ,

where a0 is the Bohr radius. This result follows from the two relationships: 2πe2 = 2πr0 , mc2 λC 2π~2 = = 2πa0 = λdB . α me2

αλC =

Now define λZ = 2πr0 . Whereas the wavelength λdB = λC /α represents a wavepacket with group velocity vg < c, the phase velocity vφ > c is associated with the Zitterbewegung of wavelength λZ = α · λC ; vg vφ = c2 [15]. This argument relates to two problematic parameters: α and the classical electron radius r0 which still awaits quantum-mechanical definition. The fine-structure constant appears firmly associated with the wave nature of an electron, seen as a standing wave that results from the superposition of diverging and converging spherical components. The internal wave structure of the electron is observed as high-frequency Zitterbewegung while the macroscopic effects in an electromagnetic field are fixed by the spread of the wavepacket, conveniently defined as a de Broglie wavelength. Trapped in the field of a proton the de Broglie wavelength is quantized to avoid self-destruction, such that e2 λC = αn = . λdB n~c For an effective charge separation of rn , the ratio αn may be considered the ratio of two energies: 2 e2 1 e = · n~c rn hν an electrostatic and a quantum-mechanical factor. The constant c = λ/τ = λν describes the virtual photon that occurs as a standing wave (nλ = 2πr) between the charge centres. The balance between the classical coulombic attraction and the quantum-mechanical repulsion (the quantum potential) defines the fine-structure constant with a value, fixed by the de Broglie wavelength of the virtual photon. In a strong field the size of an electronic wavepacket may be compressed below the Compton radius to an absolute minimum of λZ , which describes the minimum size to which an electron may be compressed, measuring r0 = λZ /2π, for an electron defined as an electric charge −e distributed over a sphere of radius r0 . The potential energy E = e2 /r0 corresponds to r0 = e2 /mo c2 , as measured classically.

4.2.

Environmental Effects

Schrödinger’s solution for the hydrogen electron serves as the starting point for the qualitative discussion of all chemical effects in quantum formalism. It is routinely forgotten that the simple hydrogen solution ignores all interactions that the electron would experience in

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Jan C.A. Boeyens

a chemical environment. Even the use of hydrogen energy levels to rationalize the structure of the periodic table is of limited value. A useful approach to simulate environmental effects was pioneered by Sommerfeld on solving Schrödinger’s equation under modified boundary conditions. Non-zero environmental pressure was introduced by assuming that ψ(r) → 0 as r approaches some finite value rc , rather than infinity. All energy levels move to higher values with decreasing rc , until the ground level reaches the ionization limit at rc = r0 , the ionization radius. It is noted that on reaching the ionization limit by uniform compression the electron that becomes decoupled from the nucleus finds itself confined to a spherical cavity at zero potential and kinetic energy. However, the non-zero energy of a free electron in a hollow sphere, must therefore be interpreted as quantum potential energy. The Helmholtz equation for such an electron: p ∇2 + k 2 ψ = 0 , k = 2mE/~2 p has the radial solutions, R = 2kr/π · kl (kr). At the first zero of the spherical Bessel function 0 = sin(kr)/(kr), kr0 = π, and hence (compare 21) E0 =

h2 = Vq . 8mr02

(22)

The Fourier transform of 0 is the box function f (r) =

√

2π/2r0 if |r| < r0 , 0 if |r| > r0 .

(23)

It follows that the decoupled (valence) electron of the hydrogen atom, compressed to r0 is uniformly spread across the ionization sphere.

4.3.

Emergent Properties

Chemical theory requires insight into more than atomic stability. One-electron quantum theory provides no guidance beyond the hydrogen atomic ground state and the structure of many-electron atoms must be inferred from the empirically known periodic table of the elements. However, the superficial correspondence between the calculated quantum states of hydrogen and the observed elemental periods strongly suggests a functional relationship between the two sets. To better appreciate the relationship it is noted that both sets can be generated by convergent sequences of Fibonacci or Lucas fractions as shown below. 4.3.1.

Periodicity

A modular pair of rational fractions

h1 k1

,

h2 k2

has the property:

h1 h2 k1 k2 = ±1.


209

Such a pair is geometrically represented by two Ford circles with radii and y-coordinates of 1/2k 2 at x-coordinates of h/k. A series of rational fractions with all neighbouring terms in unimodular relationship is represented by a set of tangent Ford circles [4]. Examples of such modular series are the Farey sequences, Fn and the converging Fibonacci and Lucas fractions on the segment ( 12 35 32 ) of F5 :

Despite a number of uncertain half-lives, a reasonable estimate of 264 divides the stable (non-radioactive) nuclides into 11 periods of 24. Plotting the ratio of protons:neutrons (Z/N ) for all isotopes as a function of atomic number, the hem lines that separate the periods of 24, intersect a reference line, at Z = τ , in Z-coordinates which correspond to well-known ordinal numbers that define the periodic table of the elements [2, 3]. Remarkably, the same hem lines intersect a reference line at Z/N = 0.58 in atomic numbers that correspond to the closure of the calculated wave-mechanical energy levels for hydrogen. Noting that the radii of unimodular Ford circles are inversely proportional to the number (2k2 ) of atoms in elemental periods, we look for converging circles that match the two forms of periodicity. The primary circle

at x = 0 or 1, rF = 21 , is flanked by two tangent circles at x = 0.5 and 1.5 ≡ −0.5, 1 1 and x = ± 34 , rF = 32 . This rF = 81 ; further converging pairs are at x = ± 23 , rF = 18 arrangement mimics the periodic table:

210

Jan C.A. Boeyens

What is probably the aesthetically most pleasing form of the periodic table is obtained by rearrangement in circular array, as shown in Figure 5 for the hundred naturally occurring cosmic elements. Interpreted in terms of electronic distribution it implies twelve 8-fold and three 2-fold energy levels, with all closed-shell elements grouped together. These are not hydrogen-like energy levels, but they agree with the valence levels, calculated for compressed atoms in Hartree-Fock-Slater approximation [14]. The hypothetical arrangement based on the hydrogen solution is recognized in the nested set of Ford circles at x = 1, predicting consecutive periods of 2n2 , n = 1, 2, . . . , arranged as follows:

If n is interpreted as Schrödinger’s principal quantum number, periods of the correct length (2n2 ) are predicted. Each of the periods consists of n subshells for subsidiary quantum numbers 0 ≤ l < n. The number of elements per subset equals 2(2l + 1), l ≤ ml ≤ l. This result provides the basis of Pauli’s exclusion postulate, which defines an emergent property, not of quantum-mechanical origin. The hypothetical and observed versions of the periodic table are in agreement for elements 1 to 18. The superficial agreement (e.g. for elements 28, 46 and 78; and 29–36) beyond that point is purely accidental. We conclude that the wave-mechanical hydrogen model fails to account for elemental periodicity mainly because it ignores all interactions apart from the central-field unitary electrostatic attraction. The common thesis of chemistry textbooks that Schrödinger’s equation, with due allowance for interelectronic effects, accounts for the periodic table, fails on two important counts. It predicts transition series of ten elements, compared to the observed eight. The guiding principle, known as the Aufbau procedure, is valid only for the alkaline-s and p blocks. Less than two-thirds of the nominal transition elements obey an Aufbau rule. The correct periodic system occurs in an environ-


211

Figure 5. The Periodic Table of the elements in circular form. ment that requires the convergence of stable nuclear composition, Z/N to the golden ratio, τ , and subject to an emergent exclusion principle. Further new properties are expected to emerge in the analysis of chemical affinity, cohesion and conformation, at a higher hierarchical level.

4.3.2.

Electronegativity

Chemical affinity is the intuitive qualitative concept that guided experimental chemistry for centuries. The first quantitative measure of affinity was discovered by Lothar Meyer as the atomic volume of an element – his basis of periodicity. It served to differentiate between electropositive and electronegative elements, with a natural affinity between them. The concept was generalized by Pauling, Mulliken and others, by placing all elements on

212

Jan C.A. Boeyens

Figure 6. Electronegativity as quantum potential of the valence state a single empirical electronegativity scale. By demonstrating the equivalence of electronegativity and the quantum potential of the valence state [16] it was finally recognized as an emergent atomic property, readily reduced to fundamental quantum theory. Like hydrogen, an atom is said to be in its valence state when ionized by environmental pressure. The energy of the electron, decoupled from the nucleus but confined to the ionization sphere, is given by (22). Characteristic ionization radii, r0 , are obtained by numerical Hartree-Fock-Slater calculation [14] with boundary conditions modified as for H. Redefined on this basis, electronegativity, χ, is calculated as χ2 =

h2 , 8mr02

expressed in eV, such that χ relates to Pauling electronegativities on a linear scale and χ2 √ to the Mulliken scale. A plot of χ = E0 reveals the same periodicity as Figure 5 and as Lothar Meyer atomic volumes. The uniform electron density of the valence state, from (23): 1 ρ = ψ 2 (r), ψ(r) = (φ/V0 ) 2 exp {−(r/r0 )p } , p >> 1. (24) The scale factor, φ, which compensates for an inaccessible core, is proportional to r0 and varies inversely with the number of nodes as defined by an effective principal quantum number n. Hence, φ = cr0 /n. The wave function p 1 exp {−(r/r0 )p } (25) ψ(r) = 3c/4πn r0 describes the interaction of an atom with its chemical environment.

Emergent Properties in Bohmian Chemistry 4.3.3.

213

Covalence

Chemical cohesion has for many years been the main topic of theoretical chemistry, conducted as an exercise in computational quantum physics, described by one practitioner [17] as if repeatedly ′′ ...validating Schrödinger’s equation!′′ . There is a curious conviction that the Born-Oppenheimer scheme enables molecular structure to be computed ab initio. An initially assumed structure is treated only as a device to kickstart the calculation. Once the electronic density has been obtained, the nuclear framework is computed theoretically, without assumption. It always comes out miraculously close to the assumed structure. To the uninitiated the procedure appears to be circular and unlikely to produce anything of physical significance beyond the assumed molecular structure. An obvious alternative is to model the electron exchange that constitutes atomic pairwise interactions, known as covalent bonds, before assembly into a three-dimensional structure is attempted. The computational details for this procedure, which requires atomic wave functions, have been documented as the well-known Heitler-London method. Appropriate wave functions (25) are obtained from empirically adjusted ionization radii that compensate for steric factors [4]. H–L calculations predict both dissociation energy and equilibrium interatomic distance for any first-order covalent interaction. High-order interaction results from the valence-level screening of the internuclear repulsion.

Figure 7. Covalent binding energy curve for homonuclear diatomics in dimensionless units. The same set of characteristic atomic radii (r) can be used to model covalent electron exchange by point-charge simulation, as a function of interatomic separation (d) only. It

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Jan C.A. Boeyens

is found that with the ratio d/r and binding energy E, expressed in dimensionless units, all homonuclear diatomic interactions are described by a single interaction curve, shown in Figure 7. The curve turns where d/r = τ and E = −2τ , at the point where exactly two electron waves are concentrated in the interatomic region. The condition is seen to reflect the exclusion principle for fermions. Its relationship to the golden ratio defines the origin of the exclusion principle as the curvature of space-time. Without this emergent property there is no understanding of covalent interaction. 4.3.4.

Molecular Shape

The inability to derive molecular structures from fundamental quantum theory identifies molecular shape as another emergent property. Although it cannot be inferred from basic theory it is readily reduced to the conservation of orbital angular momentum. Conventional computational schemes, designed to minimize energy, with total neglect of orbital angular momentum, must, by definition converge to a spherically symmetrical arrangement. To prevent this from happening a potential field of lower symmetry is imposed by assuming a fixed nuclear framework. Instead of imposing an experimentally determined structure, conservation of orbital angular momentum provides a theoretically more satisfying algorithm to generate such a structure from first principles. Polarization of mutually approaching reactants resolve local angular-momentum vectors, just like an applied magnetic field. During the formation of a molecule, the alignment of reactants that minimizes angular momentum in the local polar direction is favoured. In many reaction systems there is sufficient symmetry for the orbital angular momentum in the polar direction to become quenched completely. Where the quenching in low-symmetry (chiral) systems is incomplete, the residual angular momentum will couple to the magnetic field of polarized light, causing optical activity. Should quenching be possible only for a specific angular alignment of neighbouring fragments, a rigid system, which resists torsional deformation, is obtained. So-called double bonds and aromatic systems are common examples. The empirical stereochemical rules, pioneered by Kekulé, van’t Hoff and others, are consistent with the principles outlined here and these have been generalized into empirical computational schemes, collectively known as molecular mechanics. There is no more fundamental procedure to predict molecular structure.

References [1] H. Primas, Chemistry, Quantum Mechanics and Reductionism, 2nd ed., SpringerVerlag, Berlin, 1983. [2] J. C. A. Boeyens and D.C. Levendis, Number Theory and the Periodicity of Matter, Springer.com, 2008. [3] J. C. A. Boeyens, Periodicity of the stable isotopes, J. Radioanal. Nucl. Chem., 2003 (257) 33–41.


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[4] J. C. A. Boeyens, Chemistry from First Principles, Springer.com, 2008. [5] M. Wolff, Beyond the Point Particle – A Wave Structure for the Electron, Galilean Electrodynamics, 1995 (6) 83–91. [6] J. C. A. Boeyens, Angular Momentum in Chemistry, Z. Naturforsch. 2007 (62b) 373– 385. [7] R. C. Jennison and A. J. Drinkwater, An approach to the understanding of inertia from the physics of the experimental method, J. Phys. A, 1977 (10) 167–179. [8] J. C. A. Boeyens, Quantum theory of molecular conformation, C.R. Chimie, 8 (2005) 1527 – 1534. [9] E. Schrödinger, Collected Papers on Wave Mechanics, (Translated from the second German edition), 2nd ed., Chelsea, NY, 1978. [10] D. Bohm, A Suggested Interpretation of the Quantum Theory in Terms of ′′ Hidden′′ Variables, Phys. Rev., 1952 (85) 166–179, 180–193. [11] D. W. Belousek, Einstein’s 1927 Unpublished Hidden-Variable Theory: Its Background, Context and Significance, Stud. Hist. Phil. Mod. Phys., 1996 (27) 437–461. [12] P. R. Holland, The Quantum Theory of Motion, University Press, Cambridge, 1993. [13] A. Sommerfeld, Atombau und Spektrallinien, 4th ed., Vieweg, Braunschweig, 1924. [14] J. C. A. Boeyens, Ionization radii of compressed atoms, J. Chem. Soc. Faraday Trans., 1994 (90) 3377–3381. [15] J. C. A. Boeyens, New Theories for Chemistry, Elsevier, Amsterdam, 2005. [16] J. C. A. Boeyens, The Periodic Electronegativity Table, Z. Naturforsch., 2008 (63b) 199–209. [17] N. C. Handy, in: R. Broer, P.T.C. Aerts and P.S. Bagus, New Challenges in Computational Chemistry, University of Groningen, 1994.



Chapter 10

THE ALGEBRAIC CHEMISTRY OF MOLECULES AND REACTIONS Cynthia Kolb Whitney* Galilean Electrodynamics, 141 Rhinecliff Street, Arlington, MA 02476-7331, USA

Abstract A new line of research generally characterized as ‘Algebraic Chemistry’ is here applied to the problem of modeling energies involved in molecule formation and in chemical reactions. The approach is based on algebraic scaling laws that allow one to estimate energies of interest by evaluating simple algebraic expressions, without recourse to computer calculations based on detailed quantum mechanical formulations and phase-space integrations, such as found in traditional Quantum Chemistry. The simplicity of the algebraic approach means that it can address molecules and reactions involving more atoms than the ones that are presently convenient for traditional Quantum Chemistry. In fact, there is no complexity-related limit on the atom count amenable to molecule or reaction analysis with the algebraic method. Algebraic Chemistry is a simple tool always suitable for hand calculations. In the cases of many real molecules and reactions, data is available to test the algebraic approach, and thus build some confidence about it. This is important because the scaling laws used come from new, and not yet broadly known, theoretical extensions to the traditional quantum mechanics of atoms, and even to special relativity theory and classical electromagnetic theory. These extensions of traditional theory are briefly summarized in Appendices and detailed further in the References.

Keywords: Algebraic Chemistry, Quantum Chemistry, heat of molecule formation, heat of chemical reaction.

*

E-mail address: [email protected]

218

Cynthia Kolb Whitney

1. Introduction The present work develops a technique for quantitative analysis of molecules and reactions of arbitrary complexity. The approach is called ‘Algebraic Chemistry’, because it is based on simple multiplicative scaling laws. The present work extends the development of Algebraic Chemistry [1-4] from the discussion of individual elements to the discussion of complete molecules and reactions. The objective is to make Algebraic Chemistry more ready for full adoption into Chemistry, and less likely to remain a disruptive stepchild within Physics. Much of the earlier work in the development of Algebraic Chemistry concerned ionization potentials of atoms. It showed that all the information necessary to specify ionization potentials of arbitrary order for all the elements is embodied in first-order ionization potentials for all the elements, and that, in fact, all the information necessary to specify the first-order ionization potentials for all the elements is embodied in the first-order ionization potential of Hydrogen, and that, in fact, this number is predictable from theory. Here the idea of scaling laws is used to extend the known information about ionization potentials of neutral atoms to estimate ionization potentials for ‘already-ionized’ atoms. This new information is not to be confused with ‘higher-order’ ionization potentials. The ‘higherorder’ ionization potentials of neutral atoms, apparently [1], describe multiple ionizations that occur simultaneously, whereas the ionization potentials of ‘already-ionized’ atoms describe single ionization events that occur sequentially. The ‘one-at-time’ physical process is gentler than the ‘all-at-once’ physical process, and is more typical of events that actually occur throughout most of normal Chemistry. Like the previously known information about the higher-order ionization potentials of neutral atoms, the new information about ionization potentials of already-ionized atoms is entirely derivable from first-order ionization potentials of all the elements. Section 2 shows exactly what the scaling laws have to be, given the algebraic model under consideration. The new information is then used in subsequent Sections to estimate energies involved in molecule formation and chemical reactions.

2. Scaling Laws The basis for having scaling laws at all lies in imagining atoms to consist of a nucleus and an electron ‘cluster’ sufficiently well defined that the atom as a whole is similar to a twobody system. Being imagined as similar to a two-body system, all atoms are then similar to Hydrogen, and scaling laws based on Hydrogen follow. This idea is developed in [1], and then applied and extended in [2-4], and is summarized for the present applications in Appendix 3. The key results are as follows: The magnitude of the potential energy for the one electron in the Hydrogen atom is:

e2 (re + rp ) = 3c 2 me2 25 mp

(1)

The Algebraic Chemistry of Molecules and Reactions

219

where e is electron charge, r is orbit radius, c is light speed, m is mass, and subscripts e and p distinguish electron from proton. Suppose we want to model the system potential energy, not for Hydrogen per se, but for its isotopes, Deuterium and/or Tritium. The proton mp and rp then need to be replaced with a more generic nuclear mass M and its orbit radius rM . The magnitude of potential energy for this more massive atom is:

e2 (re + rM ) = 3c 2 me2 25 M .

(2)

Next, suppose we want to deal with a neutral atom with nuclear charge number Z , as well as the generic nuclear mass M . Then we have Z electrons as well. For the more charged system, the magnitude of the potential energy becomes:

Z 2e2 (re + rM ) = Z 2 3c 2 me2 25 M .

(3)

This scaled-up expression represents the magnitude of the total potential energy of the system involving Z electrons. What is then comparable to the ionization potential for removing a single electron is:

(

)

(

Z e2 (re + rM ) = Z × 3c 2 me2 25 M ≡ (Z / M ) × 3c 2 me2 25

)

(4)

Thus we see that Z / M scaling that is predicted for measured ionization potentials. This is cancelled out by M / Z scaling to produce the IP ’s collected in Appendix 1 and used in following Sections. More generally, if the atom is in an ionized state, we have a distinct electron count Ze and proton count Z p . For the baseline nuclear-orbit part, we have for the total system:

)(

(

Z p Ze e2 (re + rM ) = Z p Ze M × 3c 2 me2 25

)

(5)

What is then generally comparable to the nuclear-orbit part of the ionization potential for removing a single electron? Appendix 3 shows that it is as if all factors of e changed to

Z p Ze e . What is then comparable to the ionization potential for removing a single electron is:

Z p Ze e2 (re + rM ) =

(Z Z

p e

)(

M × 3c 2 me2 25

)

(6)

220


In the present work, all scaling laws are based on first-order ionization potentials of neutral elements. Removing the Z / M scaling by multiplying raw ionization data by the inverse, M / Z , produces IP ’s that are all comparable to the IP of Hydrogen. In this paper, the symbol IP always means ‘ M / Z -scaled ionization potential to compare with that of Hydrogen’. Let the M / Z -scaled first-order ionization potential for Hydrogen be represented by IP1,1 . The first subscript 1 means ‘first ionization potential’, and the second subscript 1 means ‘first element’; i.e. Hydrogen. The M / Z scaled first-order IP for the element Z is then represented by IP1,Z . In the earlier work [1], the problem addressed was modeling IP ’s of all orders higher than unity for all elements - the ‘all-at-once’ problem. The results showed that we could enlist information from one element to develop information about an ion of another element. That idea is exploited here to infer the first-order IP ’s of already-ionized atoms – the ‘one-at-atime’ problem. The IP1,Z for removing an electron separates into a baseline, nuclear-orbit part, IP1,1 , and a deviation, electron-cluster part, ΔIP1,Z = IP1,Z − IP1,1 . The baseline part is independent of Z , but the deviation part is a complicated function of Z . The present paper takes the deviation part as input data. But there exists a basis for future deeper analysis and modeling of the deviation part. The development of an Expanded SRT, detailed in [1] or [2], allows for superluminal speeds, which in turn allows for same-charge systems. Rings of multiple charges rotating at superluminal speeds are analyzed in [3], and such rings stacked like little magnets are used to model the electron populations in atoms. The deviation ΔIP1,Z can be positive or negative, depending on where the element is in the Periodic Table. The deviation generally tends to zero at mid period, between noble gasses. It is maximally positive at a noble gas, and maximally negative just after a noble gas. For example, [see Appendix 1] the deviation term for Helium is

ΔIP1,2 = IP1,2 − IP1,1 = 49.875 − 14.250 = 35.625 , whereas the deviation term for Lithium is

ΔIP1,3 = IP1,3 − IP1,1 =Ê12.469 − 14.250 = −1.781 . +

−

Let symbols like IP1,Z and IP1,Z represent first-order IP ’s for already-ionized atoms. The superscript + means positively charged due to previous electron removal, and multiple +’s would mean multiple electrons removed. Superscript − means negatively charged due to previous electron addition, and multiple − ’s would mean multiple electrons added. The

Z n Ze M scaling maps the IP1,1 baseline part of the IP1,Z for the neutral element into


221

+ − the baseline part of the IP1, Z or IP1, Z or whatever. The increment part is different; it depends only on the number of electrons involved in the charge cluster, so the scaling that

applies to it is just Ze / M , and what it applies to is the deviation term for the element whose Z matches the Ze needed. The IP1,1 and the ΔIP1,Z are the basic data used below to describe first-order IP ’s like

IP1,+Z and IP1,−Z for already-ionized atoms.

3. Example Atoms In Physics, it is traditional to begin any discussion about atoms with the simplest possible atom, Hydorgen 1 H . I am going to depart from this tradition: I am starting this discussion with Carbon 6 C . Hydrogen is just too simple; with Hydrogen too many important parts of the ionization problem disappear as invisible zero’s. Carbon has no such degeneracies, and yet it is still a lot like Hydrogen; after Hydrogen, it is the next ‘keystone’ element [see Appendix 4]. Like all keystone elements, it is just as willing to give as to take electrons, until the number reaches a ‘noble-gas’ number: 2 or 10 for Carbon (as compared to 0 or 2 for Hydrogen). For Carbon 6 C , IP1,6 = IP1,1 + ΔIP1,6 times Z / M = 6 / M 6 represents the work that must be supplied to take one electron off the neutral Carbon atom. That exercise produces + a positively charged ion 6 C . The transition also returns some heat, because after the ionization, the electron cluster is different: 5 electrons instead of 6, and all resting at a different energy. I call it ‘heat’, not ‘work’, because it is uncontrollable; Nature simple ‘does’ this. The heat returned is evidently (ΔIP1,6 × 6 − ΔIP1,5 × 5) / M 6 . Observe how the ‘5’ for 5 B information enters here.

+ + The ionization potential of the singly ionized 6 C , IP1,6 , times 6 / M 6 represents the work that must be supplied to remove another electron from the already singly ionized

(

Carbon atom. It is IP1,1 ×

)M

6 × 5 + ΔIP1,5 × 5

6 . Observe how the

6 × 5 due to the

+ previous ionization enters here. It means that IP1,6 is certainly not the same thing as the so-

called ‘second ionization potential’ of the neutral Carbon atom, IP2,6 . As before, heat is also returned as the electron cluster readjusts. This time, the amount of heat returned is

(ΔIP1,5 × 5 − ΔIP1,4 × 4) M6 . Observe how the 4 for 4 Be information enters here.

222


Proceeding to the next step, the work for removing another electron from the now doubly ++ ++ ionized 6 C , IP1,6 , times 6 / M 6 is IP1,1 × 6 × 4 + ΔIP1,4 × 4 M 6 . The heat

(

(

)

) M6 . The now triply ionized 6 C+++ takes

returned this time is ΔIP1,4 × 4 − ΔIP1,3 × 3 +++ work IP1,6 × 6 / M 6 =

(IP

1,1 ×

)M

6 × 3 + ΔIP1,3 × 3

(

The M / Z -scaled heat returned is ΔIP1,3 × 3 − ΔIP1,2 × 2 negative number. This means that getting from 6 C environment.

And

the

(

now

quadruply

ionized

++++ × 6 / M 6 = IP1,1 × 6 × 2 + ΔIP1,2 × 2 IP1,6

+++

)M

6C

6 to remove another electron.

) M6 . This turns out to be a

to 6 C ++++

++++

cools the local

would

take

work

ΔIP1,2 is a large

6 . Because

number, this last ionization is not very likely to happen. That is, like the 2 He atom, the ++++ ion is quite stable. 6C With all this information, one can construct an energy tally for all the electron removal + ++ +++ , and scenarios that Carbon invites: 6 C → 6 C , 6 C → 6 C , 6 C → 6 C ++++ : 6C → 6C 6C → 6C

+

6C → 6C

++

6C → 6C

+++

takes IP1,1 × 6 / M 6 and (ΔIP1,6 × 6 − ΔIP1,5 × 5) / M 6 ;

( 6 × 5 ) M and (ΔIP takes IP (6 + 6 × 5 + 6 × 4 ) M

takes IP1,1 6 +

1,6 × 6 − ΔIP1,4 × 4) / M 6 ;

6

1,1

6

and (ΔIP1,6 × 6 − ΔIP1,3 × 3) / M 6 ; 6C → 6C

++++

(

takes IP1,1 6 +

6×5+ 6×4 + 6×3

)M

6

and (ΔIP1,6 × 6 − ΔIP1,2 × 2) / M 6 . Carbon also allows the addition of electrons, producing negatively charged ions. To −−−− quantify these, let use the pattern for electron removal, which means we start with 6 C . The

(IP

1,1 ×

work

to

remove

6 × 10 + ΔIP1,10 × 10

an

)M

electron 6 . Like

must

be

−−−− IP1,6 × 6 / M6 =

ΔIP1,2 , ΔIP1,10 is a large number, meaning

The Algebraic Chemistry of Molecules and Reactions that, like 10 Ne , 6 C

−−−−

is quite stable. The heat returned in this de-ionization is

(ΔIP1,10 × 10 − ΔIP1,9 × 9) M6 .

(IP

−−− × 6 / M6 = IP1,6

1,1 ×

The

another

−−

electron

1,1 ×

removal,

6 × 7 + ΔIP1,7 × 7

(

ion

and

)M

−−−

,

takes

work

6 to remove the next electron. The

(IP

1,1 ×

this −

6C

And

)M

6C

down,

(ΔIP1,9 × 9 − ΔIP1,8 × 8) M6 . Then the next ion

−− , takes work IP1,6 × 6 / M 6 =

(ΔIP1,8 × 8 − ΔIP1,7 × 7 ) M6 .

(IP

next

6 × 9 + ΔIP1,9 × 9

heat returned in this de-ionization is down, 6 C

223

6 × 8 + ΔIP1,8 × 8

de-ionization takes

work

)M

6 for

returns −

heat

IP1,6 × 6 / M 6 =

6 for the last electron removal, and this final de-

ionization then returns heat ΔIP1,7 × 7 − ΔIP1,6 × 6

) M6 .

The process of adding electrons to an atom is just opposite to the process of removing electrons. So the energy tally for all the electron addition scenarios is: 6C → 6C

−

6C → 6C

−−

takes − IP1,1 × takes − IP1,1

6 × 7 M 6 and −(ΔIP1,7 × 7 − ΔIP1,6 × 6) / M 6 ;

( 6×8 +

6×7

)M

6

and −(ΔIP1,8 × 8 − ΔIP1,6 × 6) / M 6 ; 6C → 6C

−−−

takes − IP1,1

( 6×9 +

6×8 + 6×7

)M

6

and −(ΔIP1,9 × 9 − ΔIP1,6 × 6) / M 6 ; 6C → 6C

−−−−

takes − IP1,1

( 6 × 10 +

6×9 + 6×8 + 6×7

)M

6

and −(ΔIP1,10 × 10 − ΔIP1,6 × 6) / M 6 . With Carbon now well in tow, we are ready to look back to Hydrogen. The transition + 1 H → H takes IP1,1 / M1 and (ΔIP1,1 − ΔIP1,0 ) / M1 , but the latter two IP data items are zero, so the pattern being followed isn’t well revealed by them. The transition 1 H → H

−

takes − IP1,1 1 × 2 M1 and (−ΔIP1,2 × 2 + ΔIP1,1 × 1) / M1 , but ΔIP1,1 is zero, and so doesn’t fully reveal the pattern. That is why it was better to violate tradition and start with Carbon.

224

Cynthia Kolb Whitney Since hydrocarbons are so important in the discipline of Chemistry, it will also be useful

to have at hand the corresponding results for Oxygen. Being so close to 10 Ne , 8 O is + probably characterized thoroughly enough by just four transitions: 8 O → 8 O , ++ − −− and 8 O → 8 O , 8 O → 8 O . 8O → 8O + 8 O → 8 O takes IP1,1 × 8 / M 8 and (ΔIP1,8 × 8 − ΔIP1,7 × 7) / M8 , 8O → 8O

++

8O → 8O

−

8O → 8O

−−

(

takes IP1,1 8 + 8 × 7

)M

8 and (ΔIP1,8 × 8 − ΔIP1,6 × 6) / M 8 ;

takes − IP1,1 × 8 × 9 M8 and −(ΔIP1,9 × 9 − ΔIP1,8 × 8) / M 8 , takes − IP1,1

( 8 × 10 +

8×9

)M

8 and

−(ΔIP1,10 × 10 − ΔIP1,8 × 8) / M8 .

4. Application to Analysis of Molecules The basic idea exploited in this work is that a molecule consists of ionized atoms, some positively ionized and some negatively ionized, exerting Coulomb attraction for each other. It is therefore possible to learn something about a molecule by determining what its constituent ions are, and modeling the energy requirements for creating those ions from the neutral atoms involved. The first part of that question was addressed in Ref. [1], which gave the following two mirror-image propositions: Proposition 1: Molecules that are relatively stable have total electron counts such that every atom present can be assigned an electron count equal to that of a noble gas, or else zero. Proposition 2: Molecules that are highly reactive have total electron counts such that not every atom present can be assigned an electron count equal to that of a noble gas, or else zero. Proposition 1 is very often fulfilled, and was illustrated by molecules from small ( NH 3 , NaOH )

to

larger

( CH 3CO 2C10 H17 ,

(CH 3CO2 )2 Pb ⋅ 3H 2O ,

(C17 H 35CO2 )2 Ca ). Proposition 2 was illustrated with some common atmospheric gasses ( O 2 , O3 , or NO ). So the Propositions are probably true, but we really need to know much more. We need not just a binary division into stable / reactive; we need numerical rankings within those categories. That would mean modeling the energy requirements for creating the ions involved. Furthermore, it is often true that a ‘stable’ molecule admits more than one possible set of noble-gas electron assignments, or that a ‘reactive’ molecule admits more than one possible set of not-quite noble-gas electron assignments. When multiple assignments are possible, which one is the one Nature picks? Again, we need numerical rankings.


225

The present work is aimed at developing such quantitative rankings, based on the algebraic modeling concepts developed in the previous Section. There are several preliminary remarks to be made: 1) In all of the following examples, the IP data used come from the algebraic model developed in [1] and summarized in Appendix 1 and Appendix 2. Occasional IP data points differ detectably from M / Z -corrected raw data. So be it. I wish to test the concept of algebraic modeling over all, and if errors that are manifest in molecule modeling actually arise from errors in atom modeling, they are nevertheless errors, and I want them to be seen; 2) The calculations reported carry several more digits than can be justified as ‘significant’. This level of numerical detail is provided only to make the calculations easier to follow and to reproduce, and not to imply an amazing level of precision; 3) Comparison to reported data is generally desired, but not always possible, for reasons discussed below. One problem is that the energies calculated here are just for the formation of the ions in a single molecule. Some of this energy will be consumed in actually forming the molecule; i.e. giving the ions enough energy to stay at some stand-off distance from each other. Some more energy will be consumed in getting to the ‘state of matter’ for which data are reported. That means many molecules, in bulk matter – a mole – endowed with collective attributes. Molar heats of formation are generally reported for ‘standard conditions’, i.e. temperature (typically 25o C), pressure (for gasses, one atmosphere), dilution (for solutions, infinite dilution), etc. A single molecule can hardly be said to possess such properties. Indeed, not even the ‘state’ of matter – solid, liquid, or gas – seems meaningful for a single molecule. So it is clear that the energy calculated here for forming the ions in a single molecule in isolation cannot be said to correspond directly to reported data on the heat of formation for a molecule as a constituent of bulk matter. The energies calculated here are generally are generally less than reported heats of formation. Why less? Because, for molecules that actually form spontaneously, thus releasing heat, the reported heats of formation are by convention negative. That means the energies for forming the ions in molecules should also come out negative, and indeed more negative than the heats of formation reported for bulk matter. “Less is more,” the saying goes! Another problem, all too pervasive throughout Chemistry, is the need for conversion among many different systems of units. That is why the edition of Lang’s Handbook that provided most of the data in this paper has some 42 pages devoted to specifying conversion factors. And even at that generous page count, it does not have the particular conversion factor needed here. Heats of formation are quoted there in ‘kilogram-calories per mole’ (here abbreviated as ‘Kg-cal’s’), whereas ionization potentials are quoted in ‘electron volts per atom’ (here abbreviated as ‘eV’s’). Our energy-tally results are reported in ‘electron volts per molecule’, and with ‘molecule’ being just a generalization on ‘atom’, those units too are abbreviated as ‘eV’s’. So for almost everything in this paper, we need the conversion from Kg-cal’s to eV’s. The meaning of ‘kilogram-calories’ seems somewhat ambiguous, but the relevant conversion information provided appears to be: kilogram-calories to joules: 4186; 7 −12 23 ; gm-mole to molecules: 6.0228 × 10 . joules to ergs: 10 ; eV’s to ergs: 1.602 × 10 The needed conversion factor is then probably

4186 × 107 Kilogram-calories to joules × joules to ergs = ≈ 0.043 eV's to ergs × gm mole to molecules 1.602 × 10−12 × 6.0228 × 1023

226


This interpretation can be checked by successful use with enough example molecules. Many examples will be presented the following Sections.

5. Analyses of Some Small Molecules A Molecule with Two Atoms H 2 . Reported heat of formation: 0 Kg-cal’s, times conversion faction 0.043 yields 0 eV’s out (meaning no energy is released in forming this molecule). Probable electron assignments: one 1 H atom, 0 electrons; the other 1 H atom, 2 electrons. Relevant Model Data about 1 H : The transition 1 H → 1 H The transition 1 H → H

+

−

takes work IP1,1 / M1 = 14.250 / 1.008 = 14.137 eV’s. takes negative work for the extra electron falling into nuclear

orbit: − IP1,1 × 1 × 2 M1 = −14.25 × 1.414 / 1.008 = −19.990 eV’s, and takes negative heat

for

the

formation

of

an

electron

ΔIP1,2 = IP1,2 − IP1,1 = 49.875 − 14.250 = 35.625 , −35.625 × 2 / 1.008 = −70.685 eV’s.

The

−ΔIP1,2 × 2 / M1 ,

cluster:

sum

so

−ΔIP1,2 × 2 / M1 =

that of

where

energies

taken

is

−19.990 − 70.685 = −90.675 eV’s. Observe that this 2-electron negative state, H − , is + very much favored over the neutral state, 1 H , or the zero-electron state positive state, 1 H .

Thus

the

formation

of

the

ions

in

the

H2

molecule

takes

14.137 − 90.675 = −76.538 eV’s, the minus meaning that heat is released in forming this molecule. So there is a lot of energy available, which can allow H 2 molecules to form, and H 2 bulk matter to vaporize; i.e., become a gas.

Another Molecule with Two Atoms O 2 . Reported heat of formation: 0 Kg-cal, times conversion factor 0.043 is 0 eV’s. Probable electron assignments: 8 O , 6; 8 O , 10. Note: it is not possible for both 8 O atoms to get a ‘Noble gas’ electron count. Relevant model data about 8 O : The transition 8 O → 8 O

++

(

takes IP1,1 8 + 8 × 7

)M

8 , or


(

)

(

227

)

14.250 8 + 8 × 7 15.999 = 14.250 8 + 7.483 15.999 = 13.791 eV’s, and

(ΔIP1,8 × 8 − ΔIP1,6 × 6) M8 , or

(13.031 × 8 − 7.320 × 6) 15.999 = (104.248 − 43.920) 15.999 = 3.771eV’s. ++ takes altogether 13.791 + 3.771 = 17.562 eV’s. So the transition 8 O → 8 O −− The transition 8 O → 8 O takes − IP1,1 8 × 10 + 8 × 9 M8 , or

(

(

)

)

−14.250 8.944 + 8.485 15.999 = −15.524 eV’s,

(

and − ΔIP1,10 × 10 − ΔIP1,8 × 8

(

)

) M8 , or

(

)

− 29.391 × 10 − 13.031 × 8 15.999 = − 293.910 − 104.248 15.999 = −11.855 eV’s. So the transition 8 O → 8 O

−−

takes altogether −15.524 − 11.855 = −27.379 eV’s.

++ Interpretation concerning O 2 : Transforming the neutral 8 O atom to the positive 8 O −− ion takes ion takes 17.103 eV’s, and transforming 8 O to the negative 8 O

−27.379 eV’s,

so

forming

the

ions

in

the

O2

molecule

takes

17.562 − 27.379 = −9.817 eV’s. This molecule readily forms, and there is excess energy available to vaporize the bulk matter formed.

A Third Molecule with Two Atoms CO . Reported heat of formation: −26.42 Kg-cal as a gas, multiplied by conversion factor 0.043 yields −1.136 eV’s. Probable electron assignments: 6 C , 4; 8 O , 10. Note: it is not possible for both atoms to get a ‘Noble gas’ electron count. Relevant model data about 6 C : ++ The transition 6 C → 6 C takes IP1,1 6 +

(

(

)

6×5

)M

(

6 , or

14.250 6 + 5.477 12.011 = −13.617 eV’s, and ΔIP1,6 × 6 − ΔIP1,4 × 4

) M6 ,

or

(7.320 × 6 − 9.077 × 4) 12.011 = (43.920 − 36.308) 12.011 = 0.6338 eV’s.

Interpretation concerning 6 C : Transforming a neutral 6 C atom to a positive 6 C++ ion takes −13.617 + 0.6338 = −12.983 eV’s.

228


Relevant model data about 8 O (part of the information given for O 2 ): −− takes −27.379 eV’s. The transition 8 O → 8 O Interpretation concerning CO : Forming the ions in the CO molecule takes −12.983 − 27.379 = −40.362 eV’s. This molecule readily forms, and as bulk matter it readily becomes a gas. Indeed, it looks even more favorable than CO 2 , analyzed next. This may explain why CO is a frequent, though unwelcome, product of combustion.

A Molecule with Three Atoms CO 2 . Reported heat of formation: −94.05 Kg-cal as gas, or −98.69 Kg.cal as aqueous solution (either way meaning energy is released in forming this molecule) multiplied by the conversion factor 0.043 yields −4.044 -eV’s for gas or −4.244 eV’s for solution. Probable electron assignments: 6 C , 2; 8 O ’s, 10 each. Relevant Model Data about 6 C : The transition 6 C → 6 C

++++

(

takes IP1,1 6 +

(

)

6×5+ 6×4 + 6×3

)M

6 , or

14.250 6 + 5.477 + 4.899 + 4.243 12.011 = 24.463 eV’s, and ( ΔIP1,6 × 6 − ΔIP1,2 × 2) / M 6 , or

(7.320 × 6 − 35.625 × 2) / 12.011 = (43.920 − 71.250) / 12.011 = −2.275 eV’s. So the transition 6 C → 6 C

++++

takes altogether 24.463 − 2.275 = 22.188 eV’s.

Relevant model data about 8 O : (same as for CO above); −− The transition 8 O → 8 O takes −27.379 eV’s. Interpretation concerning CO 2 : Forming the ions in the CO2 molecule takes 22.188 − 2 × 27.379 = −32.570 eV’s; that is, this molecule easily forms, and there is plenty of energy to make the bulk matter into a gas. Observe that just looking at eV’s for ion formation, without the complications embedded in Kg-cal’s for getting to the gaseous state at prescribed conditions, shows CO 2 at −32.570 eV’s to be less favorable than CO at −40.362 eV’s. The −26.42 Kg-cal’s for CO as a gas, vs. −94.05 Kg-cal’s or −98.69 Kgcal’s for CO 2 as gas or aqueous solution, obscures this situation.


229

Another Molecule with Three Atoms H 2O . Reported heat of formation −57.80 Kg-cal’s as gas, −68.32 Kg-cal’s for liquid,

multiplied by the conversion factor 0.043 yields −2.485 eV’s as gas, −2.938 eV’s for liquid. Probable electron assignments: 8 O , 10, 1 H , both zero. Relevant Model Data about 1 H (part of the Information given for H 2 above): The transition 1 H → 1 H

+

takes 14.137 eV’s.

Relevant model data about 8 O (same as for CO 2 above): −− takes −27.379 eV’s. The transition 8 O → 8 O Interpretation concerning H 2O : The formation of the ions in the H 2O molecule takes 2 × 14.137 − 27.379 = 0.895 eV’s. This is very near zero, but it is positive, and so seems puzzling: it suggests that water takes some net energy to form, rather than yielding some + energy. But one more phenomenon can occur with water. The two H ions are really naked protons, extremely tiny, and so able to form a positive binary charge cluster similar to the − negative binary charge cluster that two electrons form in an H ion. If the two naked protons indeed do that, the process yields some energy. That energy would be related to, the −70.685 eV’s that the two electrons in the H − ion take [see H 2 ]. We do not at this time have a scaling law to express the relation between clusters of electrons and clusters of protons. What we do have is the comparable data for heavy water (made with deuterons): −59.56 Kg-cal for gas, −70.41 Kg-cal for liquid. These results are not so different from those for regular water. So the yet-to-be-articulated scaling from electrons to protons, and to deuterons, does not strongly involve the mass of protons vs. deuterons. So perhaps it does not strongly involve the mass of either one vs. the mass of electrons. If that is so, the resulting heat could be very similar to, possibly even equal to, the −70.685 eV’s for the two electrons in the H − ion. Taking this value as an estimate, the proton clustering would easily make the energy tally for making the ions in the water molecule appropriately negative, at 0.895 − 70.685 = −69.790 eV’s. This energy yield would allow water to both form its molecule and then melt into a liquid. Note too that proton clustering would also make a water molecule polarized – which indeed it definitely is. As a result of this polarization, a lot of other molecules dissolve in water, meaning they dissociate into positive and negative ions. This dissolution behavior 7 includes even the pure water itself: at any given moment in time, about one out of 10 of the + − molecules in a sample of pure water is dissociated into H and OH ions. Hence we have the phenomenon of ‘pH’ with ‘neutral’ set at 7.

230


A Molecule with Four Atoms Co3C . (Like H 2 , the Co3C molecule involves only atoms of ‘keystone’ elements [see Appendix 4]. Unlike the other molecules treated so far, Co3C is a crystalline solid at standard conditions, so no part of the heat released in forming its constituent ions is used in getting to liquid or gas state.) Reported heat of formation +9.5 Kg-cal’s (the ‘+’ meaning heat is consumed in making this molecule) times conversion factor 0.043 yields

+0.4085 eV’s. Probable electron assignments: 27 Co , one 25, two 30’s; 6 C , 2. Relevant model data about 27 Co : The transition 27 Co → 27 Co

(

++

(

takes IP1,1 27 +

)

27 × 26

)M

27 , or

14.250 27 + 26.495 58.933 = 12.935 eV’s, and

(ΔIP1,27 × 27 − ΔIP1,25 × 25) M 27 , or

(1.980 × 27 − 1.289 × 25) 58.693 = (53.460 − 32.225) 58.693 = 0.362 eV’s. ++ takes altogether 12.935 + 0.362 = 12.573 eV’s. So the transition 27 Co → 27 Co −−− takes The transition 27 Co → 27 Co

− IP1,1

( 30 × 27 +

29 × 27 + 28 × 27

(

)

)M

27 , or

− 14.25 28.460 + 27.982 + 27.495 58.933 = −20.296 eV’s, and

(

− ΔIP1,30 × 30 − ΔIP1,27 × 27

(

)

) M 27 , or

(

)

− 4.242 × 30 − 1.980 × 27 58.933 = − 127.260 − 53.460 58.933 = −1.252 eV’s. So the transition 27 Co → 27 Co −21.548 eV’s.

−−−

takes altogether −20.296 − 1.252 =

Interpretation concerning 27 Co : Transforming 3 neutral 27 Co into one 27 Co −−− ions takes 12.573 + 2 × (−21.548) = −30.524 eV’s. and two 27 Co Relevant model data about 6 C : (same as in CO 2 ): ++++ takes altogether 22.188 eV’s. The transition 6 C → 6 C

++

ion


231

Interpretation concerning Co3C : Making the ions in the Co3C molecule takes −30.524 + 22.188 = −8.336 eV’s. Evidently, all this energy, and a slight bit more, is consumed in forming the molecule and its bulk-matter crystal structure. It is encouraging that the calculation here produces a result that is reasonably interpretable. It is cautioning that the calculation involves some rather small differences between rather large numbers. This circumstance may be even worse for some molecules. If so, related numerical problems would probably occur with traditional Quantum Chemistry as well. Nature is a challenge for us all!

A Molecule with Five Atoms CH 4 (Methane; another ‘keystone-only’ molecule, like H 2 and Co3C ) Reported heat of formation: −17.89 Kg-cal’s, times conversion factor 0.043 yields −0.769 eV’s. Probable electron assignments: 6 C , 10; 1 H , all zero. Relevant model data concerning 6 C : The transition 6 C → 6 C

− IP1,1

( 6 × 10 + (

6×9 + 6×8 + 6×7

)

)M

−−−−

takes

6 , or

−14.250 7.746 + 7.348 + 6.928 + 6.481 12.011 = −14.250 × 28.503 / 12.011 =

(

−33.816 , and − ΔIP1,10 × 10 − ΔIP1,6 × 6

(

)

(

) M6 , or

)

− 29.391× 10 − 7.320 × 6 12.011 = − 293.91− 43.920 12.011 =

−20.813 eV’s. Interpretation concerning 6 C : The transition 6 C → 6 C −33.816 − 20.813 = −54.629 eV’s.

−−−−

takes

Relevant model data concerning 1 H (from Sect. 5): The transition 1 H → 1 H 14.137 eV’s.

+

takes

+ Interpretation concerning 1 H : Turning 4 neutral 1 H atoms into 4 positive 1 H ions takes 4 × 14.137 = 56.548 eV’s. These four ions are really four naked protons. Recall the

case of two naked protons in the H 2O : they apparently formed a positive binary charge cluster. So consider that the four naked protons in CH 4 could form two binary clusters, or one four-fold cluster. The two-binaries configuration seems favored on energetic grounds, and it seems to be confirmed on polarization grounds. Observe that the four-fold configuration would produce a polarized molecule, maybe four times as strongly polarized as

H 2O , whereas the two-binary configuration allows the two binaries to seek opposite sides of

232


the central C

−−−−

ion, and so produce no net polarization of the CH 4 molecule. And note

that CH 4 is indeed not a polarized molecule. The energy taken to make two binary proton clusters is related to, and probably very similar to, twice the −70.685 eV’s estimated for making the one binary proton cluster in

H 2 O ; i.e. −141.320 eV’s. Interpretation concerning CH 4 : Forming the ions in the CH 4 molecule takes something like −54.629 + 56.548 − 141.320 = −139.401 eV’s. This molecule readily forms, and as bulk matter becomes a gas.

A Molecule with Six Atoms CH 4O (Methyl alcohol). Reported heat of formation −48.10 Kg-cal’s for gas, times conversion factor 0.043 yields −2.0683 eV’s. Probable electron assignments: 6 C , 2; 1 H , one at 0, three at 2; 8 O , 10. Relevant Model Data about 6 C (same as in CO 2 ): ++++ takes altogether 22.188 eV’s. The transition 6 C → 6 C Relevant model data about 1 H (same as in H 2 ): + − The transition 1 H → 1 H takes 14.137 eV’s and the transition 1 H → 1 H takes −90.675 eV’s. Relevant model data about 8 O (part of the information given for O 2 ): −− takes −27.379 eV’s. The transition 8 O → 8 O Interpretation concerning CH 4O : Forming the ions in the CH 4O molecule takes 17.040 + 14.137 + 3 × (−90.675) − 27.379 = −268.227 eV’s. This molecule readily forms, and as bulk matter becomes a gas.

6. Analysis of a Much Larger Molecule With Algebraic Chemistry, there is no real impediment to analyzing large molecules. This can be demonstrated here with some suitably larger molecule. A socially significant one is sucrose; C12 H 22O11 ; we do eat a lot of that one! It has 45 atoms and 182 electrons. That qualifies it as ‘much larger’, I do believe. The probable electron assignments are: C ’s, 6 at 2 , 6 at 10 ; H ’s, all 0 ’s; O ’s, all 10 ’s. The relevant model data all comes from Sect. 5:


233

++++ takes 22.188 eV’s; from Concerning 6 C : From CO 2 , the transition 6 C → 6 C CH 4 , the transition 6 C → 6 C−−−− takes −54.629 eV’s. + Concerning 1 H : From H 2 , the transition 1 H → 1 H takes 14.137 eV’s. −− takes −27.379 eV’s. Concerning 8 O : From O 2 The transition 8 O → 8 O

Interpretation concerning 6 C : Transforming 6 neutral 6 C atoms into 6 positive ++++ ions takes 6 × 22.188 = 133.128 eV’s, and transforming 6 neutral 6 C atoms to 6C 6 negative 6 C

−−−−

ions takes 6 × (−54.629) = −327.774 eV’s.

Interpretation concerning 1 H : Transforming 22 neutral 1 H atoms into 22 positive + ions takes 22 × 14.137 = 311.014 eV’s. [Do these then form binary clusters? 1H Probably not; where would they go?] Interpretation concerning 8 O : Transforming 11 neutral 8 O atoms into 11 negative −− ions takes 11 × (−27.379) = −301.169 eV’s. 8O Interpretation concerning C12 H 22O11 : Forming the ions in the C12 H 22O11 molecule takes

133.128 − 327.774 + 311.014 − 301.169 = −184.801 eV’s . I wanted to compare the total heat of ion formation modeled here and the heat of bulk matter formation reported in Lang’s Handbook of Chemistry. However, the latter information was not available in the rather old edition that had provided all the other heat data used in this paper. The modern-day web revealed reason for the omission: the presumed direct synthesis reaction for sucrose, 1

12C(s) + 11H 2 (g) + 5 O2 (g) → C12 H 22O11(s) + heat , 2 was never accomplished in the lab. Note that all species on the left side were reported to have zero heat of formation, and that sucrose did form naturally, and would have been expected to have a negative heat of formation. The assumed condition for the reaction to work would likely have been that the heats of formation be less negative on the left than on the right, so the reaction would have looked feasible. So its actual non-feasibility would have been a big surprise. Algebraic Chemistry reveals the reason for the failure. The prerequisite for the reaction to work is really about the energy taken to form ions. From Sect. 5, forming all the ions in all the species on the left side of the sucrose synthesis reaction takes 12 × 0 + 11 × (−76.538) + 5.5 × (−9.817) = −841.918 − 53.994 = −895.912 eV’s.

234


This is more negative than the estimated −184.801 eV’s to form the ions in the sucrose molecule on the right. This situation indicates that the presumed direct synthesis reaction for sucrose does not go to the right, as it is depicted. The web also provided a sucrose heat of formation instead inferred from its combustion reaction,

C12 H 22O11(s) + 12O 2 (g) → 12CO 2 (g) + 11H 2O + heat Since this reaction definitely works, Algebraic Chemistry should be able to confirm that fact. Forming the ions on the left side of the reaction is estimated to take −184.801 + 12 × (−9.817) = −302.605 eV’s, whereas forming the ions on the right side is estimated

to

take

12 × (−32.570) + 11× (−70.685) =

−390.840 − 779.535 =

−1168.375 eV’s, which is indeed more negative. This says the sucrose combustion reaction does indeed run to the right as depicted. But the reported heat of formation inferred from the sucrose combustion reaction is not reported in the Kg-cal/mole units that the heats of formation for the other molecules quoted earlier were; it is instead reported in Kjoules/mole (here abbreviated Kj’s). So the needed conversion factor changes, from

Kilogram-calories to joules × joules to ergs 4186 × 107 = ≈ 0.043 eV's to ergs × gm mole to molecules 1.602 × 10−12 × 6.0228 × 1023 to

103 × 107 Kjoules to joules × joules to ergs = ≈ 0.0104 eV's to ergs × gm mole to molecules 1.602 × 10−12 × 6.0228 × 1023 The reported −2226.1 Kj’s is −23.25 eV’s. This is much less in magnitude than the −184.801 eV’s here calculated to form the ions in sucrose. This means that a very large portion of the energy generated making the ions in sucrose is then tied up in making the molecule and its crystal structure. Being so energy-packed, it is no wonder that sucrose crystals dissolve so easily in water!

7. Analyses of Current Benchmark Reactions The history of the sucrose problem highlights the importance of studying not only molecules, but also reactions. There was recently a status report on efforts in Quantum Chemistry in the form of a collection of papers collectively titled “Challenges in Theoretical Chemistry” [Science Magazine, October 2008]. Among the papers included was one entitled “Quantum Dynamics of Chemical Reactions”, by D.C. Clary [5].


235

A Reaction Involving Four Atoms In

[5]

Clary

set

as

a

benchmark

a

reaction

involving

four

atoms:

OH + H 2 → H 2O + H . He cited extensive calculations on this reaction using the ‘wave packet’ method [6,7]. We can also analyze this same reaction from the viewpoint of Algebraic Chemistry, as follows:

Analysis of the Left Side of the Reaction, OH + H 2 : Analysis of OH : Probable electron assignments: 8 O , 9; 1 H , zero. − From Sect. 2, the transition 8 O → 8 O takes − IP1,1 × 8 × 9 M8 , or

−14.250 × 8.485 / 15.999 = −7.558 eV’s, and −(ΔIP1,9 × 9 − ΔIP1,8 × 8) / M8 , or −(20.254 × 9 − 13.031 × 8) / 15.999 = −(182.286 − 104.248) / 15.999 = −4.878 eV’s. + From Sect. 5, H 2 , the transition 1 H → 1 H takes 14.137 eV’s. So forming the ions in the OH radical takes −7.558 − 4.878 + 14.137 = 1.701 eV’s.

Analysis of H 2 : From Sect. 5, the formation of the ions in the H 2 molecule takes −76.538 eV’s. Forming all of the ions involved in the left side of the reaction, OH + H 2 , takes

1.701− 76.538 = −74.837 eV’s.

Analysis of the Right Side of the Reaction, H 2O + H : Analysis of H 2O . From Sect. 5, the formation of the ions in the H 2O molecule is estimated to take −69.790 eV’s. Analysis of H : None required. So formation of the ions involved in the H 2O + H right side of the reaction is estimated to take −69.790 eV’s. This is less negative than the −74.837 eV’s on the left side. So this reaction does not run to the right, as it is depicted, and so is not a realistic target for analysis by Quantum Chemistry. But note: The OH on the left side of the reaction, OH + H 2 , is not usually seen that − way, as a neutral species; it is usually seen as a negative ion, OH . So perhaps we should

look at some variations on the stated reaction OH + H 2 → H 2O + H . For example, − − consider OH + H 2 → H 2O + H + e .

236


Analysis of the New Left Side of the Reaction, OH − + H 2 : − Analysis of OH : Probable electron assignments: 8 O , 10; 1 H , zero. From Sect. 5, O 2 , the transition 8 O → 8 O−− takes −27.379 eV’s. From Sect. 5, H 2 , the transition + − takes 14.137 eV’s. So forming the ions in the OH radical takes 1H → 1H −27.379 + 14.137 = −13.242 eV’s.

Analysis of H 2 : From Sect. 5, the formation of the ions in the H 2 molecule takes −76.538 eV’s. − So formation of the ions involved in the left side of the reaction, OH + H 2 , takes

−13.242 − 76.538 = −89.780 eV’s. This is even worse than the −74.837 eV’s for the original left side, OH + H 2 .

But note: the H on the right side of either OH + H 2 → H 2O + H or the alternative OH − + H 2 → H 2O + H + e− is not usually seen like that, as a neutral atom; it is more + − often seen as H or H . So perhaps we should also look at two further variant reactions: OH − + H 2 → H 2O + H + + 2e− and OH − + H 2 → H 2O + H − .

Analysis of the Common Left Side of these two Teactions, OH − + H 2 From above, formation of the ions involved takes −89.780 eV’s.

Analysis of the Two Right Sides of the Two Reactions, H 2O + H + + 2e − and

H 2O + H − Analysis of H 2O . From Sect. 5, the formation of the ions in the H 2O molecule is estimated to take −69.790 eV’s. − + and H : From Sect. 5, H 2 , the transition 1 H → 1 H takes 14.137 eV’s, and the transition 1 H → 1 H − takes −90.675 eV’s. − Analysis of 2e : None required.

Analysis of H

+

+ − So in the first case, the right side of the reaction H 2O + H + 2e is estimated to take −69.790 + 14.137 = −55.653 eV’s, whereas in the second case, the right side of the


237

− reaction H 2O + H is estimated to take −69.790 − 90.675 = −160.465 . The first case still does not run to the right as depicted, but the second case does, and very strongly so. So only this one last variant form of the reaction involving these four atoms appears to be a realistic target for analysis by Quantum Chemistry.

A Reaction Involving Six Atoms In [5], Clary also cited a six-atom reaction, printed as H + CH 4 → CH 3 + H , but probably really meaning H + CH 4 → CH 3 + H 2 , for which the ‘wave-packet’ calculations require significant approximations [8], but for which some recent progress has been achieved by exploiting permutations of identical atoms [9]. We can also look at this reaction from the viewpoint of Algebraic Chemistry.

Analysis of the Left Side of the Reaction, H + CH 4 : Analysis of H : None required. Analysis of CH 4 : From Sect. 5, forming the ions in the CH 4 molecule takes something like −139.401 eV’s. So forming the ions involved in the left side of the reaction, H + CH 4 , takes something

like −139.401 eV’s.

Analysis of the Right Side of the Reaction, CH 3 + H 2 : Analysis of CH 3 : Probable electron assignments: 6 C , 9; 1 H , all zero. −−− takes Relevant model data on 6 C : From Sect. 2, the transition 6 C → 6 C

− IP1,1

( 6×9 +

6×8 + 6×7

(

)M

6 or

)

−14.250 7.348 + 6.928 + 6.481 12.011 = −24.626 eV’s,

(

and − ΔIP1,9 × 9 − ΔIP1,6 × 6

(

)

) M6 , or − (20.254 × 9 − 7.320 × 6) 12.011 =

− 182.286 − 43.92 12.011 = −11.520 eV’s. Relevant model data on 1 H : From Sect. 5, H 2 , the transition H → H 14.137 eV’s, so creating 3 H + ions takes 3 × 14.137 = 42.411 eV’s. Interpretation concerning CH 3 : Forming the ions in the CH 3 molecule takes

+

takes

238


−24.626 − 11.520 + 42.411 = 6.265 eV’s. Observe that this is significantly positive, meaning this molecule is not easy to make. Analysis of H 2 : From Sect. 5, the formation of the ions in the H 2 molecule takes 14.137 − 90.677 = −76.538 eV’s. So forming the ions involved in the right side of the reaction, CH 3 + H , takes 6.265 − 76.538 = −70.273 eV’s. This is less negative than the −139.401 eV’s for forming the

ions

on

the

left

side

of

the

reaction,

H + CH 4 ,

so

the

reaction

H + CH 4 → CH 3 + H 2 does not run to the right as it is depicted. But again, some aspects of the reaction H + CH 4 → CH 3 + H 2 are not realistic: the

H seen on the left is usually seen as H + or H − , and the CH 3 seen on the right is usually − seen as (CH 3 ) . Since making a reaction that can run to the right requires less negative on the left and/or more negative on the right, the best candidate reaction appears to be: H + + CH 4 + 2e − → (CH 3 )− + H 2 . + Change on the left side: From Sect. 5, H 2 , the transition H → H takes 14.137 eV’s. So forming all the ions on the left side of the reaction now takes 14.137 − 139.401 = −125.264 eV’s. − Change on the right side: Analysis of (CH 3 ) ; Probable electron assignments: 6 C ,

10; 1 H , all zero. Relevant model data on 6 C : From Sect. 5, CH 4 , the transition −−−− takes −54.629 eV’s. So forming all the ions on the right side of the 6C → 6C reaction now takes −54.629 + 42.629 − 76.538 = −88.538 eV’s. This is not enough change to make a reaction that runs to the right as depicted. So this family of reactions appears not very promising for study with the techniques of Quantum Chemistry.

8. Results and Discussion At the present time, many of our most talented people, armed with our most powerful computing capabilities, are committed to applying of traditional Quantum Mechanics to the problems of interest for Quantum Chemistry. In this book, you are likely to see many current accomplishments reported, along with future agendas laid out. So Quantum Chemistry is today a work in progress. The exercises documented in the present paper demonstrate that while the work in Quantum Chemistry continues to develop, we can at the same time accomplish some preliminary analyses by applying the techniques of Algebraic Chemistry. Such exercises can be useful, for example, in planning computational and experimental investigations. We can


239

calculate the amount of heat that the formation of an atomic ion in a molecule will consume (negative heat for natural formation). We can predict if a hypothetical chemical reaction may fail to transpire naturally (ion-formation heat more negative on the left side of the reaction than on the right side). Extensions of the present work are also possible. One that is important, although beyond the scope of the present paper, is the calculation of energies for lots more ions, in lots more molecules, involved in lots more reactions. Another extension is less obvious, but just as necessary: the calculation of energies, not just for formation of the ions in molecules, but also for the settlement of ions into molecules, and the organization of molecules into the particular states of matter for which their data are quoted: solid state, liquid state, or gas state. And sometimes data might be quoted for even more specifically described states, such as: ‘solid crystal’, ‘amorphous solid’, ‘super-conducting’, ‘polymer’, ‘plastic’,… ‘pure liquid’, ‘aqueous solution’, ‘super fluid’,…, ‘molecular beam’, ‘plasma’,…etc. So there is a lot of scope for future development in the line of work reported here.

Appendix 1. Essential Data about First-Order Ionization Potentials The following Tables capture the input data that one needs to conduct analyses of the type introduced in this paper. Table 1. Periods 1, 2 and 3

Element H He

Charge Z 1 2

Mass M 1.008 4.003

Ionization IP = Ionization Model Model IP Potential Potential × M / Z ΔIP 14.250 13.718 13.610 0 49.875 35.625 49.244 24.606

Li

3

6.941

5.394

12.480

12.469

−1.781

Be B C N O F Ne

4 5 6 7 8 9 10

9.012 10.811 12.011 14.007 15.999 18.998 20.180

9.326 8.309 11.266 14.544 13.631 17.438 21.587

21.011 17.966 22.551 29.101 27.260 36.810 43.562

23.327 17.055 21.570 27.281 27.281 34.504 43.641

9.077 2.805 7.320 13.031 13.031 20.254 29.391

Na Mg Al

11 12 13

22.990 24.305 26.982

5.145 7.656 5.996

10.753 15.506 12.444

10.910 −3.340 16.565 2.315 14.923 0.673

Si P S Cl Ar

14 15 16 17 18

28.086 30.974 32.066 35.453 39.948

8.154 10.498 10.373 12.977 15.778

16.357 21.677 20.790 27.063 35.017

18.874 4.624 23.871 9.621 23.871 9.621 30.192 15.942 38.186 23.936

240


The data are separated into blocks corresponding to the periods in the Periodic Table. A few useful comments are interleaved with the blocks to complete the display. The IP Model to which the Tables refer is detailed in Appendix 2. Observe the M / Z scaling that is introduced to convert the raw ionization-potential data into the IP data to be modeled mathematically. The reason for the M / Z scaling emerges from the physical model in Appendix 3. Basically, the physical model shows that the measurable ionization potentials of elements do, to first approximation, scale with Z / M . So raw ionization-potential data is very element specific, and in fact it is very isotope specific. To create information that is not so element/isotope specific, we remove that Z / M scaling by applying its inverse M / Z . Observe finally that these Tables refer only to neutral atoms. The scaling to similar information for ions is worked out in Appendix 3. Table 2. Period 4 Charge

Mass

Ionization

IP = Ionization

Model

Model

Element K

Z 19

M 39.098

Potential 4.346

Potential × M / Z 8.944

IP 9.546

ΔIP −4.704

Ca Sc Ti V

20 21 22 23

40.078 44.956 47.867 50.942

6.120 6.546 6.826 6.743

12.265 14.013 14.851 14.934

13.057 −1.193 13.057 −1.193 13.638 −0.612 14.244 −0.006

Cr

24

51.996

6.774

14.676

14.877

0.627

Mn Fe Co

25 26 27

54.938 55.845 58.933

7.438 7.873 7.863

16.345 16.911 17.163

15.539 15.539 16.229

1.289 1.289 1.980

Ni Cu

28 29

58.693 63.546

7.645 7.728

16.026 16.934

16.951 17.705

2.701 3.455

Zn Ga Ge As

30 31 32 33

65.390 69.723 72.610 74.922

9.398 6.006 7.905 9.824

20.485 13.509 17.936 22.303

18.492 14.494 17.860 22.007

4.242 0.244 3.610 7.757

Se Br Kr

34 35 36

78.960 79.904 83.800

9.761 11.826 14.015

22.669 26.998 32.623

22.007 7.757 27.116 12.866 33.412 19.162

One key to the task of modeling ions is to separate the IP ’s into two parts: one being a baseline amount corresponding to the Hydrogen IP , IP1, Z − 14.250 , and the other being the increment from the Hydrogen IP ; i.e., ΔIP1, Z = IP1, Z − IP1,1 = IP1, Z − 14.250 eV’s. For neutral atoms, the two parts both scale with Z . For ions, the Z -scaling generalizes differently for the two parts. The baseline 14.250 eV’s scales with

Ze Z M , where Ze is


241

the electron count and Z M is the nuclear charge. (Appendix 3). The ΔIP just scales with

Ze . The way all these data are used to work out information for ions is to sum up the information for the desired number of single-electron removals. In the case of the baseline 14.250 eV contributions, that results in a sum of various square roots. In the case of the ΔIP contributions, a lot of cancellations occur, leaving just two terms, one from the beginning state and one from the end state. Observe that the Periods differ in length, and the remaining ones will be much longer than these first three. The periods also differ in dips from the end of one period to the start of the next period. The dips go: 1 / 4 , 1 / 4 , 1 / 4 , and then 2 / 7 (the inverse of the 7 / 2 ) thereafter. That is, Period 2 is 7 / 2 × 1 / 4 = 7 / 8 below Period 1, Period 3 is 7 / 8 below Period 2, and Period 4 is 7 / 8 below Period 3. All the rest of the periods will start, and end, just where Period 4 did. The result of the dips is that the elements mid period generally have IP ’s that are very similar to that of Hydrogen. There are other functional similarities as well, so that the whole list of elements that are exactly mid period is called out and featured by the terminology ‘keystone elements’ (Appendix 4). Table 3. Period 5 Charge

Mass

Ionization

Element Rb Sr

Z 37 38

M 87.620 88.906

Potential 5.695 6.390

IP = Ionization

Model

Model

Potential × M / Z IP ΔIP 9.657 9.546 −4.704 13.132 13.057 −1.193 −1.193

Y

39

91.224

6.846

14.567

13.057

Zr Nb Mo Tc

40 41 42 43

92.906 92.906 95.940 98.000

6.888 6.888 7.106 7.282

15.614 15.608 16.232 16.597

13.638 −0.612 14.244 −0.006 14.877 0.627 15.539 1.289

Ru Rh Pd Ag Cd

44 45 46 47 48

101.070 102.906 106.420 107.868 112.411

7.376 7.469 8.351 7.583 9.004

16.942 17.080 19.319 17.403 21.087

15.539 16.230 16.951 17.705 18.492

1.289 1.980 2.701 3.455 4.242

In Sn

49 50

114.818 118.710

5.788 7.355

13.563 17.462

14.494 17.860

0.244 3.610

Sb Te I

51 52 53

121.760 127.600 126.904

8.651 9.015 10.456

20.655 22.120 25.037

22.007 7.757 22.007 7.757 27.116 12.866

Xe

54

131.290

12.137

29.508

33.412 19.162

242

Cynthia Kolb Whitney Table 4. Period 6 Charge Mass Ionization IP = Ionization Model Model M IP Z Element Potential Potential × M / Z ΔIP 132.905 9.546 −4.704 55 9.425 Cs 3.900 137.327 13.057 −1.192 56 12.796 Ba 5.218 138.906 12.393 −1.857 57 13.600 La 5.581 140.116 12.583 −1.667 58 13.232 Ce 5.477 140.908 12.776 −1.474 59 12.957 Pr 5.425 144.240 12.972 −1.278 60 13.217 Nd 5.498 145.000 13.171 −1.079 61 13.192 Pm 5.550 150.360 13.374 −0.876 62 13.660 Sm 5.633 151.964 13.579 −0.671 63 13.687 Eu 5.674 −0.671 −0.463 −0.251

Gd Tb Dy

64 65 66

157.250 158.925 162.500

6.141 5.851 5.934

15.089 14.305 14.609

13.579 13.787 13.999

Ho Er

67 68

164.930 167.260

6.027 6.110

14.836 15.029

Tm Yb Lu

69 70 71

168.934 170.040 174.967

6.183 6.255 5.436

15.137 15.463 13.395

14.213 −0.037 14.431 0.181 14.653 0.403 14.878 0.627 17.860 3.610

Hf Ta

72 73

178.490 180.948

7.054 7.894

17.487 19.568

18.755 19.696

4.505 5.446

W Re Os

74 75 76

183.840 186.207 190.230

7.988 7.884 8.714

19.844 19.574 21.811

20.684 21.721 21.721

6.434 7.471 7.471

Ir Pt

77 78

192.217 195.076

9.129 9.025

22.788 22.571

22.811 23.955

8.560 9.705

Au Hg

79 80

196.967 200.530

9.232 10.446

23.019 26.184

Tl Pb Bi

81 82 83

204.383 207.200 208.980

6.110 7.427 7.293

15.417 18.768 18.361

25.156 10.906 26.418 12.168 16.515 2.265 19.696 5.446 23.490 9.240

Po At

84 85

209.000

8.423

20.958

210.000

Rn

86

222.000

10.757

27.769

23.490 9.240 28.015 13.765 33.412 19.164

There are many tantalizing facts modeled but not yet understood: • •

Observe that the IP Model rise on all periods is 7 / 2 . The meaning of the universal factor 7 / 2 is not yet known; it is at present just a fact of Nature. Observe that, except for Period 1, the ΔIP always start negative at the beginning of a period, and ends positive at the end of the period. And the positive increments are larger than the negative ones. The reason is that all this data falls into a simple


•

243

pattern when plotted on a log scale, as shown in Appendix 2. On the log scale, the increments up and down are similar in magnitude. Observe too that within each period, there are sub periods. These correspond to runs of nominal single-electron quantum states being filled. On the log scale, the sub periods form straight-line segments. That means they follow power laws. And the slopes are simple functions of the quantum numbers of the nominal single-electron states being filled. The physical meaning of these functions is not yet known. Table 5. Period 7 Ionization IP = Ionization Potential Potential × M / Z

Charge Z

Mass M

Fr Ra Ac

87 88 89

223.000 226.000 227.000

5.280 6.950

13.560 17.727

Th Pa

90 91

232.038 231.036

6.089 5.892

15.699 14.959

U Np

92 93

238.029 237.000

6.203 6.276

16.050 15.994

Pu Am Cm

94 95 96

244.000 243.000 247.000

6.068 5.996 6.027

15.752 15.337 15.507

12.972 −1.277 13.171 −1.079 13.374 −0.876 13.579 −0.671 13.579 −0.671

Bk Cf

97 98

247.000 251.000

6.234 6.307

15.875 16.154

13.787 13.999

Es Fm Md

99 100 101

252.000 257.000 258.000

6.421 6.504 6.587

16.345 16.716 16.827

No Lf

102 103

259.000

6.660

16.911

14.213 −0.037 14.431 0.181 14.653 0.403 14.877 0.627 17.859 3.610

Rf Db Sg

104 105 106

18.755 19.696 20.684

4.505 5.446 6.434

Bh Hs

107 108

21.721 21.721

7.471 7.471

Mt Uun

109 110

22.811 23.955

8.561 9.705

Uuu Uub ???

111 112 113

??? ???

114 115

25.156 10.906 26.418 12.168 16.515 2.265 17.019 2.769 23.490 9.240

??? ??? ???

116 117 118

23.490 9.240 28.015 13.765 33.412 19.162

Element

Model

Model ΔIP

IP 9.546 −4.704 13.057 −1.193 12.393 −1.857 12.583 −1.667 12.776 −1.474

−0.463 −0.251

244


There are two more periods remaining, and their Tables have some blank spaces where this author did not find raw data available. Predictions, however, are no problem to provide, and are listed in anticipation that the real data will eventually emerge for comparison.

Appendix 2. The Algebraic Model for Ionization Potentials 10000

1000

100

10

1 0

10

20

30

40

50

60

Figure A2.1. Ionization potentials, scaled by

70

M/Z

80

90

100

110

120

and modeled algebraically.

Figure A2.1 depicts the behavior of IP ’s for all elements (nuclear charge Z = 1 to Z = 120 shown). Element Z actually allows Z ionization potentials, but for larger Z , many IP ’s are not so easy to measure. Readily available data go only to seventh order, so that is how many orders are shown here. The points on Figure A2.1 are measured IP eV’s, scaled for comparison with each other as indicated by the new theory summarized in the next Appendix. The scale factor M / Z , where M is nuclear mass number and Z is nuclear charge, is in no way indicated by traditional QM. The lines on Figure A2.1 represent the algebraic model for IP ’s, rendered in its current best state of development. The model is capable of producing plausible estimates for all M / Z -scaled IP ’s for all IO ’s, even beyond those measured, and all Z ’s, even beyond those known to exist. The model-development approach is an example ‘data mining’. Figure A2.1 has less than 400 out of approximately 5000 desired data points. But that is enough data points to reveal a


245

pattern. The work involved is a good example of continuing positive feedback between theory and experiment. Theory shows what to look for; experiment shows what to try to understand. The first step in model development was fundamentally observational: for IO = 1 , with M / Z scaling, there are consistent rises on periods, and consistent mid-period similarity to Hydrogen ( Z = 1 ). For IO > 1 , there is consistent scaling with IO . There are several ways that the scaling can be described, and the simplest way found so far is summarized as follows: 1) First-order IP ’s contain ALL the information necessary to predict ALL higher-order IP ’s via scaling; 2) Every ionization potential IP of any order IO can be expressed as a function of at most two first-order IP ’s; 3) For a given ionization order IO > 1 , the ionization potentials for all elements start at element Z = IO , and follow a pattern similar to the IP ’s for IO = 1 , except for a shift to the right and a moderation of excursions. Details are given in [1].

Figure A2.2. First-order

IP ’s: map of main highways through the periods.

Only the first-order IP s are needed in the present work. They follow a definite pattern, detailed below. For every period, the rise is 7 / 2 , and the drops from one period to the next start at 7 / 8 and go to unity. Figure A2.2 shows the data used and the pattern inferred concerning periods. Within each period, there are internal rises keyed to the nominal quantum numbers of single-electron states being successively filled. Eqs. (A2-1a-1b) and Figure A2-3 give this level of detail.

incremental rise = total rise × fraction

(A2-1a)

246


fraction = ⎡⎢(2l + 1) / N 2 ⎤⎥ ⎡⎣( N − l) / l ⎤⎦ ⎣ ⎦

N

l

1 2 2 3 3 4 4

0 0 0 0 0 0 0

fraction 1 1/2 1/3 1/4 1/4 1/4 1/4

(A2-1b)

l

fraction

l

fraction

l

fraction

1 1 2 2 3 3

3/4 3/4 5/18 5/18 7/48 7/48

1 1 2 2

2/3 2/3 5 / 16 5 / 16

1 1

9 / 16 9 / 16

Figure A2.3. First-order

IP ’s: map of local roads through the periods.

The pattern described above has been inferred from the data. Let us be quite clear: knowledge about the pattern is presently wholly ‘ontic’ (about what the observable facts are); we also need knowledge that is ‘epistemic’ (about why the facts are that way). We definitely do not have this yet.

Appendix 3. Hydrogen, the Basis for all Scaling Laws The work reported in this paper derives from a sequence of earlier works [1-4] that are reviewed very briefly in this Appendix. The story really goes all the way back to Maxwell [4]. The differential equations that he bequeathed us admit a large diversity of particular solutions. In any application, the trick lies in choosing the right particular solution to fit the problem. An imperfect choice was made around the turn of the twentieth century. The problem was to describe an electromagnetic signal as the basis for developing special relativity theory (SRT). Einstein imagined a signal pulse, presumably of finite energy, that would propagate like a wave, infinite in extent and infinite in energy, forever undistorted at speed c = 1

ε 0μ 0 ,

where ε0 and μ 0 are electric permittivity and magnetic permeability of free space. Actually, that scenario isn’t possible. We were given a warning in the well-known phenomenon of diffraction: when limited in directions transverse to the nominal propagation direction, electromagnetic waves always develop a spread in propagation directions. We should have wondered what would happen if the electromagnetic wave were also confined in the longitudinal direction – as it would have to be, to make a wave packet of finite energy. Would some kind of spread result? Indeed, wouldn’t there have to be some kind of longitudinal spread in order for the concept of ‘wavelength’ to ever to become applicable? Well yes, it turns out that, following emission from a source, a pulse expands in the longitudinal direction, and preceding absorption into a receiver, the spread-out energy distribution contracts to pulse again.


247

Redeveloping SRT to use the more realistic model for an electromagnetic signal leads to a new theory that is expanded with respect to the original one. I call it ‘Expanded SRT’. It includes all the symbols and formulae that the original SRT contains. But it also includes an additional symbol, and additional formulae, and more precise interpretations of the old ones. The additional symbol is V for old-fashioned Galilean speed, distinguished from Einsteinian speed v , which is limited to light speed c . Galilean speed is not limited. With SRT thus extended, it is appropriate to review decisions that were based on the earlier ideas. One of these concerned the applicability of Maxwell’s electromagnetic theory (EMT) to problems at the small scale of atoms. Traditional QM was developed on the presumption that Maxwell theory could not apply. The reasoning was that an electron orbiting around a nucleus should produce radiation, which would rob the orbit of energy, causing collapse of the atomic system. But the Expanded SRT shows that the system also has a second physical process going on; namely, internal torquing. This one provides an energy gain mechanism. So it is possible to have the atomic system persist with a balance between the two physical processes. The new internal torquing mechanism is characterized by an energy gain rate

PT = (e4 / mp ) c(re + rp )3 .

(A3-1)

The more familiar radiation mechanism is characterized by an energy loss rate

PR = (25 e6 / me2 ) 3c3(re + rp )4 .

(A3-2)

4 This is enhanced by a factor of 2 over the totally familiar Larmor dipole formula. The enhancement is caused by Thomas rotation of the atomic system, which is in turn caused by non-central forces, which were absent from Newtonian physics, and were not originally noticed in Maxwellian physics.

The balance between the two mechanisms occurs when PT = PR . At the balance point for the Hydrogen system, we have Eq. (1) in the main text; i.e.,

e2 (re + rp ) = 3c 2 me2 25 mp .

(A3-3)

for the magnitude of the potential energy of the one electron in the Hydrogen atom. Eq. (A3-3) provides the basis from which to build. The extension to Deuterium and/or Tritium requires that the proton mass mp be replaced with a more generic nuclear mass M and that rp be replaced by rM . Then we have Eq. (2) in the main text; i.e.,

e2 (re + rM ) = 3c 2 me2 25 M .

(A3-4)

248


for the magnitude of the potential energy of this more massive system. The extension for a neutral atom with nuclear charge number Z , involves Z electrons as well. To develop an opinion on this question, we must return to Eqs. (A3-1) and (A3-2). 2 2 2 2 2 2 All the factors of e change to Z e , and the factor of me changes to Z me . PT = PR 4 6 2 4 becomes Z PT = ( Z / Z ) PR = Z PR . So nothing happens to the equality (A3-4). But for the more charged system, the magnitude of the potential energy becomes Eq. (3) in the main text; i.e.,

Z 2e2 (re + rp ) = Z 2 3c 2 me2 25 M .

(A3-5)

This scaled-up expression represents the magnitude of the total potential energy of the system involving Z protons and Z electrons. What is then comparable to the ionization potential for removing a single electron is Eq. (4) in the main text; i.e.,

(

)

(

Z e2 (re + rM ) = Z × 3c 2 me2 25 M ≡ (Z / M ) × 3c 2 me2 25

)

.

(A3-6)

Thus we see the Z / M scaling that is predicted for raw-data ionization potentials, and so is cancelled out by M / Z scaling to produce the IP ’s documented in Appendix 1 and used in Appendix 2. More generally, if the atom is in an ionized state, we have a distinct electron count Ze and proton count Z p . For the baseline nuclear-orbit part, we have for the total system Eq. (5) in the main text; i.e.,

)(

(

Z p Ze e2 (re + rM ) = Z p Ze M × 3c 2 me2 25

)

.

(A3-7)

What is then generally comparable to the nuclear-orbit part of the ionization potential for removing a single electron? To develop an opinion on this question, we must return again to 2 2 Eqs. (A3-1) and (A3-2). Clearly, all of the factors of e change to Z p Ze e . It is as if all factors of e changed to

Z p Ze e . Removal of one electron is then like removal of one

Z p Ze e charge. What is comparable to the ionization potential for removing a single electron is then Eq. (6) in the main text; i.e.,

Z p Ze e2 (re + rM ) =

(Z Z

p e

)(

M × 3c 2 me2 25

)

.

(A3-8)


249

Appendix 4. The Periodic Arch Figure A4.1 shows a new and convenient presentation of the Periodic Table from [1]. It is called the ‘Periodic Arch’ (PA).

Figure A4.1. The Periodic Arch (PA).

The ‘keystone’ elements referred to in the main text are the ones that fall at the keystone positions of the successive layers of the PA. They are important as facilitators of molecule formation because they can give or take electrons in equal numbers. The terminology ‘keystone element’ is appropriate for many other reasons as well. Hydrogen 1 H is certainly the ‘keystone’ for all of present-day physical analysis of atoms. Carbon 6 C is certainly the ‘keystone’ for all of organic chemistry and biological life. Silicon 14 Si is certainly the ‘keystone’ for present-day technological life. Cobalt 27 Co is not so famous, but it lies between Iron 26 Fe and Nickel 28 Ni , and is functionally better than either of them: harder, more corrosion resistant, and more heat resistant. And, like Iron, it is much strengthened by the addition of a trace of Carbon – that other keystone element. We humans are currently more than three millennia into our ‘Iron Age’, which, with the help of Carbon and other trace additives, has morphed into our ‘Steel Age’. Had Cobalt been more plentiful on this planet, this might have been our ‘Cobalt/Steel Age’. Rhodium 45 Rh is also not so famous, mainly

250


because it is not so plentiful, but it is good for plating and alloying. The remaining keystone elements, Ytterbium 70Yb , and Nobelium 102 No , remain to be exploited very much.

Acknowledgments A number of individuals have played important roles in fostering the development of this whole line of research. I am especially grateful to Michael C. Duffy, Ruggero M. Santilli, and Mihai V. Putz.

References and Notes [1] [2] [3] [4] [5] [6] [7] [8] [9]

Whitney, C.K. Closing in on Chemical Bonds by Opening up Relativity Theory. Int. J. Mol. Sci. 2008, 9, 272-298. Whitney, C.K. Single-Electron State Filling Order Across the Elements. Int. J. Chem. Model. 2008, 1, 105-135. Whitney, C.K. Visualizing Electron Populations in Atoms. Int. J. Chem. Model. 2008, 1, 245-297. Whitney, C.K. Recent Progress in Algebraic Chemistry. Computational Chemistry: New Research, Frank Columbus, Ed., Nova Science Publishers, in press. Clary, D.C. Quantum Dynamics of Chemical Reactions. Science 2008, 321, 789-791. Zhang, D.H., Zhang, J.Z.H., J. Chem. Phys. 1994, 101, 1146. Zhang, D.H., J. Chem. Phys. 2006, 125, 133102. Yang, M.H.; Zhang, D.H., Lee, S.Y., J. Chem. Phys. 2002, 117, 9539. Zhang, X., Braams, B., Bowman, J.M., J. Chem. Phys. 2006, 124, 021104.



Chapter 11

QUANTUM AND ELECTRODYNAMIC VERSATILITY OF ELECTRONEGATIVITY AND CHEMICAL HARDNESS Mihai V. Putz* Laboratory of Computational and Structural Physical Chemistry, Chemistry Department, West University of Timişoara, Str. Pestalozzi No.16, Timisoara, RO-300115, Romania

Abstract Aiming to affirm specific physical-chemical quantities of electronegativity and hardness as the major electronic indicators of structure and reactivity, their systematic density functional, as well their quantum and electrodynamic field formulations are presented; it may serve for analytical studies of chemical bonding, reactivity, aromaticity, up to the biological activity modeling of atoms in molecules and in nanostructures.

1. Introduction In the last years, the first rate scientific research was mainly focused on the synergistic approaches of the structure and properties of the natural complex systems at the quantum chemical level [1-7]. Yet, while the pure physics still struggles on the great unification paradigm through the fundamental forces in nature, being in the last decades subject to a continuous reform, a similar attitude is now emerging in chemistry, at the quantum level of representation, related with the existing natural chemical bonds: the ionic, covalent, metallic, hydrogenic and the van der Walls (as driven, induction and diffusion) ones. Because the types of chemical bonds coexist in various degrees and combination through the matter organization, only a unitary quantum treatment, based on the first physical-chemical principles, can release an estimation of the structure-properties correlations across the complex natural nano-systems: metals, clusters, fullerenes, liquid crystals, polymers, ceramics, biomaterials, metaloenzymes. As a *

E-mail addresess: [email protected], [email protected]. Tel: +40-256-592633; Fax. +40-256-592620,

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Mihai V. Putz

synergistic field of research, the nanosystems have received many connotations. When the spatiality of the chemical bond is studied, e.g., when the atomic systems condensate into a smaller volume than of the isolated components, the arising composite nanosystems display exceptional properties of coherences, used afterwards in processing, storing and communication of the quantum information [8-10]. On the other hand, when dealing with the chemical concentration of elements, it has already been proven that the range of nano-molar better reflects the complex bio –organic and –inorganic combinations, especially when focusing on the doze zones of responses for an essential element, with a role in selection or inhibition of a certain biological function in organisms, with effects on the growth and reproduction of cells and living organisms [11]. This way, a unitary picture to link and flexibly adapt the quantum mechanical formalisms at the chemical bonding level was intensively searched [12]. In such studies, it was recently established that for an adequate treatment in the quantum space of the polyatomic combinations stays the electronic density ρ (r ) rather than the already historical wave function ψ (r1 ,...rN ) as the main variable for a system with N electrons. This because, on the contrary to the wave function, the electronic density is an experimentally detectable quantity, is defined in the real three-dimensional space, and not within a 3N Hilbert abstract one, being also directly related with the total number of electrons in the concerned system through the functional relation [13-20]:

∫ ρ (r)dr = N

(1)

Therefore, the electronic density receives the central role within the newest quantum paradigm of matter, the Density Functional Theory (having Walter Kohn as its father, Nobel laureate in Chemistry for this theory in 1998) [15]. On the other hand, the reactivity indices’ studies are essential for indicating the propensity of a multielectronic system to participate in a chemical reaction. At the molecular level, these indices are defined in order to quantitatively measure the chemical reactivity, while at the biomolecular level they are associated with the biological activity. Thus, as the reactivity indices are informationally placed at the interface between the electronic systems’ stability and their tendency to transform and combine, they are mathematically introduced as the integral functions of the electronic density function, releasing the so-called electronic density functionals as the efficient tool for the global prediction of the electronic properties of the investigated nanosystems [21]. Although the Density Functional Theory offers concepts, e.g., the density functionals, the reactivity indices or the localization functions, with an exact formal character, still the computational effort to evaluate the electronic densities for the polyatomic systems is often immense and not without being susceptible to errors from the numerical recipes used. Also, many times, the chemical intuition is totally hidden within the routines and the basis sets chosen for implementations [12]. Faced with such programmatic problems, a closed form solution is searched, with a proper phenomenological impact relative to the qualitative-quantitative predictive character of the chemical reactivity and the biological activity when characterizing the complex natural nanosystems.

Quantum and Electrodynamic Versatility of Electronegativity…

253

However, there was recently realized a suitable revisited point of view that we should recall the basic atomic structure, and from that only treating the valence shell as the main ingredient of the chemical interactions. Then, it was searched for an intermediate concept and quantity between the single- and polyatomic systems such that an iterative construction, that adequately describe the hierarchy of matter organization, to be possible in reflecting the specific structure and interaction. In this context, it was recently established that the complete description of a polyatomic system both at equilibrium and in interaction may be realized through a minimal set of quantum observables, containing an electronic density functional derived from energy functional expansion [22,23]:

1 ⎛ ∂2E ⎞ ⎛ ∂E ⎞ ⎟ (ΔN )2 + ... E[ N + ΔN ] = E0 [ N ] + ⎜ ⎟ ΔN + ⎜⎜ 2 ⎟ 2 ⎝ ∂N ⎠V ⎝ ∂N ⎠V

(2)

When restrained to the second order, as the most common perturbative approach, eq. (2) provides the variation of energy relationship with the charge variation

ΔE = − χΔN + η (ΔN )

2

(3)

by means of the chemical electronegativity [24-28]

⎛ ∂E ⎞ ⎟ ⎝ ∂N ⎠V

χ = −⎜

(4)

and chemical hardness [29-37]

1 ⎛ ∂2E ⎞ 1 ⎛ ∂χ ⎞ η = ⎜⎜ 2 ⎟⎟ = − ⎜ ⎟ 2 ⎝ ∂N ⎠V 2 ⎝ ∂N ⎠V

(5)

indices, in the field of external applied potential V. It appears that this simple energy-charge correlation furnishes the basics of the chemicalphysical phenomenology either in isolate and reactive state [38-44], as well in bonding modeling [45-52] up to the most recent reactive biological activity (ReBiAc) principles [53]. In this context, the present work likes to review few of the most intriguing forms of electronegativity and chemical hardness emphasizing the versatility of these reactivity indices in reflecting various energetic (energy) density functionals, of the second quantized or electrodynamical fields.

2. Density Functionals of Electronegativity and Hardness 2.1. Absolute Electronegativity and Hardness Aiming to elaborate the implications of eqs. (3)-(5), if one likes to model the energy consumed in forming the AB bonding through the gauge equilibrium reactions [54,55]:

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Mihai V. Putz

A − + B + ↔ AB ↔ A + + B −

(6)

may equivalently write

ΔE = E NA0 −1 + E NB0 +1 − E NA0 +1 − E NB0 −1

(

) (

) (

) (

= E NA0 −1 − E NA0 + E NA0 − E NA0 +1 − E NB0 −1 − E NB0 − E NB0 − E NB0 +1

)

= IP A + EA A − ( IP B + EA B ) = 2( χ A − χ B )

(7)

thus establishing two important facts: •

The so called Mulliken electronegativity considered in (7) in terms of ionization potential (IP) and electronic affinity (EA) [56]

χM ≡

IP + EA 2

(8)

may be viewed as the (chemical) finite difference approximation of the differential electronegativity (4)

χM ≡

IP + EA ( E N 0 −1 − E N 0 ) + ( E N 0 − E N 0 +1 ) ⎛ ∂E ⎞ = ≅ −⎜ N ⎟ ≡ χ , 2 2 ⎝ ∂N ⎠V

(9)

while furnishing as well the more general form of absolute electronegativity by means of the integral [57,58]: N +1

1 0 χ A = − ∫ dE N 2 N 0 −1 •

(10)

Eq. (7) prescribes that in order to proper describe the reactive propensity of the chemical systems the change in electronegativity dχ P has to be also considered, eventually averaged against the interval (N0-1, N0+1)

1 = 2

1 N 0 +1

∫ dN

N 0 −1

(11)


255

in accordance with above (5) introductory definition, towards the so called absolute hardness: N +1

1 0 η A = − ∫ dχ 2 N 0 −1

(12)

that unfortunately cannot give particular information as far χ(N) remains unknown. However, worth noting that when considering the eq. (11) as the factor for eq. (12) one accounts for the average of the acidic (electron accepting, N0 ≤ N ≤ N0+1) and basic (electron donating, N0–1 ≤ N ≤ N0) behaviors, being therefore an inherent part of the hardness definition, although equal arguments for skipping it may be as well considered; this is at the end only a scaling factor and remains to be chosen as per whish for the given problem at hand (and somehow depending on the taste of the investigator) [59-61]. Now, unlike the differential expressions (4) and (5), the absolute electronegativity and hardness take the symmetrical forms of (10) and (12) in terms of their “potential” (i.e. their cause), the (total – for ground state or valence – for excited states) energy and (differential) electronegativity, respectively. Moreover, eqs. (10) and (12) highly advocates the energy and electronegativity total differentials as furnishing the main thermodynamical (quantum) equation for achievement a systematic development of chemical reactivity in terms of changing charges and applied potential, N and V(r), respectively. Nevertheless, this electronegativity-hardness symmetric formulation produces an elegant unification of the χtheories and the hard-and-soft-acids-and-bases (HSAB) rules beyond the ionization potentials and electronic affinities [43,44].

2.2. Systematic Electronegativity and Hardness Since the established absolute electronegativity and hardness dependence on energy and differential electronegativity with the forms (10) and (12), for an N-electronic system, being under the external potential influence V(r), their respective functionals E = E[ N ,V (r )] and

χ = χ[ N ,V (r)] may be next employed either as total differential equation or as perturbative expansions in order to systematically obtain electronegativity and chemical hardness density functionals. For total differential equation of energy and electronegativity one has the working forms:

⎛ δE ⎞ ⎛ ∂E ⎞ ⎟⎟ δV (r)dr = − χ dN + ∫ ρ (r )δV (r )dr , dE = ⎜ ⎟ dN + ∫ ⎜⎜ δ V ( r ) ⎝ ∂N ⎠V ⎝ ⎠N

(13)

⎛ δχ ⎞ ⎛ ∂χ ⎞ ⎟⎟ δV (r ) dr = −2ηdN + ∫ f (r )δV (r )dr dχ = ⎜ ⎟ dN + ∫ ⎜⎜ ⎝ ∂N ⎠V ⎝ δV (r ) ⎠ N

(14)

being the last relation introducing the Fukui index [45-50]

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Mihai V. Putz

⎛ δχ ⎞ ⎛ ∂ρ (r ) ⎞ ⎟⎟ = ⎜ − ⎜⎜ ⎟ = f (r ) ⎝ δV (r ) ⎠ N ⎝ ∂N ⎠V

(15)

based on the Cauchy property of the total differentiable quantities of eq. (13), playing the role in locally revealing the reactivity nature of the system being a descriptor which measures how sensitive a system’s electronegativity is to an external perturbation at a particular space point. These equations may be now employed to furnish various systematic realizations of electronegativity and hardness, either under chemical or absolute forms. For instance, refereeing to the energy equation (13), while considering Δ as the change in electrons restricted to the valence shell, it can be appropriately integrated: E ( N v ± Δ) − E ( N v ) = −

{

Nv ±Δ

}

∫ χ dN + ∫ ρ (r) ∫ δ [V (r)] dr = − Nv

Nv

Nv ±Δ

∫ χ dN + ∫ ρ

Nv

(r )V N v (r )dr (16)

Nv

to give the electronic affinity and ionization potential working expressions

− EA = −

N v +1

∫ χdN + C A ,

(17)

Nv

IP = −

N v −1

∫ χdN + C A

(18)

Nv

assuming the variation of electronic charge to be unity ( Δ = 1 ) around the number of concerned valence electrons, Nv; note the appearance of the so called chemical action [18, 62-64]

C A = ∫ ρ N v (r )V N v (r )dr

(19)

with a major role in chemical bonding and reactivity, see [52] and Chapter 1 of this edited monograph. Further on, the ionization potential and electron affinity of eqs. (17) and (18) may be combined amomg them so that when replaced into the finite difference approximations of derivative forms of electronegativity and hardness of eqs. (4) and (5) to produce their chemical forms

χC =

N +1

IP + EA 1 v = χdN , 2 2 N v∫−1

IP − EA ηC ≡ = χC − 2

N v +1

∫ χdN + C A

(20)

(21)

Nv

which may be regarded as working forms of the absolute definitions (10) and (12), respectively.


257

Table 1. Different electronegativity (left column) and hardness (right column) in the absolute (first two rows) and chemical (last two rows) formulations relating the local and global softness contributions [18]

χ Nv

χ=−∫ 0

η 1 ⎛ ∂χ ⎞ ⎟ 2 ⎝ ∂N ⎠V

1 1 dN − ∫ s ( x)V ( x)dx S S

ηχ = − ⎜

1 2S 1 ⎛ ∂χ ⎞ η Cχ = − ⎜ C ⎟ 2 ⎝ ∂N ⎠V

ηS = χC =

N +1

1 v χdN 2 Nv∫−1

η CCA = χ C −

N v +1

∫ χdN + C A

Nv

This step, however, succeed in expressing both the chemical electronegativity and hardness as uniformly depending on single “kernel” electronegativity, which can be, instead, expressed from employing the above electronegativity total differential equation (14). To this end, one makes use of the so called local softness definition [33,36] ⎛ ∂ρ (r ) ⎞ s (r ) = −⎜⎜ ⎟⎟ ⎝ ∂χ ⎠V ( r )

(22)

which through integrating to the global softness

⎛ ∂N S = ∫ s (r ) dr = −⎜⎜ ⎝ ∂χ D

⎞ 1 ⎟⎟ = ⎠ V ( r ) 2η

(23)

allows for rewriting of eq. (14) as

dχ = −

1 s (r ) dN − ∫ dV (r )dr S S

(24)

followed by the formal integration: Nv

χ =−∫ 0

1 s (r ) ⎛⎜ V (r ) ⎞⎟ dN − ∫ dV dr = − ⎟ S S ⎜⎝ ∫0 ⎠

Nv

∫ 0

1 1 dN − ∫ s (r )V (r )dr S S

(25)

From practical implementation point of view, one can identify the form (25) as the realization of the absolute electronegativity (10) to be inserted in the chemical forms of

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Mihai V. Putz

electronegativity and hardness as prescribed by eqs. (20) and (21), with the analytical advantage that now the dependence was moved on the knowledge of local softness rather than on the total energy. To summarize, all working absolute and chemical levels of electronegativity and hardness are collected in the Table 1. The final step in this algorithm regards the way in which the softness influence is considered. For that one consider the complete softness hierarchy from the global to local to kernel contributions

1 ≡ S = ∫ s (r )dr = ∫∫ s (r , r ' )drdr ' 2η

(26)

and to asses for local and non-local effects the specific density related terms; as such under three quantum mechanical constraints, namely the translational invariance condition, the Hellmann-Feynman theorem, and the normalization of the linear response function, it leaves with the approximate formulathere was derived the approximate formula [16]:

s(r, r ' ) = L(r)δ (r − r ' ) + ρ (r ) ρ (r' )

(27)

with the local response function

L(r ) =

being

∇ρ (r) ⋅ [− ∇V (r )]

[− ∇V (r)]2

(28)

δ (r − r ' ) the delta-Dirac function. Therefore, going back by performing the

successive integrations of (27) one gets for the local and global softness the respective results:

s (r ) = L(r ) + Nρ (r ) ,

(29)

S = a+ N2

(30)

a ≡ ∫ L(r )dr

(31)

where the short-hand notation:

was considered. Now, while supplementing the reactivity softness related index (31) by the associate one which appears as the last term in integration (25),

b ≡ ∫ L(r )V (r )dr

(32)

the various explicit density functional of absolute and chemical electronegativity and hardness are obtained, within various approximation levels of charge and applied potential order, as systematized in Table 2 [54].

Table 2. The absolute electronegativity and hardness density functionals as results from electronegativity and total energy expansions within different combination between the first and second order of charge and first order of external potential variations, respectively. The notations a, b, and CA corresponds to integrals of eqs. (31), (32), and (19), respectively [54] Electronegativity Sources

dχ

dE

Absolute (Softness)

N +1

Nv

χ = ∫ dχ

χ =−

0

1 − dN S

1 − dN S ⎡ ∂ ⎛ 1 ⎞⎤ − ⎢ ⎜ ⎟⎥dNdN ⎣∂N ⎝ S ⎠⎦

1 dN =− a + N2 N dNdN + 2 a + N2

(

)

− χdN

⎛N ⎞ arctan⎜⎜ v ⎟⎟ − a ⎝ a⎠

− χdN

⎛N ⎞ 1 − arctan⎜⎜ v ⎟⎟ a ⎝ a⎠

1 ⎛ ∂χ ⎞ − ⎜ ⎟ dNdN 2 ⎝ ∂N ⎠V

1

+

Hardness Absolute

Absolute (Energy)

(

Nv − a arctanNv / a 2a

)

1 v dE 2 N∫v −1

N +1

η

[1] A

1 v = − ∫ dχ 2 N v −1

⎧ ⎡ ⎛ Nv −1⎞ ⎤⎫ ⎟⎟ ⎥⎪ ⎪ ⎢(Nv −1) arctan⎜⎜ ⎝ a ⎠ ⎥⎪ ⎪1 ⎢ ⎪ ⎪ 1 ⎪ a ⎢⎢− (N +1)arctan⎛⎜ Nv +1⎞⎟⎥⎥⎪ v ⎨ ⎜ ⎟⎬ 2 ⎪ ⎢⎣ ⎝ a ⎠⎥⎦⎪ ⎪ ⎪ 1 ⎡a + (N +1)2 ⎤ v ⎪ ⎪+ ln⎢ ⎥ ⎪⎭ ⎪⎩ 2 ⎢⎣ a + (Nv −1)2 ⎥⎦

⎡ ⎛ Nv +1⎞ ⎤ ⎟⎟ ⎥ ⎢arctan⎜⎜ 1 ⎢ ⎝ a ⎠ ⎥ 2 a⎢ ⎛ N − 1 ⎞⎥ ⎢− arctan⎜⎜ v ⎟⎟⎥ ⎢⎣ ⎝ a ⎠⎥⎦

⎫ ⎧ ⎡a + (Nv +1)2 ⎤ ⎪ ⎪4Nv + 3aln⎢ ⎥ 2 ⎪ ⎪ ⎣ a + (Nv −1) ⎦ ⎪ ⎪ 3 ⎪ ⎡ ⎛ Nv −1⎞ ⎤⎪ ⎨ ⎟ ⎥⎬ ⎢( Nv −1)arctan⎜ 16a ⎪ ⎝ a ⎠ ⎥⎪ ⎢ ⎪ ⎪+ 6 a ⎢ ⎛ N +1⎞⎥ ⎪ ⎢− (Nv +1)arctan⎜ v ⎟⎥⎪ ⎪⎩ ⎝ a ⎠⎦⎥⎪⎭ ⎣⎢

⎧ ⎛ N + 2⎞ ⎫ ⎛ Nv − 2 ⎞ ⎟ + arctan⎜ v ⎟ ⎪ ⎪arctan⎜ a ⎝ a ⎠ ⎪ ⎠ ⎝ ⎪ ⎪ 1 ⎪ ⎡ ⎛ Nv −1⎞ ⎛ N ⎞⎤⎪⎪ ⎟ + arctan⎜ v ⎟⎥⎬ ⎨ ⎢arctan⎜ 4 a⎪ ⎢ ⎝ a ⎠ ⎝ a ⎠⎥⎪ −2 ⎪ ⎢ ⎥⎪ ⎛ N +1⎞ ⎪ ⎢− arctan⎜ v ⎟ ⎥⎪ ⎪⎩ ⎣⎢ ⎝ a ⎠ ⎦⎥⎪⎭

Table 2. Continued Electronegativity Sources

1 dN S s (r ) −∫ δV (r )dr S

−

1 − dN S ⎡ ∂ ⎛ 1 ⎞⎤ − ⎢ ⎜ ⎟⎥dNdN ⎣∂N ⎝ S ⎠⎦ −∫

s(r) δV(r)dr S

− χdN + ∫ ρ (r )δV (r )dr

Absolute (Softness)

⎧ ⎡ ⎛ Nv −1⎞ ⎤⎫ ⎟ ⎥⎪ ⎪ ⎢(b + Nv −1)arctan⎜ ⎝ a ⎠ ⎥⎪ ⎪1 ⎢ ⎛ Nv +1⎞⎥⎪⎪ 1 ⎪⎪ a ⎢ ⎟⎥⎬ ⎨ ⎢− (b + Nv +1)arctan⎜ 2 ⎪ ⎢⎣ ⎝ a ⎠⎥⎦⎪ ⎪ ⎪ C −1 ⎡a + (N −1)2 ⎤ v ⎪ ⎪+ A ln⎢ 2⎥ ⎪⎭ ⎪⎩ 2 ⎣a + (Nv +1) ⎦

1 ⎛N ⎞ arctan⎜ v ⎟ a ⎝ a⎠ b + NvCA − a + N v2

−

− χdN

−

1 ⎛ ∂χ ⎞ − ⎜ ⎟ dNdN 2 ⎝ ∂N ⎠V

+

+ ∫ ρ (r)δV (r)dr

Absolute (Energy)

1 ⎛N ⎞ arctan⎜ v ⎟ a ⎝ a⎠

(

N v − a arctan N v / a 2a b + NvC A − a + N v2

)

⎫ ⎧ ⎪ ⎪ ⎪ ⎪4Nv ⎪ ⎡ ⎤⎪ ⎪ (2b + 3Nv − 3)arctan⎛⎜ Nv −1⎞⎟ ⎥⎪ ⎢ 3 ⎪ ⎝ a ⎠ ⎥⎪ = ⎬ ⎨+ 2 a ⎢⎢ 16a ⎪ ⎛ N + 1 ⎞⎥ − (2b + 3Nv + 3)arctan⎜ v ⎟⎥⎪ ⎢ ⎪ ⎝ a ⎠⎦⎥⎪ ⎣⎢ ⎪ ⎪ ⎪ ⎪ ⎡ a + ( Nv − 1)2 ⎤ ⎪ ⎪+ a(2CA − 3) ln⎢ 2⎥ ⎣ a + ( Nv + 1) ⎦ ⎭ ⎩

[

Hardness Absolute

]

⎧ 2 CA (1 + a − Nv2 ) − 2bNv ⎫ ⎪ 2 2 4⎪ ⎪(1 + a) + 2(a −1)Nv + Nv ⎪ ⎡ 1 ⎪⎪ ⎛ Nv +1⎞ ⎤ ⎪⎪ arctan ⎜ ⎟ ⎥⎬ ⎨ 2 ⎪ 1 ⎢⎢ ⎝ a ⎠ ⎥⎪ + ⎪ a⎢ ⎛ N −1⎞⎥ ⎪ ⎪ ⎢− arctan⎜ v ⎟⎥ ⎪ ⎪⎩ ⎝ a ⎠⎦⎥ ⎪⎭ ⎣⎢ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 4 C 1+ a − N 2 − 2bN ⎪ A v v ⎪ ⎪ 2 2 4 ⎪(1+ a) + 2(a −1)Nv + Nv ⎪ ⎪ ⎪ ⎡ ⎛ Nv − 2 ⎞ ⎤ ⎪ ⎪ arctan ⎜ ⎟ ⎢ ⎥ 1 ⎪⎪ 1 ⎢ ⎝ a ⎠ ⎥ ⎪⎪ − ⎨+ ⎬ ⎥ 4⎪ a ⎢ ⎛ N + 2⎞ ⎪ + arctan⎜ v ⎟⎥ ⎢ ⎪ ⎪ ⎢ ⎝ a ⎠⎦⎥ ⎣ ⎪ ⎪ ⎪ ⎡ ⎤⎪ ⎛ Nv −1⎞ ⎟ ⎪ ⎢arctan⎜ ⎥⎪ ⎝ a ⎠ ⎪2 ⎢ ⎥⎪ ⎪ a⎢ ⎛ Nv +1⎞⎥⎪ ⎛ Nv ⎞ ⎪ ⎢+ arctan⎜ ⎟ − arctan⎜ ⎟⎥⎪ ⎝ a ⎠⎦⎥⎭⎪ ⎝ a⎠ ⎩⎪ ⎣⎢

[ (

)

]


261

By analyzing the Table 2 there is remarked that the chemical electronegativity of eq. (20) is obtained as a special case of the absolute one of eq. (10) when the total or valence energy is restrained itself as to the first order variation in charge, i.e. the fourth row of Table 2, while the total differential equation of electronegativity and energy corresponds to the sixth row of Table 2, providing a reasonable complex electronegativity and hardness density functionals to be further used in modeling chemical bonding and reactivity [55]. Yet, although with such variety of quantum formulations for electronegativity and hardness, depending of the chemical reactivity framework, appears the fundamental question whether the basic definitions (4) and (5), which opened all the above exposed analytical phenomenology, are of intrinsic quantum nature to be then implicitly subsisting in any other related formulation. This matter will be in next addressed.

3. Electronegativity and Chemical Hardness by Second Quantization 3.1. Affinity and Ionization Fields by Second Quantization Starting from the bi-dimensional unitary operator on the Fock electronic space F{ 0

1ˆ = 0 0 + 1 1 = aâˆ + + aˆ + aˆ = {aˆ,aˆ + }

,1

}

(33)

one may easily introduce the annihilation and creation operators respectively as:

aˆ = 0 1 ,

(34a)

aˆ + = 1 0

(34b)

aˆ + 0 = 1 0 0 = 1 ,

(35a)

noting their fundamental actions

aˆ 1 = 0 1 1 = 0

(35b)

throughout fulfilling the fundamental ortho-normal rules

0 1 = 1 0 = 0,

(36a)

0 0 = 1 1 =1

(36b)

for the vacuum and single electronic states, respectively.

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Mihai V. Putz

Going now to treat the electronegativity and hardness by means of the second quantization, one will consider the “valence state” reality as characterized by the unperturbed stationary state ψ 0 with associated eigen-energy E0 :

Hˆ ψ 0 = E 0 ψ 0

(37)

Yet, the normalization constrain for the valence wave-function allows the unitary operator decomposition on the vacuum and uni-particle occupancies as:

1 = ψ 0 ψ 0 = ψ 0 1ˆ ψ 0 = ψ 0 (aâˆ + + aˆ + aˆ ) ψ 0 = ψ 0 aâˆ + ψ 0 + ψ 0 aˆ + aˆ ψ 0 2

= 0ψ0

+ 1ψ0

2

= (1 − ρ 0 ) + ρ 0 , ρ 0 ∈ [0,1]

(38)

and for further identifying the wave function projections:

0 ψ 0 = ψ 0 0 = 1 − ρ0 ,

(39a)

1 ψ 0 = ψ 0 1 = ρ0

(39b)

In these conditions, the valence wave-function may be modified such that the affinity and ionization chemical states are written as corrections or perturbations added towards the occupancy or vacuum quantum states, respectively as:

(

)

ψ λA = 1 + λaˆ + aˆ ψ 0 = ψ 0 + λ 1 0 0 1 ψ 0 = ψ 0 + λ ρ0 1 ,

(40a)

ψ λI = (1 + λaâˆ + ) ψ 0 = ψ 0 + λ 0 1 1 0 ψ 0 = ψ 0 + λ 1 − ρ0 0

(40b)

These quantum chemical affinity and ionization electronic states are to be used to closely characterize the electronegativity and hardness fields in a way to resolve the question of their observable character.

3.2. Observability of Electronegativity and Hardness Aiming to evaluate the electronegativity and hardness “response” to the affinity and ionization perturbations of eqs. (40) one employs their basic definitions (4) such that to include the perturbative effect [65,66]:

χλ = −

δ Eλ ∂ E λ ∂λ =− , δρ λ ∂λ ∂ρ λ

(41a)

Quantum and Electrodynamic Versatility of Electronegativity… 2 1 ∂ Eλ 1 ⎧⎪⎡ ∂ ⎛ ∂ E λ ηλ = = ⎨⎢ ⎜ 2 ∂ρ λ2 2 ⎪⎩⎣⎢ ∂λ ⎜⎝ ∂λ

⎞⎤ ∂λ ∂ E λ ⎡ ∂ ⎛ ∂λ ⎟⎥ ⎜ ⎟ ∂ρ + ∂λ ⎢ ∂λ ⎜ ∂ρ ⎠⎦⎥ λ ⎣ ⎝ λ

⎞⎤ ⎫⎪ ∂λ ⎟⎟⎥ ⎬ ⎠⎦ ⎪⎭ ∂ρ λ

263

(41b)

by iteratively employing the chain-derivation rule

∂ • ∂ • ∂λ = ⋅ ∂ρ λ ∂λ ∂ρ λ

(42)

From expressions (41) appears that the main components of electronegativity and hardness are the density and energy derivatives respecting the perturbation factor λ . Therefore they both will be unfolded respecting the affinity and ionization chemical states (40) in a special way so that to record their reciprocal transition (no matter in which temporal order), while normalized at their scalar product for limiting occupancy 0

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