Multi-Component Molecular Orbital Theory

MULTI-COMPONENT MOLECULAR ORBITAL THEORY 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.

MULTI-COMPONENT MOLECULAR ORBITAL THEORY

TARO UDAGAWA AND

MASANORI TACHIKAWA

Nova Science Publishers, Inc. New York

Copyright © 2009 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. 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. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA ISBN: 978-1-60741-912-9 (E-Book) Available upon request

Published by Nova Science Publishers, Inc.  New York

CONTENTS

Preface

vii

Chapter1

Introduction

1

Chapter 2

Multi-Component Molecular Orbital Methods

5

Chapter 3

Application of the Multi-Component Methods

21

Chapter 4

Conclusions

49

References

51

Index

57

PREFACE We report the multi-component first-principles methods which can take account of the quantum effect of light particles such as proton and positron, as well as electron. In particular, we introduce our multi-component molecular orbital (MC_MO) methods and multi-component “hybrid type” density functional theory (MC_DFT). Using these multi-component procedure, we can analyze many chemical phenomena, H/D isotope effect, positronic systems, and so on. We will show some examples of MC_MO and MC_DFT works.

Chapter1

INTRODUCTION The conventional molecular orbital (MO) methods which based on BornOppenheimer (BO) approximation[1] have been developed to describe the electronic motions. In fact, these conventional MO methods successfully describe the electronic structures and provide us useful information for some chemical phenomena. However, nuclear motion is known to be very important to elucidate the hydrogen-bonded systems and hydrogen/proton transfer reactions[2,3]. Furthermore, H/D isotope effects which are caused by difference of the mass between proton and deuteron induce many chemical phenomena. Geometrical parameters are changed by H/D isotope effect, and this geometric isotope effect (GIE) induces drastic change of phase transition temperature in ferroelectric materials. According to such background, the quantum mechanical description of nuclear motion is a problem of central interests in physics, chemistry, and interdisciplinary fields[4-7]. To obtain the nuclear motion or nuclear wave function within the framework of BO approximation, it is necessary to get a potential energy hypersurface. Many theoretical methods considering quantum nature on hydrogen nucleus are developed in the framework of BO approximation, such as path integral molecular dynamics method[7,8], surface hopping trajectory simulation[9,10], and grid-based BO analysis[11]. However, since the potential energy hypersurface corresponds to the eigenvalues with the electronic Hamiltonian for all possible nuclear configurations, this treatment is possible to only few-atoms containing system. On the other hand, nuclear motion is also described by extension of the MO method, in which both electrons and nuclei are simultaneously treated quantummechanically. Generalization of MO to a multi-component (MC) system can

2

Taro Udagawa and Masanori Tachikawa

include the non-adiabatic coupling between electronic and nuclear motions, as well as the quantum nature of the nuclei. The pioneering work of this theory was presented by Thomas[12-15] in 1969. He added the kinetic energy operators of nucleus to the electronic Hamiltonian and calculated HF, H2O, NH3, and CH4 molecules with fixed Slater-type basis functions. Monkhorst[16] derived the coupled-cluster formulation for MC systems. Recently, we have proposed “Multi-component MO (MC_MO)”, and successfully analyzed various chemical and physical phenomena, such as phase transition of ferroelectric materials[17], geometrical H/D isotope effect of C-H...O hydrogen bonds[18,19], kinetic H/D isotope effect in hydrogen transfer systems[20], and so on. Our MC_MO method can determine both nuclear and electronic wavefunctions simultaneously, and expresses the isotope effect including coupling effects between nuclei and electrons directly. In the conventional MO method, only the electronic state is described under the condition of fixed nuclei, i.e., the motion of electrons is evaluated in the field of fixed nuclear charges. It is noted that this electronic Hamiltonian expresses only the electronic state, and no operator terms for the nuclear kinetic energy are taken into account. As mentioned above, in order to describe the isotope effect of nuclei, one must solve the nuclear motion of the adiabatic potential under the BO approximation. Although the nonadiabatic effects are very small and the adiabatic approach suffices for most chemical systems, the coupling effect between electronic and nuclear motions and the nonadiabatic effect would be very important in the case of a system containing the hydrogen atom; therefore, the lightest nuclei should be treated as a quantum wave by appropriate method, such as our MC_MO. Several groups also studied by using this multi-component MO method. Nakai and coworkers[21-24] have also developed the nuclear orbital plus molecular orbital (NOMO) method, where translational and rotational motions are eliminated from corresponding Hamiltonian[24]. Hammes-Schiffer and coworkers[25-32] have developed the nuclear-electronic orbital (NEO) method and extended it to CI[25,31], MP2[30], multiconfigurational self-consistent-field (MCSCF)[25], and explicitly correlated Hartree-Fock (XCHF) method[32]. Shigeta and coworkers[33,34], and Gross[35] and coworkers have developed the multi-component density functional theory (MCDFT), and analyze the isotope effect of hydrogen molecule and its isotopomers. Adamowicz and coworkers[3646] have developed a explicit correlated Gaussians (ECG) for nonadiabatic calculations. They have calculated polarizability of several small molecules within ultra-precision with their ECG, and they also have presented the improved nonadiabatic ground-state energy upper bound for several small molecules.

Introduction

3

Nowadays, many methods of multi-component MO have been developed. Details of each theory and work would be introduced in next chapter.

Chapter 2

MULTI-COMPONENT MOLECULAR ORBITAL METHODS 2-1. PIONEERING WORKS BY I. L. THOMAS In 1969, the first study of multi-component systems in terms of molecular orbital method was reported by Thomas[12-15]. He added the nuclear kinetic term to conventional “nuclear fixed” Hamiltonian. Then Hamiltonian has a form

H =∑ a

Z Z Z 1 1 1 Δ a − ∑ Δ i − ∑∑ a + ∑ + ∑ a b 2m a i 2 a i rai i < j rij a j r μ >ν r ij μν ⎠ ,

(5)

where the i and j indices refer to the electrons, μ and ν to the nuclei and μ represents the nuclear charge. In a conventional molecular orbital (MO) calculation, the time-independent Schrödinger equation of the electronic Hamiltonian is solved approximately using the variational method with the fixed nuclei. Another way of saying, the motion of electrons is evaluated in the field of fixed nuclear charges[51,52]. This electronic Hamiltonian expresses only electronic states and includes no nuclear kinetic energy operator terms. To analyze the nuclear motion in the framework of Born-Oppenheimer (BO) approximation[1], we have to solve the electronic Hamiltonian for all possible nuclear configurations. Of course, this procedure is not practical and only applicable to few atom-containing systems.

Z

Multi-Component Molecular Orbital Methods

7

According to these backgrounds, we have extended a concept of MO to light particles such as proton, deuteron, positron, and muon, to analyze the nuclear motions. In order to obtain both the electronic and nuclear wavefunctions simultaneously, we use the total Hamiltonian which includes not only the electronic part but also the quantum-treated nuclear part: M Z ⎛ 1 2 M Z μ ⎞ Ne 1 N p ⎛ ⎟ + ∑ + ∑ ⎜ − 1 ∇ 2p + ∑ μ H tot = ∑ ⎜ − ∇ i − ∑ ⎜ 2 ⎟ i> j r ⎜ 2m i =1 ⎝ p =1 ⎝ μ =1 riμ ⎠ μ =1 r pμ ij p Np Z p Z q N e N p Z p M Z μ Zν +∑ − ∑∑ +∑ μ >ν rμν p > q r pq i p rip Ne

⎞ ⎟ ⎟ ⎠

,

(6)

where p and q indices refer to the quantum-treated nuclei. The total wavefunction could be expressed as the full-configuration interaction (CI) form: ⎛ Ψ = ∑ Φ L C L = Φ 0A Φ 0B Λ Φ 0M C0 + ⎜⎜ ∑∑ Λ L ⎝ LA LB

'

∑ LM

⎞ A ⎟ Φ L Φ BL Λ Φ ML C( L ,L ,Λ L ) A B M A B M ⎟ ⎠

(7) where the Φ s are symmetrized wavefunctions for bosons, or antisymmetrized wavefunctions for fermions, respectively. The superscript of Φ s refers to the type of particles, and the subscripts indicate the chosen configuration for each type of particle. The prime on parenthesis means the exclusion of

L A = LB = Λ LM = 0 . In the MC_MO-HF method, the total nuclear wavefunction

φ

Φp

is expressed

Φ

by the nuclear MO( p )s, as well as the total electronic wavefunction e . For simplicity, we here treat one kind of nuclear species as the quantum mechanical wave and other nuclei as the point charges. Of course, to treat lightest nucleus, such as protons, is better approximation. The zeroth-order wave function in Eq.

Ψ = Φe ⋅ Φ p

0 0 0 corresponds to the HF wavefunction. Thus the HF (7), wavefunction is expressed as simple product of the electronic and nuclear wavefunctions. The HF wavefunctions of electrons and nuclei are given by antisymmetric or symmetric products of the electronic and nuclear MOs,

8

Taro Udagawa and Masanori Tachikawa

Φ e0 = χ i χ j Λ χ k Φ

p 0

,

= χ p χq Λ χr

(8) ,

(9)

χ

χ

where i and p are the spin MOs of an electron and a nucleus. The energy of this system after integration of the spin coordinates is given as Ne

Ne

i

i, j

{ (

)

(

)}

Np

E = ∑ nie hiie + ∑ α ije φiφi φ jφ j + β ije φiφ j φiφ j + ∑ n pp h ppp Np

{ (

p

)

(

)}

(

Ne N p

p + ∑ α pq φ pφ p φqφq + β pqp φ pφq φ pφq − ∑∑ nie n pp φiφi φ pφ p p ,q

where

i

p

) ,

(10)

p φi and φ p are the spatial MOs of an electron and a nucleus. hiie and h pp φiφi φ jφ j φiφ j φiφ j

are one-electron and one-nucleus integrals,

(

Coulomb and exchange integrals of electrons, those of nuclei, and

(φ φ p

p

) and ( ) are the (φ φ φ φ ) are φφ ) and q q

p q

p q

(φ φ φ φ ) is the Coulomb integral between an electron and i i

p

p

p nie and n p are the occupation numbers of φi and φ p , N N the α and β are Coulomb and exchange coupling constants and e and p

a nucleus. The coefficients

are the number of electrons and nuclei, respectively. Then we can derive the HF equations for electrons and quantum-treated nuclei by the variational method as

f eφi = ε iφi , f φ p = ε pφ p

(11)

p

where

,

(12)

Multi-Component Molecular Orbital Methods Ne / 2

∑ (2 J

f = 2h + e

e

i

− Ki ) −

i

f

p

= 2h p +

∑ (2 J

Ne / 2

∑ 2J

p

p

Ne / 2

p

± K p )−

p

9

,

(13)

Ne / 2

∑ 2J

i

i

.

(14)

In Eqs. (13) and (14), J and K are the Coulomb and exchange operators,

φ

respectively. We can also see that the effective field of the electronic MO i is due to the motion of the nuclei and other electrons in Eq. (13). Similarly, that of

φ

nuclear MO p is due to the motion of the electrons and other nuclei. To solve these Fock equations, we use linear combination of Gauss type functions (LCGTF) approximation for both electronic and quantum nuclear MOs, which is follows

φi = ∑ Crie χ re r

φ p = ∑ Cν χν p

,

ν

where GTF

(15)

p

,

(16)

χ is

χ σ ( x, y, z ) = (x − X σ ) ( y − Y σ ) (z − Z σ ) l

{

(

× exp − α σ x − X σ

m

n

) + (y − Y ) + (z − Z ) }. 2

σ 2

σ 2

(17)

As mentioned above, electronic and nuclear MOs have three kinds of

(

e

parameters such as LCGTF coefficients C , C

(

e

and GTF centers R , R

(

e

p

)

p

p

) , GTF exponents (α

e

)

,α p ,

). In conventional MO calculations, only the LCGTF

are determined by the variational theorem with the other coefficients C , C parameters fixed. Using electronic and quantum nuclear basis functions,

φ

φ

electronic MO i and quantum nuclear MO p are obtained by solving the electronic and quantum nuclear Roothaan equations simultaneously. We have

10

Taro Udagawa and Masanori Tachikawa

implemented these schemes and gradient routine of energy with respect to the classical nuclear coordinate to GAUSSIAN03 program package[53] and our original program FVOPT[48,49,54]. The detail of the gradient routine will be shown in below. Then we encounter the problem of determination of nuclear GTF exponent. Since electronic GTF centers are settled on the nuclear point charge in the conventional BO calculations, we can restrict the electronic and nuclear GTF centers are identical. Of course this restriction is only approximation, however we think this approximation is allowed in many cases. In contrast to the case of nuclear GTF centers, there is no indicator for determination of nuclear GTF exponent, because the all nuclei are treated as point charge in the conventional BO calculations. In addition, there are no well known nuclear basis sets, while large amount of electronic basis sets are well known. Thus we will introduce the Fully variational molecular orbital procedure which can determine the not only LCGTF coefficients but also GTF exponents and centers variationally. B. Fully variational molecular orbital (FVMO) method Recently we have proposed the fully variational MO (FVMO) method[48,49,54], in which all parameters in the basis functions are optimized under the variational principle. We note here the advantages of FVMO method. First, the total energy is improved under the variational principle with the small number of basis functions due to the extension of variational space. The extension of variational space also provides the drastically improved values for the properties, such as dipole moments and polarizabilities. In addition, the basis sets of quantum light particles, such as positron, proton, muon, and other nucleus can be variationally determined. By optimizing the GTF centers[55-58] and exponents[59], Hellmann-Feynmann[60,61] and virial[62] theorems are completely satisfied in the FVMO method. It is noted that these two theorems are satisfied in the exact wavefunction, but not always obeyed in the conventional procedure.

N

e electrons and a Here, we consider a system which is constructed by proton for simplicity. Then HF energy of this system is given as Ne / 2

E HF = 2 ∑ hiie + i

∑ {2(φ φ φ φ ) − (φ φ

Ne / 2

i i

i, j

j

j

i

j

)}

Ne / 2

(

φiφ j + h ppp − 2 ∑ φiφi φ pφ p i

) .

(18)

Multi-Component Molecular Orbital Methods

11

{ } {χ }

Each molecular orbital is expanded in the basis set χ r or ν . Hereafter we denote the parameters, orbital exponents, and orbital centers of electronic and p

e

nuclear basis sets as a whole, by Ω . In order to optimize the HF energy of Eq. (18) with respect to these parameters, the analytical formulas of the HF energy derivatives are required. The first derivative of the HF energy of Eq. (32) with respect to parameter Ω is given as follows:

{(

Ne / 2 Ne / 2 ∂ E HF = 2 ∑ hiie (Ω ) + ∑ 2 φiφi φ jφ j ∂Ω i i, j Ne / 2

(

+ h ppp (Ω ) − 2 ∑ φiφi φ pφ p i

)( ) − (φ φ φ φ )( ) } Ω

)( ) − 2 ∑ S ( )ε Ne / 2

Ω

Ω

i

Ω ii

i

i

j

i

j

(Ω ) − S pp εp

,

(19)

where AO

S ij(Ω ) = ∑ Cμi Cνj μ ,ν

∂S μν ∂Ω

(20) (Ω )

h is the skeleton overlap derivative[63] integral with respect to parameter Ω , ii (Ω ) h pp

and

are the skeleton derivative of one-electron and one-proton integral,

(φ φ φ φ )( ) (φ φ φ φ )( ) Ω

i i

j

j

(φ φ φ φ )( ) Ω

and

i i

p

p

are that of Coulomb integrals, and

Ω

i

j

i

j

is that of exchange integral. In addition, the derivatives of GTF with respect to the GTF exponent α and x component of GTF center are given as

{

∂ χ = (x − X )l + 2 ( y − Y )m (z − Z )n + (x − X )l ( y − Y )m+ 2 (z − Z )n + (x − X )l ( y − Y )m (z − Z )n+ 2 ∂α

[ {

× exp − α (x − X ) + ( y − Y ) + ( z − Z ) 2

2

2

}],

{

∂ χ = 2α (x − X )l +1 ( y − Y )m (z − Z )n + l (x − X )l −1 ( y − Y )m (z − Z )n ∂X

[ {

× exp − α (x − X ) + ( y − Y ) + ( z − Z ) 2

2

2

}].

}

(21)

} (22)

12

Taro Udagawa and Masanori Tachikawa

The derivatives with respect to the y and z components of GTF center are also can be expressed in the similar way to that of x axis. By using FVMO method, we can determine the “unknown” optimum parameters such as GTF exponent and GTF center for nuclear GTFs variationally. C. Configuration interaction (CI) Møller-Plesset perturbation theory treatment of MC_MO method We have introduced that the total wavefunction of the multi-component system is given in the CI formalism, Eq. (7), and have derived the Hartree-Fock equation by treating only the first term of Eq. (7). The CI matrix element is calculated by modification of the GUGA technique for the multi-component system as M

H LL ' = ∑ Φ

HI Φ

I LI

M

I L 'I

I

∏δ K ≠I

M

LK , L 'K

+ ∑ Φ ILJ Φ ILI VIJ Φ IL 'I Φ LJ ' J I >J

M

∏δ

K ≠I ,J

LK , L ' K

,

(23)

where M is a number of kinds of quantum particles, the first term denotes the contribution of intra-correlation and the second term denotes that of intercorrelation. After diagonalization of Eq. (23), one obtains the CI coefficient and its energy expressed as IMO M ⎛ IMO M IMO JMO I (φiφ j φkφl )l ⎞⎟⎟ + ∑ ∑ ∑ ΓijklIJ (φiφ j φkφl )IJ ECI = ∑ ⎜⎜ ∑ γ ijI hijI + ∑ Γijkl i , j ,k ,l∈I I ⎝ i , j∈I ⎠ I > J i , j∈I k ,l∈J ,

where

γ ijI

and

I Γijkl

(24)

are the one- and two-body reduced density matrices (RDM)

in the MO base and

hijI

and

(φ φ i

φ k φl )

I

j

are the one- and two-particle MO

integrals of the I th and J th kinds of particle, and IMO and JMO are the number MOs of the I th and J th kinds of particle, respectively.

E The first derivative of CI with respect to the parameter Ω (e.g., GTF exponent and GTF center) is expressed as

Multi-Component Molecular Orbital Methods

(

M ⎛ IAO IAO ∂ I I (Ω ) I ECI = ∑ ⎜⎜ ∑ γ μν hμν φ μφν φ ρφσ + ∑ Γμνρσ ∂Ω I ⎝ μν ∈I μνρσ M

I > J μν ∈I

where

I γ μν

(AO) basis,

and

I Γμνρσ

∑ Γμνρσ (φμφν φρφσ ) ρσ

I Ω)

−

IAO

Wμν S μν( ∑ μν I

∈I

I Ω)

⎞ ⎟ ⎟ ⎠

IJ (Ω )

IAO JAO

+∑ ∑

)(

13

IJ

∈J

,

(25)

are the one- and two-particle RDM in the atomic orbital

I (Ω ) I (Ω ) S μν hμν

,

( ( φφ φφ ) , and

I Ω)

μ ν

ρ σ

are the skeleton derivatives[58] of

overlap, one-, and two-particle integrals of the I th kind particle. IAO is the numbers of AOs, and as

WμνI

is the energy-weighted density matrix which is given

IMO

WμνI = ∑ C μI i CνIj xijI ij∈I

.

(26)

Note that Eq. (25) also satisfies the multi-configuration self-consistent field (MCSCF) wavefunction as well as the full-CI wavefunction. Next, we introduce the second-order Møller-Plesset (MP2) perturbation theory of MC_MO method (MC_MO-MP2). Since MC_MO method treats light nuclei as quantum mechanically, we must consider three types of the MP2 (2 )

(2 )

E E energies, electron-electron ( (e−e ) ), electron-nucleus ( (e− p ) ), and nucleusE (2 )

nucleus ( ( p − p ) ). The explicit formulae of these schemes at closed shell system are given as

ij ab ijab ε i + ε j − ε a − ε b

E((e2−)e ) = ∑

E((e2−) p ) = ∑

iapp '

2 ip ap'

,

(27)

2

ε i + ε p − ε a − ε p'

,

(28)

14

Taro Udagawa and Masanori Tachikawa

E((p2−) p ) =

∑

pqp 'q '

pq p' q' (2 p' q' pq − p' q' qp ε p + ε q − ε p' − ε q'

) ,

(29)

where i, j and a, b indices refer to occupied and unoccupied electronic orbitals, and p, q and p ' , q ' are those for nuclear orbitals, respectively. We have implemented these schemes of MC_MO-CI and also MC_MO-MP2 to our original program FVOPT[48,49,54]. Of course, the FVMO procedure is available for these MC_MO-CI and MC_MO-MP2 methods. D. Multi-component “hybrid” density functional theory. Recently, we have developed the multi-component “hybrid” type of density functional theory (MC_MO-(HF+DFT))[64] by combining the multi-component technique and density functional theory (DFT). As mentioned above, our MC_MO has been already extended to Møller-Plesset perturbation and configuration interaction methods beyond the mean field approximation and several groups have also studied such non-BO treatment based on this MC_MO concept to evaluate the many-body effect. However, the disadvantages of enormous computational costs are still remained in these correlated multicomponent methods. According to this background, we focus on DFT[65,66] which provides quantitative results with reasonable computational costs. In particular, recent hybrid functionals[67-69] have provided a significant contribution for analyzing many chemical phenomena, such as molecular geometries, vibrational frequencies, and single-particle properties within chemical accuracy. Although DFT had several problems for van der Waals interaction energies, Tsuneda and Hirao[70-72] have already reproduced such weak interaction by modification of the exchange-correlation functional. As mentioned above, we again note that DFT will be more and more promising and powerful tool by the suitable density functionals. Several studies of MC_DFT have been already reported[33-35,73]. In 1982, the concept of MC_DFT was first presented by Capitani and Parr, et al.[73], who have established the existence theorem of Hohenberg-Kohn for multi-component system. Shigeta and coworkers[33,34] have shown the non-BO-DFT scheme based upon the real-space grid method, and applied to hydrogen molecule and its isotopomers. They have concluded that even though the electron-nucleus and

Multi-Component Molecular Orbital Methods

15

nucleus-nucleus correlation terms are completely neglected, isotope effects among these species can be discussed at least qualitatively. Kreibich and Gross[35] have also presented MC_DFT where electron-nucleus correlation term was approximation by the classical electrostatic interaction of the corresponding mean field charge distributions. To our knowledge, however, there are no reports of multi-component “hybrid” DFTs, despite a remarkable success of the conventional hybrid density functionals. First, let us introduce the concept of hybrid density functional theory briefly. Nowadays, several types of hybrid functionals are proposed. In particular, “Becke’s half and half” (BHandH) concept[67] is one of the most popular schemes, where the integrand of 1

λ E XC = ∫ U XC dλ 0

(30)

is approximated to

1 0 1 E XC ≅ U XC + U 1XC 2 2 ,

(31)

λ U XC is the potential energy of U0 exchange-correlation at intermediate coupling strength λ , XC is the exchange-

using a linear interpolation. In Eqs. (30) and (31),

correlation potential energy of the noninteracting reference system, and

U 1XC is

0 U XC is the exchange-correlation U0 potential energy of the noninteracting reference system, XC can be replaced by

that of the fully-interacting real system. Since

the Kohn-Sham (KS) exchange energy E X ,

E XC ≅

1 1 LSDA E X + U XC 2 2 .

(32)

At the practical point of view, Hartree-Fock orbital is used for the E X term in Eq. (32) instead of KS orbital. Becke has also developed the “Becke’s threeparameter hybrid functional” (B3)[68] by relaxing the linear

λ dependence of

16

Taro Udagawa and Masanori Tachikawa

the “half and half” theory. Today, B3LYP has become the most popular hybrid exchange-correlation functional. Next, we would extend our MC_MO scheme to multi-component hybrid density functinonal theory (MC_MO-(HF+DFT)) by combining the conventional DFT procedure. In our MC_MO-(HF+DFT) approach, the Kohn-Sham operators for electron and quantum nuclei are derived by adding the hybrid type exchangecorrelation potentials to Fock operators of MC_MO-HF scheme as Ne

Np

e

p

Np

Ne

p

e

( HF + DFT ) + DFT ) f e( HF + DFT ) = he + ∑ J e − ∑ J p + VXC + VC( HF ( e −e ) (e − p )

( HF + DFT )

fp

where

,

(33)

,

(34)

( HF + DFT ) ( HF + DFT ) = h p + ∑ J p − ∑ J e + V XC ( p − p ) + VC (e− p )

( HF + DFT ) V XC ( e −e )

( HF + DFT ) V XC = ( e −e )

is given as ( HF + DFT ) δE XC ( e −e ) δρ .

(35)

K

At this stage, the exchange term between two quantum nuclear particles p in Eq. (14) is ignored, and we take account of only electron-electron correlation by using conventional functionals because it accounts a major part of many-body

V ( HF + DFT )

V ( HF + DFT )

in Eqs. (33) effect. This corresponds to the neglect of XC ( p − p ) and C (e− p ) and (34), so the quantum nuclear KS operator of MC_MO-(HF+DFT) in Eq. (34) has a similar manner of the quantum nuclear Fock operator of MC_MO-HF in Eq. (14). Hereafter, thus we will denote the

( HF + DFT ) V XC (e −e )

in Eq. (33) as

N

( HF + DFT ) V XC .

e electrons and a For simplicity, let us consider the system containing proton. The energy of this system after integration of the spin coordinates in the framework of MC_MO-(HF+DFT) is given as follows:

Multi-Component Molecular Orbital Methods ( HF + DFT )

Etotal

Ne / 2

Ne / 2

i

ij

= 2 ∑ hii +

∑ 2J

( HF + DFT )

ij

+ E XC

17

Ne / 2

+ h pp − 2 ∑ J ip i

,

(36)

E ( HF + DFT )

the exchange-correlation energy contribution by hybrid type of where XC exchange-correlation functional. We have developed and implemented the two MC_MO-(HF+DFT) versions of BHandHLYP (MC_BHandHLYP) and B3LYP (MC_B3LYP) functionals by combining the MC_MO technique with conventional BHandHLYP and B3LYP hybrid density functionals, BHandHLYP E XC = 0.5E XHF + 0.5(E XSlater + ΔE XB 88 ) + ECLYP ,

(37)

B 3 LYP E XC = 0.2 E XHF + 0.8E XSlater + 0.72ΔE XB 88 + ECLYP + 0.81ΔECLYP .

(38)

HF

The notation E X

Slater

is Hartree-Fock exchange, E X

the Slater exchange

ΔE LYP

C the Becke’s gradient correction, and and the functional, ΔE local- and non-local correlation provided by the LYP expression, respectively. To optimize the molecular geometry, we have also implemented the first order energy derivative of Eq. (36) with respect to classical nuclear coordinate B 88 X

E

LYP C

“ R ”. The explicit formulae is as follows Ne / 2 Ne / 2 Ne / 2 ∂ ( HF + DFT ) (R ) Etotal = 2 ∑ hii( R ) + 2 ∑ J ij( R ) + h pp − 2 ∑ J ip( R ) ∂R i ij i Ne / 2 ∂ ( HF + DFT ) (R ) + E XC − 2 ∑ S ii( R )ε i − S pp εp ∂R i .

(39)

We have implemented these schemes to GAUSSIAN03 program package[53]. We mention that non-hybrid functionals (e.g., BLYP, BOP, etc.) are, of course, usable in the GAUSSIAN03 modified for our MC_MO-(HF+DFT) scheme. We will show several applications of our multi-component methods in the next chapter.

18

Taro Udagawa and Masanori Tachikawa

2-3. OTHER GROUP’S APPROACH Nakai and coworkers[21-24] have also developed the multi-component method which is named as nuclear orbital and molecular orbital (NOMO) methods. Original NOMO method is equivalent to our multi-component molecular orbital (MC_MO) method. They have extended NOMO method to configuration interaction single theory[21], many-body perturbation theory[23], and coupled-cluster method[23] in a similar way of extends the conventional HF method to these methods. Recently, they have developed the translation- and rotation-free (TRF) NOMO method by eliminating translational and rotational contributions from NOMO Hamiltonian. In the non-BO theory, Gauss type function (GTF) is frequently used as nuclear basis function. The GTF is proper to represent a vibrational motion, while it is not proper to represent translational- and rotational motion. They have proposed a scheme to eliminate the effect of the translational motion from the NOMO calculation. The Hamiltonian of the translational motion is the kinetic term of the center of mass which is given by

TT (x ) = −

1 2M

∑μ ∇(x μ )

2

−

1 M

∇(x μ ) ⋅ ∇(xν ) ∑ μ ν