Use of biorthogonal orbitals in calculation by perturbation of molecular

Use of biorthogonal orbitals in calculation by perturbation of molecular interactions J. F. Gouyet Citation: The Journal of Chemical Physics 59, 4637 (1973); doi: 10.1063/1.1680674 View online: http://dx.doi.org/10.1063/1.1680674 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/59/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear molecular properties using biorthogonal response approach J. Chem. Phys. 101, 4914 (1994); 10.1063/1.467413 Calculation of the dispersion interaction energy by using localized molecular orbitals J. Chem. Phys. 94, 5565 (1991); 10.1063/1.460492 Calculation of molecular polarizabilities using a semiclassical Slater‐type orbital‐point dipole interaction (STOPDI) model J. Chem. Phys. 79, 2256 (1983); 10.1063/1.446075 Extrapolation in configuration interaction calculations using perturbation theory natural orbitals: Applications to H2O and HeH2 + J. Chem. Phys. 71, 124 (1979); 10.1063/1.438110 Use of biorthogonal orbitals in perturbation calculation of molecular interactions from a multiconfigurational unperturbed state. II J. Chem. Phys. 60, 3690 (1974); 10.1063/1.1681590

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VOLUME 59,

THE JOURNAL OF CHEMICAL PHYSICS

NUMBER 9

1 NOVEMBER 1973

Use of biorthogonal orbitals in calculation by perturbation of molecular interactions J. F. Gouyet Groupe de Physique Moleculaire. Ecole Polytechnique. 17. Rue Descartes. Paris- V, France (Received 9 June 1972)

A Rayleigh-Schrooinger type of perturbation expansion, using a partition of the Hamiltonian into non-Hermitian parts, and biorthogonal orbitals, is developed to study excited molecular states. In particular an expansion of the molecular interactions is defined in which all intramolecular terms cancel exactly.

directly expressed in a nonorthogonal basis which can be applied to different kinds of problems, and secondly to show that in the RS expansion for the supersystem (A+ B) the diagrams belonging to intramolecular terms of A and B can be completely eliminated.

I. INTRODUCTION

The use of biorthogonal orbitalS in perturbation techniques was discussed by des Cloizeaux1 and also by Brandow2 in order to transform the nonHermitian operators which appear in the linkedcluster expansion in the case of degenerate zero order states, into new hermitian effective operators.

II. GENERAL REMARKS

Let us consider two interacting systems A and B, atoms or molecules. Their Hamiltonians will be denoted HA and HB and the total Hamiltonian will be

More recently Moshinsky and Seligman3 combined the biorthogonal viewpoint with the second quantization procedure. Their method was associated with a group theoretical treatment to select definite spin symmetry, and was applied to the resolution of the secular equation.

(1 )

Each individual system A and B can be studied by perturbation, USing a partition of HA (resp HB)

On the other hand, a treatment was recently given for calculation of the intermolecular energies by a perturbation ab initio method. 4 This treatment, which tried to build a bridge between the usual long-range description of intermolecular forces and the ab initio Molecular Orbitals Calculations, uses a hierarchized orthogonalization of LOwdin type 5. 112 procedure and an Epstein-Nesbet Rayleigh-Schrodinger perturbation expansion. But some difficulties subsist in the elimination of the properly intramolecular terms (due in particular to the expansion of the basis of a system A in terms of the orbitals of the other system B).

(2)

with lfJA=6h A(i) • I

The unperturbed solutions are known, r A • • • are the orbitals, r A a· .. the spin orbitals of system A (resp B) and A

h IrA) = ErA IrA) ,

(3)

hOAlfA)=~JfA)

In opposition with preceding works (see for instance a review in recent papers 5 - 7 where the zero order function does not possess directly the proper symmetry with respect to the intermolecular electron permutation, the last work4 uses properly antisymmetrized functions. In the same way, a very recently published work 8 chooses an unperturbed Hamiltonian so that it possesses the correct symmetry properties with respect to electron permutations. The Hamiltonian includes non-orthogonality when the basis set is orthogonalized, but only the small overlap region of intermolecular interactions can be studied. The purpose of this paper is firstly to give a Rayleigh-Schrodinger perturbation expanSion,

for system A and equivalently for B. The excited configurations of A are fA and the fundamental will be written ¢~. There exists an overlap between orbitals of A and B so that (r A ISA)

=

Or A s A (r B ISB) = Or B s B

,

(4)

but (rAlsB) =Sr A s B

(5)

The two bases {rA} and {r B} are theoretically two complete bases for the Hilbert space, and the set {rA}e{rB} gives an overcomplete basiS set; but in practice one uses finite basis sets and the two bases will be taken as a total basis set.

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F.

J.

4638

Ill. INTRODUCTION OF THE BIORTHOGONAL BASIS

These states can be built from the second quantization formalism as

We now denote by Ir), Is),··· orbitals belonging either to {r A } or to {r B }. The overlap matrix is: (6)

with 1A=lIoTAsAIi; 1B=IIOTBSB II; SAB=II(r A ls B)1I and then define ,1-3 the biorthogonal basis I! Is)··· by

>,

Ir / = -

"61 s ) (s 1r) s - -

0'

6 Is) s

5;;

(7)

and conversely (8)

The biorthogonal basis has the following important property: (rAlsB/=(rAlsB)=OT -

-

S

A B

(9)

and in our special case (the orbitals of each individual system are orthogonal) IrA )=

IrA )+6I s B)(sBlrA ) -

GOUYET

t t t ••• a pAap at t a l vac ) . IK) -- alAala2Aa2 lBPl •••

(13)

Naturally, because of the nonorthogonal character of the orbitals, the 1K) are not normalized to unity:

These states will be considered as the unperturbed states of the supersystem. Let us now examine the different choices for the unperturbed "Hamiltonian" of the supersystem. V. NON-HERMITIAN ZERO-ORDER "HAMILTONIANS" OPERATOR FOR THE SUPERSYSTEM

A caret is used to distinguish the non-Hermitian character of operators: A. A "Miiller-Plesset" Type of Ifl

The operators whose eigenvectors are the IK) previously defined are necessarily non-Hermitian because of the nonorthogonality of these IK). Then let us consider

jjoA

S-

oBP•

6

t TAa = T,a€TAaTA a!!:.

' (14)

and so on.

if =j{JA + j{JB ,

The equivalent polyelectronic operators, in the second quantization formalism have been explicitly given by Moshinsky and Seligman. 3 The mono- and bielectronic operators can be written Fequ

="0 (ra If Isp) arcrt aS~ r,s cr,

p

Gequ -- '" u

"I g Isp s ") p arat ar'a,ta s' p' a sp -

(10)

( rar a

rr'ss' au' pp'

-

(11)

but {!!

Ta

t 1 ,ar'a' r

;:::

ifIK/=E~IK/ with IK)=IIAAJB )

,

(15)

The unperturbed fundamental for the supersystem is (16)

where ra, sp, ... are spin orbitals, a rat creates the spin orbital Ira) and aSP annihilates the spin orbital Isp). The anticommutation relations between these operators are ttl {ra T'a', 0 { aya,ar'CI'r;:::!!. ,!!:, r==

which verify the unperturbed Schrodinger equations

~r ~cr

uri u(J' ,

while {a Ta , a~,cr'} = ( r 1r' ) 0aa'. Hence with the introduction of annihilators asP the usual commutation relations are preserved, as well as all the diagrammatic rules. IV. BUILDING UP OF THE UNPERTURBED STATES OF THE SUPERSYSTEM (A + B)

Let us consider the antisymmetrized products of the unperturbed states of A and B: (12)

and its energy E60)

=

6

2€kA +

kAOCC

6

ka oce

2€k

B

(17)

Naturally it is possible to define a particle-hole formalism, necessary introduction to diagrammatic expansions. I
k·/ :A / / / / -- / / /

equal zero in the perturbation expansion for the supersystem, because ls is an occupied orbital of

B. But it is easy to give examples of nonzero diagrams which are in fact intramolecular terms. Thus in the second order energy in the perturbation of a monoexcited I (¢g)l~S > the diagram

is in fact a part of a second order intramolecular diagram of A using the vacant ls as virtual orbital.

+

exchange terms.

The contribution of the last diagrams is simply the first order correction to the intramolecular energies of subsystems A and B as can be seen later. The other diagrams represent the repulsion energy Erep at the limit of zero overlap, which can be noted:

B. The Elimination of Intramolecular Diagrams

To examine the intramolecular terms we must consider the extension of the basis due to the orbitals of the subsystem B for the terms of A (and of the orbitals of A for the terms of B). The best solution for introducing the same kind of diagrams as in the expansion of the supersystem is to con-

belongs to E(2)(A). Thus the intramolecular diagrams are easily eliminated in the perturbation expansion. To characterize an intramolecular diagram of A (resp B) one considers for a given diagram the excitated unperturbed part of B as new vacuum state for B. Then all the hole lines of the transformed diagram must belong to occupied spin orbitals of A. For instance a diagram such as

is not an intramolecular term.

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CALCULATION OF MOLECULAR INTERACTIONS

4641

VIII. DETAILED DEVELOPMENT OF INTERMOLECULAR ENERGY IN THE "EPSTEIN-NESBET" PARTITION

CONCLUSION

Let us examine here the case of Epstein-Nesbet partition, up to the second order. The zero, first and second order of the intermolecular energy are:

This paper develops a method which, firstly, uses directly nonorthogonal orbitals but also, secondly, gives exact cancellation of intramolecular terms. This represents the greatest difficulty in the expression of intermolecular effects.

Ei~i = (pO I HI rfJ°) - E1°) - E~O)

Ei!i = 0 , E(2) In! = L,0 (rfJ°IHII)UIHlrfJ°)/(E(O)-E(O») _ _ 0 1 1*

_ E12) _ E~2) . We now use the following notation: A label A (B) for propagating lines of system A (B) which are drawn on the right (left) of the diagrams, to separate excitations of A from excitations of B. Then the zero order energy is

ACKNOWLEDGMENTS

+ exchange terms.

These terms are considered as the repulsion energy between A and B, and exchange contributions. Now the second order energy gives a greater number of terms: (1) Single excitation in the intermediate state (a) Diagrams contributing to the classical polarization energy like -,...:--

VA

ill--

The author wishes to express his thanks to Dr. J. P. Malrieu and P. Claverie for discussions, and to S. Diner for having pOinted out to him the existence of papers by D. J. Newman who developed second quantization formalism in connection with crystal field theory, 11 for nonorthogonal orbitals and has also established a RS development and the linked cluster theorem.

-

[]}_ UA

VA IA

Two problems remain. Can the range of application of this intermolecular treatment bridge over with ab initio molecular orbital calculation and what is the effect of a non negligible overlap between orbitals. Secondly the use of non-Hermitian effective operators gives nonorthogonal approximate eigenfunctions which have to be projected. The method is appropriate for studying intermolecular effects between excited molecules (exciplex' .. ); then another simplfication can occur in the expression of the difference between intermolecular effects for excited states and for the ground state .10

UA IA

(b) Diagrams corresponding to a charge transfer contribution. For instance

(2) Double excitation in the intermediate state. This gives in particular a dispersion energy contribution like

A notation of a vertical bar to separate the two subsystems appears in the newly published paper of Basilevsky and Berenfeld. B

IJ. des Cloizeaux, Nuc!. Phys. 20, 321 (1960). 2B. H. Brandow, Rev. Mod. Phys. 39, 771 (1967). 3M. Moshinsky and T. Y. Seligman, "Group theory and second quantization for nonorthogonai orbitals," submitted for publication in the "Amos de Shalit Memorial Volume" of the "Annals of Physics." 4J. P. Daudey, P. Claverie, and J. P. Mairieu, "Perturbative ab initio calculations of intramolecular energies. I. Method" (unpublished); J. P. Daudey, J. P. Malrieu, and Olivia Rojas, "II. The He ... He problem" (unpublished). 5J. O. Hirschfelder and P. R. Certain, Int. J. Quantum Chern. 25, 125 (1968). 6A. T. Amos, Chern. Phys. Lett. 5, 587 (1970). 7R. Boehm and R. Yaris, J. Chern. Phys. 55, 2620 (1971). This paper utilizes a linear response theory based on the many-body Green's function techniques of Martin and Schwinger, to study van der Waals forces including exchange in the small overlap region. 8M. V. Basilevsky and M. M. Berenfeld, Int. J. Quantum Chern. 6, 23 (1972). 9 J. F. Gouyet, Phys. Rev. A 2, 139 (1970); Phys. Rev. A 2, 1286 (1970); J. Math. Phys. 13, 745 (1972) and thesis, Paris, 1971. lOS. Diner, P. Claverie, and J. P. Malrieu, Theor. Chim. Acta 8, 390 (1967). liD. J. Newman, Chern. Phys. Lett. 1, 684 (1968); J. Phys. Chern. Solids 30, 1709 (1969); J. Phys. Chern. Solids 31, 1143 (1970).

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