Perturbation Approach to a Molecular Orbital Theory of Interaction

J. Phys. Chem. 1982, 86, 1141-1146

+

and for three-dimensional H H2 on two different potential energy surfaces. This is encouraging, but we should still remember the weaknesses of the model. The model is not expected to work as well in general for very small skew angles, for very exothermic reactions, or for higherenergy resonances. Within its domain of validity though, it appears to be a powerful tool.

1141

Acknowledgment. This work was supported in part by the National Science Foundation through grant no. CHE80-25232 and by the U.S.Army through the Army Research Office under contract no. DAAG29-81-C-0015. The authors are grateful to Dr. Robert B. Walker and Edward F. Hayes for stimulating discussions of resonances in reactive collisions.

Perturbation Approach to a Molecular Orbital Theory of Interaction Energies? Danlel M. Chlpman

*

Downloaded by UNIV OF LETHBRIDGE on September 7, 2015 | http://pubs.acs.org Publication Date: April 1, 1982 | doi: 10.1021/j100396a019

Radlethm Lahatmy, Unhrersny of Notre Dam, Notre D a m , Indlana 46556 (Recelved: July 27, 198 1; I n Final F m : October 19, 1981)

In order to provide a framework for better understanding of phenomena such as reaction mechanisms and photoelectron spectra that can be semiquantitatively described by molecular orbital theory, a formalism is developed to expresa the orbitals of a molecule in terms of orbitals of various fragmentsthat make up the molecule. The total energy and individual orbital energies of the interacting fragments are obtained through second order in perturbation theory. Electrostatic interactions, orbital overlap effects, and self-consistency refinements are all considered explicitly and simultaneously to obtain the final results.

Introduction Professor Hirschfelder has been a major figure in the theory of intermolecular forces.' It is now widely recognized that the forces that determine a wide class of reaction mechanisms, molecular properties, spectroscopic parameters, and numerous other experimental observables of molecules can be semiquantitatively described by selfconsistent-field molecular orbital calculations within finite basis sets. In many such cases, it would be useful for better understanding to be able to describe the orbitals of a molecule in terms of the interacting orbitals of various fragments that make up the molecule (the fragments may themselves be smaller molecules). Perturbation theory, another of Professor Hirschfelder's major interests? is used in this paper to accomplish this goal. Attention is focused on the total energy, which can be used to characterize reaction mechanisms, and also on individual molecular orbitals and orbital energies that, through Koopmans' t h e ~ r e m can , ~ provide a framework for understanding photoelectron spectra. Early attempts along these lines were made by Dewar: Klopman et al.? and Salem6using Huckel type models that ignore self-consistency effects and utilize semiempirical estimates of integrals. The extension of Salem's approach by Devaquet' to include self-consistency effects was still limited to 'R systems, used a rough model for electrostatic interactions, and invoked semiempirical integrals. Sustmann and Binsch8presented a theory which applied to all electrons in a system and included self-consistency effects. However, they made the theoretically unjustified assumption of orthogonality between orbitals of different fragments. Also, their formalism was based on the density matrix and so provided only the total energy, and not the t This document No. NDRL-2270 from the Notre Dame Radiation Laboratory. 'This paper is dedicated to Professor J. 0. Hirschfelder on the occasion of his 70th birthday celebration. As a scholar, a teacher, and a friend, Joe has had a dominant influence on my own development.

0022-3654/82/2086-1141$01.25/0

individual orbital energies. Applications of the method by these8 and othergworkers have, as yet, been made only with the further simplifications of semiempirical parametrizations. Basilevsky and Berenfeld'O treated both the nonorthogonality problem and the self-consistency effect in a diagrammatic formulation, but here again only the total density matrix was considered, and not individual molecular orbitals. These authors also presented a semiempiricalversion of their theory. More recently, Stone and Erskinel' have implemented this method for ab initio calculations. The method of Daudey et a1.12 begins with fragment molecular orbitals, but is different in focus beyond that point. It seeks to obtain the total interaction energy, including correlation effects, and so cannot be used to obtain molecular orbitals of the interacting system. Mention should also be made of a somewhat different approach to determination of the various contributions to the molecular orbital energy of an interacting system. While the specifics of the methods proposed by workers such as Morokuma,13Epiotis et al.,14 Whangbo et al.,16 and (1) Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. 'Molecular Theory of Gases and Liauids": Wilev: New Ork. 1964. (2) Hirschfeldk, J. 0.;By& Brown, W:; Epstein, S. T., Adu. Quantum Chem. 1964, I , 255. (3) Koopmans, T. Physica 1933, I , 104. (4) Dewar, M. J. S. J . Am. Chem. SOC.1952, 74,3341. (5) Klouman. G.: Hudson. R. F. Theor. Chim. Acta 1967., 8.. 165. Klopman,-G. J.'Am.. Chern. soc. 1968,90, 223. (6) Salem, L. J . Am. Chem. SOC.1968,90, 543, 553. (7) Devaquet, A.; Salem, L. J . Am. Chem. SOC.1969,91,3793. Devaquat, A. Mol. Phys. 1970,18, 233. (8)Sustmann, R.; Binsch, G. Mol. Phys. 1971,20, 1,9. (9) Bachler, V.; Mark,F. Theor. Chim. Acta 1976,43, 121. (10) Basilevsky, M. V.; Berenfeld, M. M. Znt. J . Quunturn Chem. 1972, 6, 555. (11) Stone, A. J.; Erskine, R. W. J. Am. Chem. SOC.1980,102, 7185. (12) Daudey, J. P.; Malrieu, J. P.; Rojas, 0. Int. J . Quantum Chem. 1974,8, 1. (13) Morokuma, K. J . Chem. Phys. 1971,55,1236. Kitaura, K.; Morokuma, K. Znt. J . Quantum Chern. 1976,10, 325. (14) Epiotis, N. D.; Cherry, W. R.; Shaik, S.; Yates, R. L.; Bernardi, F. Top. Curr. Chem. 1977, 70. (15) Whangbo, M.-H.; Schlegel, H. B.; Wolfe, S. J . Am. Chem. SOC. 1977,99, 1296.

0 1982 American Chemical Society

1142

The Journal of Physical Chemistty, Vol. 86, No. 7, 1982

Chipman


Bernardi et al.16 (among others) differ considerably from one another, they all have in common the feature of only being suitable for analyzing a posteriori the results of a complete supermolecule SCF calculation on the full interacting system. By contrast, the approach taken here and taken by workers mentioned in the previous paragraph is synthetic in nature and, as such, may be applicable to somewhat larger and more complex systems. In this paper the molecular orbitals, orbital energies, and total energy of a system are approximated by perturbation theory in terms of the molecular orbitals, orbital energies, and total energies of the interacting fragments. Both electrostatic effects and overlap effects (characterized by order of smallness parameters V and S, respectively) are treated by considering them simultaneously as the perturbation. All energy terms are obtained correct through second order, i.e., through and including terms of O(V), O ( m , and O(V).O(s>. Self-consistencycorrections are also treated in a single unified formalism.

Theory Isolated Fragments. Consider the interaction of two fragments A and B to form the composite molecule AB. The Hamiltonian operator describing the M A nuclei and N A electrons in fragment A is

If we arrange the MO’s in q A 0 so that the first N A are occupied, the normalized MO electronic wave function for A can be written as The total MO energy of isolated fragment A is then given by

It consists of the sum of all occupied orbital energies, each corrected by a term that prevents double counting of electron repulsion interactions in the net sum, and the nuclear repulsion energy. Analogous expressions, which need not be written out here, obtain for the molecular orbital description of isolated fragment B. Composite System. The Hamiltonian operator for the M = M A + M B nuclei and N = N A + N B electrons in the composite system AB is

with with 2,

1

M

2

, rpt

h(t) = --vt2 - c (atomic units are used throughout this paper). It is assumed that the restricted Hartree-Fock (RHF) equations” (or unrestricted Hartree-Fock (UHF) equations18 in case of an open-shell fragment) have been solved within a finite basis set of KAfunctions centered on A. The resulting self-consistent canonical molecular orbitals (MO’s) of A can be collected into a row vector PAo

=

(d)lod)ZO***d)K:)

(3)

that satisfies the orthonormality conditions =

(4) where lAA is a K A X KAunit matrix. All functions will be assumed real throughout this paper. The Fock operator for fragment A is given by ((PAOIVAO)

?Ao

=

hA

~ A A

+ dAo

(5)

+

The finite set of K = K A K B occupied and virtual MO’s of the two fragments A and B can be merged into a single row vector PO

I

(8)

is diagonal, the diagonal elements being equal to the orbital energies elo, €2,..., CK,” which are collected into the diagonal matrix cAO: (16) Bernardi, F.; Bottoni, A. Theor. Chim. Acta 1981, 58, 245. (17) Roothaan, C. C. J . Rev. Mod. Phys. 1951,23,69. (18) Pople, J. A.; Nesbet, R. K. J . Chem. Phys. 1954,22, 571.

((PAOVBO)

(14)

and used as a basis set to expand MO’s of the interacting system AB. Although the basis MO’s within each separate fragment are mutually orthogonal, they will generally overlap basis MO’s of the other fragment. Thus, the overlap matrix of qo is not diagonal:

where

SAB3

(16) For a weakly interacting system, each basis MO qowill be similar to some final self-consistent canonical MO cp of AB. The latter are assumed to be arranged into a row vector cp in the same order as the corresponding similar MO’s in cpo. Let C be the transformation matrix taking cpo into cp: SBA+ 1

1 j,O(1) - IZ,o(l)= I d 7 2 d)j”(2)-(1 - Pl2)@(2) (7) r12 The summation in eq 6 runs over all occupied spin orbitals so that both the RHF closed-shell and UHF open-shell cases can be treated with a single expression. The matrix of the Fock operator in the MO basis

=

(cpAOlcp~O)

$9 = c poc (17) Le., each column of C gives the expansion of one final AB MO in terms of all the fragment basis MO’s. In the limit of no interaction between A and B, C then passes into the K X K unit matrix 1. Orthonormality of the final MO’s of the interacting system AB can be expressed by (cpplcp) = CfSC = 1 (18)

The Fock operator for the composite system AB is

The Journal of Physical Chernisrry, Vol. 86, No. 7, 7982

Molecular Orbital Theory of Interaction Energies

Let us now cata1,og the various fragment basis matrix elements according to their order of smallness in V and S. Letting a and b be generic symbols for MOs in (pAo and pB0, respectively, we have

U

where j uand K, are evaluated with the exact final MO's. The matrix of this Fock operator in the fragment MO basis is

F

E ((pofi(po)

1143

-

(alb)

(alv21b)

OW)

-

(32)

OW)

(33)

(21)

We wish to solve the matrix Hartree-Fock equation"^'* FC = SCc (22) for the MO expansion coefficients C and the diagonal matrix c of orbital energies. When this is done, we wizl be able to form the MO wave function of AB as *'AB

[aalab]

= A [&(I)-.hA(NA)dKA+l(NA+ 1)*-'#'KA+NB( N )1 (23)

[aalbb]


Perturbation Expansion. We now seek to obtain an approximate solution of the matrix Hartree-Fock eq 22, together with the normalization conditions in eq 18, by means of perturbation theory. To this end we introduce the perturbation expansions

+ F2 + ... c = 1 + C' + c2 + ...

(25)

S=l+S1

(27)

F = Fo + F'

c

= €0

+ €1 + €2 + ...

The zero-order part of eq 18 is automatically satisfied, and the zero-order part of eq 22 is just

O(V)

-

-

O(S)

+ OW2)

O(S2)

(36) (37) (38)

Some matrix elements contribute to more than one order-in fact some of the above also have contributions of O(V)*O(S2)and O(S3). In the following we classify a matrix element according to the lowest order contribution contained in it. Thus, the results obtained below are correct as advertised through first order (O(V), O(S))and O(V).O(S)), and, in through second order ( O ( V ) ,O(S2), addition, contain some higher-order contributions that would be very difficult to analytically separate out. The first-order contribution to the overlap matrix S is of O(S) and given explicitly by the off-diagonal matrix

tAA "3

(26) (28)

[bblba]

[ablab]

and calculate the total MO energy of AB from

(24)

-

(39)

SBA OBB

where the nonzero elements have already been defined in eq 16. Different expansions of the Fock operator 3 will be used for each block of the matrix F

FO = €0 (29) This is easily satisfied by the two noninteracting fragments with the definitions

F=

(:s)I:

In the AA block FAA E ((pAofi(pAo) = FAo

+ FAA' + FAA2 + ...

(40)

(41)

the most appropriate expansion is

f = ?Ao

+ $A1 + $A1 + d A 2 +

(42)

which gives

FAA'= I t is not so straightforward to define the higher orders of perturbation. We identify two types of interaction, according to the kinds of matrix elements over fragment basis MOs that can enter. The first is due to electrostatic effects, characterized by an order of smallness parameter V. This arises from matrix elements of long range, varying as R-I where R is a measure of the distance between the two fragments. These terms combine to describe the interaction energy, varying as some power of R-', of the permanent and induced multipole moments of the two fragments (dispersion interactions, of course, are absent from a MO theory). The second type of interaction is due to overlap of the fragment basis MO's, and is characterized by an order of smallness parameter S. These terms are recognized by matrix elements that fall off exponentially with R.

((PAOI$A'

+ ~A'IvAO)

Fu2= ((PA~I~A~IPA~)

(43) (44)

Here

(45) describes the electrostatic potential seen by A due to the undistorted charge distribution of B. The terms i ~ lj , ~ ~ , arise from the exchange potential and from self-consistency corrections to the MO's and are given by

...

1144

The Journal of Physical Chemistty, Vol. 86, No. 7, 1982

Chipman

with an analogous expression for elements of CBA'. But for the orbital energies and for the diagonal blocks of F', the above equations require further analysis, since in these cases F' on the right-hand side depends on C' through the operators &' and &' that arise from self-consistency corrections to the MO's. Treating explicitly only the AA block (analogous expressions obtain from the BB block) we have occ virt

where

ei' =

-

Cij' =

r12

1

(49)

Each term of ZA1J is second order since, at least after integration over basis MO's belonging to A, it contains a product of two first-order terms. The first- and secondorder orthonormality conditions C'+ S' C' = 0 (50)

+ + C2++ C'+sl + C1+C1+ S'C' + C2 = 0

(51) that follow from the perturbation expansion of eq 1s have been used to simplify eq 46-48. An expansion off analogous to eq 42 but appropriate for the BB block is used to obtain similar expressions for the perturbation expansion of FBB, which need not be written out here. The off-diagonal blocks FBA and Fm are only required through first order to obtain final energies through second order. For these blocks FBA+ E F A B a ((pAoficpBo) = Fm* + ... (52)

(62) Although the first-order corrections to virtual MO's 1 are not of interest in themselves, it is seen in eq 61 and 62 that the portions of them described by occupied basis MO's 12 are required to calculate orbital energy corrections ti1and MO coefficients Cijl for the occupied MO's i of A. The required LA E (NA - KA)NA coefficients of virtual MO corrections must then first be obtained by solving the LA linear simultaneous inhomogeneous equations obtained from eq 62 with i E A virt, j E A occ. These can be solved by standard methods, such as Cramer's rule or Gauss elimination, to obtain exact numerical results. To gain further insight into the solution for the diagonal blocks of C', it is instructive to consider more explicit, albeit approximate, solutions of eq 62 obtained by iterative methods. For example, one might neglect all terms arising from self-consistency effects to obtain a first estimate of

we use an expansion off that treats A and B on the same footing: FBA'+

kEA lEA

(61)

1 4u0(2)-(1 - P12)4t0(2)


(4;OIfi~'l4iO> + C C (4iO1&'14iO>Cik1 i E A

= FAB' =

((PAOIPIVBO)

ej'

(4;01fiA11$/')

i

EA

(63)

(53)

with

fQ=L+p

(54)

gJ = yt (j,o- K,o)

(55)

First-Order Perturbation Equation. The first-order component of eq 22 is FOC' F' = C1to t' S'to (56) which is to be solved subject to the normalization conditions in eq 50. Consideration of the diagonal elements of eq 56 gives the orbital energy corrections et' = F tt (57)

+

+ +

This first estimate can be equivalently regarded as either an uncoupled Hartree-Fock approximation or as the perturbation solution for the first iteration in the usual iterative solution of the full SCF equation. An improved estimate can then be obtained by using eq 64 to approximate the self-consistency terms, leading to t:l x

'

while the off-diagonal elements determine the MO coefficients

and, from eq 50 C,,' = 0 (59) Since the off-diagonal blocks of F1are known explicitly, see eq 53, the off-diagonal blocks of C' are immediately obtained from eq 58, e.g., for CAB1

Further iterations could be performed to obtain a still more accurate representation of the self-consistency effects, but we will not write out any more here. Second-Order Perturbation Equation. The second-order component of eq 22 is F°C2 F'C' + F2 = C2co+ S'C'e' + C't' + S't' + t2

+

(67)

which is to be solved subject to the normalization conditions in eq 51. Our primary interest is in the orbital energy correction

The Journal of Physical Chemistty, Vol. 86, No. 7, 7982 1145

Molecular Orbital Theory of Interaction Energies et2

+

= (F2 F'C' - S'C' cO)tt

(68)

which, using eq 58, can be rewritten as

The MO coefficient corrections are given by C t 2 = (€2 - tt") (F2 F'C' - S'd - C'd - S'C'tO),, (70) and, from eq 51

C

-' +

Ct? = l/ZC(StUIStul- CtulCtu')

(71)

U


The expressions in eq 69 and 70 both have an implicit dependence on C2through the F2terms on the right-hand side, e.g., for the AA block

( d'?10A'14ko) ( 4k01v^A'l@'r) (e?

kEA (k#ij)

C

(4?P- q14po)(4poP- tjOIuj9 (tjO - t P O )

PEB

+

- tho) i

# j

E A (77)

An improved estimate that includes self-consistencyeffects

can be obtained by using eq 66 and 77 to approximate CAA1 and CAA2on the right-hand side of eq 72 and 73, with possibly further iterations as necessary to obtain the desired accuracy of solution. A word should be mentioned about the source of the terms in eq 76 and 77 that involve explicit overlap integrals. These come from the higherorder terms in the expansion of 2 that are due to nonorthogonality of the fragment MO's. Total Energy. We now can evaluate the total molecular orbital energy of the interacting system AB with eq 24 by using the perturbation solutions correct through second order for the orbital energies and MO coefficients. It is found that many terms involved in the orbital energy expressions cancel out while other new types of terms appear, leading to the final results through second order E EAo + EBo E' E2 + ... (78) with the interaction energies given by OCC 2, occ E' = C (&"I - C - I&") + C

+ +

iEA

i

# j

NEB ru

PEB

E A (74)

Analogous expressions obtain for the BB block, and the off-diagonal blocks are not required to obtain energies in eq 74 can be evaluated with through second order. FAA1*' the solution for C' obtained previously, so the linear simultaneous equations for the elements of CAA2given in eq 73 can be solved exactly and the solution used to evaluate the second-order orbital energies in eq 72. As in the first-order case, it is also instructive to consider more explicit but approximate iterative solutions for CM2. For a first estimate, we neglect all contributions arising from self-consistency effects in both first and second order, leading to

(80) Particularly notable is the fact that the first-order MO coefficients required to obtain the individual first-order orbital energies cancel out and are not needed for the total first-order energy. Similarly, the second-order MO coefficients required to obtain the individual second-order orbital energies are not needed for the total second-order energy. Thus, the total energy correct through (at least)

1148

J. Phys. Chem. 1982, 86, 1146-1 149

second-order can be obtained from MO’s correct through only first order, as expected from ordinary perturbation theory.2

the general case, somewhat more difficult than in ordinary perturbation theory and lies outside the scope of the present work. To summarize, we have obtained explicit formulae for the canonical orbital energies and total molecular orbital energy, including self-consistency effects, of a composite system AB through second order in electrostatic and overlap interactions in terms of (nondegenerate) molecular orbitals of the fragments A and B. The expressions are somewhat cumbersome, but can be evaluated in a simple straightforward manner. It is hoped that the results, or possibly some simplified semiempirical version of them, will prove useful in gaining an improved understanding of molecular orbital interactions.


Concluding Discussion Two final points should be mentioned about the formalism of this paper. First is that all quantities have been developed in terms of orthonormal canonical molecular orbitals of the isolated fragments. If desired, the expressions could be easily transformed to be put in terms of more elementary nonorthogonal Slater- or Gaussian-type basis functions on the fragments. Second is that the formalism, as it stands, applies only to nondegenerate interactions. If two or more orbitals are degenerate, either within one fragment because of high symmetry or between Acknowledgment. This research was supported by the the two fragments if they happen to be identical, then Office of Basic Energy Science of the Department of Enmodifications are required to prevent terms from blowing ergy. up due to vanishing energy denominators. Due the nonlinear nature of the Hartree-Fock equations, the proper treatment of degeneracylSz1or near d e g e n e r a ~ y is, ~ ~in- ~ ~ (20) Hirschfelder, J. 0. Int. J . Quantum Chem. 1969, 3, 731. (19) Dalgamo, A. “Quantum Theory”;Bates, D. R., Ed.; Academic Press: New York, 1961; Vol. I.

(21) Silverstone, H. J. J. Chem. Phys. 1971,54, 2325. (22) Kirtman, B. J. Chem. Phys. 1968,49, 3890, 3895. (23) Certain, P. R.; Hirschfelder, J. 0.J. Chem. Phys. 1970,52,5977. (24) Hirschfelder, J. 0. Chem. Phys. Lett. 1978, 54, 1.

Quantum Cell Model for Monolayer Sollds L. W. Bruch* Physics Department, University of Wisconsin--Madison,

Madlson, Wisconsin 53706

and J. M. Phllllps Physics Department, University of Missouri-Kansas City, Kansas Ciiy, Missouri 5644 1 (Received: August 3, 1981)

The quantum mechanical form of the cell model approximation of Lennard-Jones and Devonshire is presented for monolayer two-dimensional (2D) solids. A perturbation series is developed and applied to a 2D Lewd-Jones solid of argon. The corresponding application to a 2D neon solid fails because the monolayer solid of neon is close to the quantum solid limit.

I. Introduction Monolayer solid phases have been observed in the adsorption of all of the inert gases.l As for the three-dimensional solids,2 helium forms an extremely quantum solid while xenon forms a nearly classical solid. The difference in dimensionality, however, makes the monolayer solid of neon3 closer to the quantum solid limit than its three-dimensional solid is. For adsorption on substrate surfaces with appreciable corrugation, such as the basal plane of graphite, commensurate and orientationally aligned phases of the monolayer solid occur. There are also very smooth surfaces, such as the (111)face of silver, for which these registry effects are not observed and for which the monolayer solids are effectively two-dimensional (2D) solid^.^ In spite of some subtleties in defining the nature of the order in 2D solids: thermal properties of these solids (1) See, for instance, the reviews in ‘Phase Transitions in Surface Films”, J. G . D d and J. Ruvdds, Ed., Plenum, New York, 1980. (2) J. 0. Hirschfelder, C. F. Curtias, and R. B. Bird, ‘Molecular Theory of Gases and Liquids”, Wiley, New York, 1954. (3) S. Calisti and J. Suzanne, Surf. Sci., 105, L255 (1981). (4) M. B. Webb and L. W. Bruch, to be published. 0022-365418212086-1146$01.25/0

can be evaluated with rather crude approximations in many cases? The purpose of this note is to develop the 2D version of the quantum mechanical cell model7 for a closepacked monatomic solid. Approximate forms for the quantum cell model are given which are well suited to the 2D solid of argon and which show that the 2D solid of neon has large quantum effects. Thermal properties of 2D solids have been calculated with lattice dynamical evaluations of the free energy of a harmonic solid, with cell model approximations, and with computer simulations.6 For 2D solid xenon, the Lennard-Jones and Devonshire (LJD) cell model2 provides a simple and fairly accurate method for evaluating the thermal properties between the lowest temperatures, where collective motions are the dominant thermal excitations, and the melting region, where large-scale lattice defect structures become important. For xenon, quantum effects (5) D. R. Nelson and B. I. Halperin, Phys. Reu. E, 19,2488 (1979), and references contained therein. (6) J. M. Phillips, L. W. Bruch, and R. D. Murphy, J. Chem. Phys., 75, 5097 (1981), and references contained therein. (7) J . M. H. Levelt and R. P. Hurst, J. Chem. Phys., 32, 96 (1960).

0 1982 American Chemical Society

Perturbation Approach to a Molecular Orbital Theory of Interaction

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