INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. VIII, 777-782 (1974)

Virtual Orbital Transformation Prior to Configuration Interaction with

Localized Orbitals PATRICK COFFEY *

Department of Chemistry, St. Louis University, St. Louis, Mo. 63156, U.S.A.

Abstracts

A virtual orbital transformation is proposed involving pairing of localized occupied orbitals with virtual orbitals. The virtual orbitals are transformed so that each virtual orbital is as “close” as possible to its occupied counterpart, where closeness is the inverse of the particular definition of localization. The appropriate transformation is derived for the special case of Foster-Boys localization, and an illustrative C N D O / ~ calculation on HNO is presented. INDO CI results on the series N, , CO, BF indicate that use of this trans- formation may reduce the number of energetically significant configurations.

On propose une transformation d’orbitales virtuelles qui engendre une certaine com- binaison d’orbitales occupkes localistes avec des orbitales virtuelles. Les orbitales virtuelles sont transformtes de faqon A crter des paires d’orbitales virtuelles et occuptes aussi “voisins” que possible, oh le “voisinage” est l’inverse de la dtfinition particulitre de localisation. La transformation correspondante au cas sptcial de la localisation de Foster-Boys a Ctt obtenue, et un c a h l C N D O / ~ ihstratif pour HNO est prtsentt. Des rtsultats INDO CI pour la strie N, , CO, BF indiquent que l’emploi de cette transformation-ci peut rCduire le nombre de configurations significatives du point de vue tnergetique.

Eine Transformation der virtuellen Orbitale wird vorgeschlagen, die zu einer Paarung lokalisierter besetzer Orbitale mit virtuellen Orbitalenfuhrt. Die virtuellen Orbitale werden so transformiert, dass jedes virtuelles Orbital so “nahe” wie moglich seinem besetzten Gegenstiick kommt, wobei “Nahe” das inverse der besonderen Definition von Lokalisierung ist. Die der Foster-Boys’schen Lokalisierung entsprechende Transformation wird her- geleitet, und eine illustrative c~~o/2-Berechnung fur HNO wird prasentiert. INDO-CI- Ergebnisse fur die Reihe N, , CO, BF deuten an, dass die Anwendung dieser Transformation die Anzahl der energetisch bedeutsamen Konfigurationen reduzieren kann.

1. Introduction

A wave function formed from an antisymmetrized product of orbitals is invariant under any unitary transformation. In the two-orbital case, this may be ~

* Present address: Department of Chemistry, Vanderbilt University, Nashville, Tenn. 37203 U.S.A.

777

0 1974 by John Wiley & Sons, Inc.

778 COFFEY

written as

(5; = q51 cos 8 - rP2 sin 8

4; = sin 8 + +2 cos 8

where 41 and cj42 are orbitals before transformation, and 4; and 4; are orbitals after transforma tion.

In molecular orbital calculations based on the Roothaan equations, the orbitals arising from diagonalization of the SCF matrix are in general delocalized over the entire molecule. A number of localization methods have been proposed, differing only in the particular definition of “localization.” Depending on the definition, the angle 6 in Equation (1) is determined so as to maximally localize the two orbitals. In the many-orbital case, localization is achieved by an iterative series of two-orbital transformations until the desired degree of convergence is attained. Although the various localization methods differ conceptually and in numerical complexity, they all seem to lead to orbitals closer to intuitive concepts of bonds and lone pairs than do canonical orbitals. They are consequently often useful in the interpretation of calculations.

Localized orbitals also have certain advantages in a limited configuration interaction calculation. Since the electrons are localized in different regions of space, interorbital correlation effects will hopefully be smaller than for canonical orbitals, and the number of energetically important configurations may thereby be reduced. There is some evidence however that interorbital correlation may not always be reduced by using localized orbitals; Ahlrichs and Kutzelnigg [l] have found a counter example in LiH.

The construction of unoccupied orbitals for use in such a calculation poses a problem, however. The motion of electrons in a certain region of space is best correlated by excitation to an unoccupied orbital in the same region of space. This was dramatically demonstrated by Shull and Lowdin [2], who showed that a CI calculation on the ground state of helium using only the discrete hydrogenlike functions is not complete, but that a similar calculation using associated Lagurre polynomials of order (21 + 2) is complete and converges fairly rapidly. The latter set is much more compact than the hydrogenlike functions; functions of high principal quantum number are still largely localized in the same region as the ground state. Diner et al. [3] have developed a perturbational technique based on the concept that the excited orbitals should be in the same region as the occupied orbitals. They build simple bonding and antibonding orbitals in the bond regions using hybrids of atomic orbitals. As a general method for developing suitable excited orbitals, Boys [4] has proposed multiplying the occupied orbitals by functions of the x,y, and z spatial coordinates, orthogonalizing, and normalizing. This is unsuitable for calculations based upon a Roothaan-type expansion, because functions not included in the basis set will be generated. Simple localization of the virtual orbitals in the same fashion as the occupied orbitals is also unsuitable, since

VIRTUAL ORBITAL TRANSFORMATION 779

there is no guarantee that a given occupied orbital will have an unoccupied orbital nearby.

2. Method

We propose a general method for the construction of suitable unoccupied orbitals, applicable to any localization procedure. For a set of n orbitals, of which m are occupied, choose q of the occupied orbitals, where q 5 n - m, and pair one of the n - m virtual orbitals with each of the q-occupied orbitals. Then trans- form all n - m virtual orbitals among themselves so that the q-pairedvirtualorbitals are as close as possible to their occupied counterparts. In practice, one chooses the q-occupied orbitals to be paired so as to include those localized in the bond regions. The definition of closeness is the inverse of the definition of localization for the particular method. Thus for the Edmiston-Rudenberg [5] method, electrons in the transformed orbitals would exhibit maximal repulsion with electrons in their occupied partners. For the Foster-Boys [4,6] method, the dipoles of the transformed virtual orbitals would be as close as possible to the dipoles of their assigned occupied partners. We have chosen the latter scheme for a demon- stration.

Foster and Boys localize the occupied orbitals so that

be maximized, where

Simple trigonometric manipulation leads to

8 = $ arctan (4A . B/(4B2 - A2 )>

for the transformation angle for the two-orbital case in Equation ( l ) , where

A G R, - R2

If 4B2 - A2 is positive, 48 lies in the second or third quadrants for equation (2) to be maximized, and if 4B2 - A2 is negative, in the first or fourth quadrants. Once the occupied orbitals have been constructed, we can transform the virtual orbitals so that

(3) Q

J E 2 (Ri-RRi_,J2 i=m+l

is minimized, where the occupied orbitals have been ordered so that the first q orbitals are those that are to be paired. Use of Equation (3) for the two-orbital

780 COFFEY

case leads to the equation

(4)

where

(4B2 - A2) sin 48 - 4A. B cos 48 - 2C (A sin 28 + B cos 28) = 0

C RZ-,,, - R1-m

Equation (4) may be solved numerically, and the roots tested for an absolute minimum.

Convergence of the many-orbital transformation may be improved by taking full advantage of the symmetry of the system. I n the case of C,, symmetry, for example, if the z-axis is chosen as the symmetry axis, basis functions that transform as the x- andy-axes either are equivalent or may be simply made so. In addition to the transformation [Equation (1 )] between molecular orbitals, a transformation between the equivalent basis functions is also possible :

(5) xi = xr cos w - xz sin w

2; = xw sin w + xz cos w

where xz and xr are basis functions before transformation through the angle w, and xk and xb are basis functions after transformation. Subsequent to the transformation of Equation ( l ) , a second transformation through the angle w of Equation (5) may be performed. w is determined from Equation (3), giving

w = arctan (DIE) where

D = (XlIY IXI>(xl-mI x 1x1-m> + (X21Y IX2)(X2-ml x 1x2-m)

E = (Xll x IXI>(Xl-ml x 1x1-m) + (x2l x IXz>(Xz-ml x 1x2-m)

- (Xll x IXl)(XI-mlY IXI-m> - (xz l x IXZ)(X2-mlY 1x2-m)

+ (XlIY IX1)(Xl-mlY 1x1-m) + (X21Y IX2)(X2-mlY 1x2-m>

The quadrant in which w lies is chosen so that D sin w + E cos w is positive. Convergence may also be improved by judicious ordering of the orbitals

prior to transformation. We order the occupied orbitals so that those orbitals that are to be paired are placed first. The virtual orbitals are then ordered so that the orbital initially paired with each occupied orbital is most similar to it. Our definition of similarity is as follows: take the scalar product of each occupied orbital with each unoccupied orbital, but consider all LCAO coefficients as positive. Those pairs of orbitals with the largest such products are most similar. An initial iterative series of transformations between virtual orbitals of the same symmetry only has been found to help convergence. For the Foster-Boys scheme, trans- formation of both the occupied and virtual orbitals is extremely fast, and takes only a small fraction of the time required for either the SCF or CI processes.

VIRTUAL ORBITAL TRANSFORMATION 78 1

TABLE 1. Foster-Boys localized C N D O / ~ orbitals and transformed virtual orbitals for HNO at the geometry of Figure 1.

4 O r b i t a l C e n t e s o f C h a r g e

V i r t u a l fi

Occupied

O r b i t a l X Y 2 O r b i t a l X Y 2

1 1 . 2 8 1 6 4 0 . 6 1 3 1 3 - 0 . 2 2 9 2 1 7 1 . 2 3 7 4 1 0 . 5 5 6 5 2 -0 .17306-- 2 1 . 2 8 1 6 4 0 . 6 1 3 1 1 0 . 2 2 9 2 1 8 1 . 2 3 7 4 1 0 . 5 5 6 5 2 0 . 1 7 3 0 1 3 0 . 3 8 1 0 3 - 0 . 0 1 4 0 6 0.00000 9 0 . 3 9 6 7 4 0 . 0 2 1 1 1 0.00000 4 1 . 2 5 7 4 5 - 0 . 3 1 2 9 1 0.00000 5 1 . 2 3 3 6 6 1 . 3 7 0 6 4 0.00000 6 1 . 7 4 3 4 2 1 . 1 9 0 7 5 0.00000

O r b i t a l s

1 2 3 4 5 6 7 8 9 0 . 0 0 2 6 0 . 0 0 2 6 0 . 6 8 0 7 0 . 0 9 9 5 0 . 0 5 1 8 0 . 0 6 2 3 0 . 0 2 8 3 0 . 0 2 8 3 0 . 7 2 0 1 is 0 . 3 0 5 6 0 . 3 0 5 6 0 . 3 9 2 7 - 0 . 6 8 4 4 0 . 0 1 2 3 - 0 . 0 7 5 3 - 0 . 2 4 5 2 - 0 . 2 4 5 2 - 0 . 2 5 3 9

Ns 0 . 1 6 5 4 0 . 1 6 5 4 - 0 . 6 1 6 2 - 0 . 3 7 3 9 0 . 0 7 7 5 0 . 0 5 3 7 - 0 . 0 9 8 2 - 0 . 0 9 8 2 0 . 6 3 0 5 . -.- 0 . 3 4 6 3 0 . 3 4 6 3 - 0 . 0 4 7 3 0 . 6 0 6 2 0 . 0 4 6 0 - 0 . 0 7 6 5 - 0 . 4 3 7 3 - 0 . 4 3 7 3 - 0 . 0 0 3 9 NX

N; - 0 . 4 7 6 2 0 . 4 7 6 2 0.0000 0.0000 0.0000 0 .0000 0 . 5 2 2 7 - 0 . 5 2 2 7 0.0000 0 0 . 3 2 8 2 0 . 3 2 8 2 - 0 . 0 0 0 1 0 . 0 7 0 0 - 0 . 5 7 8 5 0 . 5 8 3 4 0 . 2 2 7 2 0 . 2 2 7 2 - 0 . 0 3 8 6 0' - 0 . 1 2 1 7 - 0 . 1 2 1 7 0 . 0 2 0 8 - 0 . 0 2 2 8 0 . 5 3 9 6 0 . 7 8 6 8 - 0 . 1 5 3 2 - 0 . 1 5 3 2 - 0 . 1 1 0 5 Ox - 0 . 3 6 9 8 - 0 . 3 6 9 8 0 . 0 0 2 4 - 0 . 0 9 4 8 - 0 . 6 0 2 7 0 . 1 4 9 5 - 0 . 4 0 3 9 - 0 . 4 0 3 9 0 . 0 7 5 6 0: - 0 . 5 2 2 7 0 . 5 2 2 7 0 .0000 0.0000 0.0000 -0 .0000 - 0 . 4 7 6 2 0 . 4 7 6 2 0 .0000 -

An example of a CNDO/:! [7] calculation on HNO at the experimental [8] geometry is given in Table I. The dipole matrices were constructed using Pople's [9] approximation of neglecting all two-center integrals. The occupied orbitals were localized by the Foster-Boys method, and the virtual orbitals transformed according to equation (3). Orbitals 1 and 2 may be considered as the N=O

(1.4496,1.1487,0) 0 P "

Figure 1. Experimental geometry of d N 0 , including the coordinates used in the calculation of Table I. The center of charge of each orbital is denoted by the orbital number within a circle; bold circles are above the x-J plane, dotted circles

below the plane, and circles of normal thickness are in the plane.

782 COFFEY

double bond, orbital 3 as the H-N bond, orbital 4 as the lone pair on N, and orbitals 5 and 6 as lone pairs on 0. The virtual orbitals 7, 8, and 9 were trans- formed so that their centers of charge were as near as possible to the centers of orbitals 1, 2, and 3, respectively. Results are shown in Table I and Figure 1. A comparison shows that the orbitals are quite close considering that the N, and 0, orbitals are almost fully occupied and therefore unavailable to the virtual orbitals.

We have also performed INDO [lo] calculations on the isoelectronic series N,, CO, BF at experimental geometries and have localized and transformed as above. In each case, three occupied orbitals were localized between the two atoms, with a lone pair on each atom. The virtual orbitals were paired with the three bonding orbitals. A configuration interaction calculation entailing a double excitation from each bonding orbital to its paired unoccupied orbital was com- pared to a more extensive calculation including an additional twelve configura- tions. The additional configurations consisted of all double excitations to a different occupied orbital, and all double split excitations where one electron went to the paired orbital and the other to a different orbital. The smaller calcula- tion provided 96.8, 95.3, and 97.5% of the energy improvement of the larger calculation for N,, CO, and BF, respectively, indicating the effectiveness of the virtual orbital transformation in limiting the number of important configurations.

Bibliography

[l] R. Ahlrichs and W. Kutzelnigg, J. Chem. Phys. 48, 1819 (1968). [2] H. Shull and P. 0. Lowdin, J. Chem. Phys. 30, 617 (1959). [3] S. Diner, J. P. Malrieu, and P. Claviere, Tbeoret. Chim. Acta 13, 1 (1969). [4] S. F. Boys, in Quantum Theory of Atoms, Molecules, and the Solid State, P. 0. Lowdin, ed. (Academic

[5] C. Edmiston and K. Rudenberg, Rev. Mod. Phys. 35, 457 (1963). [6] J. M. Foster and S. F. Boys, Rev. Mod. Phys. 32, 300 (1960). [7] J. A. Pople and G. A. Segal, J. Chem. Phys. 44, 3289 (1966). [8] F. W. Dalby, Can. J. Phys. 36, 1336 (1958). [9] J. A. Pople and G. A. Segal, J. Chem. Phys. 43, S136 (1965).

Press, New York, 1966), p. 253.

[lo] J. A. Pople, D. L. Beveridge, and P. A. Dobosh, J. Chem. Phys. 47, 2026 (1967).

Received August 15, 1973. Revised April 2, 1974.