INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. VIII, 263-266 (1974)

Potential Energy Integrals in Semiempirical MO Methods

PATRICK COFFEY * Department of Chemistry, St . Louis University, S t . Louis, Missouri. 63156, USA

Abstracts

The two-center core-electron attraction integral VAB in zero-differential overlap semiempirical MO methods is examined. I t is concluded that core-valence orthogonality and valence symmetrical orthogonalization effects must be considered, and that these effects provide justification for the CND0/2 approximation VAB = ZByAB .

L’inttgrale d’attraction VAB B deux centres pour les Clectrons de coeur a CtC examinte dans la mkthode MO semi-empirique avec I’approximation de recouvrement diffkrentiel nul. On en conclut qu’il faut tenir compte des effets de l’orthogonalitt coeur-valence et de l’orthogonalisation symCtrique des tlectrons de valence, et que ces effets-ci fournissent une justification pour l’approximation VA, = ZByAB employde en CND0/2.

Das Zweizentren-Anziehungsintegral VAB fur die Rumpfelektronen in semiempirischen Mo-Methoden mit Null-Differentialuberlappung wird untersucht. Es wird daraus gesch- lossen, dass die Effekte der Rumpf-Valenz-Orthogonalitat und der symmetrischen Ortho- normierung der Valenzelektronen in Betracht gezogen werden mussen, und dass diese Effekte eine Rechtfertigung fur die in der c~~o /2 -Methode verwendente Naherung VAB = ZByAB liefern.

C N D O / ~ [I], the original formulation of the CNDO method, predicts equilibrium bond lengths that are often too small; co is predicted to have an equilibrium bond length of 0.691 A, for example, far shorter than the experimental value of 1.128A. To remedy this situation Pople and Segal [2] presented a modified version, C N D O / ~ . The most significant modification lies in the approximation of two-center core-electron attraction integrals of the form

Z B VaB (a1 - l a ) , orbital a on atom A

I B

where Z , is the core charge of atom B, consisting of the nucleus and inner shell electrons.

In order to preserve rotational and hybridizational invariance, V,, is taken as the integral over the s-orbital for all orbitals on center A in C N D O / ~ :

(2) v~~ = vssbB

* Present address: Department of Chemistry, Vanderbilt University, Nashville, Tenn. 37203, USA.

263 0 1974 by John Wiley & Sons, Inc.

264 COFFEY

In presenting the C N D O / ~ method Pople and Segal comment that use of (2) results in an overemphasis of penetration of the shell of atom B by the electrons in orbital a, where such penetration terms may be written as

yAB is the s-s Coulomb integral between atoms A and B. Pople and Segal remedy this situation by simply neglecting penetration integrals in CND0/2 , i.e., taking

C N D O / ~ predicts bond lengths in good accord with experiment. The CNDO methods, along with most other semiempirical methods, use a basis

set of Slater valence orbitals and take no explicit account of core orbitals. If the MO’S resulting from a full ab initio calculation are classified as core and valence orbitals, the valence orbitals are orthogonal to the core orbitals. Neglect of the effects of this constraint will cause any approximate method to behave improperly. While one-center core-valence orthogonality effects are implicitly included in CNDO/ 1 by use of experimental ionization potentials, two-center effects have not been accounted for. Zerner [3] has recently published a pseudopotential method designed to account for this lack of orthogonality, of which the simplest form is

( 5 ) V; = -C SiaFaaSaj a

where summation is over all core orbitals, Si, is the overlap integral (i I M), and F,, is the diagonal element of the Fock matrix.

Coffey and Jug [4] have pointed out that there is another significant effect if the basis is interpreted as symmetrically orthogonalized [5], and have suggested that this effect may be responsible for the success of the CND0/2 approximation (4). In the homopolar case, the diagonal terms of the core Hamiltonian matrix should be modified approximately as

Haa is the core Hamiltonian integral over orthogonalized orbitals and Ha, the integral over nonorthogonal orbitals.

Both terms in (5) and in (6) may be partitioned and added to the integrals (1) or (2) to give an effective potential integral for a valence-only method. In keeping with the original CNDO methods, we preserve rotational invariance by averaging all interactions as the integral over the s orbital:

(7)

POTENTIAL ENERGY INTEGRALS IN SEMIEMPIRICAL MO METHODS 265

where S,, is the integral (25, I 25,). We have also calculated the effects on a p , orbital on A and averaged over all three P-orbitals on A. The results were quite similar to (7 ) . For equation ( 5 ) , accounting for core-valence orthogonality, the p , value is approximately three times as great as the s-value in the bonding region, giving the s-value after averaging over all orbitals. In equation ( 6 ) , accounting for symmetrical valence orthogonalization, there are four interactions that must be considered: 25, - 2s,, 2sA - 2pbB, 2pUa - 2sB, and 2paA - 2@,B. All are of comparable magnitude in the bonding region, so that the 2s, - 2sB value may be taken as an appropriate average over the four orbitals on atom A.

Figure 1. Various approximations for VaB in C,: A-ZByAB (eq. (4)); B-VF$a"f"d (eq. (7)) ; C-Vyf i f ied neglecting symmetrical orthogonalization effects; D-ValaS~ (eq. (2)).

We present the integrals V,,, , ZByAB , and VTiaifid for C, in Figure 1. These are the integrals resulting from use of equations (2), (4), and (7) , respec- tively. Slater rule exponents were used, and the value for FISClsC was taken as 11.31 a.u., as given by Zerner [3]. The Vriaifiea curve is nearly coincident with ZByaB, explaining the success of that approximation in the C N D O / ~ method. We have also included in Figure 1 a curve neglecting the last term of Vzaifi" (equation (7) ) , showing that the core-valence and symmetrical orthogonality effects are of approximately equal importance. We conclude that neglect of these effects in a valence-only Slater basis method employing the zero-differential- overlap approximation is theoretically unsound, and that either explicit (e.g., equation (7)) or implicit (e.g., equation (4)) inclusion of such effects is necessary.

Bibliography [l] J. A. Pople and G. A. Segal, J. Chem. Phys. 43, S136 (1965). [Z] J. A. Pople and G. A. Segal, J. Chem. Phys. 44,3289 (1966)-

266 COFFEY

[3] M. C. Zerner, Mol. Phys. 23, 913 (1972). [4] P. Coffey and K. Jug, J. Am. Chem. SOC. 95,7575 (1973). [5] P. 0. Lowdin, J. Chem. Phys. 18, 365 (1950).

Received August 30, 1973. Revised October 22, 1973.