INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XV, 745-750 (1979)

Improved Uncoupled Hartree-Fock (IUCHF) Perturbation Methods and Bounds for the Second-Order Energy in

Coupled Hartree-Fock Perturbation Theory

H. VOGLER Organisch-Chemische.F Institut, Universitat Heidelberg, Im Neuenheimer Feld 270,D-6900 Heidelberg,

West Germany

Abstracts

Different kinds of improved uncoupled Hartree-Fock methods are proposed for the calculation of second-order perturbation energies. Using these methods inequalities are derived for the error of the uncoupled procedure with geometric approximation.

On a obtenu des methodes des theories des pertubations de Hartree-Fock non-accouplee de I'energie du second ordre. Avec ces methodes des inegalites pour I'erreur dans l'approximation giometrique sont derivtes.

Es werden verschiedene verbesserte ungekoppelte Hartree-Fock storungstheoretische Methoden 7ur Rerechnung von Energien 2. Ordnung abgeleitet. Diese Methoden liefern untere Schranken fur den Fehler des ungekoppelten Verfahrens mit geometrischer Approximation.

1. Introduction

Based on the Hartree-Fock (HF) approximation one has the closed-shell wave function +: = lq5?& . . . q5:cc&ccl, where q5: is a real and orthonormal orbital satisfying the e equation F('q5: = ~ y q 5 I ) with the operator Fo= Ho+ G" 111. In the case of an external perturbation (e.g., magnetic or electric fields) which can be expressed as the sum of one-electron operators, the wave function will be =

Iq51$l . . . q 5 0 L C ~ O C C / , where 4, satisfies the equation Fq5, = E&. The operator F and d l can be expanded in a power series of the parameter of the perturbation v : F = Cy=O v P F P with FP = H P + G P and 4I = q5: + vq5f +. . . . For p > 1 we will put H" = 0 since we are not interested in this paper in terms arising from these operators. All operators are to be Hermitian and the following consideration will be restricted to the case where H' is purely imaginary. The extension to the real case is straightforward. The Hamiltonian of the system is b = xz H ( i ) + I E < , r ; * . For the expectation value of b one has E = ( + o l b 14") = E: + v2E: +. . . with the HF closed-shell energy E: [l] and the exact second-order energy (SOE) of the coupled HF (cHF) perturbation theory 121 E: = 2 xp" (q5:lH11q5:), where the imaginary first-order orbitals 4: are obtained from the equation (FO- E : ) q5: = - ( H ' + G1)q5Y. GI is the electron interaction operator which is first order in Y :

G'(1) = --y d w ; ; lq5:(2)*P,2q5p(2)+q5:(2) PI&: (2)l

3.Q 1979 John Wiley KL Sons, Inc. 0O20-7608/79/001S-O74S$Ol .OO

746 VOGLER

Since G' contains all the first-order orbitals, solving the CHF equations is a tedious numerical calculation. To avoid this, the uncoupled HF (UCHF) procedure [3] can be used. In this case the orbitals are taken as q5j = 4: + v p 4 ! +. . with a scaling factor p [4,5]. The first-order orbitals are obtained from the equation (Po- E ? ) # ! = -H'4:. The expectation value of b is then Eo = E: + v'E; +. . . with the SOE

E: = (211. - 2 ) ~ 0 2 + cL 2 ~ ' 2

where

( i k ) I k

Since the CHF theory is essentially a variational method in that it is based on the best single determinant wave function for the perturbed Hamiltonian b [6], we have E: sE; . Minimizing E i with respect to p leads to the SOE of the UCHF

theory with geometric approximation (GA) [ 5 ] , I?; = po502 with po = (1-E12/Eo2)-'. I?: will be a minimum only if d'E;/dp Iw=w.o>O and, consequently, - Eo2 + El2 > 0. Since E"' 5 0 it follows that the use of the UCHF

theory with GA will give useful results only if po 2 1.2; -E? = e is the error of the SOE of the UCHF theory with GA.

2. Improved Uncoupled Hartree-Fock Perturbation Methods

A better approximation to the CHF theory is possible if configuration inter- action with the singlet monoexcited configurations @rk is included: I+!J = +bo + x ( t k ) C l k 4 1 k . b can be arbitrarily divided as b = b'+ A ( b - bo) with the dummy parameter h = 1 . b " has to be chosen in such a way that it is diagonal with respect to CLo and all (Ltk. Expanding 4 and the expectation value E = (t+bIbl$) and applying the Rayleigh-Schrodinger perturbation theory we arrive at the energy E = E,+Ez, which is correct up to second order in A with

IUCHF PERTURBATION METHODS 147

The various matrix elements are expressed as follows:

A possible choice of bo is (a = 1)

with

EIk =($8kIt)I$lk)=El)i +U2E;i + ' " , EY; 'EPk" +2KiOkO-JiOkO

[7]. This leads to U:k = 2-"* [ p v , k + ( p - 1) w,k] and the configuration-inter- action contributions are zero in the framework of the CHF method. The SOE with respect to u is

E 2 1 - - - p 2 ( E o 2 - E E 2 - E { - E : - E : ) + 2 p ( E o 2 - E ( - E f / 2 ) + E :

Extremizing E: with respect to p leads to E ; = E ? ( p l ) with

(Eo2 - E{ - Ey/2) ( E O ~ - E ' ~ - E'; - E : - E ; ) p1 =

Since ( U : k ) 2 ~ 0 and El'; > O as long as the HF approximation is meaningful we have the inequality e: 5 Ef$pO) 5 I?;. The restriction to monoexcited configura- tions guarantees that E: 5 E l . Therefore, this method is only meaningful if g; is a minimum of E:. This will be the case only if a2E:/ap21 ~ =1"1 > 0, thus leading to the condition

- E ( ) ~ + E ~ ~ + E { +E:+E:>o or - E " * + E ~ ~ > ~ E { + E ~ + E : ~

This improved UCHF (IUCHF) method (called IUCHF-I1 in Ref. 7) provides lower bounds to the error e by

-2 2 -2 e ? e l = En -El =-ei - = E , - E : ( ~ ~ )

= - p i ( E { + E f +E; )+2po(E[ + E : / 2 ) - E [

A disadvantage of these bounds is that they require the calculation of the Coulomb and exchange integrals JIOkO and &k". This difficulty can be avoided by another definition of b n (a = 2):

748 VOGLER

with

G'=C(H+GO) ( i ) , E:k =(gl/,kIt)'lgl/,k)=EPk2+yZE:k2++. . . and EP,' = EP;

We obtain the same formula for the SOE Eg as above; however, a = 1 has to be replaced by a = 2. Since E: = Eo2 and E5 = - 2E" we obtain by extremizing E i with respect to p g; = E"'+ p2Et2 with p2 = (1 - E;/E")-'. This UCHF method is called IUCHF-I in Ref. 7 where, however, there was given no rigorous derivation. Only if El2 - E," < 0 is gi the minimum of E: and can provide an approximation to the SOE of the CHF procedure. Now we have another lower bound for the error e :

I

ez=,$':, -gi =[(I -E'2/E02)~'-(1-E2h/E12)-1]E12

The replacement of g: by E: (po) would result in the meaningless lower bound e ; = 0. We expect that el 2 e 2 since in general E,k 5EYf .

It is possible to choose G o as in the IUCHF-I method (a = l ) , but we shall put EZ = E "= EO" + v2E2". . . for all (ik). Then the formula for the SOE E; is the same as for E i if one replaces a = 2 by the appropriate a value. We shall call this method IUCHF-I11 (IV) if one fixes

01

0 1 04 - 0 Eo3 = min Elk (E - E LuMo - E ioMo) ( i k )

The extremum E: is, however, a minimum or a maximum of E i and it might be a lower or an upper bound to E: due to the choice of E'". Burrows [S, 91 has given an upper bound eB for the error e. It turns out that

e B = g i - E : ( p O ) = -pi(E:+E;+E3h)+2p0(E:+E3R/2)-E$

This bound has been used to prove the reliability of the UCHF theory with respect to the CHF approach [9, 101. However, the derivation of eB in Ref. 8 is not correct and we shall see that it is indeed possible to find cases where e > eB. That eB cannot be an upper bound for e can also be visualized by the following reasoning. The minimum ZB of eB with respect to p is obtained with p B = (Ef3 + E2/2)/(E{ +E: + E!) . Since in general po # p B we find that the error of E? calculated with p = 2p.8 -pO is not greater than that of l?; although Ei(2pB - EL") 2 Eo. - 2

3. Numerical Examples

To demonstrate the utility of our lower bounds for e and the two new IUCHF methods we made calculations on the nonlocal r-electron contributions to the magnetic susceptibility based on Amos and Roberts [ll] within the PPP approxi- mation [12] (resonance integrals -2.39 eV, electron repulsion integrals due to Nishimoto and Mataga [13] with a monocentric repulsion integral 11.13 eV). In all cases all possible monoexcited configurations have been taken into account. In Table I the comparison is made between the values obtained with the UCHF,

IUCHF PERTURBATION METHODS 749

TABLE I. Second-order energy contributions E: and ,!?: to the magnetic susceptibilities, errors e, and lower bounds e, for e and eB (all values in - cm3 mol-I).

Compound E: BE B: E; F : ~ B: e e l e; e 2 eg -.

nnphthn l e n e 12 .4 1 2 . 4 1 2 . 4 12 .4 12 .4 1 2 . 4 0.02 0 . 0 2 0 .01 0 . 0 2 0 .05

e n t h r a c e n e 46.2 45.7 46.1 46 .0 46 .3 46 .4 0 .49 0.42 0.42 0.29 0 .92

t e t r a c e n e 109 .7 107 .5 1 0 9 . 3 1118.7 1 1 0 . 3 1 0 5 . 3 2 .19 1.82 1 . 8 0 1 .21 3.90

p e n t a c e n e 211.0 205 .0 209.8 209.1 211.8 206 .5 5.94 4 .72 4.64 3 .10 10.18

h e x a c e n e 358.1 3 4 5 . 6 355.1 3 5 1 . 8 3 5 9 . 8 350 .6 12 .47 9.51 9.21 6 .17 20.73

h e p t a c e n e 568 .9 5 3 6 . 4 552 .6 547.0 562.0 546 .7 22 .47 1 6 . 1 9 15 .70 10 .63 36 .38

a z u l e n e 1 1 . 5 11 .1 11 .4 1 1 . 2 1 1 . 5 11 .4 0 .34 0 . 2 6 0 . 2 6 0.12 0 . 2 9

c y c l o p e n t a z u l e n e 35 .9 31.3 35 .2 33 .0 35 .2 3 4 . 5 4.51 3.91 3 .90 1 . 6 2 3 .92

n n t h r a z u l e n e 47.6 43 .6 45.9 4 4 . 3 48.1 44 .2 3 . 9 5 2.32 2.31 0.71 2 . 2 3

a In all cases we have miqLk, EYi = EEoMo LUMO.

IUCHF-I, 11,111, and IV, and the CHF method. As was already shown in Ref. 7 the IUCHF-I, I1 values gf and E ; approximate the exact E: values considerably better than the IJCHF values si, which are very unsatisfactory for the larger polyacenes. The absolute errors of the new IUCHF-IV procedure are comparable to those of the IUCHF-I one. However, IUCHF-IV does not provide any compu- tational advantage over IUCHF-I. The best agreement between CHF and UCHF

values is obtained with the IUCHF-111 method which avoids the calculation of the many KiOkO and J l o k o integrals. The errors e and their bounds are given in Table I. We find that eB < e for the polyacenes but not for the nonalterant hydrocarbons. Therefore, eB cannot be used as an upper bound to e. The lower bounds el and e; are not much smaller than e. The differences between el and ei which are easily computable are negligible. The lower bound e2 is not as good as e l ore; . However, e2 may be calculated without significant additional computational effort together with E i and can show in which cases gi will be not a reliable approximation to the CHF value,

Acknowledgments

The author is greatly indebted to the Deutsche Forschungsgemeinschaft for financial support and to the Universitatsrechenzentrum Heidelberg for computer time.

Bibliography [ l ] C. C. J. Roothaan, Rev. Mod. Phys. 23,69 (1951). [2] A. Dalgarno, Proc. Roy. SOC. A 251,282 (1959); G. G. Hall and A. Hardisson, Proc. Roy. SOC.

[3] A. T. Amos and J. I. Musher, Mol. Phys. 13, 509 (1967). A 268, 328 (1962).

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[4] A. T. Amos, J. Chem. Phys. 52, 603 (1970). [5] D. F. Tuan, Chem. Phys. Lett. 7, 115 (1970). [6] H. Nakatsuji, J. Chem. Phys. 61, 3728 (1974). [7] G. Ege and H. Vogler, Chem. Phys. Lett. 31, 516 (1975). [8] B. L. Burrows, Int. J. Quantum Chem. 7, 345 (1973). [9] H. G. F. Roberts, Theoret. Chim. Acta (Berl.) 33, 269 (1974).

[lo] P. Lazzeretti, R. Zanasi, and B. Cadioli, J. Chem. Phys. 67, 382 (1977). [ l l ] A. T. Amos and H. G. F. Roberts, J. Chem. Phys. 50,2375 (1969). [12] J. A. Pople, Trans. Faraday SOC. 49, 1375 (1953). [13] K. N. Nishimoto and N. Mataga, Z. Phys. Chem. (Frankfurt) 12, 335 (1957).

Received August 22, 1978 Revised November 13, 1978 Accepted for publication December 14, 1978