J. Phys. Chem. 1982,86,3135-3139
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Optimal Geometric Approxlmation of the Perturbed Uncoupled Hartree-Fock Method for Physical Propertles. 2. Quadrupole Polarizabilitiest Debble Fu-tal Tuan' and Gary W. Loar LMparhnent of Chemlsby, Kent State University, Kent,
Ohio 44242 (Received: Ju& 21, 1981; In Final Form: March 2, 1982)
The optimal geometric approximation of the perturbed uncoupled Hartree-Fock (optimal GPUCHF) method is applied to the study of the quadrupole polarizabilities of the He and Be isoelectronic sequences. Results are compared with those from the coupled Hartree-Fock (CHF) method and the less accurate uncoupled Hartree-Fock (UCHF) methods. Dependences of the optimal variational parameters, 70p, on the choice of the zeroth-order orbitals, the forms of the first-order orbitals, and the systems are studied. Applications of the optimal GPUCHF method here and to dipole polarizability in paper 1indicate that results from the optimal GPUCHF method are the most accurate approximations to result from the CHF method. Since the optimal GPUCHF method requires no iterative procedure, it is particularly promising for larger atomic or molecular systems which are cumbersome to treat by CHF or other theoretical methods.
Introduction It has been known that the coupled Hartree-Fock (CHF) method' is one of the most satisfactory perturbation methods for computing physical properties of atoms and molecules.2 This method can give reasonably accurate resulta. However, ita iterative procedure requires laborious computational effort. Different uncoupled Hartree-Fock (UCHF) schemeslaJ have been proposed to simplify the CHF procedure but with a certain trade-off of accuracy. Better numerical resulta of some UCHF schemes have been attributed to the neglecting of certain coupled terms in the firsborder perturbation equation and explained as due to the absence of the "self-potential".N The perturbed uncoupled Hartree-Fock (PUCHF) method4 is Dalgarno's UCHF method,la formulated with a zeroth-order manyelectron Hamiltonian, improved to include the first-order correction. The geometric approximation of the PUCHF method, the GPUCHF method, was proposed by Schulman and Musheras Amos has proveds that the geometric approximation is actually a Feenberg-type scale transformation' of the PUCHF method. It has also been shown that the results from the GPUCHF method should be approximations to those from the CHF method.8 In order to gain a deeper understanding of the self-potential and to study the accuracy of various methods at different-UCHF, PUCHF, GPUCHF-levels of approximation, we have proposed three different GPUCHF meth~ds.~JOThese three methods differ in that the zeroth-order Hamiltonian, Ho,,includes all of, a half of, or none of the self-potential term, T ( & - cyii), when 7 = 1,1/2, and 0, respectively. We have applied these methods to the study of dipole polarizabilities and dipole shielding factors of atomic systems. Results showed that the method with no self-potential term offered the best approximation at the UCHF level, but the method with half of the self-potential term gave the most accurate result at the GPUCHF level. The above results suggested the following: (a) One should not rigidly associate the self-potential term with some fictitious physical effect-self-potential. Instead, it is merely the mathematical expression indicating the dependency of the perturbed orbital i' on the unperturbed 'This paper is baaed on part of a thesis that was presented by Gary W. Loar to the Graduate School of Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
orbital ios in the presence of the nucleus, other electrons, and the external perturbation. (b) There should be an optimal amount of self-potential, T ~ ~ (-&cyii), which should correspond to the most proper representation of the dependency of 'i on io. Since'i and the dependency of i' on '?i reflect the readjustment of the orbitals in the system due to the perturbation, T ~ ~ -( aii) @ ~should ~ depend on the system, the physical properties, and the choice of ios. In paper 1'' of this series, we proposed a variational scheme to obtain the optimal T , T~ . Physical properties obtained by the GPUCHF metho3 corresponding to T , should give the optimal approximation to those from the CHF method and should be the upper bound of the exact values. We called the method the optimal geometric approximation of the perturbed uncoupled Hartree-Fock method, or simply the optimal GPUCHF method. The merits of this method and the dependence of T~ on the system, on the physical properties, and on the cKoice of io and forms of i' are investigated in this series of papers. Results of dipole polarizabilities for the He and Be isoelectronic sequences were presented in paper 1.'' Those of quadrupole polarizabilities are investigated in this paper. The studies of shielding factors, which require the use of triple perturbation theory, will be given in paper 3. A
~
(1)(a) A. Dalgarno, Proc. R. SOC.London, Ser. A, 251,282(1959);(b) L. C. Allen, Phys. Rev., 118,167(1960);(c) A. Kaneko, J. Phys. SOC.Jpn., 14, 1600 (1959); (d) H. Peng, Proc. R. SOC.London, Ser. A , 178,499 (1941). (2)Other perturbation methods for computing physical properties are described in the following: (a) J. Goldstone,Proc. R. SOC.London, Ser. A, 239,267 (1957);(b) H. P. Kelly, Phys. Rev.,131, 684 (1963);(c) H. P.Kelly, ibid., 136,B896 (1964);144,39 (1966);(d) E.S. Chang, R. T. Pu, and T. P. Das., ibid., 174, 1, 16 (1968); (e) T.C. Caves and M. Karplus, J. Chem. Phys., 50,3649 (1969);(0 J. M. Schulman and D. N. Kaufman, ibid., 53,477 (1970). (3)(a) M.Karplus and H. J. Kolker, J.Chem. Phys., 38,1263 (1963); 39,2011 (1963);(b)M.Yoshmine and R. P. Hurst, Phys. Rev.,135,A612 (1964);(c) P. W. Langhoff and R. P. Hurst, ibid., 139,A1415 (1965);(d) P. W. Langhoff, M. Karplus, and R. P. Hurst, J . Chem. Phys., 44,505 (1966);comparison of different coupled and uncoupled Hartree-Fock methods is given here too; (e) the various UCHF methods and their perturbation corrections have been considered formally by B. R. Riemenscheider and N. R. Kestner, Chem. Phys. Lett., 5,381 (1970). (4)(a) D. F.-t. Tuan, S. T. Epstein, and J. 0. Hirschfelder, J. Chem. Phys., 44,431(1966);(b) S.T. Epstein and R. E. Johnson, ibid., 47,2275 (1967);(c) D.F.4. Tuan and K. K. Wu, ibid., 53, 620 (1970). (5)J. M. Schulman and J. 1. Musher, J. Chem. Phys., 49,4845(1968). (6)A. T. Amos, J . Chem. Phys., 52,603 (1970). (7)E. Feenberg, Phys. Rev., 103, 1116 (1956);E. Feenberg and P. Goldhammer, ibid., 105,750(1957). (8)D. F.-t. Tuan, Chem. Phys. Lett., 7,115 (1970). (9)D. F.-t. Tuan and A. Davidz, J. Chem. Phys., 55, 1286 (1971). (10)D. F.-t. Tuan and A. Davidz, J. Chem. Phys., 55, 1294 (1971). (11)D. F.-t. Tuan, J . Phys. Chem., 83,1520 (1979).
QQ22-3654/82/2Q86-3735$01.25/Q 0 1982 American Chemical Society
The Journal of Physical Chemistty, Vot. 86, No. 16, 1982
3136
procedure for treating open-shell systems will be given later.
Theory and Calculation The theory of the optimal GPUCHF method for calculating second-order physical properties based on the Hellmann-Feynman theorem,12 the Hylleraas variation principle,13 the double perturbation theory,14the interchange theorem,16and the geometric approximation5 was given in paper 1. Here, we only outline the main features of it to facilitate our presentation and discussion. Optimal GPUCHF Method for Second-Order Physical Properties. For an N-electron atomic system with a closed-shell configuration, when the Hartree-Fock (HF) approximationle is used for the unperturbed system, the zeroth-order wave function, $Ova, is in the form of a Slater determinant: A
$/o,o
fj io i=l
where io is the orthonormal HF spin orbital. The zerothorder Hamiltonian, HO, is of the form
H,O = N
N
Tuan and Loar
The first-order orbital il in Qo and Q1 are obtained by minimzing the following functional: = (il,[(hi- ):e + T ( & - aii)]i') N
C
((e;
- e/')(i1,j0)2
- (1
j#i
- T)(il, (pii- aii)jo)+
N
j#i
where :e is the HF orbital energy, and ti1 = (io,wiio). (3) Use the GPUCHF method to calculate the geometric approximation of Q, Qgeom,according to for 0 5 7 I1 (4) Qgeom = Qo(1 - Qi/Qo)-' (4) Use the optimal GPUCHF method to find the optimal Qgeom,Qop,by variational method. Namely, using T as a variational parameter, solve (aQgeom/a~) = 0
for an optimal T = T , ~ , and find the Q,. Based on the variation principle, Qop should be the upper bound of Q. Qop has also been shown to be an approximation of QCHP Calculation. For the quadrupole polarizability aq = -2Q; W due to an electric field in the z direction is N
N
i=l
i=l
W = C wi = -C [(1/2)r?(cos2 ei - I)] where the parameter T designates the amount of self-potential term included in H,O, the one-electron Hamiltonian N
and pij
- aij = (jOg')l(l - Pij)/FijPO(j))j
None of the quantities of the Hartree-Fock approximation for the unperturbed system will be affected by the choices of H,O.9,11 When the external perturbation pertaining to the second-order physical property, Q, is the s u m of one-electron operators, MW= pCi=lNWi,the principal steps of using the optimal GPUCHF method to calculate Q are the following: (1) Use the UCHF method to calculate the zeroth-order approximation of Q, according to N
Qo =
C (il,wiio)
for 0 IT I1
i=l
(1)
(2) Use the PUCHF method to calculate the property corrected to first order, Qo + Q1, with N
Q1 =
i=l
{ ~ [ ( i ' i ~ l -i ~(i1i0$'io)] i') +
N [(iljlliOjO)
- (iljlb9iO) + (iOj91iljl) - (iOjObli1) +
i
