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complex-coordinate procedure. Thanks also to Professor Peter D. Robinson, Dr. Jim Diamond, and Dr. Nimrod Moiseyev for collaboration and discussion. Financial support in the form of National Science Foundation Grant CHE76-22760 and a Science Research Council Senior Visiting Fellowship at Bradford University, where portions of this work were carried out, is gratefully acknowledged.
References and Notes (1) E. B. Wilson, Pure Appl. Chem., 47, 41 (1976). (2) R. G. Parr, f r o c . Nail. Acad. Sci. (U.S.A.), 72, 763 (1975). (3) See particularly the early papers of W. P. Reinhardt and co-workers:
(4) (5) (6) (7)
E. J. Heller, W. P. Reinhardt, and H. A. Yamani, J . Comp. Phys., 13, 536 (1973); W. P. Reinhardt, “L2 Discretization of Atomic and Molecular Electronic Continua: Moment, Quadrature, and J-Matrix Techniques”, University of Colorado preprint, 1978. J:Aguilar and J. M. Combes, Commun. Math. Phys., 22, 269 (1971). E. Balslev and J. M. Combes, Commun. Math. fhys., 22, 280 (1971). B. Simon, Ann. Math., 97, 247 (1973). (a) N. Moiseyev, P. R. Certain, and F. Weinhold, Mol. fhys., 36, 1613 (1978). (b) Int. J. Quantum Chem., 14, 727 (1978). (c) For references to the extensive literature on complex-coordinate calculations by the variation method, see the special complex-scaling issue (October
Debbie Fu-tai Tuan
1978) of Int. J . Quantum Chem., as well as C. L. Huang and E. R. Junker, fhys. Rev. A, 18, 313 (1978);G. D. Doolen, J. Nuttall, and C. J. Wherry, fhys. Rev. Letf., 40, 313 (1978); C. W. McCurdy and T. N. Rescigno, lbkf, 41, 1364 (1978);T. N. Rescigno, C. W. McCurdy, and A. E. Orel, fhys. Rev. A, 17, 1931 (1978). (8) See, e.g., S. Seung and E. 6.Wilson, J. Chem. fhys., 47, 5343 (1967). (9) This simplified notation implicitly assumes that wave functions associated with the unrotated Hamiltonian are chosen to be real, as may be done without loss of generality. The more general situation is discussed in F. Weinhold, University of Wisconsin Theoretical Chemistry Institute Report WIS-TCI-590 (1978). (10) Bounding properties of these and related functionals have been investigated by P. D. Robinson and F. Weinhold (in preparation). (1 1) An alternative perturbation approach, leading to corrections in alternating orders for the position and width, was outlined by P. Froehlich, M. Hehenberger, and E. Brandas, Int. J . Quantum Chem., S11, 295 (1977). Perturbation theory was employed in a related context by P. Winkler and R. Yaris, J . fhys. 8 ,in press; see, however, N. Moiseyev and P. R. Certain, Mol. fhys., to be published [University of Wisconsin Theoretical Chemistry Institute Report WIS-TCI-595 (1978)l. For an application of perturbation theory to autoionizing resonances in the conventional (unrotated) framework, see, G. W. F. Drake and A. Dalgarno, Roc. R . SOC.London, Ser. A, 320, 549
(1971). (12) A. K. Bhatia and A. Temkin. fhys. Rev. A, 11, 2018 (1975).
Optimal Geometric Approximation of the Perturbed Uncoupled Hartree-Fock Method for Physical Properties. 1. Dipole Polarizabilities$ Debbie Fu-tai Tuant Department of Chemktry, Kent State University, Kent, Ohio 44242 (Received October 27, 1978)
A variational scheme for obtaining the optimal geometric approximation of the perturbed uncoupled Hartree-Fock (GPUCHF) method for physical properties is proposed. Physical properties obtained by this variational GPUCHF method should give the best geometric approximation to those obtained from coupled Hartree-Fock method. This optimal GPUCHF method is applied to the study of the dipole polarizabilities of He and Be isoelectronic , choices of the zeroth order orbitals sequences. The dependence of the optimal variational parameter, T ~on~ the and on the systems are studied.
Introduction The perturbation methods of calculating physical properties that can be applied to many-electron systems1 are (1) perturbative procedures based on Hartree-Fock theory2 using double perturbation t h e ~ r y or ~ ?using ~ Brueckner-Goldstone many-body perturbation theory5 and (2) the coupled Hartree-Fock theory6 or various approximations to it.7-9 The coupled Hartree-Fock (CHF) method has given reasonably accurate results. The iteration procedure, however, requires a laborious computational effort. Consequently, different uncoupled Hartree-Fock (UCHF) approximations have been sought. Within the framework of the one-electron first-order perturbation equation, the alternative uncoupled Hartree-Fock method of Langhoff, Karplus, and Hurst (method b’ of LKH)’ and Dalgarno’s uncoupled Hartree-Fock method1a’6aamount to the neglect of different coupled terms of the CHF formulation. Better results from method b’ have been explained by the absence of the self-potential in the first-order equation of method b’.’ Since Dalgarno’s UCHF method is formulated with a zeroth order many-electron Hamiltonian, one can obtain the correction of it by perturbation t h e ~ r y . Dalgarno’s ~,~ t T h i s research is supported b y a Summer Faculty Research Fellowship of Kent State University.
0022-3654/79/2083-1520$01 .OO/O
UCHF method with first-order correction-perturbed uncoupled Hartree-Fock (PUCHF) methodlOJ1and the geometric approximation based on it12-14have also been found to be simple and accurate approximations for the CHF method. In order to gain a deeper understanding of the “selfpotential” and to study the accuracy of various methods a t different levels of approximation, we have p r o p o ~ e d l ~ , ~ ~ three different perturbed uncoupled Hartree-Fock methods. These three methods differ in that the zeroth order Hamiltonian, H,O, includes all of, half of, or none of the self-potential. They are called PUCHF methods Co, CIl2, C1 according to I = 0,1/2,1 in the self-potential term T ( & - ail). At zeroth order level, method C1 is Dalgarno’s UCHF method, and method Co equals method b’ of LKH only for special cases.I5 We have applied these methods to the study of dipole polarizabilities and dipole shielding factors of atomic systems. Results showed that geometric approximation of PUCHF method Cljz gave the best approximation to the CHF method, whereas a t the zeroth order level method Co offered the best approximation. The above results further suggested the following. (a) One should not rigidly associate the self-potential term with some fictitious physical effect. Instead, it is merely the mathematical expression indicating the dependency of the perturbed orbital i’ on the unperturbed orbital ios 0 1979 American
Chemical Society
The Journal of Physical Chemistry, Vol. 83, 'No. 11, 1979
Approximation of Perturbed Uncoupled Hartree-Fock Method
=
in the presence of the nucleus, other electrons, and external perturbation. (b) There should be an optimal amount of , which should correspond to the self-potential, T ~ ~ ( P-- , a,J, most proper representation of the dependency of i' on io. Since il and the dependency of il on io reflect the readjustment of the orbitals in the system due to the perturbation, - aLL) should depend on the system, the physical property, and the choice of ios. In this paper we propose using a variation method to obtain the optimal 7, T~,,. Physical properties obtained by the geometrnc approximation of the PUCHF method corresponding to 70pshould give the optimal approximation t o those found from the CHF method, and should be the upper bound of the exact values. In order to distinguish from the norioptimal version of studies,15J6we call the present methiod the optimal geometric approximation of the perturbed uncoupled Hartree-Fock method, or simply the optimal GPUCHF method. The dependence of T , on ~ the system, on the physical property, and on the choice of io are investigated in this and succeeding papers. As prototypic studies, results of dipole and quadrupole polarizabilities for the He and Be isoelectronic sequences are presented in this and the following papers. Those shielding factors which require the use of the triple perturbation theory will be given in paper 3. A procedure for treating open-shell systems will be given later.
J2
Theory a n d Calculation" Physical Properties i n Terms of Single and Double Perturbation, Theories. Consider a system with Hamiltonian H, energy EO, and wave function \ko under a perturbation p W. The Schrodinger equation of the perturbed system is defined by (H + pW)\k = E\k
and
Based on perturbation theory, the perturbed energy and wave function, E and \k, can be expanded as
E = C y m E m and m=O
\k= Cpm\km m=O
N
H = Ho + XV = - 1/2C[V:+ Z / r J c=1
$Os0
Q
where \ k 0 v 0 aind \kOJcorrespond to the zeroth- and firstorder perturbed wave functions of an approximate system: @\kW
,
