which defines the (fi; + Z,(E))ij of section 111. References and Notes. (1) See, for example, A. Streitwieser, Jr., âMolecular Orbital Theory for. Organic Chemistsâ ...
The Journal of Physical Chemistty, Vol. 83, No.
Complex Eigenvalues of Resonance States
the valence space x MO’s. Thus if x p and X/ are valence x MO’s
xX’ =
CCi”i
(14) In this sense the work is similar in spirit to that of E. B. Wilson, Jr., and P. S.C. Wang, Chem. Phys. Lett., 15, 400 (1972), and to the more formal work of P.-0. Lowdin, J . Math. Phys., 3, 1171 (1962). (15) See, for example, P. Nozieres, “Theory of Interacting Fermi Systems”, McGraw-Hill, New York, 1971; G. Csanak, H. S. Taylor, and R. Yaris, Adv. At. Mol. Phys., 7, 287 (1971); J. Linderberg and Y. Ohm, “Propagators in Quantum Chemistry”, Academic Press, London, 1973. (16) We follow the development of J. D.Doll and W. P. Reinhardt, J. Chem. Phys., 57, 1169 (1972), which contains an extended set of references. (17) Recent quantum chemical work is typified by W. von Niessen, G. H. F. Diercksen, and L. S. Cederbaum, J . Chem. Phys., 87, 4124 (1977); K. F. Freed and D. L. Yeager, Chem. Phys., 22, 401 (1977); B. T. Pickup and 0. Goscinski, Mol. Phys., 26, 1013 (1973); G. D. Purvis and Y. Ohrn, J . Chem. Phys., 60, 4063 (1974); J. Simons and W. D. Smith, ibid., 58,4899 (1973); F. S. M. Tsui and K. F. Freed, Chem. Phys., 5,337 (1974). (18) We are assuming that r includes spin, which is denoted by “s” when made explicit: (7) denotes a coordinate vector in 3-space. Similarly, di is a coordinate spatial differential, dr includes an additional spin space volume differential. (19) Expressions with no explicitly shown coordinate space dependence will represent matrices. (20) A. J. Layzer, Phys. Rev., 129, 897 (1963). (21) Discussions of the existence of such an operator appear in A. Klein and R. Prange, Phys. Rev., 112, 995 (1958); see also A. Klein in “Lectures on the Many Body Problem”, E. R. Caianiello, Ed., Academic Press. New York. 1962. D 279. (22) For example, P.-O. Lowdin, J . Chem. Phys., 19, 1396 (1951); J. Math. Phys., 3, 969 (1962). (23) H. Feshbach, Ann. Phys. (New York), 19, 287 (1962). (24) For example, K. Rudenbera, J . Chem. Phvs.. 34, 1878 (1961). (25) An aiternate approach would-be to use an anakgue of the partitioning technique to essentially fold next-nearest neighbor interactions into effective a , @, and S’s. While this clearly can be done, we do not pursue it here. (26) We are again denoting the 2pa orbitals on the various carbons by i , j = 1, 2, .3, .. The u and a MO’s entering the perturbatlve expansions of L,(€)will be denoted uw,uu,... a,,,a”for spin orbitals occupied in a one-configuration reference state, and up,uq..., aP, aq... for the “excited” MO’s. (27) For example, K. Ruedenberg, J. Chem. Phys., 66, 375 (1977); T. Annu and Y. Sakai, ibid., 67, 4771 (1977); M. A. Whitehead, ibid., 69, 497 (1978). (26) L. S. Cederbaum, W. Domcke, J. Schirmer, N. von Nicssen, G. H. F. Diercksen, and W. P. Kraemer, J. Chem. Phys., 69, 1591 (1978). (29) For example, (a) H. C. Longuet-Higgins and J. A. Pople, Proc. Phys. SOC.London, 58, 591 (1955); (b) A. D. McLachlan, Mol. Phys., 2, 271 (1959). (30) R. Pariser and R. G. Parr, J . Chem. Phys., 21, 767 (1953); J. A. Pople, Trans. Faraday Soc., 49, 1375 (1953). Both are reprinted in R. G. Parr, ref 1.
(A.2a)
k
X/ = CC#j
(A.2b)
1
where 4i is a 2px A 0 on carbon i, and
which defines the
11, 7979 1517
(fi; + Z,(E))ij of section 111.
References and Notes (1) See, for example, A. Streitwieser, Jr., “Molecular Orbital Theory for Organic Chemists”, Wiley, New York, 1961; R. G. Parr, “The Quantum Theory of Molecular Electronic Structure”, W. A. Benjamin, New York, 1963; L. :Salem, “The Molecular Orbital Theory of Conjugated Systems”, W. A. Benjamin, New York, 1966; M. J. S. Dewar, “The Molecular Orbital Theory of Organic Chemistry”, McGraw-Hill, New York, 196!3; W. T. Borden, “Modern Molecular Orbital Theory for Organic Chemists”, Prentice-Hall, Englewood Cliffs, N.J., 1975. (2) See, however, M. Newton, F. P. Boer, and W. N. Lipscomb, J . Am. Chem. So~c.,88, 235.3 (1966), and subsequent papers. (3) J. Linderberg and Y. Ohrn, J . Chem. Phys., 49, 716 (1968). (4) P. G. Lykos and P;. G. Parr, J . Chem. Phys., 24, 1166 (1956). (5) R. A. Harris, J. Chem. Phys., 47, 3967 (1967); 48, 3600 (1968). (6) K. F. Freed, Chem. Phys. Lett., 13, 181 (1972); 15, 331 (1972); 24, 275 (1974); and in “Modern Theoretical Chemistry”, G. A. Segal, Ed., Plenurn Press, New York, 1976. (7) P. Westhaus, E. G. Bradford, and D. Hall, J. Chem. Phys., 62, 1607 (1975); P. Westhaus and E. G. Bradford, J . Chem. Phys., 63, 5416 (1976). (8) S. Iwata and K. F. Freed, Chem. Phys. Lett., 28, 176 (1974); Chem. Phys., 11, 433 (1975); J . Chem. Phys., 65, 1071 (1976); 61, 500 (1974). (9) V. KvansniEka, Ph.vs. Ref. A , 12, 1159 (1975); D. Mukherjee, R. K. Moitra, and A. Mukhopadhyay, PramEina, 9, 545 (1977). (10) 0. Sinanoglu, Proc. Natl. Acad. Sci. U.S.A., 47, 1217 (1961); Adv. Chem. Phys., 8, 315 (1964); K. F. Freed, phys. Rev., 173, 24 (1968). (11) B. H. Brandow, Adv. Quantum. Chem., 10, 187 (1977); 1978 preprint. (12) B. H. Brandow, Rev. Mod. Phys., 39, 771 (1967). (13) Early discussions of the present work appear in J. D. Doll, Thesis, Harvard University, 1971 (unpublished) and J. D. Doll and W. P. Reinhardt in “Energy, Structure and Reactivity: Proceedings of the 1972 Boulder Conference on Theoretical Chemistry”, D. W. Smith and W. B. McRae, Ed., Wiley, New York, 1973, pp 390 and 393.
Variation-Perturbation Approach to Complex Eigenvalues of Resonance States Frank Weinhold+ Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706 (Received November 13, 1978) Publication costs assisted by the National Science Foundation
A complex analogue of the Hylleraas variation-perturbation procedure is suggested for estimating the complex eigenvalues which determine the position and width of quasi-bound “resonance” states in the complex-coordinate formalism. This approach is illustrated by a simple application to the lowest IS autoionizing resonance of the helium atom.
The bound states of atoms and molecules are described by square-integrable solutions of Schrodinger’s time-independent eigenvalue equation H(s’)*k(r‘) = Ek*k(r’) (1) and the technology for approximating these solutions at a chemically useful level of accuracy has advanced impressively in recent years.1,2However, some closely related states of quasi-bound (“resonance”) character are asso-
0022-3654/79/2083-1517$01 .OO/O
ciated with non-normalizable solutions of eq 1, and thus have seemed to require the quite dissimilar techniques of quantum scattering theory. Because the boundary conditions for the “scattering solutions” are of a more difficult form, comparatively little is known about the quasi-bound states. The quasi-bound states bear a striking resemblance, both theoretically and experimentally, to their bound-state counterparts. Some states show both types of behavior; 0 1979 American
Chemical Society
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The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
for example, the ground state of the HP-molecular anion is an ordinary bound state when R k 1.7 A, but becomes quasi-bound a t shorter bond lengths, where the spontaneous autodetachment process H2- Hz + e- becomes possible. Perhaps the most familiar example from molecular spectroscopy is the phenomenon of predissociation, in which upper vibrational states of a rotating molecule broaden near a dissociative threshold. The width rk of the broadened line reflects the finite lifetime 7 k (a rk-l) of a species rendered unstable because its energy Ek permits dissociative tunneling through the centrifugal barrier. Another familiar example occurs in the ordinary Stark effect, where the bound-state energy levels are both shifted and broadened, their lifetimes limited by the possibility of field-induced ionization. While such “broadened” states undoubtedly require additional theoretical considerations, it seems evident that they are in some sense “close” to ordinary bound states. Theoretical attention has thereby been drawn to methods which could describe states of finite width within the framework of Hilbert where the techniques of bound-state approximation theory are available. Significant stimulus to such studies was given by the r e a l i ~ a t i o n ~that - ~ these broadened states correspond (rigorously!) to square-integrable solutions of an analytically continued Schrodinger equation
-
HO(r‘)*ks(?) = Wk*kO(?) (2) where the complex eigenvalue wk incorporates both the energy Ekand width rk of the broadened state in the form wk = Ek - ( i / 2 ) r k (3) The analytically continued Hamiltonian H,(R results formally from the ordinary H(Rby “rotation” of the radial coordinates r‘ (complex scaling of radial distances) into the complex plane by an angle 0 r‘-
?eLo= (reLs)?
H(7)
-
0
>O
H(7eLs) H,(?)
(44
Frank Weinhold
particularly small. There is physical reason to expect the perturbation method to be effective for describing narrow resonances. Such states arise from the weak coupling of a bound state *bound with the embedding background continuum \kcontinuurn *resonance
+
*(O)
*(I)
*continuum
+
*(2)
+ ...
where an “unperturbed” having @bound as its eigenfunction, will often be clear from the physical context. Perturbation theory permits one to incorporate such information into a systematic procedure for calculating the resonance position and width. The perturbation theory of eq 2 begins with the customary expansions H s = H(0)+ H(1)
+ \k(’) + + ... wk = w“’ + w(l)+ w‘’) + ... *ks
=
*(’)
(6)
@(”
leading to the usual perturbation equations of various orders, the lowest examples being (‘0) - w(O))q(O) =0 (74 - w(o))\k(l) + (@)
- w(l))*(O) =
0
(7b) Expressions for the perturbation energies W n )will differ slightly from the familiar ones, however, because of the non-Hermitian symmetry of the operator Ho. For real multiplicative potentials (e.g., of Coulomb form), this symmetry is
HJ = H,*
(8) where the dagger denotes the Hermitian adjoint, and the asterisk the complex conjugate. When the unperturbed H(O)also has this symmetry, the expressions for the lowest few energy corrections are readily found to be (in intermediate normalization)
W(0)= (*‘O’*I“O’\k(O))
(4b)
The “rotated” eigenvalue problem (2) has all the same bound-state levels as does the unrotated eq 1,i.e., all levels with r k = 0, but includes as well the complex discrete levels of finite width associated with resonance states. These latter, of course, do not occur in the spectrum of the unrotated (Hermitian) operator H(r‘) of eq 1, unless one admits functions outside of Hilbert space (with “scattering boundary conditions”) which break the Hermiticity. In this sense, the complex-rotated Schrodinger equation (eq 2) provides the more satisfactory basis, within the framework of Hilbert space, for the unified description of both bound and quasi-bound states. Previous studies7 have shown that the non-Hermitian Hamiltonian operator H, has properties closely analogous to those of the unrotated H, so that one recovers complex analogues of the virial theorem, Hellmann-Feynman theorem, and the like. Unfortunately, the Rayleigh-Ritz variational principle for the complex eigenvalues fails to have its usual extremum character, so that methods for upper and lower bounds are not directly applicable. This lack of rigorous bounding principles has impeded the variational approximation of resonance wave functions, and indicated the desirability of alternative approximation techniques. We have investigated the use of perturbation and variation-perturbation techniques to solve the complexrotated eigenvalue problem ( 2 ) for resonance states. Such techniques have proven useful for bound states? even when (as in the 2-l perturbation theory) the perturbation is not
*bound
(94
(*(O)*IH(l)*(O)) (9b) = (*(O)*lH(1)*(1))= -( \k(I)*I(“(O)- w(O))\kCl)) (gc) w(1)=
&‘(‘I
w(3)=
(q(O)*l~(1)q(z)) = (\k(l)*I(H(l) - w(U)$U)
(gd)
where (Pig) = Jf(3g(7) d37(ie., omit complex conjugation of the first factor in the integrandg) is equivalent to the “c-product” of ref 7a. The two forms in (9d) reflect a complex analogue of the “2n 1rule”, which allows energy corrections up to WZn+l) to be calculated from the solution of the nth order equation. Thus, when \ k ( O ) is known, solution of the first-order equation (eq 7b) determines the resonance position and width through third order, which should suffice for many cases of interest. The solutions of the first-order equation (eq 7b) may be approximated by a variation-perturbation procedure based on the functional J ( @ )= (@*I(H(O)- W(O’)@) 2(@*1(“” - W(l))*(O)) (10) which is the complex-rotated generalization of the usual Hylleraas-type functional. This functional is stationary about the true W(’),with second-order error” J(W 6 W ” ) = W(2)+ O ( P ) (11) By expanding the trial in a set of square-integrable basis functions (Xk} @ = CCkXk (12)
+
+
+
k
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The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
Complex Eigenvalues of Resonance States
and choosing the (clomplex) coefficients ck to make J(Q,) stationary, one obtains an approximate first-order wave function = aOpt which can be used to calculate an from eq 9c and 9d. As sugapproximate @@)and gested by the discussion surrounding eq 5, HC0)should be rotated by the same angle as H Bitself, so that the imaginary part of the complex eigenvalue (which reflects the coupling of *bound with the background continuum) first appears in second order. We illustrate this, procedure by an application to the lowest lying ( “ 2 ~ ~I”S )autoionizing resonance of the helium atom. The “2!s2”excited state, which would be rigorously bound in the independent-particle approximation, decays spontaneously to thLe lsks continuum state of He+ (with which it is degenerate) via the rlT1 inlterelectron repulsion in the helium Hamiltonian. Under transformation 4, the complex-rotated Hamiltonian becomes, in this case
I
I
I
-.0051-
e :.225 Wy’. -0.00505 O.U. I
-.0099
,
1
-.0100
w;’
I
I
-.0102
-.0101
-.0103
(a.u.1
Figure 1. Complex second-order energy fi2) plotted on “trajectories” as functions of rotation angle 0 (dashed line) or scale factor a (solid line) near the optimum values, ’6 = 0.225, a = 1.226. Tight bends or cusps correspond to stationary values of the second-order correction (see ref 7).
Unfortunately, the corresponding third-order energy was where rl, r2, and rI2are the usual radial and interelectron coordinates, and 2 =: 2 is the nuclear charge (all quantities being expressed in Hartree atomic units). For the unperturbed H(O),one might choose a (rotated) operator of Hartree-Foclk type, but we took a simpler operator of screened hydrogenic form’’ H(0) =
hlfl + h2O
(14)
with
~(3)(aoptrOopt) = +0.00396
+ i0.00770 au
@(3)
(21c)
indicating that low-order perturbation theory is not giving useful results in this case. and However, with values of W(O),“(I), @@), ( W1)*l@l)), one can obtain a corresponding uariational estimate of the resonance eigenvalue, based on trial functions \kvarof the form qva,= W) C W (22)
+
with The effective nuclear charge Z,ff was chosen to make the first-order correction W(I)vanish; this occurs for Z,ff = 1.754095 (16) slightly greater than the value (1.6875) characteristic of the ls2 ground state. With the value from eq 16, the unperturbed energy is
W0)= -0.76912 BU (17) and \ k ( O ) is thle appropriate combination of the degenerate (rotated) 2sB2and 2p02(’S) configurations of screened hydrogenic orbitals q ( O ) = 0.8796(2~:)
+ 0.4756(2pg2)
(18)
as determined by degenerate first-order perturbation theory. For the trial Q, of eq 12, we took as basis functions the lowest 19 Hylleraas-type functions of the form Xlk
+
= r11rZmr12ne-a(P~+r2) exchange
(19)
with the restrictions 3 1 1 5 m 1 0 and 1 1 n 2 0. The variational approxiination W@)= unlike the true W2),will generally depend on the rotation angle 8 and scale factor a in the basis functions. It is therefore desirable, following a similar procedure for variational approximations,I to _choose0 and a to make the real and imaginary parts of W2)simultaneously stationary d@(2)/do = dW@)/da = 0 (20) With the chosen baeis set, condition 20 was satisfied in the neighborhood of aopt 1.226 Oopt = 0.225 (2W (see Figure l),leading to the estimate @(2)(aopt,80pt) = -0.01005 - i0.00505 au
(21b)
In this case, perturbation theory is employed only to suggest the “basic functions” for a simple two-term approximation to the resonance eigenfunction. For the values in eq 21a of a and 8, the optimum coefficient c was 0.56855 - i0.15540, leading to W,,, = -0.77571 - i0.00131 au. However, the values in eq 21a are not optimal for the estimate in eq 23, and one should instead choose values of LY and b’ to make W,, stationary. Stationary behavior with respect to 8 was found to occur in the neighborhood of aopt= 1.48, Oopt = 0.14 and with the associated optimal coefficient Copt = 0.63395 - i0.11572, the variational estimate is WV, = -0.77755 - i0.00214 au This is in reasonable agreement with the accurate value12 W,,,, = -0.77881 - i0.00228 au and with full variational calculation^^^ using comparable basis sets. It seems possible that simple perturbation theory could still be useful for other autoionizing resonances, whose widths are narrower than that of the 2s2 state. However, the combination of variation and perturbation techniques represented by eq 22 and 23 is able to deal also with broader resonances that would not have a useful low-order perturbation expansion. Some such hybrid strategy should prove to be a useful compromise in applications to larger systems.
Acknowledgment. It is a pleasure to join in this tribute to Professor E. Bright Wilson for his distinguished contributions to theoretical and experimental chemistry, and his wise and inspiring personal example. Bill Reinhardt and Bill Miller, fellow alumni of the Wilson research group, stimulated my interest in resonance states and the
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The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
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 sequences. The dependence of the optimal variational parameter, T ~on~ the , choices of the zeroth order orbitals 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
