Virtual orbitals for obtaining rapid convergence in configuration interaction calculations

Chemical Physics 10 (1979) 301-309 0 North-Holland Publishing Company

VIRTUAL ORBITALS IN CONFIGURATION William

FOR OBTAINING RAPID CONVERGENCE INTERACTION CALCULATIONS

L- LUKEN

Depnrrrnetu of Chemistry, Duke University,

Durham, North Curolino 27706.

USA

Received 17 August 1978

A perturbatiomal method is used to generate virtual orbitals intended to yield rapid convergence in conliguration interaction calculations. II is shown that Ihe problem of finding the orbit& and pair functions which maximize the interaction with the Harrree-Fock waverunction can be solved exactly. The resulting first-order semi-internal virtual orbitals are determined by u simple closed expression. All further semi-internal excitations vanish to lirst order. ASan example, the method of this work is applied to the semi-inicmal orbit& of the ground slate of the carbon + 1 ion. The resulting first order orbitals compare very well to corresponding variational virtual orbit&.

1. Introduction

one-electron spin-orbitals { 4,. $Z, . . _, 4,). where tn > N. The various Cl methods differ from one

There are a number of methods for calculating many electron wavefunctions which can be collect-

another in the means used to determine the set oT spin-orbitals, as well as in the means used to select the set of N-electron functions. In this paper, however, only the problem of how to determine the set of orbitals will be considered, and it will be assumed that a suitable set of N-electron functions has been selected by appropriate means. Furthermore, it wilf be assumed for simplicity that the N-electron functions a, are each individual Slater determinants, although the methods developed here can also be applied to functions Q, composed of linear combinations of Slater determinants. The methods used to determine the sets of spinorbitals used in various CI calculations can be divided into two categories: those which use orbitals which are optimized with respect to the many electron wavefunction, and those which use orbit& which are not optimized with respect to the many electron wavefunction. This latter category includes, for example, use of the occupied Hartree-Fock orbitals of the electronic state of interest, supplemented by all or some of the “Hartree-Fock virtual orbitals” determined by the unoccupied eigenvectors of the Hartree-Fock matrix based on a finite set of orbital basis functions [I]. Alternatively, the set of

ively referred to as ‘-configuration interaction” (CI) methods. Each of these methods yields an N-electron wavefunction W’(r I, r ,,...,r,)oftheform Jf Y’“‘(rl,

rz, . __, rK) =

C C~“‘Q~,(r,, r2, __ _, rJ_

(1)

I-0

in which the N-electron functions Q, are either N-electron Slater determinants or linear combinations of N-electron Sister determinants (e.g. contigurational state functions), and the coeflicients CJ’“’are elements of an eigenvector 0”’ of the M + 1 by M + 1 hamiltonian matrix H(“” having the N-electron functions Qo, ‘I+, _ . . , @_,ras a basis set. That is, H,.%CJI) = ,+“@f)

7

where the elements

of H(“‘j are defined as

*HGf’ = IfI, = <@,I A I@,>,

(2)

(3)

for the N-electron hamiltonian operator 8. Each Slater determinant contained in the set of N-electron functions {oo, aI. . . _, @‘n) is an antisymmetrized product of N members of a set of m 301

302

It:L. Lukrr~/Cl calcularions of oirrual orbirds

accupied_Hartree-Fock orhitals may be supplemented by a set of”virtua1 orbitals” determined by orthogonalizing an arbitrary set of orbital basis functions to the occupied set of orbitals [2,3]. These and related methods are characterized by the very small amount ofcomputational effort (beyond the initial Hartree-Fock calculation) required to determine such a set of orbitals. CI calculations based on such calculations, however, tend to be very slowly convergent, requiring large numbers (often many thousands) of configurations. Consequently, in spite of the ease with which such a set of orbitals can be obtained, the CI calculations based on such orbitals tend to be costly and difficult to interpret. In the optimized orbital types of CI calculations, the number of configurations required to obtain an accurate many electron wavefunction is greatly reduced by carefully determining the functional form of each of the orbitals. Such optimized orbital methods include the multi-configurational selfconsistent field (MC SCF) method 14, 51, as well as the various natural orbital (NO) methods [6-l 11. Another type of an optimized orbital Cl method makes use of the occupied Hartree-Fock orbitals of an electronic state, fixes these orbitals, and supplements this set with a set of variationally determined virtual orbitals [U-14]. Each of these methods allows accurate many-electron wavefunctions to be calculated using much smaller numbers ofconfigurations (or Slater determinants) than are required by the non-optimized orbital CI calculations. The ultimate CI calculation performed in each of the optimized orbital CI methods, however, is only a small part of the overall calculation, which may include a number of preliminary CI calculations, as well as other elaborate preparatory calculations. Consequently, these calculations, like the non-optimized orbital CI calculations, tend to be costly and difiicult to interpret. In this work, an alternative method for determining a set of orbit& for use in a CI calculation is dcvcloped. This set will consist, in Ijart, of the occupied restricted Hartree-Fock (RHF) orbit& of the state of interest, as well as any additional vacant RHF Orbitak which are degenerate or nearly degenerate with any of the occupied RHF orbitals. This se1 corresponds to the “&rtree-Fock sea”

defined by Sinanoglu [12, 15, 161. The remaining

(--virtual”) orbit&

arc determined

by means of a

perturbational method described below. The object of this method is the generation of an approximately optimized set of orbitals which can be determined with a very small amount of computational effort, and which allow accurate many electron wavefunctions to be calculated using a small number of configurations (or Slater determinants). Consequently, this method combines the advantages of both the non-optimized orbital methods and the optimized orbital methods. In addition, this method also has advantages regarding the physical interpretation of virtual orbitals for CI calculations. In the following, the methods for determining the orbitals of this work tvill be presented, followed by the results of an application to the ground state of the carbon + 1 ion. In this work, it will be assumed that the RHF wavefunction of the state of interest has previously been calculated, and the resulting RHF orbitals will be regarded as known functions. Consequently, the following presentation will be concerned only with the determination of the additional (virtual) orbitals required to perform a CI calculation.

2. First-order virtual orbitals The energy of the CI wavefunction eq. (1) is given by

defined by

where E, E (a01 fi ~$,>/ is the energy of the N-electron wavefunction o0 = a{q5,, &, __., 4,3, 2 is the N-electron anti-symmetrizer, and the coeficients C,, I = 0 to M, are determined by eq. (2). The energy E(“’ is _-d; -.fbnctional of the set of spin-orbit& { &1, &, ___, 4,) used to construct the CI wavefunction ‘I?‘)_ An energy optimized set of such orbitals is one which minimizes E(“‘) as a functional of all of these orbitals. Subject to symmetry restrictions, this is the MC SCF set of orbitals for this set of configurations. The first term in eq. (4), E,, depends only on the spin-orbitals {d,, dZ, ___, 4,) which are occupied in $,_ The set of orbitals which minimizes EO, subject to symmetry restrictions, is the set of RHF orbitals

IKL. Lukerl/CI cnlcdoriorrs vf~;irr~~l for this conliguration. These orbitals are usually very similar to the corresponding MC SCF orbit&. The remaining spin-orbitals Qx+ ,, . __, q&, include both vacant members of the Hartree-Fock sea (if 4,, is not a closed shell wavefunction), as well as additional “virtual” orbit&. The orbitals of the Hartree-Fock sea, both occupied and vacant, will be assumed to form an orthonormal set. The virtual orbitals will each be assumed to be normalized to unity and orthogonal to all members of the HartreeFock sea. The virtual orbitals. however, are not necessarily orthogonal to one another. If the orbitals of the Hartree-Fock sea are fixed as e.g. the corresponding RHF orbitals, then E,-, is a constant and the virtual orbitals which minimize E’“” will also minimize the difference

where D:_“’ E H,,Cy’/Cb;\“. Because of the constraints imposed on the orbitals of the HartreeFock sea, the resulting virtual orbit& will not be identical to the corresponding MC SCF orbitals (unless the members of the_Hartree-Fock sea are identical to the corresponding MC SCF orbit&). However, if the RHF orbitals are very similar to the corresponding MC SCF orbit&, then the virtual orbitals which minimize E$: should also approximate the corresponding MC SCF orbitals well. With the orbitals of the Hartree-Fock sea fixed, the interaction integral H,, depends only on the virtual orbitals in 0,. Furthermore, for H,, to be non-zero, aD, can contain at most two virtual orbit& [ 171. Therefore, Ho1 depends on at most two members of the set of virtual orbitals. The contribution Dy”, however. depends on the composition of the entire set of virtual orbit& through the coeflicients Cl;“’ and Cr’). If the ratio of coefficients in Di’” is replaced by the first-order approximation Cp’)/Cb”” s H[,,/(E,

-

E,),

(6)

where E, G <@,I fi I@,), then the contribution D’-‘n by r is approximated Dr = IHc,,lZ/(E,

-

W.

Unlike Di”‘, D, depends only on one or two virtual orbit&. The approximate contribution D, depends

(7)

orbirds

303

on these virtual orbitals through the integral H,, as well as through E,. However, if the orbit& which minimize D, fall in a region (of Hilbert space) where eq.(6) is a good approximation (i.e. IHo, < I E, - E,I), then the variation of 15, as a function of these orbit& tends to be small compared to the magnitude of E, - E,, and the dependence of DI on the orbit& of @, is dominated by the behavior of the integral HoI. In this case, rhe orbitals which minimize Dr will be well approximated by those which maximize the absolute value of H,, (assuming E, > E,). Likewise, if conditions are tvorable, the orbitals which maximize IH~,I will be good approximations to those which minimize E$_ as well as to those which minimize Et-\‘) (i.e. the corresponding MC SCF orbitals). Consequently, the following work will seek to maximize the integral H,, subject to constraints of normalization and orthogonality to the HartreeFock sea. Others. for example, Bender and Davidson [NJ and Whitten [19] have previously proposed that virtual orbitals be determined‘by maximizing similar integrals. This work, however, differs from the earlier work [lS, 191 in several important

respects. In this work, unlike the earlier work [lS, 191, it is demonstrated that the problem of maximizing H,, can be solved exactly without respect to any basis sets. Furthermore, it is shown here that the resulting solutions define a set of specific mathematical functions with certain useful properties [see e.g. eq. (14) below]. Unlike the earlier work [18. 191, the virtual orbitals of this work are determined separately for each specific excitation (e.g. “occupied” orbit& i and j replaced by “vacant” orbitals u and b). In the earlier work, the virtual orbitals were optimized with respect to i’ to uZ excitations only (i.e. the special case where the spatial parts of i and (I equal those ofi and 6 respectively), in which case H,, is an exchange integral
304

WL. Luken/CI calculatiom ofvirrual orbitals

Because the virtual orbit& of this work are optimized separately with respect to each excitation, this work will yield larger numbers of such orbitals than the previous work [lS, IS], and the resulting virtual orbitals will not be orthogonal to one another. Each of these orbitals, however, is required to be orthogonal to the orthonormal set of HartreeFock sea orbitals. Consequently the use ofa partially non-orthogonal basis set does not introduce serious complications [20, 213. Likewise the greater number of virtual orbitals does not increase the dimension of the CI calculation because each virtual orbital is used only in a single specific Slater determinant. If a fully orthonormal basis set is desired, fully optimized orbitals are not possible, and some compromises must be made. In this case, the nonorthonormal set of orbitals determined by the means given below can be inspected to find which are most important and which ones are most similar to one another. Those with large overlaps may be averaged together. Those with very small values of H,, may be deleted, and the resulting smaller set of virtual orbitals may be made orthonormal by, e.g. the Schmidt process. This will require an increase in the number of Slater determinants in the basis set. This increase in the number of Slater determinants is a general consequence of requiring an orthonormal basis set. Another limitation of the method presented here is that it is strictly first-order in the sense that only direct interactions with the zero-order (e.g. HartreeFock) wavefunction are used to determine the forms of these orbitals. Higher order or indirect effects such as triple and quadruple excitations, etc. can often be significant. The most important of these can be treated by using symmetry adapted conligurational state functions instead of Slater determinants. and by including unlinked clusters [22] in the wavefunction. It also helps to have a “good” zero-order wavfunction. In this case, if any configuration is found to have a coefficient greater than 0.1 or 0.2, it should be combined with the main conliguration in the form of an MC SCF zero-order wavefunction. Higher order effects can also be treated by using the results of the first-order treatment given here as a new zero-order wavefunction.

3. Pair excitations The determinant 0, is related to determinant @, by the pair excitation i, j + u, b if CD,can be constructed by replacing spin-orbitals 4i and 4j of determinant Q,, with spin-orbitals 4, and &. As indicated by Sinanoglu and co-workers, these excitations can be described as internal, semiinternal, and all-external types of pair excitations [12, 15, 16,23,24]. In the case of the internal excitations, +,, and r#+,are both vacant members of the Hartree-Fock set of orbitals. These excitations involve no virtual orbitals and the Hartree-Fock set of orbitals are assumed to be previously determined. Therefore there are no unknown functions to be determined for the case of internal excitations. In the case of semi-internal excitations, orbital 4, is a vacant member of the Hartree-Fock set of orbitals and $b is a semi-internal orbital orthogonal to all members of the Hartree-Fock set. The interaction of a semi-internal determinant with a0 is given by H,0 = (- 1)’ (bti(l)&(2)/

1/r,214i(1)9j(2)-~,(l)~j(~)),

(8)

where p is the number of permutations needed to align the matching orbitals of @, and a,,, and 1 and 2 indicate both the space and spin coordinates of electrons 1 and 2. By means of the definition Fij.o(2)

-

-

<4.(1)1

<@cI(~)~

l/r12

l/r,2

]4i(l)>l@,J(3

[4jf1))l

(9)

4it2L

it can be seen that Hlo has the form of an overlap integral: H,, = (-UP

(10)


Consequently, the form of 4b which maximizes the magnitude of H,, is given by 4, = Mij,OFij,,, where M,, is a constant given by

(11)

Mij.0 E C(Fij.0 I Fij.a>] - I”-

To be used as a virtual orbital & must be orthogonal to all members of the Hartree-Fock set. This can be accomplished by the Schmidt process to yield +k

<+kI

Fij.o>

1 I

(12)

WL. LukenfCI calcdations oJoirtuo1 orbit&

where the index k runs over all members of the Hartree-Fock set. The factor Nij.,, normalizes the resulting function to unity. As a consequence of the normalization and the orthogonality of &., it can be seen that the interaction of @, with Q0 is given by HIO = (-

(13)

~lpINij.m

for this choice of the virtual orbital (Pb. It should be noted that the function $rj,. detined by eq. (I 3) is an exact solution to the problem of maximizing Ho, for a semi-internal excitation. This function is determined entirely by the Hartree-Fock orbitals of the state of interest and does not depend upon any choice of basis functions. (Although ambiguities may arise in regard to the vacant members of the Hartree-Fock sea, these problems can usually be resolved easily [14].) Consequently, the long range form, short range form, nodal structure, etc. of these orbitals are determined by the physics of the atom or molecule, and not by basis set restrictions. The function &. appears to represent the effects of a collision between one electron in orbital i and a second electron in orbital j, weighted by the amplitude of vacant orbital a and the strength of the interaction (I/r,J between orbitals i and j. Examples of such orbitals are given below and compared to variationally determined semi-internal orbit&. Because of the various approximations which have been made to arrive at the orbital ~ij,, defined by eq. (12), this will not be identical to the exact semi-internal orbital defined by Sinanoglu and Oksuz [12]. Any corrections to 9ij.., however, are expected to be relatively small because the integral H,, vanishes for any semi-internal pair excitation ij + a, c, where #I, is any orbital orthogonal to 4ij.a (as well as all members of the Hariree-Fock set). This is seen by noting that HJe = (-l)P<$~#~I =

( -

1)’

<@cl

tlrIZ 4ij.a>/Nij.a

Ih$j

-

=

0.

4j@i>

(14)

Consequently, &ja will be correct to first order, and any additional excitations of the same type can enter only as higher order (indirect) effects. The actual analytical form of the orbitals determined by eqs. (9) and (12) may be excessively cumbersome for practical purposes. These orbitals

305

may be put into a more convenient form by expressing &, as linear combinations of a set of orbital basis functions x,,,: 4ij.a

z

C

dnJnr

(15)

The set of basis function z,,, may be the same set of basis functions used to determine the Hartree-Fock set of orbitals, but this set is not limited to this choice. The coeflicients d, which provide the best tit in a least squares sense are determined by d = S-lx,

(16)

where &I” = .

(17a)

and xtn E



(1W

The approximation given by eqs. (15) to (17) unlike that of eq. (12), depends upon the choice of basis functions. Inspection of the function detined by eq. (12), however, can help to prevent the use of basis functions which have the wrong long range, short range, or nodal structure.

4. All-external pair excitations For an all external pair excitation, i,j -+ II,b, the interaction integral HI0 is given by eq. (S),just as for a semi-internal pair excitation. In the all-external case, however, orbitals $J, and & must both be optimized, unlike the semi-internal case in which only a single orbital need be optimized. Consequently, the function dij., defined by eq. (12) does not immediately yield an optimized form for either 4, or &. Given an initial guess, @‘r, for orbital 40 optimized forms for both I$, and & can be determined iteratively using a procedure in which the forms of Cp.and & after tr iterations are given by

and

directly because of the factor of I/r, L_Given an arbitrary set of one-electron functions {~1 a leastsquares approximation to ej can be determined as a sum of products of the functions x,, of the form

where fij$’ is defined as in eq. (9) for @“, and N$E’ is the corresponding normalization factor. F,IJ can also be determined by the method used in eq. (15). Unlike the semi-internal case, the all-external correlation of orbitals i and j cannot be represented by a single product function such as I$~&, or even a linite number of such products. Therefore, an equation analogous to eq. (14) is not possible for the results of eqs. (18). Consequently, it may be necessary to consider further excitations of the type i, jlo a’, b’, where orbitals CI’and 6’ are orthogonal to orbitals N and/orb as well as all members of the Hartree-Fock set. Alternatively, all possible all-external pair excitations i,j + n, b for orbitals i and j can be combined into a normalized all-external pair iunction Pij defined by P,,(l, 2) = M,ti,(l,

2),

(19)

where ti,( 1,2) is the all-external pair function of Sinanoglu and co-workers [ 12, 15, 16,23,24] and M, is a normalization factor defined by ‘M, = ((li;j~fiij>)-“~.

(20)

In this case, the interaction of@, with the function QD,obtained by replacing orbitals i andj with function P, is given by HI0 =

(pCj(1~2)1

l/r,,

(4it1M,(2)

The form of Pij which maximizes function Qij defined by

&Cl, 2) E

[A(1M_d2)

-

~,,(lhbi(2)l/r~2.

-

dj(l)bi(z)>-

(21)

H,, is given by the

(23)

and Nii is a normalization factor- This form of M$,, like eq. (9) for F,, is likely to be dificult to use

(24)

W,(L 2) = 1 DIiZli,(l)%h.a(2). K

where the coefficients DK are elements of the vectors D given by D=T-‘U,

(25)

the elements of the matrix Tare defmed by

and the elements

of the vector LI are defined by

uIi E <%Kl%liZlllr-IZ

IdAj

-

djdi>-

(27)

The set {I]. which need not be orthonormal, may be chosen as the atomic basis functions used to obtain the Hartree-Fock orbit& Other possible choices for { ,Y}include symmetry adapted linear combinations of these functions, the “Hartree-Fock virtual orbitats”, or orbit& determined using eqs. (15) to (IS). Each of these sets, ii used in its entirety, is actually equivalent to each of the others, although certain choices may have computational advantages. Because the sum in eq. (24) involves only a finite number of products, the resulting function cannot represent Kj exactly. In favorable cases a small number of terms will give an adequate approximation. Because the calculations involved in eqs. (24) to (27) are much simpler than a full CI calculation, it should be possible to explore the possible choices of basis sets needed in eq. (24) more carefully than in a direct CI calculation. It should also be noted, the N-electron function DI determined by eqs. (21) to (27) is not a single Slater determinant, as is assumed in the preceding portions of this paper. The number of Slater determinants in @, is determined by the numbe; oiorbital producis included in eq. (24). Because the resulting multi-determinantal function can be used as single basis function in a subsequent CI calculation, the use of such functions can permit a large reduction in the dimension oiC1 calculations.

WL. Lukrn/CI culcdatiornofuirtd

5. Single excitations The interaction of 4e with a determinant @i created by exciting occupied orbital & to orbital 4, outside the Hartree-Fock set of orbitals is given by

where it, is the one-electron L, 3 -*VP

- T Z,/lr,

operator

defined by

- t-1,

(30)

h-~dk(4*IL>-v [

1

where the functionf;,

‘PO state of the carbon + 1 ion. This is a non-closed shell state for which the Hartree-Fock sea consists of the ten Is, 2s, and 2p spin-orbitals. For this state, L-shell semi-internal orbitals arise from 2s’ + Zp’f,: 2s2p + 2p’&; and 2s2p + Zp’f, semi-internal excitations. where 2s and 2p are occupied RHF orbit&, 2p’ is a vacant RHF orbital, and f,, f,, and fd are semi-internal orbitals [12, 15, 161. The RHF orbitals for this state have been calculated by Clementi and Roetti [26] in the forms of linear combinations of Slater type orbitals (STO’s):

(29)

in which r, is the position of a nucleus with nuclear charge 2,. The sum overj in eq. (28) runs over all orbitals occupied in both G+, and @,. The number of permutations needed to align these matching orbitals is given by p. In this case, the form of the . . orbttal 4, which maxtmizes H,, is given by 4azNi

307

orbitals

is defined by

Approximate semi-internal orbitals were determined by substituting these expansions into eqs. (9) and (E), projecting out the desired angular symmetry (i.e. s, p or d), and normalizing the resulting function to unity. The radial amplitudes of the resulting s and p symmetry orbitals are shown in figs. 1 and 2 (solid Iines). The radial amplitudes of variationally determined s and p symmetry semi-internal orbitals [ 143

X(1) E rf~(l)+i(l) - C C<+ji2)1 WIZ )dj(2)>2 &i(l) J - <9j(2)I l/r,,

(di(2))Z d,(l)]-

(31)

The sum over k runs over all members of the Hartree-Fock set of orbitals, both occupied and vacant. The normalization factor’Ni is used to normalize 4, to unity. If, for a closed shell state, the Hartree-Fock ser of orbitals are taken to be the canonical Hartree-Fock orbitals of this state, thenf;- = e,&, where er is the orbital energy of Hartree-Fock orbital #ti In this case, 4JNi vanishes everywhere, as required by Brillouin’s theorem [25]. For cases where q&/Ni does not vanish, & can be approximated by a linear combination of basis functions using a procedure equivalent to that given previously for semi-internal orbitals.

I

-1 I :

cl

2-t

2

:

$

L\

5 I,;

I1 : Ii Ol

i

‘,

i/F \ 7r.y,._... ..”

___. ___._...., _.,-‘.--.-...I r(au)

-I AJ --_-.___.__ 6. Application to the carbon + 1 ion As an example, the preceding work was used to determine the semi-internal orbitals of the ‘Isz2sz2p

Fig. 1. Radial functions for s-symmetry orbitals of the Is’2s22p zP” state of the carbon -I-1 ion. Solid line: f, semi-internal orbital (this work); dashed line: variational fS semi-internal orbital [t4]: dotted line: restricted HartreeFock Zs orbital [26].

308

WL. Luken/CI calculuriom of oirrual orbirals orbital, which has the largest value of Ho,, is shown in fig. 3 (solid line). A variational d symmetry orbital [ 141 is given for comparison. This orbital is determined by a single 3d ST0 with an exponential factor determined as for the s and p symmetry variational

semi-internal orbitals. This d-symmetry orbital is compared to the fd orbital of this work because (a) the variational calculation is expected to emphasize the orbital which makes the largest

contribution to the total energy, and (b) this orbital, like the fd orbital of this work, has no radial nodes (unlike the fi and r; orbitals). The d’ semi-internal orbital of Luken and Sinanoglu [ 143 is not given here because this orbital contributes only about 3 % of the amplitude contributed by the corresponding d orbital. Consequently, the influence of this orbital is not significant compared to the difference between the fd orbital of this work and the variational d-symmetry semi-internal orbital shown in lig. 3. Fig. 2. Radial functions for p-symmetry orbitals of the ls’2s’2p ‘P” state of the carbon f 1 ion. Solid line: $ semi-internal orbital (this work); dashed line: variational f, semi-internal orbital [14]: dotted line: restricted HartreeFock 2p orbital [26]_

are given for comparison (dashed lines). In addition, the radial amplitudes of the RHF 2s and 2p orbitals for this state are also given (dotted lines). The variational semi-internal orbitals were determined by orthogonalizing a single 3s ST0 and a single 3p ST0 to the RHF Is, 2s, and 2p orbitals. The ST0 exponential factors <, and &, were determined so as to minimize the energy E”f) of a CI wavefunction which included all possible L-shell semi-internal excitations (including the d-symmetry orbitals described below).

Considering the limited flexibility of the variational semi-internal orbitals, the semi-internal orbitals of this work are seen to compare very well with the corresponding variational semi-internal orbitals. The position of the peaks and nodes of the corresponding p-symmetry orbitals are especially remarkable. It should be noted that approximately 15 to 20 CI

calculations were required to determine the optimized exponential factors of the variational s, p and d symmetry semi-internal orbitals shown in figs. 1,2 and 3. The corresponding orbitals of this

Three types of d-symmetry orbitals are possible for this state, depending on the spins of the 2s and 2p spin-orbitals involved in the excitation. These are, e.g. 2@, 2px --t 2p/l, f,a,

2@, 2pr + 2lW fiP,

and 2sLX,2pr + 2p3, f$, types of excitations. The s orbital is a linear combination of fd and f& The fi orbital has a radial node coincident with that of 2s orbital. The fd

Fig. 3. gadial functions for d-symmetry orbitals of Is’Zs’Zp ‘PO state of the carbon -I-1 ion. Solid line: fd semi-internal orbital for (2s/?, 2px; 2pj?, 6%) excitation (this work); dashed line: variational fd semi-internal orbital [14]

work, by contrast, were determined directly using closed analytical expressions. Evaluation of these expressions involved a computational effort comparable to a single small CI calculation. Consequently, use of the orbitals of this work in a CI calculation on this state would require nearly an order of magnitude less computational effort than was required by the use of variational virtual orbitals. The similarity of the lirst order virtual orbit&

of this work with the variational

virtual orbitals

shown in figs. 1, 2 and 3 does not necessarily imply that these orbitals will perform better than or as well as the variational virtual orbitals. However, this similarity, coupled with the consequences of eq. (14): leaves little doubt that these orbitals will perform better than conventional choices such as “HartreeFock virtual orbi:als’*. A more conctusive test wit1 come from the results of actual CI calculations using these orbitals. Such calculations have not yet been performed, but they are currently being planned and the results of such calculations will be published later. Additional calculations are also planned to investigate the application gf this method to allexternal correlation functions.

Acknowledgements

Acknowledgement is made to the donors of the Petroleum Research Fund, administered by the American

Chemical

this research.

Society,

for partial support

of

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