Proc. Indian Acad. Sci. (Chem. Sci.), Vol. 106, No. 2, April 1994, pp. 91-102. Printed in India.

Density approximation to the average Hartree-Fock exchange potential for atoms

O V GRITSENKO, A RUBIO, L C BALBAS and J A ALONSO* Departamento de Ffsica Te6rica, Universidad de Valladolid, E-47011 Valladolid, Spain

Abstract. A non-local density-based approximation to the average Hartree-Fock (HF) exchange potential is developed. The new potential is formulated within the spin-dependent version of the weighted density approximation, and is based on a novel form of the (exchange- only) pair-correlation function for electrons in finite systems. The results for total energies and one-electron orbital energies of atoms are reasonably accurate in comparison with those obtained using the exact average HF potential, or the exact orbital-dependent HF potential.

Keywords. Density approximation; Hartree-Fock exchange potential; weighted-density approximation.

1. Introduction

The Hartree-Fock (HF) method is often used for the study of atoms and molecules (Froese-Fischer 1977). In this method one has to solve self-consistently one set of one-electron equations of the form (Hartree atomic units will be used through the paper).

- + Vn(r) + V,(r) + v~i~(r) ~i~(r) = ~iOi~(r). (1) I 1-v2 1 In these equations V~ is the attractive nuclear potential, Vc is the classical electrostatic potential of the electronic cloud

V~(rx)= Idr2 p(r2), (2) J r12 and vxi~ is the spin- and orbital-dependent HF exchange potential for an electron in the spin-orbital ~i:

vx, trl) = 1 Idr2 % re)p (rl, re). (I)i~(r 1 ) ,J r12 On the other hand, p is the electron density, built from the occupied one-electron

* For correspondence 91 92 0 V Gritsenko et al spin-orbitals,

p(r) = ~ &~(r)Ot~(r), (4) i,r and p~(rx, r2) is the first-order density matrix component with spin index a:

p~(r,, r2) ---- ~ &~(rx )~,(r2). (5) i The fact that the HF exchange potential vxi~ is orbital-dependent led first Slater (1951, 1974) and then other authors (Karsch and Rennert 1972; Alonso and Girifalco 1978) to propose a simplified alternative to the HF method where equations (1) are replaced by others,

[- ~ V2 § Vn(r) + V.(r) + W~.~(r)30,~(r) f ,,~,~(r ), (6) containing an orbital-independent exchange potential W~. This Slater (S) potential is obtained by averaging the orbital-dependent HF exchange potentials, namely

14~,~(r) = ~162r162 ,., (7)

Incidentally, it is also worth recalling that equations like (6), with an orbital-independent exchange potential V,o, also arise in the standard formulation of the density functional (DF) formalism (Kohn and Sham 1965). In this case the exchange potential is the functional derivative of the exchange energy functional E~[p]:

DF OE,~] v (r)-- ap~(r) " (8)

The Slater potential of (7) can also be written in an alternative form which is more convenient for some purposes as

W~(rl) = f dr2P2(r2) C"(rl' r2), (9) r12 where Cr is the ~ component of the Fermi (or exchange) pair-correlation function Cr = p2(rl'r2) (10) p~(rl) p~(r2)" More recently, working within the framework of the optimized effective potential (OEP) method (Sharp and Horton 1953; Talman 1989; Wang et al 1990), Krieger et al (1992) have set up a method leading to one-electron equations similar to (6), although containing an optimized exchange potential. The Krieger-Li-Iafrate (KLI) optimal potential is given by the expression ~i0" (r) O/q(r)( Vix~tcr-- Vxia)int VXL](r~xr . . = WS~(r) + p~(r) ' (I1) Density approximation to the average HF exchange potential 93

where pint i~ and V,+~i"t are evaluated from the following integrals

vi".XlO't = fdr**(r) vx,` (r)~,(r) ) (12)

vintx,o= f dr ~*(r) vgLn'r'(l),io t ) i~tr),- , (13)

The optimal potential from (11) yields one-electron orbitals whose Slater determinant minimizes the expectation value of the HF Hamiltonian. The feature we want to stress is that w..S is a component of the KLI potential. Furthermore, it has been shown (Krieger et al 1992) that the exchange potential of Harbola and Sahni (HS; 1989) and Li et al (1989) can be obtained from (11) by making a plausible approxima- tion for the second term of the r.h.s, of this equation. Calculations for closed-shell atoms (Krieger et al 1992) have shown that I4/~, is the dominant term of the KLI and HS potentials at all electron-nucleus distances. The

values of various electronic properties calculated with the Slater potential W,as were found to be close to those obtained from the KLI and HS potentials. Moreover, WS(r) provides the correct asymptotic behavior for the KLI and HS potentials (Ortiz and Ballone 1991; Engel et al 1992). For instance, for a neutral atom

WS(r)_~ _ 1, for r ~ ~, (14) r

and this induces the same behavior in VTM and Vus At the nucleus, W s has a finite Xr Xd " X~ value, specific for each atom. The features presented above generate new interest in the problem of calculating the average HF exchange potential WS. Equations (9) and (10) show that Vr o can be evaluated from the first-order density matrix of the system. To our knowledge, this has only been made for atoms. But for the purpose of applications to more complex systems, it should be desirable to derive accurate approximations to WS from the knowledge of the electron density only (that is, the diagonal part of the first-order density matrix). In this paper we propose an approximation to W~, developed within the weighted-density approximation (WDA) scheme (Alonso and Girifalco 1978; Gunnarsson et al 1979). Our approach is based on a novel form of the pair-correlation function C,(rl,r2) recently developed for finite systems (Gritsenko et al 1993). In section 2 we introduce the WDA approximation to the average HF exchange potential and we compare the exact and approximate potentials for Ne and Cu § A comparison is then made of orbital and total energies in w1673 and 4, this time for all atoms from He to Ar and for selected heavier atoms.

2. Density-based approximations to the Slater potential

Various density-based approximations to the Slater potential WS have been proposed in the literature (Slater 1951, 1974; Karsch and Rennert 1972; Gopinathan et al 1976; Alonso and Girifalco 1978; Malkina et al 1990), mainly in the context of the local density (LDA) (Kohn and Sham 1965) or local spin-density (LSDA) (Hedin and 94 0 V Gritsenko et al

Lundqvist 1971; Von Barth and Hedin 1972) approximations, namely

wLSDAIr ~ ~ 1,=pr 3 ~ dr 2 Cr162 (15) 3 r12 Here C~(rl2;Pr is the HF pair-correlation function for a spin-polarized homo- geneous electron gas with constant density precisely equal to pr the true spin- density of the inhomogeneous system at r~. Inserting the HF pair-correlation function, calculated by Dirac (1930), we arrive at the LSDA approximation to wSxr (Slater 1951):

wLSDA(r)xa.. -- - 3 pl/a (r). (16)

First of all we compare WX~LsDA to the exact Ws Xff" It is well known that the local density approximation is unable to reproduce the correct long-range asymptotics. This can be observed in figures 1 and 2, for Ne and Cu + respectively, which show that the LSDA underestimates electron exchange, compared to the exact Slater potential, at low densities (large r). The LSDA potential decays, in fact, exponentially. On the contrary, the LSDA substantially overestimates exchange at high densities (small r), while at intermediate densities it provides a fair approximation to wS~. The LSDA potential plotted in these figures was constructed from the HF wave functions of Clementi and Roetti (1974). On the other hand the average HF potential wS has been taken directly from Kim and Gordon (1974) and Hartree (1958) for Ne and Cu § respectively. One obvious comment about the potentials plotted in figures 1 and 2 is that the exchange potentials are the same for both spin components (a = T, ~) since Ne and Cu § have dosed subshells. Some years ago, an approximation to WSx,the non-spin-polarized analogue of wS, was proposed by Alonso and Girifalco within the context of WDA (Alonso and Girifalco 1978; Gunnarsson et al 1979). Contrary to the LDA, the true density p(r2) is retained in the integrand of (9) in this scheme, and only the pair-correlation function is approximated, namely

wWDA(rl) = fdr 2p(r2) cWDA(r12;P(rl)). (17) r12 This equation shows that the pair-correlation function C(rl,r2), which is a symmetric function to the interchange of the electron coordinates rl,r 2, and in general is not spherically symmetric with respect to the exchange-hole center rl, has been replaced by cWDA(r12;~(rl) ), which, in contrast, is spherically symmetric around rl and is not symmctric with respect to the interchange of r~ and r 2. cWDA(r12; ~(r~)) is the Dirac (or HF) pair-correlation function for a homogencous electron gas (Dirac 1930) with constant density equal to the weighted density ~(r 1). This weighted-density parameter is determined at every point rl from the sum rule f dr2p(r2)CWVA(rl2;~(rl))= -- 1, (18) which sets the normalization of the exchange-hole charge. The WDA approximation to the Slater potential was tested by Alonso and Girifalco (1978) by performing Density approximation to the average HF exchange potential 95

| |

-- Sloter --- LDA r -- ~VDA -2.o

Slater 6 -7.0 LDA -..I WDA -1.5 Z ILl rlB Ne

Z -1.0 ,,,\\ Z X \\ W \ \\\N~ \x\ -0.5

o o.1 0.2 oJ 0.4 as 0.6 0.6 1.o ts .o 2.s o 35 ~.o r(a.u.) r (a.u.) Figure 1. Comparison of the average HF exchange potential (or Slater potential) Ws for Ne, and two approximations to this one: the local-density(LDA) and the weighted-density (WDA) approximations. non-selfconsistent calculations of this potential using HF wave functions, but only a qualitative comparison of W woA and W s was performed. In this paper we extend the test further, by comparing orbital energies and total energies. In addition we propose an improved WDA approximation to wS, by incorporating the spin dependence. Apart from the spin dependence, another difference with respect to the work of Alonso and Girifalco (1978) is the use of a novel form of the pair-correlation function CWDA(rl,r2). Since the functional form of the pair-correlation function in a finite system like an atom or a molecule may not be well represented by the corresponding expression for a homogeneous gas, we use a novel model pair-correlation function, with a simple exponential form, recently developed and already tested for atoms (Gritsenko et al 1993): 1 cWDA(r12 , rx.(r 1)) = - ~exp[ - (r12/r~(r I ))3/2 3. (19)

In this formula r~.(rl) is the radius of the exchange (or Fermi) hole around an electron at r 1. This hole radius is fixed by a sum-rule equivalent to (18):

fdr2p.(r2)CWSVA(r12; rxo(rl )) = -- 1. (20) 96 0 V Gritsenko et al

Siater \ \ LDA

-28 % -- @

=. -24 O

-.J ~- -2o z ,\\ w I-- ~ -16 W Z -12 %%'N~ "1" t.) x LU -8

-4

i t t i i i i I t I I I I I I I I I I / 0.02 0.04 0.06 0.08 0.1 0.12 0,14 "OJ.6 0.18 0.20

Slater -8 LDA ---. WDA (~ -7

-6 ~l Cu*

O=. -s --.I

z w I.-- 0 -3 (2. LU (D Z -2 "I" (0 X LU -I

, ~ ~ , , , . , . , 1 . J , , , , , , I 02 0.4 0.6 o.e 1.o 12 1.4 1.6 1.e 2~ r(a.u.)

Figure 2. Comparison of the average HF exchange potential (or Slater potential) Wsx for Cu +, and two approximations to this one: the local-density (LDA) and weighted-density (WDA) approximations. Density approximation to the averaoe HF exchanoe potential 97

C wsD^ fulfils the following rigorous limiting conditions (Rajagopal 1980)

C~WSDA (0,. rx~(rl )) = - 1, (21) dCWSDA(rl2; r~(rl)) = 0, (22) drl2 r1~=o and is free from empirical parameters. The corresponding exchange potential W wsDA is obtained by introducing C wsDA in (9):

--P rWSDAIr "r IF ~ Wx~WSDA ('lJ_ ~ ~ ] dr2p~(r2)~o t 12, ~ 111. (23) J r12 Figures 1 and 2 show that wW.sDA gives a good approximation to the exact W~ (the WSDA potential was evaluated using electron densties calculated from HF wave functions, Clementi and Roetti 1974). Due to the sum rule (20), the WSDA potential reproduces the correct asymptotics at large r. Near the nucleus it comes much closer than the LSDA potential to the exact Slater potential. At intermediate distances W wsD^ oscillates around W~,s providing a good overall approximation. A more quantitative test of the WSDA potential is presented in the next sections. We close this section by stressing a rather obvious point. Although we have introduced LDA and WDA approximations to the average HF exchange potential, these LDA and WDA potentials are not the LDA and WDA exchange potentials usually employed in the density functional formalism (Kohn and Sham 1965). The exchange potential of the density functional formalism is obtained, as shown in (8) of the Introduction, by performing the functional derivative of the exchange energy Ex[p]. If one. applies the LDA or WDA approximations to this energy (namely E~DA[p] or EWDA[p]; Jee for instance Alonso and Girifalco, 1978) and then performs the corresponding functional derivative, one arrives at the LDA or WDA exchange potentials of the density functional formalism. These are, evidently, not the same as the LDA or WDA potentials obtained here from the average HF exchange potential.

3. Atomic orbital energies

3.1 Highest occupied atomic orbital (ttOAO) Equations (6) have been self-consistently integrated for all atoms between He and Ar and for selected closed-shell heavier atoms using three different approximations to the average HF potential Ws xo" namely the local spin-density approximation W xG~DA of (16), and two nonlocal approximations: the WSDA of (23) and its non-spin-polarized counterpart. The last one is evidently obtained by using in (17) the non-polarized version of the pair-correlation function (19) (Gritsenko et al 1993b). The results obtained for the energy (Sh) of the highest occupied atomic orbital (HOAO) are given in table 1; - eh will be referred to below as the binding energy of the HOAO. These results are compared with those obtained by Krieger et al (1992) for closed-shell atoms with the exact average HF potential wS~. The HOAO binding energy provides an important test for the exchange potential, especially in the outer valence region. For instance, using Koopmans's theorem (Koopmans 1933), which is valid in the HF method, -eh equals the HF ionization 98 0 V Gri~senko et al

Table 1. Highest occupied orbital energies of atoms (given with opposite sign and in a.u.) obtained from self-consistent calculations using the average Hartree-Fock (or Slater) exchange potential (Krieger et a11992) and its LSDA, WDA and WSDA approximations. Exact spin-unrestricted Hartree-Foek results (Li et al 1993) and experimental ionization potentials (West 1973) are given as reference.

Atom IP (exp.) HF Slater LSDA WSDA WDA

He 0.903 0.918 0.735 0'920 0"920 Li 0'198 0.196 0"167 0'271 0.262 Be 0"343 0.309 0.326 0"256 0'396 0-396 B 0.305 0'311 0-231 0.374 0"356 C 0.413 0.436 0.350 0-491 0.452 N 0.534 0.571 0"473 0'617 0.555 O 0.500 0-679 0'338 0'590 0-663 F 0.640 0'674 0-511 0.740 0-777 Ne 0.792 0.850 0.912 0.682 0-898 0"898 Na 0"189 0-182 0-160 0"274 0"265 Mg 0.281 0.253 0-284 0'216 0'348 0'348 AI 0.220 0-210 0'165 0'262 0-253 Si 0-299 0.297 0"252 0"340 ff320 P 0.385 0.392 0'340 0"424 0'390 S 0"381 0.364 0-275 0-432 0.464 CI 0'478 0.473 0"391 0"525 0-542 Ar 0.579 0-591 0"638 0-500 0-623 0'623 Ca 0-225 0.196 0.225 0.172 0'267 0"267 Zn 0.345 0.292 0"357 0'271 0.406 0.406 Kr 0"515 0.524 0'567 0"452 0'545 0.545 Sr 0"209 0.178 0-209 0.159 0-241 0"241 Cd 0-330 0"265 0"329 0.251 0"361 0"361 Xe 0-446 0.457 0-496 0.399 0"469 0"469 Rn 0.376 0"430 0-430

potential without orbital relaxation. Furthermore, in exact density functional theory -e~ equals the ionization potential obtained as the difference between the ground- state energies of atom and ion (Perdew et al 1982) (unfortunately this theorem is not true for approximate versions of the density functional theory; it is well known that the local density approximations lead to values of -e h which underestimate the ionization potential). For these reasons we have also included in table 1 the HF results for e h (Li et al 1993) as well as the experimental ionization potentials (West 1973). To make the comparison meaningful the HF results are from spin-unrestricted calculations (Li et al 1993). First of all we can observe that the energy of the HOAO obtained with the Slater potential is rather close to the energy obtained with the exact "orbita'-dependent" HF potential. The HOAO of the exact HF calculation are a little less bound, which indicates that the HF potential for this level is slightly less attractive than the average HF potential. This seems rather obvious. We now turn to the approximations to wS,. By comparing the Slater and LSDA columns (for atoms with closed shells) we observe that the LSDA underestimates the binding energy of the HOAO (this can also be seen in the comparison of the LSDA and HF columns). The WSDA leads to better results. Only for Be and Mg are the Density approximation to the average HF exchange potential 99

LSDA and WSDA errors (with respect to wS,) of similar magnitude (although of opposite sign). For other atoms (we recall that the comparison is restricted to closed-shell atoms) the WSDA energy is closer to the Ws go" result. The WSDA performs particularly well for the noble gases. In this case the HOAO binding energies are only a little lower than those obtained from W s In contrast, the binding energies are overestimated for the divalent closed-sheU atoms. Similar to the exact Slater potential, its WSDA approximation overestimates HOAO binding energies compared to the HF scheme, with the exception of only oxygen. In a number of cases the error is rather small. On the other hand the LSDA systematically underestimates the binding energy. The LSDA values are closer to HF for the first few atoms of each period. However, from B (first period) and Si (second period) up to the end of the periods, the WSDA becomes closer to HF. In order to put the comparison with HF in a proper perspective we have to stress that the WSDA and LSDA attempt to approximate the average HF potential and not the HF potential of the HOAO. In other words, the best test is a comparison with the Slater potential. Let us next analyze the introduction of spin-dependent effects on the WDA. For this we have to compare the WDA and WSDA columns. For closed-shell atoms the results are, evidently, the same. For open-shell atoms the results show some differences, but these are not large. The WDA binding energies are smaller at the beginning of a subshell (B-N, AI-P) and larger towards the end of the subshells. Finally we notice that the HOAO binding energies calculated from I4~,, or from its nonlocal WDA and WSDA approximations, provide reasonable predictions-to the experimental ionization potentials. The trends for increasing atomic number in the sequences from Li to Ne or from Na to Ar are well reproduced by the three approximations. One cannot, however, expect too much accuracy from these estimates. The reason is that we have neglected Coulomb correlations beyond the Fermi (or exchange) correlations treated in this paper. But the merit of schemes in which -eh can be directly compared with the ionization potential is undeniable for practical purposes.

Table 2. Orbital energies for Ar, Kr and Xe (given with the opposite sign and in a.u.) obtained with Hartree-Fock (Clementi and Roetti 1974) and WDA exchange potentials.

Ar Kr Xe

Orbital WDA HF WDA HF WDA HF

ls 116-22 118.61 515.31 520.16 1217-23 1224-40 2s 11.63 12.32 68.08 69.90 186.44 189.34 2p 9.41 9.57 62.20 63.01 176.31 177.78 3s 1.14 1.28 9.91 10.85 38.60 40.17 3p 0.62 0.59 7.72 8.33 34.16 35.22 3d 3.65 3.84 25.66 26-12 4s 1.04 1-15 7.11 7.86 4p 0.54 0.52 5.50 6.01 4d 2"67 2'78 5s 0"85 0"94 5p 0-47 0-46 100 0 V Gritsenko et al

3.2 Other atomic orbitals In table 2 all the one-electron orbital energies of the noble gases Ar, Kr and Xe, calculated with the WDA potential, are compared with the corresponding HF values. The agreement is very satisfactory. Since these atoms have closed shells, the WDA and WSDA one-electron energies evidently coincide.

4. Total atomic energies

The average HF potential, wSxo, can be used not only to calculate one-electron states and orbital energies, but also for the evaluation of the total energies at the exchange- only level (Krieger et al 1992). By inserting into the HF expression for the total energy the functions po(r) and po(rl,r2) obtained with the Slater potential we arrive at the following expression:

E=-~ l fptr,)V.(rt)dr,

-I--~if dr lp~(r 1) fP(r2)+p,(r2)C,(r,,r2)dr 2 2 , r12

Table 3. Total ground-state energies of atoms (given with the opposite sign and in a.u.) obtained from self-consistent calculations using the average Hartree-Fock (or Slater) exchange potential (Krieger et al 1992) and its LSDA, WDA and WSDA approximations. Spin-unrestricted HF results (Li et a11993) are given as a reference.

Atom HF Slater LSDA WDA WSDA

He 2"862 2"703 2'861 2"861 Li 7.433 7.168 7.355 7.397 Be 14-573 14.562 14.193 14"485 14.485 B 24-529 24-023 24.404 24.416 C 37.690 37"057 37.511 37"562 N 54-405 56"638 54' 137 54.260 O 74.814 73"911 74"613 74.670 F 99"411 98"379 99"269 99-284 Ne 128.547 128"502 127-380 128"431 128"431 Na 161-859 160-530 161.738 161"743 Mg 199"615 199-536 198"132 199"527 199"527 AI 241.877 240-232 241.787 241"791 Si 288.854 287-048 288.747 288-768 P 340'719 338-744 340-567 340"620 S 397.506 395"367 397"403 397-430 CI 459.483 457'182 459"412 459"419 Ar 526.817 526"708 524.346 526.747 526"747 Ca 676"758 676"612 673"982 676"752 676.752 Zn 1777"848 1777'590 1773"550 1777'817 1777.817 Kr 2752.055 2751"768 2746.459 2751"610 2751.610 Sr 3131"546 3131.221 3125"580 3131-068 3131"068 Cd 5465"133 5464-714 5457"215 5464-032 5464"032 Xe 7232.138 7231'690 7222"962 7230"534 7230"534 Rn 21850.359 21862"013 21862"013 Density approximation to the average HF exchanqe potential 101 = -ldrape(h)2 f drl[V~"p(rl'r'~)]r;="+ ~ f

[ V.(rl) + 21 V~(rt)_t- 21 W~o(rl)] ' (24) where the final expression has been obtained by taking into account (2), (9) and (10).

Of course, any approximation can replace W sgO in (24), and the comparison of the corresponding total energies can be considered as an integral criterion for the quality of those approximations. Total ground-state energies for the same atoms as those in table 1 are given in table 3. The energies listed correspond to the different approxima- tions for the exchange potential: namely W~ and its local (LSDA) and nonlocal (WDA and WSDA) approximations. The exact spin-unrestricted HF energies (Li et al 1993) are given as a reference. It should be noted that the calculation of energies via (24) differs a little from that from the variational procedure which is used in the standard HF method. Our calculation is, of course, self-consistent, but is not a variational one, in the sense that the effective one-electron potential which is used for the calculation of the spin-orbitals ~tr (and thus, for the calculation of p~(r) and pr is not obtained by the variational minimization of the energy functional of (24). Judging from table 3, the WSDA approximation is definitely better than the LSDA, compared to the Slater and to the exact HF results. The LSDA underestimates substantially the absolute magnitude of the atomic energies, while the non local approximation leads to much more accurate energies. The relative error of the WSDA energy decreases progressively with atomic number, and for heavy atoms it is in the range 0.005-0.02 per cent (with respect to Slater). In some cases (Ar, Ca, Zn) the WSDA energies are closer to HF than the Slater energies. Turning to the comparison between WDA and WSDA, for open-shell atoms, we see that the WSDA energies are always lower (as it should be) and closer to HF.

5. Conclusions

Long ago Slater proposed a simplification of the Hartree-Fock method in which the orbital-dependent exchange potentials are all replaced by the "average" exchange potential. Slater himself also proposed a local density approximation to this average HF potential. In this paper we have constructed a nonlocal density approximation to the Slater potential. Self-consistent calculations using this nonlocal density exchange potential lead to rather good results for the total energies of atoms as well as for one-electron energy eigenvalues. In particulaT, we have studied in detail the highest occupied orbital, and we have verified that there is also reasonably good agreement between the negative of this eigenvalue and the experimental ionization potential. However reasonable the comparison with exact HF theory and experiment is, we stress that our only intention is to provide a density-based approximation to the average HF exchange potential. This appears to be guaranteed by the results we have obtained. 102 0 V Gritsenko et al

To connect with the usual density functional theory (Kohn and Sham 1965) or with the optimized potential method (Talman 1989) we notice that recent work by Krieger and coworkers (Krieger et al 1992; Li et al 1993) has shown that the OPM exchange potential can be written as a sum of the Slater (or average HF) potential plus a small term. The same can be said about the Harbola-Sahni potential (Harbola and Sahni 1989; Li et al 1989). This enhances the usefulness of good density approxi- mations to the Slater potential, but one has to keep in mind, of course, the additional terms.

Acknowledgements

This work has been supported by DGICYT (grant PB89-0352). One of us (OVG) is grateful to Ministerio de Educaci6n y Ciencia (Spain) for supporting his sabbatical stay at the University of Valladolid.

References

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