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Local exchange potential for atomic systems
Volume
62. number 1
CHEMICAL PHYSICS LETTERS
LOCAL EXCHANGE
POTENTIAL
M. BERRONDO
*
Imtitufo
University of Mexico.
de FZca,
15March1979
FOR ATOMIC SYSTEMS *
M&ico
20. D-F..
M&ico
and 0. GOSCINSKI
:
Quantum C7remistry Group.
University of Uppsaola.S75I 20 Uppsah. Sweden
Received 26 June 1978; in fiial form 26 October 1978
An approtimate tocal potential for the ewhanse term is found for atomic systems. A functioxul form for the density mtrix as a rezl exponential is assumed, in order to account for the nuclear attraction. The potential is obtained its the functionzl derivative of the excharige energy, imposing the idempotency condition on the density matns. The resulting local potential has the correct asymptotic behzwiour without the need of ad-hoc cut-off procedures, or additional assumptions-
1. Introduction The complications non-local
cleus r. Instead, the local potential arising from the presence of a
exchange operator in the Hartree-Fock
equa-
tions are well known [I-S] _ Local approximations for these potentials have been introduced since a long time ago [2,3], but they have been imariably deduced stsrting from a free-electron gas [2-7]_ This model is appropriate for the valence electrons in a metal, or the zelectrons in a molecuIe, but has also been used for all other electrons in aroms, molecules, clusters, and solids [7-lo]_ Corrections to the free-e!ectron gas potentials have aIso been worked out [I l-131, to try to correct for the inhomogeneities in the electron density for more realistic mode!s. In what follows we shall concentrate our attention to atomic systems. Perhaps the most dramatic evidence of the error introduced by the electron gas model in this case is the incorrect asymptotic behaviour of the local potential. For an atom, the exchange potential should behave as -1 /r at large distances from the nu* Work supported in part by CONXCYT (.\Gxico) through project PNCB-O024** Consultant at Institute Mexiuno de1 Peti6leo (Ykcico). r Supported by NFR (Sweden).
deduced
from the
electron gas decays exponentially with r. An ad-hoc cut-off introduced by Latter [14] has been adopted
in
practice [S] to cope wi:h this lack of cancellation
of charge at large dis-
the self-interaction of the electron tances [7] _ The derivation of the electron gas local potential follo\%s the main idea of the Thomas-Fermi model [4,15] _ One assumes a slowly varying electronic density in order to obtain a functional form of the local potential, and thence replace the density by its local value p(r), yielding a final expression [7] proportional to .f&r)] Iis_ This kind of behaviour has proven to be correct only in an average sense, at least for the case of two-electron ions [16]_ The details of the potential, such as its Z-dependence. and the already mentioned asymptotic behaviour, are not properly obtained in such an appro_ximation, i.e. starting from a homogeneous electron density. The main features of the electronic structure of rhe atoms depend on two basic properties, namely the central attraction by the nucleus, and the Pauli exclusion principle. The electron gas model takes into account the latter, since it starts from an idempotent density matrix albeit built up from plane waves. What we intend to do in what follows is to incorporate the central 31
Volume 62, number 1
CHEhIICAL
PHYSICS
field as well, i-e_ to include the presence of the nucleus by replacing the basic assumption of having a locally constant density by a more realistic one. The procedure we shah adopt to fmd a local exchange potential for the atoms, consists of starting off from an approximate functional form of the density matrix which decays exponentially with the distanceWe then substitute it into the exchange integral to obtain a functional form from wIiich we can find the potential. This local approximation to the exchange potential is defined as the functional derivative of the exchange integral with respect to the density. The resulting expression has the correct behaviour in the two extreme regimes of small and huge vahres of r. There is therefore no need of any further assumption or any empirical parameter. In section 2 we present the derivation of the local potential, as well as its explicit expression. Section 3 contains some further developments and remarks. The appIication to specific cases both for neutral and ionic systems, are in progress, and shall be presented in detail elsewhere_
LJZl-l-ERS
lShlarch1979
dependence of the exchange energy with respect to the density but not as numerical approximations to the density matrix or to the value of the exchange integral. We shall hence compute this integral Ex disregarding the variation of A and r) with r- Substituting in the definition:
(in atomic units), we obtain: Ex = _$A
ewri(rly:fA
e-‘l(r2+rl) drl
dr2.
(3)
taking A to be real Integrating over drr now yields: Ex = -
s
A ed3”’ e [ 1 - (I + qt-) e--7?r] 713r
dr_
(4)
Still using the same assumptions, we can rewrite this as a functional of the dens@ p(r) as:
[A
- p f+p In z]
dr,
(5)
where p(r) = p(r)/A_ The local potential obtains now as the functional derivative:
2. Local exchange potential When we took at the atomic electron density plotted versus the distance from the origin, we can readily notice [ 17, IS] that its logarithm has an almost constant slope ioc~f&, rather than being aImost constant (as is assumed in the electron gas case)- This is of course a direct consequence of the nuc1e.u attraction. The shell structure manifests itself only through sudden sIope changes. Extrapolating this result, we shall assume a functional form for the reduced (spinless) density matrix: y(rI , r2) = A e- a Or +r2),
(I)
where A, and q are sIowIy varying functions of ‘1, and rl_ For two-electron ions even taking A and n zs constant yields a very good approximation, since the Is orbitals are we11represented by a screened hydrogenie ground state function. In the general case, the screening 71should approach 2 near the origin (CC the
cusp condition for the density) and diminish monotonically. In the same spirit as in the electron gas case, we shall use eq. f I) only in order to obtain the functional 32
+(I
-&r]_
(6)
This expression can be written in terms of
JW=-_(~/P)(VP-~PWass: Vx= - ‘22 -! (1 -I-y)[l - (1 i-y)e-‘] +
$
(7) _
69
Finally, we shall compute A imposing the idempotency condition to the density matrix eq- (1): s
p(rI, r&Nr;,
‘2) d+ = p(r17 r2).
(9)
condition which gives A = q3/n. In this way, we have taken into account Pauli’s exclusion principIe, as we11as the fact that we are dealing with N electrons attracted by a nucieus, even if only in an approximate way_ The final expression for the local exchange potential
Volume 62, number 1
CHEMICAL
PHYSICS LETTERS
15 March 1979
where y is a functional of the density, defined in eq_ (7). Let us finally notice that y = 3nr for the case when p(r) =A e- znr_
tion numbers, as in electron gas potentials_ The local potential, eq. (lo), can be used in selfconsistent calculations for the atomic orbital% It can also be parametrized, and utilized as a bona fide potential [ 19.20] _ Interpolations of the Iocal exchange for the molecuiar (i.e. the multicenter) case are in progress and seem to be also very promising_
3_ ConcIuding remarks
Acknowledgement
It is easiIy verified that the value of IrX at the origin r= 0 is finite, as it should_ On the other hand. its asymptotic behaviour is given by:
We thank Professor P.-O. Lowdin for his continued interest. We have profltted from discussions with Drs. J.L. Calais and J-P. Daudey. We would like to thank the Quantum Chemistry Group (MB.) and the Instituto de Ffsica (0-G.) for their hospitality.
is thus: V,[p(r)]
vx
-+ J-e-
=-2+
-
l/r.
+$-
(1 ++yl
,
(10)
(11)
This can be traced directly from the value chosen for A, i.e. it is a consequence of accounting for the exclusion principle_ From the expression for?, eq_ (7), it is also evident that this potential incorporates the inhomogeneity of the electron density from the beginning, and not as a correction to the electron gas. Trying to improve on the initia1 functional form of the density matrix has appeared to us as a very difficult task, because of the need to invert the relation between p and p_ The derivation which we have outlined in the present paper is not unique. In particular, we can raise the question of whether to use _~(r)/Zr for the value of n in eq_ (lo), or take it as a constant_ Preliminary calculations [I91 seem to indicate that it is better to take it as a constant (e-g. the asymptotic value ofy(r)lZb), since otherwise too much weight is given to the origin. Since the present potential has the correct asymptotic behaviour, eq- (1 l), it is expected to be particuIar!y useful in the evaiuation of physical properties sensitive to large values of r such as G? or in electronatom scattering. In particular, preliminary caIculations on F- [ 191 indicate that negative ions can be described on the same footing as neutral atoms. There is no need of artificial boundary conditions (Watson spheres) or dubious limiting procedures with occupa-
References
[I] P-AN. Dime, Proc. Cknbridge Phd. Sot. 26 (1930) 376. [2] R. Gaspar, Acta Phls. Sci. IIunz_ 3 (1954) [3] J-C Shter, Phys Rev. SI (1951) 3S5.
263.
[4] W’.Kahn and L-J. Sham, Phys. Rev. l-10 (1965) Al 133. [Z] 31. Berrondo and 0. Goscinski, Phys. Re%_lS4 (1969) 10. [6] P. Gombas, The statistical theory of atoms (Springer, Berlin, i949). [7] J-C. Shter. The self-consistent field for moIecuIes and solids (McGraw-HdI. hew York, 1974) ch. l[S] F. Herman and S. SIaIlmzm, Atomic structure ulculatians (Prentice-Ha& Enk~ood cliffs, 1963). [9] I; H. Johnson, Ann. Rev. Phys. Chem. 26 (1975) 39. [lOI R.P. Messmer, in: Modem theoretical chemistry, Vol. 8, ed. G._4. Sesl (P1mt.m Press, Ne\\ York. 1977) p- 215. [ II] S K. \Ia and K.A. Brueckncr, Phys. Rev. 165 (1968) 16. [ 171 L. Hedin and B-1. Lundqvist. J. Ph>s. C 4 (1971) 2064; 0. Gunnarson and B.I. Lundqvist. Phys. Re\. B13 (1976) 4274. [13] J.L. Gkquezand J. Keller, Phqs. Rev. Al6 (1977) 135% [14] R. titter, Ph.s. Rer. 99 (1955) 510. [15] J-C_ Slater. Intern. J. Quantum Chem. S9 (1975) 7. [16] J-L_ Ghis and S_ Lasson, Chem. Ph_rs. Letters 11 (1971) [17] [IS] [19] [20]
12. G. Sperbrr, Intern. J. Quantum Chem. 5 (1971) 1S9W.-P. Wang and R G. Parr. Phys. Rev. A16 (1977) 891. M. Errrondo and J-P. Dnudey, to be published. 31. Berrondo, J.P. Daudey and 0. Goscinshi, Chem. Phys. Letters 62 (1979) 34.
33
62. number 1
CHEMICAL PHYSICS LETTERS
LOCAL EXCHANGE
POTENTIAL
M. BERRONDO
*
Imtitufo
University of Mexico.
de FZca,
15March1979
FOR ATOMIC SYSTEMS *
M&ico
20. D-F..
M&ico
and 0. GOSCINSKI
:
Quantum C7remistry Group.
University of Uppsaola.S75I 20 Uppsah. Sweden
Received 26 June 1978; in fiial form 26 October 1978
An approtimate tocal potential for the ewhanse term is found for atomic systems. A functioxul form for the density mtrix as a rezl exponential is assumed, in order to account for the nuclear attraction. The potential is obtained its the functionzl derivative of the excharige energy, imposing the idempotency condition on the density matns. The resulting local potential has the correct asymptotic behzwiour without the need of ad-hoc cut-off procedures, or additional assumptions-
1. Introduction The complications non-local
cleus r. Instead, the local potential arising from the presence of a
exchange operator in the Hartree-Fock
equa-
tions are well known [I-S] _ Local approximations for these potentials have been introduced since a long time ago [2,3], but they have been imariably deduced stsrting from a free-electron gas [2-7]_ This model is appropriate for the valence electrons in a metal, or the zelectrons in a molecuIe, but has also been used for all other electrons in aroms, molecules, clusters, and solids [7-lo]_ Corrections to the free-e!ectron gas potentials have aIso been worked out [I l-131, to try to correct for the inhomogeneities in the electron density for more realistic mode!s. In what follows we shall concentrate our attention to atomic systems. Perhaps the most dramatic evidence of the error introduced by the electron gas model in this case is the incorrect asymptotic behaviour of the local potential. For an atom, the exchange potential should behave as -1 /r at large distances from the nu* Work supported in part by CONXCYT (.\Gxico) through project PNCB-O024** Consultant at Institute Mexiuno de1 Peti6leo (Ykcico). r Supported by NFR (Sweden).
deduced
from the
electron gas decays exponentially with r. An ad-hoc cut-off introduced by Latter [14] has been adopted
in
practice [S] to cope wi:h this lack of cancellation
of charge at large dis-
the self-interaction of the electron tances [7] _ The derivation of the electron gas local potential follo\%s the main idea of the Thomas-Fermi model [4,15] _ One assumes a slowly varying electronic density in order to obtain a functional form of the local potential, and thence replace the density by its local value p(r), yielding a final expression [7] proportional to .f&r)] Iis_ This kind of behaviour has proven to be correct only in an average sense, at least for the case of two-electron ions [16]_ The details of the potential, such as its Z-dependence. and the already mentioned asymptotic behaviour, are not properly obtained in such an appro_ximation, i.e. starting from a homogeneous electron density. The main features of the electronic structure of rhe atoms depend on two basic properties, namely the central attraction by the nucleus, and the Pauli exclusion principle. The electron gas model takes into account the latter, since it starts from an idempotent density matrix albeit built up from plane waves. What we intend to do in what follows is to incorporate the central 31
Volume 62, number 1
CHEhIICAL
PHYSICS
field as well, i-e_ to include the presence of the nucleus by replacing the basic assumption of having a locally constant density by a more realistic one. The procedure we shah adopt to fmd a local exchange potential for the atoms, consists of starting off from an approximate functional form of the density matrix which decays exponentially with the distanceWe then substitute it into the exchange integral to obtain a functional form from wIiich we can find the potential. This local approximation to the exchange potential is defined as the functional derivative of the exchange integral with respect to the density. The resulting expression has the correct behaviour in the two extreme regimes of small and huge vahres of r. There is therefore no need of any further assumption or any empirical parameter. In section 2 we present the derivation of the local potential, as well as its explicit expression. Section 3 contains some further developments and remarks. The appIication to specific cases both for neutral and ionic systems, are in progress, and shall be presented in detail elsewhere_
LJZl-l-ERS
lShlarch1979
dependence of the exchange energy with respect to the density but not as numerical approximations to the density matrix or to the value of the exchange integral. We shall hence compute this integral Ex disregarding the variation of A and r) with r- Substituting in the definition:
(in atomic units), we obtain: Ex = _$A
ewri(rly:fA
e-‘l(r2+rl) drl
dr2.
(3)
taking A to be real Integrating over drr now yields: Ex = -
s
A ed3”’ e [ 1 - (I + qt-) e--7?r] 713r
dr_
(4)
Still using the same assumptions, we can rewrite this as a functional of the dens@ p(r) as:
[A
- p f+p In z]
dr,
(5)
where p(r) = p(r)/A_ The local potential obtains now as the functional derivative:
2. Local exchange potential When we took at the atomic electron density plotted versus the distance from the origin, we can readily notice [ 17, IS] that its logarithm has an almost constant slope ioc~f&, rather than being aImost constant (as is assumed in the electron gas case)- This is of course a direct consequence of the nuc1e.u attraction. The shell structure manifests itself only through sudden sIope changes. Extrapolating this result, we shall assume a functional form for the reduced (spinless) density matrix: y(rI , r2) = A e- a Or +r2),
(I)
where A, and q are sIowIy varying functions of ‘1, and rl_ For two-electron ions even taking A and n zs constant yields a very good approximation, since the Is orbitals are we11represented by a screened hydrogenie ground state function. In the general case, the screening 71should approach 2 near the origin (CC the
cusp condition for the density) and diminish monotonically. In the same spirit as in the electron gas case, we shall use eq. f I) only in order to obtain the functional 32
+(I
-&r]_
(6)
This expression can be written in terms of
JW=-_(~/P)(VP-~PWass: Vx= - ‘22 -! (1 -I-y)[l - (1 i-y)e-‘] +
$
(7) _
69
Finally, we shall compute A imposing the idempotency condition to the density matrix eq- (1): s
p(rI, r&Nr;,
‘2) d+ = p(r17 r2).
(9)
condition which gives A = q3/n. In this way, we have taken into account Pauli’s exclusion principIe, as we11as the fact that we are dealing with N electrons attracted by a nucieus, even if only in an approximate way_ The final expression for the local exchange potential
Volume 62, number 1
CHEMICAL
PHYSICS LETTERS
15 March 1979
where y is a functional of the density, defined in eq_ (7). Let us finally notice that y = 3nr for the case when p(r) =A e- znr_
tion numbers, as in electron gas potentials_ The local potential, eq. (lo), can be used in selfconsistent calculations for the atomic orbital% It can also be parametrized, and utilized as a bona fide potential [ 19.20] _ Interpolations of the Iocal exchange for the molecuiar (i.e. the multicenter) case are in progress and seem to be also very promising_
3_ ConcIuding remarks
Acknowledgement
It is easiIy verified that the value of IrX at the origin r= 0 is finite, as it should_ On the other hand. its asymptotic behaviour is given by:
We thank Professor P.-O. Lowdin for his continued interest. We have profltted from discussions with Drs. J.L. Calais and J-P. Daudey. We would like to thank the Quantum Chemistry Group (MB.) and the Instituto de Ffsica (0-G.) for their hospitality.
is thus: V,[p(r)]
vx
-+ J-e-
=-2+
-
l/r.
+$-
(1 ++yl
,
(10)
(11)
This can be traced directly from the value chosen for A, i.e. it is a consequence of accounting for the exclusion principle_ From the expression for?, eq_ (7), it is also evident that this potential incorporates the inhomogeneity of the electron density from the beginning, and not as a correction to the electron gas. Trying to improve on the initia1 functional form of the density matrix has appeared to us as a very difficult task, because of the need to invert the relation between p and p_ The derivation which we have outlined in the present paper is not unique. In particular, we can raise the question of whether to use _~(r)/Zr for the value of n in eq_ (lo), or take it as a constant_ Preliminary calculations [I91 seem to indicate that it is better to take it as a constant (e-g. the asymptotic value ofy(r)lZb), since otherwise too much weight is given to the origin. Since the present potential has the correct asymptotic behaviour, eq- (1 l), it is expected to be particuIar!y useful in the evaiuation of physical properties sensitive to large values of r such as G? or in electronatom scattering. In particular, preliminary caIculations on F- [ 191 indicate that negative ions can be described on the same footing as neutral atoms. There is no need of artificial boundary conditions (Watson spheres) or dubious limiting procedures with occupa-
References
[I] P-AN. Dime, Proc. Cknbridge Phd. Sot. 26 (1930) 376. [2] R. Gaspar, Acta Phls. Sci. IIunz_ 3 (1954) [3] J-C Shter, Phys Rev. SI (1951) 3S5.
263.
[4] W’.Kahn and L-J. Sham, Phys. Rev. l-10 (1965) Al 133. [Z] 31. Berrondo and 0. Goscinski, Phys. Re%_lS4 (1969) 10. [6] P. Gombas, The statistical theory of atoms (Springer, Berlin, i949). [7] J-C. Shter. The self-consistent field for moIecuIes and solids (McGraw-HdI. hew York, 1974) ch. l[S] F. Herman and S. SIaIlmzm, Atomic structure ulculatians (Prentice-Ha& Enk~ood cliffs, 1963). [9] I; H. Johnson, Ann. Rev. Phys. Chem. 26 (1975) 39. [lOI R.P. Messmer, in: Modem theoretical chemistry, Vol. 8, ed. G._4. Sesl (P1mt.m Press, Ne\\ York. 1977) p- 215. [ II] S K. \Ia and K.A. Brueckncr, Phys. Rev. 165 (1968) 16. [ 171 L. Hedin and B-1. Lundqvist. J. Ph>s. C 4 (1971) 2064; 0. Gunnarson and B.I. Lundqvist. Phys. Re\. B13 (1976) 4274. [13] J.L. Gkquezand J. Keller, Phqs. Rev. Al6 (1977) 135% [14] R. titter, Ph.s. Rer. 99 (1955) 510. [15] J-C_ Slater. Intern. J. Quantum Chem. S9 (1975) 7. [16] J-L_ Ghis and S_ Lasson, Chem. Ph_rs. Letters 11 (1971) [17] [IS] [19] [20]
12. G. Sperbrr, Intern. J. Quantum Chem. 5 (1971) 1S9W.-P. Wang and R G. Parr. Phys. Rev. A16 (1977) 891. M. Errrondo and J-P. Dnudey, to be published. 31. Berrondo, J.P. Daudey and 0. Goscinshi, Chem. Phys. Letters 62 (1979) 34.
33