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An approximate non-local independent fermion kinetic-energy density functional is constructed from a model for the Density Functional exchange-hole. The functional is exact for one- and two-electrons systems and gives a good approximation for a homogeneous electron gas. Reasonable results are also obtained for model systems.
In the Density Functional Formalism [1], [2], [3] the ground state energy E[n] of a system of electrons can be expressed as a functional of the ground state electron density n(r) :
where v( r) is the external potential, Ts[n] is the non-interacting kinetic-energy, U[n] is the classical electron-electron repulsion energy, and Exc[n] is the exchange-correlation energy . The exact forms of the kinetic-energy functional and the exchange-correlation functional are not known. Although the kinetic-energy can be calculated in the Kohn-Sham formalism [1], [2], [3] by using an auxiliary basis, an accurate approximation to the kinetic-energy functional will enable one to compute properties of systems of electrons by minimising Eq.(1) directly in terms of the charge density. Many approximations to the kinetic-energy functional have appeared in the literature. These include the Thomas-Fermi local density approximation, semi-local approximations [4],[5],[6],[7],[8] [9] and non-local approximations [4], [10]. As of yet, none of these approximations are of sufficient accuracy for calculations.
In this paper we examine a non-local approximation to the kinetic-energy constructed from an approximation to the Density Functional exchange-hole. The spherical average of the exchange-hole, nx av(r,r+s), is approximated by an expression of the form
where the superscript "av" indicates the spherical average of an arbitrary functions f(r,r+s) centred at r:
The kinetic-energy can be expressed in terms of the second derivative with respec to s of the spherical average of the exchange-hole, and for this approximation, Eq. (2), the kinetic-energy is then given by (in Hartree atomic units)
where the first term in Eq.(4) is the Weizsäcker term and the second
term the Pauli term (see for example discussion in [9]). It
is therefore only the Pauli term which is approximated. The function
is determined from
the sumrule,
.
which is an exact property of the exchange-hole [2], [3] and can be derived from the definition, Eq.(6). The value of
is depends on an
integration over nav (r,r+s)
and hence this approximation is non-local despite the apparent local form of
Eq. (4).
The density functional exchange-hole is defined, in analogy to the Hartree-Fock exchange-hole [2],[3], as (for convenience given here for a spin compensated system)
where the 
are the
eigenfunctions of the Kohn-Sham equations [1], [2],
[3].
The spherical average of nx (r,r+s) expanded as a series in s, retains only even order terms:
Taking the second derivative of nxav(r+s) with respect to s, evaluated at s=0, it follows after some algebra, that the kinetic-energy can be expressed as:
Here Tw[n] is the Weizsäcker energy and Tp[n] the Pauli energy. The Pauli energy is frequently less than half the Weizsäcker energy and an approximation to Tp[n] only may be more efficient than an approximation the the total kinetic-energy [9]. Note that Tw[n] appears in a natural way in this expression. Through Eq.[10] the Pauli energy is expressed in terms of the spherical average of the exchange-hole and hence a good approximation to the exchange-hole should give a good approximation to the Pauli energy. It is, however, important to note that the Pauli energy depends only on the curvature of the exchange-hole at the centre of the hole, s=0. An accurate representation of the curvature is therefore essential for a good approximation of the Pauli energy.
Writing the kinetic-energy in the form of Eq. (10) has a number of advantages. The Weizsäcker term, for example, gives the correct singularity in the first functional derivative of the kinetic-energy at the centre of a coulomb potential, which many ather approxmations fail to give [10].
Any approximation to the exchange-hole must satisfy as many of the properties that the exact hole as can be accommodated. The approximation discussed in this paper satisfy the following conditions.
where nN-1(r) is the density of a N-1 electron system or a linear combination of the charge densities if the N-1 electron system is degenerate. For the spherical average of the exchange-hole, n(r+s) is simply replaced by nav(r+s). The property implied by Eq.(12) is illustrated in Fig. (1) below for a Mg atomic configuration. The charge densities in this paper were constructed from Hydrogenic wave functions for non-interacting N-electron systems. The strength of the central potential and the charge configuration follows from the atomic symbol.
Figure 1. 2nxav(r+s)/nav(r+s) for Hydrogenic Mg centred at |r|=0.32 a.u.
Since nx(r+s) is non-positive, Eq. (12) implies that the charge density of the N-1 electron system is always less or equal to the charge density of the N electron system at the same point in space when the two systems have the same external potential. This also follows from Eq. (8).
Since the Fermi energy can be expressed in terms of the spherical average of the exchange-hole, it is only necessary to model this average. We propose an approximation of the form:
.
The function f must then satisfy
As a first attempt we take
The constants c1 and c2 are determined by fitting the Pauli energy for a number of model systems. This form for the function f(r,s) (Eq. (14)) becomes constant for large s for densities that decay exponentially, as is the case for any finite system [11]. This is illustrated in Fig. 2 below.
Figure 2. g(nav): Approximate and exact for Hydrogenic Mg for hole centred at |r|=0.16 a.u.
For this approximation to the exchange-hole the kinetic-energy is given by
Eq. (4). The function is
determined by demanding that the model exchange-hole satisfies the sumrule, Eq.
(11). For a two electron system the spherically averaged exchange-hole reduces
to Eq. (13) with f(r,s)=1. That implies that
=0 for all r
and from Eq. (4) the Pauli energy is zero. Since the total kinetic-energy for a
two electron system is contained in the Weizsäcker term the kinetic-energy
is given exactly in this case. For a one electron system there exists no
exchange-hole. However, a sumrule is still satisfied for
=0 and the total
energy is once again given exactly by the Weizsäcker term. For a
homogeneous electron gas the gradient of the charge density is zero. Only the
term depeding on nav remains in the function g. The Weizsäcker
term is zero and the Pauli term reduces to an integral over n5/3.
The prefactor, however, is too large and overestimates the correct energy by 3%.
In Fig. 2 the exact and approximate g(nav) only matches well near the origin, s = 0. Since the Pauli energy is determined by the curvature of nxav(r+s) at the centre of the hole, the total kinetic-energy is nevertheless quite accurate in this instance (see table below).
| Atom | %error | |
| Be | 10.78 | |
| Ne | -1.50 | |
| Mg | 0.64 | |
| Ar | -0.49 | |
| Ca | 0.14 | |
| Zn | -3.27 | |
| Kr | -1.32 | |
| Sr | -0.32 | |
| Pd | 0.50 | |
| Cd | 1.29 | |
| Xe | 3.5 |
The results in the table above show a scattering in the accuracy of the approximation, Be coming in as the worst example at a percentage error of 10.7% This is rather disappointing since Be is a light atom with only 4 electrons and the approximation is exact for one- and two-electron systems.
Exact kinetic energies and charge densities were generated for independent-electron Hydrogenic models of atoms for which wave functions are known analytically. Although these are not accurate models of the physical systems, the kinetic-energy is a functional of the charge density [2-3] and the exact functional is valid these systems. Any approximation must therefore also work well for these sytems. The parameters, (c1 = 77.5, c2 = 100.0) , were obtained trough a fit to the exact kinetic energy of closed shell atoms ranging from He to Xe. Apart from Be, the results are of the same order of accuracy as given by the semi-local approximations discussed in Ref. [7]. The approximation discussed in this paper is still not sufficiently accurate for doing independent calculations.
An approximate free fermion kinetic non-local energy functional has been constructed from an approximation to the Density Functional exchange-hole. Reasonable results are obtained for model calculations, but the accuracy of the approximation is not good enough for use in independent calculations.
The approximation to exchange-hole in the form nxav(r+s) = -1/2nav(r+s)f([nav];r,s), suggests that other forms of the function f([nav];r,s), with the properties listed below Eq. (13), could be explored.
Since the exchange energy can be expressed in terms of the sperical average of the exchange-hole [2], [3], the approximation, Eq. (13) and Eq. (14) can be used to approximate the exchange energy and potential. This will be part of a future study.
1. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
2. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, New York 1989).
3. R. M. Dreizler and E. K. U. Gross, Density Functional Theory (Springer-Verlag, Berlin 1990).
4. C.F.V. Weizsäcker, Z. Phys. 96, 431 (1935).
5. A.E. Depristo, J.D. Kress, Phys. Rev. A35, 438 (1987).
6. H. Lee and R.G. Parr, Phys. Rev. A44, 768 (1991).
7. D.J. Lacks and R.G. Gordon, J. Chem. Phys. 100, 4446 (1993).
8. For a discussion of properties of a number of semi-local approximations see reference [7].
9. C. Herring, Phys. Rev. A 34, 2614 (1986).
10. D.P. Joubert and M.B. Tchoula Tchokonte (to be puplished).
Daniel Joubert / University of the Witwatersrand / joubert@physnet.phys.wits.ac.za