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Physical Review A

Temperature dependence of the Slater sum: Generalization of the one-dimensional Thomas-Fermi theory

C. Amovilli, Norman Henry March

Published Sep 1, 1991• 6 citations

Abstract

The one-dimensional equation of motion for the canonical or Bloch density matrix is solved by the method of separation of variables for independent particles moving in a potential V(x). The solution is exact when third and higher derivatives of V(x) are zero, embracing thereby the examples of (a) harmonic potentials and (b) a constant electric field of arbitrary strength. The form of the Slater sum obtained involves then two functions of temperature only, to be determined. One obvious relation between these two functions is via the partition function. A further relation comes from the orthonormality of the wave functions appearing in the Slater sum: the analog of idempotency for the ground-state Dirac density matrix. These two conditions again are shown to lead back to the exact result for harmonic potentials. The results presented generalize the one-dimensional Thomas-Fermi theory to apply to a temperature range that is significantly wider than for the original method.

DOI: 10.1103/physreva.44.2846

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Slater sum for an inhomogeneous liquid of Fermions generated by a sech2xpotential in one dimension

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Journal of Mathematical Physics•Dec 1, 1987•3 citations

The one-dimensional equation of motion for the canonical or Bloch density matrix is solved by the method of separation of variables for independent particles moving in a potential V(x). The solution is exact when third and higher derivatives of V(x) are zero, embracing thereby the examples of (a) harmonic potentials and (b) a constant electric field of arbitrary strength. The form of the Slater sum obtained involves then two functions of temperature only, to be determined. One obvious relation between these two functions is via the partition function. A further relation comes from the orthonormality of the wave functions appearing in the Slater sum: the analog of idempotency for the ground-state Dirac density matrix. These two conditions again are shown to lead back to the exact result for harmonic potentials. The results presented generalize the one-dimensional Thomas-Fermi theory to apply to a temperature range that is significantly wider than for the original method.

Abstract For a one‐body potential V(r) generating eigenfunctions ψi(r) and corresponding eigenvalues ϵi, the Feynman propagator K(r, r′, t) is simply related to the canonical density matrix C(r, r}′, β) by β → it. The diagonal element S(r, r},β) of C is the so‐called Slater sum of statistical mechanics. Differential equations for the Slater sum are first briefly reviewed, a quite general equation being available for a one‐dimensional potential V(x). This equation can be solved for a sech2 potential, and some physical properties of interest such as the local density of states are derived by way of illustration. Then, the Coulomb potential –Ze2/r is next considered, and it is shown that what is essentially the inverse Laplace transform of S(r, β)/β can be calculated for an arbitrary number of closed shells. Blinder has earlier determined the Feynman propagator in terms of Whittaker functions and contact is here established with his work. The currently topical case of Fermion vapours which are harmonically confined is then treated, for both two and three dimensions. Finally, in an Appendix, a perturbation series for the Slater sum is briefly summarized, to all orders in the one‐body potential V(r). The corresponding kinetic energy is thereby accessible.

Forms of canonical (Bloch) and Dirac density matrices for free electrons with and without external applied fields are first considered. The basic tool employed is the Bloch equation with a one-electron Hamiltonian. Such an approach is used to obtain a perturbation theory to all orders for the idempotent Dirac density matrix when a common potential energy V(r) is switched on to originally free electrons. The relation to density functional theory is then considered and the exchange–correlation contribution Vxc(r) to V(r) is expressed in terms of first- and second-order density matrices following Holas and March. These latter density matrices are now for the fully interacting system and, in particular, the first-order density matrix is no longer idempotent, though it must still satisfy generalized Pauli Principle conditions. Reference is also made to a localized Wigner electron in a strong magnetic field.

In 1959, March and Young ( Nucl. Phys. 12 237) rewrote the equation of motion for the Dirac density matrix γ( x , x 0 ) in terms of sum and difference variables. Here, γ( , 0 ) for the d -dimensional isotropic harmonic oscillator for an arbitrary number of closed shells is shown to satisfy, using the variables ∣ + 0 ∣/2 and ∣ − 0 ∣/2, a generalized partial differential equation embracing the March–Young equation for d = 1. As applications, we take in turn the cases d = 1, 2, 3 and 4, and obtain both the density matrix γ( , 0 ) and the diagonal density ϱ( r ) = γ( , 0 )∣ 0 = , this diagonal element already being known to satisfy a third-order linear homogeneous differential equation for d = 1 through 3. Some comments are finally made on the d -dimensional kinetic energy density, which is important for first-principles density functional theory in allowing one to bypass one-particle Schrödinger equations (the so-called Slater–Kohn–Sham equations).

Hilton, March, and Curtis have earlier focused on the utility of the effective potential U(r,β) in determining the Slater sum Z(r,β)=Z0(β)exp(−βU(r,β)). Here an explicit, though highly nonlinear, partial differential equation is derived for determining the effective potential U(x,β) in one-dimensional problems. Direct solution of this equation by power series expansion in β leads readily to Husimi’s results obtained from off-diagonal density matrix calculations. Perturbation theory in the potential is also developed, and thereby it is shown that an infinite subseries of the Husimi expansion is readily summed, and that a scaling property is exhibited.