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Journal of Physics A Mathematical and General

Dirac density matrix and the Legendre transform of the kinetic energy generated by one-dimensional model potentials

I. A. Howard, Norman Henry March

Published Dec 21, 2005• 0 citations
Publisher

Abstract

In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix y. Here, by first taking solvable one-dimensional examples, and in particular the sech 2 (x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.

DOI
10.1088/0305-4470/39/3/009

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In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix y. Here, by first taking solvable one-dimensional examples, and in particular the sech 2 (x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.
Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential V(r)V(\mathbf{r})V(r), Stoddart and March (SM) generated a corresponding series for the kinetic energy density. While the general term of the SM series is known, a summation has not been achieved to date. A contribution to solve this problem is made here by exhibiting an explicit form of the above kinetic energy density for Fermions filling an arbitrary number of closed shells, when the confinement is harmonic. This example is of considerable current interest because of ongoing experiments on ultracold atomic gases of Fermions.
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.
The March-Suhai (MS) partial differential equation for the Dirac density matrix ${\ensuremath{\gamma}}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r},{\stackrel{\ensuremath{\rightharpoonup}}{r}}^{\ensuremath{'}})$, proved for one- and two-level occupancies, involves both the ground-state density $n(\stackrel{\ensuremath{\rightharpoonup}}{r})$, with its low-order derivatives, and the positive definite kinetic energy density ${t}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r})$. Here, we examine the relation between the equation of motion for ${\ensuremath{\gamma}}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r},{\stackrel{\ensuremath{\rightharpoonup}}{r}}^{\ensuremath{'}})$, with input now being the one-body potential of density-functional theory, and the MS equation. The important link is the differential virial theorem, which can be used to remove ${t}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r})$ from the MS differential equation. For multiple occupancy, the Pauli potential enters in an important manner. In one dimension, however, the appearance of the Pauli potential can be avoided, obtaining a necessary condition for ${\ensuremath{\gamma}}_{s}(x,{x}^{\ensuremath{'}})$ to satisfy for arbitrary level occupancy, in the form of a MS-type differential equation.
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.
The differential virial theorem of March and Young for N fermions moving in a common one-dimensional potential energy V(x) is here combined with the Euler equation of density functional theory expressing the constancy of the chemical potential throughout the entire inhomogeneous particle density. The functional derivative of the single-particle kinetic energy is thereby expressed directly in terms of the kinetic energy density; a line integral being involved in establishing the connection. This result is then used to establish a formula measuring departures from simple homogeneity of the kinetic energy functional: a matter of current interest in density functional theory. Finally, the general theory of the functional derivative of the single-particle kinetic energy with respect to the particle density is exemplified for the case of harmonic confinement of fermions in one dimension.