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Physical Review A

Differential equation for the Dirac single-particle first-order density matrix in terms of the ground-state electron density

Norman Henry March, Tamás Gál, I. A. Howard

Published Jun 22, 2010• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.81.064503

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The March-Suhai (MS) partial differential equation for the Dirac density matrix \ensuremathγs(r\ensuremath⇀,r\ensuremath⇀\ensuremath′){\ensuremath{\gamma}}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r},{\stackrel{\ensuremath{\rightharpoonup}}{r}}^{\ensuremath{'}})\ensuremathγs​(r\ensuremath⇀,r\ensuremath⇀\ensuremath′), proved for one- and two-level occupancies, involves both the ground-state density n(r\ensuremath⇀)n(\stackrel{\ensuremath{\rightharpoonup}}{r})n(r\ensuremath⇀), with its low-order derivatives, and the positive definite kinetic energy density ts(r\ensuremath⇀){t}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r})ts​(r\ensuremath⇀). Here, we examine the relation between the equation of motion for \ensuremathγs(r\ensuremath⇀,r\ensuremath⇀\ensuremath′){\ensuremath{\gamma}}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r},{\stackrel{\ensuremath{\rightharpoonup}}{r}}^{\ensuremath{'}})\ensuremathγs​(r\ensuremath⇀,r\ensuremath⇀\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 ts(r\ensuremath⇀){t}_{s}(\stackrel{\ensuremath{\rightharpoonup}}{r})ts​(r\ensuremath⇀) 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 \ensuremathγs(x,x\ensuremath′){\ensuremath{\gamma}}_{s}(x,{x}^{\ensuremath{'}})\ensuremathγs​(x,x\ensuremath′) to satisfy for arbitrary level occupancy, in the form of a MS-type differential equation.
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).
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.
From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ts{t}_{s}ts​[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ts{t}_{s}ts​[\ensuremath{\rho}] is explicitly obtained. Again ts{t}_{s}ts​[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.