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Advances in quantum chemistry

Auxiliary Field Representation of Fermion Kinetic Energy Density Functional

Jerome K. Percus

Published Jan 1, 1998• 1 citation
Publisher

Abstract

The kinetic energy, or thermodynamic potential, of a non-interacting Fermion system, as a functional of the non-uniform density pattern evoked by an external potential, serves as a crucial tool in the analysis of interacting system ground states. Examples from classical lattice gas and continuum fluid theory are cited to show that analytic tractability can be dramatically enhanced by the introduction of auxiliary density fields with respect to which a suitable thermodynamic potential is stationary. This strategy is developed for the prototype of one-dimensional spinless Fermions, resulting in an exact highly overcomplete representation. A semi-classical variational ansatz recovers the familiar Thomas-Fermi plus renormalized Weiszacker expression, as well as leading corrections, and a simple extension creates a simple modification of the former. The corresponding 3-dimensional format is set up, and evaluated by extrapolation from the 1-dimensional case.

DOI
10.1016/s0065-3276(08)60433-5

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The kinetic energy, or thermodynamic potential, of a non-interacting Fermion system, as a functional of the non-uniform density pattern evoked by an external potential, serves as a crucial tool in the analysis of interacting system ground states. Examples from classical lattice gas and continuum fluid theory are cited to show that analytic tractability can be dramatically enhanced by the introduction of auxiliary density fields with respect to which a suitable thermodynamic potential is stationary. This strategy is developed for the prototype of one-dimensional spinless Fermions, resulting in an exact highly overcomplete representation. A semi-classical variational ansatz recovers the familiar Thomas-Fermi plus renormalized Weiszacker expression, as well as leading corrections, and a simple extension creates a simple modification of the former. The corresponding 3-dimensional format is set up, and evaluated by extrapolation from the 1-dimensional case.
Although formally exact expressions can be written for thermodynamic potentials of many-particle systems in external fields, Legendre transformation to particle density as the controlling field has many virtues, insensitivity in the face of singular states being notable. It is also true that nonuniform systems, classical or quantum, which lead to simple closed-form solutions, tend to do so principally in the density functional form. However, as gleaned from the heavily investigated cases of classical lattice gas networks, this may be achieved only if additional auxiliary densities are appended, with respect to which the associated thermodynamic potential is stationary. Here, we successfully apply the overcompleteness strategy to the prototypical case of free fermion ground states in one-dimensional space, extrapolate to three-dimensional space, and apply a basic one-functional variational ansatz. The qualitative properties of the resulting free energy and potential-density profile relations are in accord with those that any valid formulation must satisfy. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 573–580, 1998
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.
We consider noninteracting Fermion system in an external field, confined to one-dimensional space. Our main objective is to take advantage of low dimension to find explicitly potential-density profile as well as kinetic energy functional in the ground state. This is accomplished by deriving the complete x-space linear response and a simple recursion representation of the gradient expansion for the kinetic energy. The results can serve as a potential test for higher dimensions, e.g., in exploring convergence properties of the gradient expansion in various regions of the energy spectrum.
For first-principles density functional theory of a many fermion system, the determination of the kinetic energy functional is important. We consider NNN independent fermions with spatially varying effective mass in two dimensions, we derive the corresponding kinetic energy density using the $\ensuremath{\hbar}$ semiclassical approach. Our result reduces, as expected, to the one obtained in the literature for a constant effective mass. We examine the analytical expressions of the position dependent effective mass terms in the kinetic energy density functional with respect to the dimensionality d=1,2,3d=1,2,3d=1,2,3 of the space.
The average-density approximation is used to construct a nonlocal kinetic energy functional for an inhomogeneous two-dimensional Fermi gas. This functional is then used to formulate a Thomas-Fermi-von Weizs\"acker-like theory for the description of the ground-state properties of the system. The quality of the kinetic energy functional is tested by performing a fully self-consistent calculation for an ideal, harmonically confined, two-dimensional system. Good agreement with exact results are found, with the number and kinetic energy densities exhibiting oscillatory structure associated with the nonlocality of the energy functional. Most importantly, this functional shows a marked improvement over the two-dimensional Thomas-Fermi von Weizs\"acker theory, particularly in the vicinity of the classically forbidden region.
A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.