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International Journal of Quantum Chemistry

Chemical potential for harmonically interacting particles in a harmonic potential

Philip Phillips, Ernest R. Davidson

Published Jan 1, 1983• 20 citations
Publisher

Abstract

Abstract The chemical potential μ, as it appears in density functional theory, is examined extensively for harmonically interacting spin‐½ fermions in three dimensions. For this system the energy and chemical potential are discontinuous functions of the particle number if the most straightforward equation is used to define the energy for a noninteger particle number.

DOI
10.1002/qua.560230118

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Abstract The chemical potential μ, as it appears in density functional theory, is examined extensively for harmonically interacting spin‐½ fermions in three dimensions. For this system the energy and chemical potential are discontinuous functions of the particle number if the most straightforward equation is used to define the energy for a noninteger particle number.
The thermodynamical properties are calculated for a three-dimensional model of NNN harmonically interacting spin-polarized fermions in a parabolic potential well. The obtained dependences of the chemical potential and of the internal energy on the complete range of the temperature and of the number of particles turn out to obey a scaling law, similar to the scaling from the continuum approximation for the density of states. The calculational technique is based on our path-integral approach of symmetrized density matrices for identical particles in a parabolic confining well.
Abstract Determination of the kinetic energy functional is one of the central problems in the density functional theory of a many‐fermion system. We consider several many‐fermion systems to be characterized by simple confining potentials: the harmonic oscillator potential in one and three dimensions, the Coulomb potential, and the potential V(x) = −Uosech2(x/L). We discuss for each of these cases the differential equation that determines the fermion density in the given potential, and the corresponding kinetic energy density. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
The thermodynamical properties are calculated for a three-dimensional model of NNN harmonically interacting spin-polarized fermions in a parabolic potential well. The obtained dependences of the chemical potential and of the internal energy on the complete range of the temperature and of the number of particles turn out to obey a scaling law, similar to the scaling from the continuum approximation for the density of states. The calculation technique is based on our path integral approach of symmetrized density matrices for identical particles in a parabolic confining well.
Open-system density functional theory may be formulated in terms of ensemble averages arising from interaction with a bath. The system is allowed to exchange particles with the bath and the states in the ensemble average are those corresponding to integer numbers of particles. The weights in the ensemble average are typically equated with time-averaged values of the occupation numbers of the various states comprising the open system. As a result, there are two constraints on the occupation numbers: (1) Their sum must be unity so that the ensemble average is a probability function and (2) The sum of the occupation numbers times the number of particles for the associated state must equal the time-averaged number of particles. By solving explicitly the first constraint we arrive at an expression for the energy having a form structurally equivalent to a Gibbs thermodynamic function. From this form follows both chemical potential equalization and the identification of the functional derivative of the energy for a particular state with respect to its integer-state density as the chemical potential. We highlight a distinction between functional derivatives of the time-averaged and the integer-state energies with respect to integer-state densities. From this we can restate the jump discontinuity behavior of the exchange-correlation functional in terms of integer-state behavior.