Identification of the Chemical Potential of an Open System at 0 K with Functional Derivatives of the Integer-State Energy Density Functionals

TL;DR

This paper formulates open-system density functional theory at zero temperature, deriving a relationship between energy functionals, ensemble averages, and chemical potential, highlighting differences in functional derivatives for integer states.

Contribution

It introduces a new formulation linking integer-state energy derivatives to chemical potential, clarifying ensemble constraints and discontinuity behavior in open-system DFT.

Findings

Derived an expression for energy analogous to Gibbs function

Identified the functional derivative of energy as the chemical potential

Clarified the discontinuity behavior of exchange-correlation functionals

Abstract

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.