Thermodynamic extension of density-functional theory. I. Basic Massieu function, its Legendre and Massieu-Planck transforms for equilibrium state in terms of density matrix

TL;DR

This paper develops a thermodynamic framework for many-electron systems using density matrices, defining characteristic functions and their transforms to extend density functional theory to finite and zero temperatures.

Contribution

It introduces a comprehensive set of thermodynamic functions and transforms based on the maximum entropy principle, facilitating extensions of density functional theory.

Findings

Defined the basic Massieu function for open systems

Derived convexity and concavity properties of the functions

Established relations between characteristic functions and traditional thermodynamic potentials

Abstract

A general formulation of the equilibrium state of a many-electron system in terms of a (mixed-state, ensemble) density matrix operator in the Fock space, based on the maximum entropy principle, is introduced. Various characteristic functions/functionals are defined and investigated: the basic Massieu function for fully open thermodynamic system (ensemble), the effective action function for the fully closed (isolated) system, and a series of Legendre transforms for partially open/closed ones - the Massieu functions. Convexity and/or concavity properties of these functions are determined, their first and second derivatives with respect to all arguments are obtained. Other characteristic functions - the Gibbs-Helmholtz functions - are obtained from previous ones as their Massieu-Planck transforms, i.e. by specific transformation of arguments (which involves the temperature) and by applying the temperature with the minus as a prefactor. Such functions are closer to traditional (Gibbs, Helmholtz) thermodynamic potentials. However, the first and second derivatives of these functions represent more complicated expressions than derivatives of the Massieu functions. All introduced functions are suitable for application to the extensions of the density functional theory, both at finite and zero temperature.