On the thermodynamical analogy in spin-polarized density functional theory

TL;DR

This paper revisits the thermodynamical analogy in spin-polarized density functional theory, highlighting the implications of nonuniqueness of the magnetic field and defining the spin chemical potential as a derivative of energy.

Contribution

It demonstrates the existence of one-sided derivatives of the energy functional with respect to spin density and clarifies the role of the spin chemical potential in the theory.

Findings

Nonuniqueness of B(r) affects energy functional differentiability.

Spin chemical potential mu_s is identified as a derivative of energy.

Ground states can be obtained without fixing spin number N_s.

Abstract

The thermodynamical analogy of density functional theory, which is an organic part of the spin-independent version of the theory, is reconsidered for its spin-polarized generalization in view of the recently uncovered nonuniqueness of the external magnetic field B(r) corresponding to a given pair of density n(r) and spin density n_s(r). For ground states, the nonuniqueness of B(r) implies the nondifferentiability of the energy functional E[n,n_s] with respect to n_s(r). It is shown, on the other hand, that this nonuniqueness allows the existence of the one-sided derivatives of E[n,n_s] with respect to n_s(r). Although the N-electron ground state can always be obtained from the minimization of E[n,n_s] without any constraint on the spin number N_s, the Lagrange multiplier mu_s associated with the fixation of N_s does not vanish even for ground states. Rather, mu_s is identified as the left- or right-side derivative of the total energy with respect to N_s. This justifies the interpretation of mu_s as a (spin) chemical potential, which is the cornerstone of the thermodynamical analogy.