Legendre transforms and Maxwell relations in density functional theory

Abstract

The Hohenberg–Kohn principle of the minimum ground-state energy density functional E[N,v]=Ev[ρ]≡F[ρ]+ ∫ v(1)ρ(1)dτ1, min{Ev[ρ′]−μN[ρ′]}, is reformulated in forms appropriate to the alternative Legendre transformed representations of the system energy. The quantities ρ and N[ρ] denote the one-electron density and associated number of electrons, while v and μ stand for the external and chemical potentials, respectively. The following natural Legendre transformed functionals are introduced: Q[μ,v]=E−(∂E/∂N)cN=Q[ρ]≡F[ρ]− ∫ [δF/δρ(1)]ρ(1)dτ1,F[N,ρ]=E− ∫ [∂E/∂v(1)]cv(1)dτ1=F[ρ], R(μ,ρ)=E− ∫ [∂E/∂v(1)]cv(1)dτ1−(∂E/∂N)cN=R[ρ], where c implies all variables kept constant except for the one in the derivative. The Maxwell relations for these representations are derived and their physical implications briefly discussed. Corresponding ’’thermodynamic’’ mnemonic square diagrams are introduced to generate the differential expressions and selected Maxwell relations. A typical Maxwell relation is [∂ρ(1)/∂v(2)]N,v≠(2)=[∂ρ(2)/∂v(1)]N,v≠(1). It is shown that, as in classical thermodynamics, the extremum principle for each of these basic functionals of state L[g,l](L=E,Q,F,R; g=N,μ; l=v,ρ) can be formulated as follows: The equilibrium value of any unconstructed internal variable of a system specified by its global g and local l parameters, minimizes the L[g,l] Legendre transform of the system energy, at constant g and ℓ. It is shown that the F[N,ρ] minimum principle is equivalent to the Levy variational principle F[ρ]=min <ψρ‖T̂+V̂ee‖Ψρ≳, which may be interpreted as a process determining the system chemical potential.