On the functional derivative of the kinetic energy density functional

The functional derivative of the kinetic energy Ts[ρ↑,ρ↓] of a noninteracting system of particles with density ρ=ρ↑+ρ↓ is evaluated in terms of the Kohn–Sham spin orbital densities ρiσ and Lagrange multipliers εiσ(σ=↑ or ↓). Of particular interest is the functional derivative of TΔ[ρ↑,ρ↓] =Ts[ρ↑,ρ↓]−Tw[ρ↑,ρ↓], where Tw[ρ↑,ρ↓] is the Weizsäcker kinetic energy functional: δTΔ[ρ↑,ρ↓]/δρσ =(1/8ρσ){𝒥Nσi=1[(∇ρiσ ⋅ ∇ρiσ)/ρiσ −(∇ρσ ⋅ ∇ρσ)/ρσ]} −(1/ρσ)(𝒥Nσi=1 εiσρiσ −μσρσ). This quantity is used to analyze the approximate kinetic energy density functional proposed by Acharya et al.