Scaling Invariance of Density Functionals

Original · EN

Based on the homogeneity (F[nλₘ]=λᵖ⁽ᵐ⁾F[n]) and invariance (F[nλₘ₀]=F[n]) properties of a functional of the electron density under uniform scaling of the coordinates in the density nλₘ(r)=λᵐ n(λr),(λ+, m), it is proven that homogeneity implies invariace and therefore all homogeneous scaling functionals have the representation F[n]=m-m₀/p(m) ∫ᵥδF[n]/δn(r)n(r)d³r. Also, the homogeneity (p(m)) and invariant (m₀) degrees of density functionals related to the Kohn-Sham theory are calculated. Besides, it is shown that the functional density and the electron density itself satisfy the general equation representing the local scaling invariance of a functional λd/dλ f([nλₘ₀],r,r') = ∑ᵢ₌₁³ d/d xᵢ [xᵢ f([nλₘ₀],r,r')] + ∑ⱼ₌₁³ d/d xⱼ' [xⱼ' f([nλₘ₀],r,r')]. The equation simplifies for cases where the functional density depends only on the density and/or its gradient, and general forms of the solutions are provided, in particular for the non-interacting kinetic energy density is shown to take the form tₛ(n,∇ n)= n(r)³ g[∂ₓ₁ n(r)n(r)², ∂ₓ₂ n(r)n(r)², ∂ₓ₃ n(r)n(r)²].