Scaling Invariance of Density Functionals

2014/04/20 by Lázaro Calderín, Calderín, Lázaro
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1404.5073

arxiv created 2014/10/15 · arxiv updated 2014/10/16

Abstract

Based on the homogeneity (F[n_λm]=λ^p(m)F[n]) and invariance (F[n_λm0]=F[n]) properties of a functional of the electron density under uniform scaling of the coordinates in the density n_λm(r)=λ^m n(λr), (λ∈ℝ⁺, m∈ℝ), it is proven that homogeneity implies invariace and therefore all homogeneous scaling functionals have the representation F[n]=(m-m₀)/(p(m)) ∫_V (δF[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 λ(d)/(dλ) f([n_λm0],r,r') = ∑_i=1³ (d)/(d x_i) [ x_i f([n_λm0],r,r') ] + ∑_j=1³ (d)/(d x_j') [ x_j' f([n_λm0],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 t_s(n,∇ n)= n(r)³ g[ \frac∂_x1 n(r)n(r)², \frac∂_x2 n(r)n(r)², \frac∂_x3 n(r)n(r)²] .

Related

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• Variational derivation of density functional theory
• The ground-state energy and external potential as functionals of the electron density and their derivatives
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• Potential Scaling in Density Functional Theory