Reduction of derivatives and simple applications of the Legendre transformed density functional theory

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Abstract
A reduction of derivatives in the Legendre transformed density functional theory (DFT) of the single-component (atomic or fixed-nuclei molecular) systems is described. It is shown that a given derivative can be expressed by an identity in terms of a set of basic derivatives. The basic partial functional derivatives for the general case of N electrons in the external field v(r) are: α(r)=[∂ρ(r)/∂N]/ρ(r), β(r, r′)=−[∂ρ(r)/∂v(r′)]N/ρ(r), and γ(r, r′)=−[∂ρ(r)/∂v(r′)]μ/ρ(r); ρ is the density and μ denotes the chemical potential. For atoms, an additional alternative set of four integral derivatives, involving only global (referring to a system as a whole) parameters, is introduced: α̃=−[∂vne/∂N]Z/vne, β̃=−[∂vne /∂Z]N/vne, γ̃=[∂N/∂Z]μ/N, and χ̃=−[∂vne/∂Z]μ/vne; Z is the nuclear atomic number and vne is the electron-nuclear attraction energy per unit nuclear charge. The reduction procedure makes free use of the DFT ‘‘thermodynamic’’ diagram equations, Maxwell relations, and mathematical manipulations on functional derivatives. Several representative applications are indicated dealing with the system responses to slight changes in one of the system parameters subject to different constraints.