The Energy-Density Functional of an Electron Gas in Locally Linear Approximation of the One-Body Potential

For a system of independent electrons moving in a common one-body potential V (r) an integral representation of Dirac's density matrix is evaluated in the approximation that V(r) at the point r is replaced by a linear potential with a gradient equal to the gradient of V at r. The particle density ᵨ, ∇ᵨ and the kinetic-energy density ε k are derived from the density matrix. After eliminating the potential and its gradient a parametric representation for ε k in terms of ᵨ and y = |∇ᵨ | ½ ᵨ -⅔ is obtained. Explicit analytical expressions are given in the limits y → 0 and y → ∞ and compared with the inhomogeneity corrections of Kirzhnits and v. Weizsäcker.