Shape wave in density functional theory

The shape wave u ( r ) is defined in such a way that the square of its absolute value gives the shape of the electron charge density for a nonrelativistic many‐electron system: | u ( r )| 2 = ρ( r )/∫ρ( r ) d r , where ρ( r ) is the total electron charge density. The u ( r ) is treated as the probability amplitude for the shape of the electron charge density in the real three‐dimensional space. The secular equation of the u ( r ) is given as L ( r ) u ( r ) = λ u ( r ), where L ( r ) and λ are a real Hermitean operator and the eigenvalue, respectively. This equation has the form of the spinless one‐particle Schrödinger equation. The potential is local and is a functional of ρ( r ), and so we need a self‐consistent field procedure to solve it. For equilibrium state shape wave u = u eq , which is temperature‐dependent, and for stationary state shape wave u = u 0 , respectively, L eq ( r ) u eq ( r ) = λ eq u eq ( r ) and L 0 ( r ) u 0 ( r ) = λ 0 u 0 ( r ). For one‐electron systems, the latter equation reduces to the Schrödinger equation for a stationary state, with the usual eigenvalue λ 0 = E 0 . In general, λ is identified with the Gibbs chemical potential μ G . Boundary conditions such as the cusp condition at a nucleus and exponential decay at an infinitely large distance from the nucleus are easily implemented. In case L ( r ) allows complex eigenfunctions, u ( r ) = A ( r ) e iS ( r ) , the hydrodynamical potential appears in the secular equation for the real amplitude A ( r ). The real phase S ( r ) satisfies the equation of continuity. By introducing «apparatus» operators [16], the excited state shape wave is also obtained. Comparisons are made with a recent discussion of Levy, Perdew, and Sahni [3].