Electron density-functional theory and x-ray structure factors

For an electronic system, assume that a single-determinant wave function Ψ is to be formed from N space orbitals which, in turn, are constructed from M real one-electron basis orbitals ${\chi }_1$,...,${\chi }_M$. It is shown that N(M-N) is the minimum number of x-ray diffraction data points that are necessary to fix both Ψ and the density n uniquely. Moreover, it is shown that N(M-N) is the minimum number of linearly independent products, ${\chi }_i$ ${\chi }_j$, that are necessary to fix Ψ uniquely from an n. Density-functional theory is invoked to put forth a prescription and formulas for approximating the exact ground-state energy, including correlation effects, and for extracting a meaningful Ψ from a ground-state n, even when Ψ is not unique. In this context, emphasis is placed upon the Kohn-Sham Ψ and its kinetic energy $T_s$ and upon the universal correlation $E_c$ and exchange $E_x$ energy-density functionals. A finite-basis procedure for the approximation of $T_s$[n] is presented. It is then shown that only $T_s$[n] and $E_c$[n] are needed to obtain the exact total ground-state energy from a ground-state n. $E_x$[n] need not be employed at all. For instance, in Coulomb systems at equilibrium E=-$T_s$[n]+$E_c$[n]-Fdr $v_c$(r)∇⋅rn(r), where $v_c$ is the correlation potential.