Electron density-functional theory and x-ray structure factors
For an electronic system, assume that a single-determinant wave function is to be formed from N space orbitals which, in turn, are constructed from M real one-electron basis orbitals 1,..., M. It is shown that N(M-N) is the minimum number of x-ray diffraction data points that are necessary to fix both and the density n uniquely. Moreover, it is shown that N(M-N) is the minimum number of linearly independent products, i j, that are necessary to fix 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 from a ground-state n, even when is not unique. In this context, emphasis is placed upon the Kohn-Sham and its kinetic energy Ts and upon the universal correlation Ec and exchange Ex energy-density functionals. A finite-basis procedure for the approximation of Ts[n] is presented. It is then shown that only Ts[n] and Ec[n] are needed to obtain the exact total ground-state energy from a ground-state n. Ex[n] need not be employed at all. For instance, in Coulomb systems at equilibrium E=-Ts[n]+Ec[n]-Fdr vc(r) rn(r), where vc is the correlation potential. Alternatively, energy formulas involving Ex[n], but not Ts[n], are also put forth. For instance, E=1/2U[n]+1/2Ex[n]+Ec[n]+1/2Fdr[vr]-vc(r0] rn(r) where U[n] is the classical repulsion energy. When the Ec functional is exact, and Ex is not exact, then the error in the latter equation is half the error of the familiar Ts[n]+Fdr v(r)n(r)+U[n]+Exc[n]. A few numerical examples are presented for atoms using nonlocal density functionals. © 1987 The American Physical Society.
Physical Review B
Levy, M., & Goldstein, J. (1987). Electron density-functional theory and x-ray structure factors. Physical Review B, 35 (15), 7887-7890. https://doi.org/10.1103/PhysRevB.35.7887