Analysis of correlation in terms of exact local potentials: Applications to two-electron systems

For a two-electron system the Kohn-Sham potential of density-functional theory is equal to the effective local potential $V_{\mathrm{eff}(x_1}$) occurring in the one-electron Schrödinger equation that is satisfied by the square root of the exact many-electron density, ${\rho }^{1/2}$($x_1$). Making use of the theory of marginal and conditional probability amplitudes, it is shown that $V_{\mathrm{eff}(x_1}$) is the sum of three potentials, each of which has a clear physical interpretation and will be studied in detail. The correlation part of the Kohn-Sham potential in a two-electron system can then be obtained by subtraction of the Coulomb and exchange potential, and it is shown how we can express this correlation potential as the sum of three physically meaningful contributions. The connection between the Kohn-Sham potential in a many-electron system and $V_{\mathrm{eff}}$ is also discussed. Calculations of the various potentials from highly accurate configuration-interaction wave functions are presented for the helium atom and for the hydrogen molecule at various distances of the two hydrogen nuclei.