Exchange-only density-functional theory in an exactly soluble model

We show that the exact Kohn-Sham wave functions and exact Kohn-Sham exchange potential $v_x^{\mathrm{KS}}$ may be calculated for any spherically symmetric system in which the electron states are described by orbitals having only two different radial wave functions. In this exchange-only theory the total energy, electron density, and maximum single-particle energy eigenvalue are all identical to the corresponding Hartree-Fock ground-state results. We present results for an external harmonic-oscillator potential with two filled subshells corresponding to a total of eight (unpolarized) electrons or a total of four (spin-polarized) electrons for a wide range of spring constants. Comparisons are made between the results obtained from the exact KS theory and those obtained from employing the local-spin-density (LSD) and Slater (SLA) exchange approximation and from the exact Hartree-Fock results with respect to wave functions, densities, single-particle eigenvalues, exchange energy, and the calculation of Hartree-Fock expectation values for total energy and single-particle energies employing single-particle density-functional wave functions. Of particular interest is the result that $v_x^{\mathrm{KS}}$ and $v_x^{\mathrm{LSD}}$ are similar in shape, but not in magnitude, which results in LSD orbitals which closely approximate the exact KS orbitals even though the single-particle eigenvalues are significantly in error and are in fact less accurate than those given by the SLA.