Studies of the Statistical Exchange Approximation in First-Transition-Row Atoms and Ions:Mn+2Ion

A study of several statistical approximations to the Hartree-Fock (HF) one-electron effective exchange potential has been carried out through a series of restricted and spin-polarized Hartree-Fock-Slater (HFS) calculations on $3d$-transition-row atoms and ions. The results of this study for the ${\mathrm{Mn}}^{+2\mathrm{}}$ ion are reported in this paper and are compared with the corresponding HF results. Of the various approximations in which only the homogeneous part of the local exchange potential is kept, the simple $\mathrm{X}\alpha$ method is shown to give quite good results. This method amounts to multiplying the exchange potential $-6\mathrm{}{\left[\right(\frac{3}{4\pi }\left){\rho }_{\xi }\mathrm{}\right(r\left)\right]}^{\frac{1}{3}}$, where $\xi =\pm \frac{1}{2}$, by a constant $\alpha$ which is usually determined by the requirement that the $\mathrm{X}\alpha$ orbitals correspond to a minimum in the HF energy. It is shown that the energy-dependent exchange schemes proposed earlier break down totally in the spin-polarized case. We examine the importance of including the inhomogeneous correction terms via the method recently proposed by Herman et al. and we discuss the possible importance of these terms in spin-polarized energy-band calculations. We conclude that the $\mathrm{X}\alpha$ method is both accurate and simple enough for practical use in carrying out energy-band calculations for magnetic solids.