PHYSICAL REVIE% 8 VOLUME 26, NUMBER 4 15 AUGUST 1982 Nonlinear ionic pseudopotentials in spin-density-functional calculations Steven G. Louie Department of Physics, University of California, Berkeley, California 94720 Sverre Froyen and Marvin L. Cohen Department of Physics, University of California, Berkeley, California 94720 and Materials and Molecul'ar Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720 (Received 28 December 1981) A new method for generating and using first-principles pseudopotentials is developed to treat explicitly the nonlinear exchange and correlation interaction between the core and the valence charge densities. Compared to existing potentials, the new scheme leads to significant improvement in the transferability of the potential. In particular, the spin- polarized configurations are well described with a single potential. The need for separate spin-up and spin-down ionic pesudopotentials is, thus, eliminated. The method can easily be implemented with minimal increase in computational effort. Results for both atoms and solids are demonstrated. I. INTRODUCTION In the past several years, the pseudopotential ap- proach coupled with the density-functional scheme has had tremendous success in describing the elec- tronic and structural properties of rionmagnetic systems. ' In this paper, we propose a method which makes it possible to extend these calcula- tions to magnetic systems. With a single spin- independent ionic potential, the method incor- porates the local-spin-density approximation ' to the exchange and correlation energy into the pseu- dopotential scheme. Problems such as those con- cerning properties of magnetic materials, spin- density waves, magnetic effects on surfaces, local- ized impurity states in defects, etc. , can now be treated with the same ease and accuracy as in the nonmagnetic case. In addition the accuracy is in many cases improved even in the paramagnetic limit. In an earlier scheme for introducing magnetic effects into pseudopotential calculations Zunger proposed constructing separate ionic potentials for the spin-up and spin-down electrons. In this spin- dependent pseudopotential approach, the ionic pseudopotential in the solid depends on the spin density of the valence electrons, and it is obtained by interpolation between the ionic potentia1 for the paramagnetic atom and that of the fully spin- polarized atom. We shall show that it is unneces- sary and, in fact, often undesirable to employ these spin-dependent ionic pseudopotentials. In the density-functional formalism, ' the total energy of the ground state is given as a functional of the total electron charge density, E. t =TIpI+E-I pI+E-Ipj+E-I pI where the various terms represent the kinetic ener- gy, the electrostatic interaction of the ions with the electrons, the electrons with the electrons, and the exchange and correlation energy, respectively. The exchange and correlation energy is usually approxi- mated by some local (nonlinear) function of the charge density, and the kinetic energy is found from the gradient of the now obtainable single- particle wave functions. Thus, E;, „= I V;, „(r)p(r)d r, ~ I p(r")p(r) d3, d3 ir' — ri E„, = J e„, [p(r)]p(r)d r . (4) The charge density, in the pseudopotential for- malism, is divided into core and valence contribu- tions, and the energy of the core is assumed to be constant and subtracted out. Furthermore, the core contribution is often completely neglected, and the total energy is given by the above expressions with the total charge density replaced by a (pseudo) valence charge density, and V;, „replaced by the pseudopotential. All interaction between the core and valence electrons is thus transferred to the pseudopotential. This implies a linearization of Q~1982 The American Physical Society