Spin contamination for Hartree-Fock, optimized effective potential, and density functional approximations

In an earlier paper [S. Thanos and A. K. Theophilou J. Chem. Phys. 124, 204109 (2006)], we found an explicit formula for the expansion of a Slater determinant |Phi(M)> in terms of eigenstates of S(2). In this paper, we use the same formula to determine the spin contamination S(con) of the unrestricted single determinant approximations, i.e., Hartree-Fock, optimized effective potential, and density functional theory. We derived an expression which gives S(con) in terms of the overlap of the spatial parts of the spin up and spin down "corresponding" orbitals. It was found that S(con) does not depend on M, the eigenvalue of S(z), at least for the lower order approximations, i.e., when || is large. In this case, the predominant coefficient of the expansion assumes its maximum value when S=M. However, for the class of solutions that || is small, the spin L of the largest coefficient increases with the number of unpaired electrons. We also derived the explicit form of the expansion states.