Violations of N-representability from spin-unrestricted orbitals in Moller-Plesset perturbation theory and related double-hybrid density functional theory

By examining the natural orbital occupation numbers, it was discovered that unrestricted Moller-Plesset perturbation theory (MP2) can violate N-representability by having natural occupation numbers greater than 2 or less than 0, even though the energy appears well behaved. Analytically, this problem stems from the fact that the MP2 effective one-particle density matrix is dependent on the inverse of the orbital hessian matrix (A'). When a molecular system goes through the point where spin-restricted orbitals become unstable to unrestriction, one of the eigenvalues of the A' matrix will become 0, making it a singular matrix, thereby causing problems in the relaxed one-particle density matrix and discontinuities in the first order properties such as nuclear forces and dipole moments. Since the new 'double-hybrid' density functionals also involve a second-order perturbation expression for correlation using Kohn-Sham orbitals, these functionals also exhibit similar issues near the unrestriction point. The unphysical orbital occupations can be eliminated and the first derivative discontinuities removed by optimizing orbitals with inclusion of the second order correlation energy. Connections to the second derivative discontinuities in Hartree-Fock and Kohn-Sham density functional theory are also discussed.