Methods of Modern Differential Geometry in Quantum Chemistry: TD Theories on Grassmann and Hartree-Fock Manifolds

ABSTRACT: Hamiltonian and Schrödinger evolution equations on finitedimensional projective space are analyzed in detail. Hartree-Fock (HF) manifold is introduced as a submanifold of many electron projective space of states. Evolution equations, exact and linearized, on this manifold are studied. Comparison of matrices of linearized Schrödinger equations on many electron projective space and on the corresponding HF manifold reveals the appearance in the HF case a constraining matrix that includes matrix elements of many-electron Hamiltonian between HF state and double excited determinants. Character of dependence of transition energies on the matrix elements of constraining matrix is established by means of perturbation analysis. It is demonstrated that success of time-dependent HF theory in calculation of transition energies is mainly due to the wrong behavior of these energies as functions of matrix elements of constraining matrix in comparison with the exact energies.