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

摘要

Hamiltonian and Schrodinger evolution equations on finite-dimensionalprojective space are analyzed in detail. Hartree-Fock (HF) manifold isintroduced 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 Schrodinger equations on many electronprojective space and on the corresponding HF manifold reveals the appearance inthe HF case a constraining matrix that involves matrix elements ofmany-electron Hamiltonian between HF state and double excited determinants.Character of dependence of transition energies on the matrix elements ofconstraining matrix is established by means of perturbation analysis. It isdemonstrated that success of time-dependent HF theory in calculation oftransition energies is mainly due to the wrong behavior of these energies asfunctions of matrix elements of constraining matrix as compared with the exacttransition energies