On the computation of excited states with MCSCF methods

We discuss the theoretical and practical problems arising when trying to compute excited states of nonrelativistic electrons in a molecular system, by multiconfiguration (MCSCF) methods. These nonlinear models approximate the linear Schrödinger theory and are a generalization of the well-known Hartree–Fock approach. Due to the MCSCF nonlinearity, a theoretical definition of what should be a MCSCF excited state is not clear at all, contrarily to the ground state case. We compare various definitions used in Quantum Chemistry. We in particular stress that some defects may lead to important computational problems, already observed in Quantum Chemistry (root flipping). We then present a definition of MCSCF excited states based on a solid mathematical ground and compare it with the most used methods. This new definition leads to a completely new algorithm for computing the first excited state, which was proposed and tested in a collaboration with Cancès and Galicher. Numerical results are provided for the simple case of two-electron systems, as an illustration of the possible issues which can arise as consequences of the nonlinearity of the MCSCF method.