A central problem in quantum chemistry is the computation of the lowest eigenvalue of the electronic Hamiltonian - an unbounded, self-adjoint operator acting on a Hilbert space of antisymmetric functions. The main difficulty in the resolution of this problem is the very high dimensionality of the eigenfunctions, being functions of 3N variables where N denotes the number of electrons in the system. A popular strategy to deal with this complexity is to use a low-rank, non-linear representation of the sought-after eigenfunction. Examples include the tensor-train-based DMRG algorithm and the coupled cluster method in which the ansatz is an exponential cluster operator acting on a judiciously-chosen reference state.
The goal of this talk is to present a new well-posedness and error analysis for the single-reference coupled cluster method. Under the minimal assumption that the sought-after eigenvalue is non-degenerate and the associated eigenfunction is not orthogonal to a chosen reference, we prove that the continuous coupled cluster equations are locally well-posed. Under some additional structural assumptions on the associated discretisation spaces, we prove that several classes of discrete coupled cluster equations are also locally well-posed, and we derive a priori and residual-based a posteriori error estimates.
This is joint work with Yvon Maday and Yipeng Wang (LJLL, Sorbonne Universite).