Using the Tensor-Train Approach to Solve the Ground-State Eigenproblem for Hydrogen Molecules

Abstract

We consider the Born--Oppenheimer approximation of the Schrodinger equation for two hydrogen atoms in the case of large separation distances. We show that the Feschbach--Schur perturbation method can be used to solve the problem for the difference between a given separation and infinite separation. This leads to a simplified problem which can be solved iteratively. We show that this iteration converges for sufficiently large separation distances and we solve the arising sequence of six-dimensional PDEs with a finite element method in combination with low-rank tensor techniques to make the computations tractable. In particular we show how the discretized problems can be represented and solved in the tensor-train format. Since the storage and computational complexity of this format scale linearly in the dimension, a very large number of grid points can be employed, which leads to accurate approximations of the ground-state energy and ground-state wavefunction. Various numerical experiments show the performa...