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

Alexander Veit, L. Ridgway Scott
2017 SIAM Journal on Scientific Computing  
We consider the Born-Oppenheimer approximation of the Schrödinger 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 dierence between a given separation and innite separation. This leads to a simplied problem which can be solved iteratively. We show that this iteration converges for suciently large separation distances and we solve the arising sequence of six-dimensional PDEs
more » ... h 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 performance and accuracy of this method. using nite dierence or nite element methods to solve the eigenvalue problem for the Schrödinger partial dierential operator are prohibitive due to the high dimension. It is well-known that the ground state of the hydrogen molecule is a singlet spin state [19] , and this allows a reduction to solving an eigenvalue problem for the Schrödinger operator without any symmetry or anti-symmetry considerations for the spatial variables [7] . In other problems, there would be a need to include certain algebraic conditions related to the Pauli exclusion principle. The state-of-the-art methods for solving the Schrödinger equation use Galerkin methods with Gaussian functions as the basis for the approximating spaces. Gaussians have special algebraic properties that make the resulting approximation algorithmically more ecient. However, the eigenfunctions of the Schrödinger operator decay like e −|x| and not e −|x| 2 , so the Gaussian approximation performs best when atoms are close. When atoms are far apart, we can imagine using a perturbation argument [7] to solve for the dierence between a given separation and innite separation. We will explore the Feshbach-Schur perturbation method [15] as an algorithm to solve such problems in a Hilbert-space setting. This approach simplies the problem substantially for large separation distances, but it does not reduce the dimensionality of the problem. Thus we also use low-rank tensor methods to make the computations tractable. In particular we will solve the simplied problems using a Finite Element Method with piecewise linear polynomials by representing the Galerkin matrix and the right-hand side in a low-parametric tensor format and by solving the arising linear system within this format. In this way also very ne six-dimensional grids and therefore a very large number of degrees of freedom are feasible and can be used to solve the simplied problems accurately. Numerical methods based on low-rank tensor representations gained a lot of attention in recent years and were applied to a variety of problems including electronic structure calculations, the Fokker-Planck equation, stochastic and parametric PDEs and high-dimensional Schrödinger equations. We refer to [14, 25, 17] for an overview of the available literature on tensor methods. In this paper we use the Tensor-Train format (TT-format) introduced in [28, 27] to represent the Galerkin matrix and the right-hand side of our problem. This format utilizes an SVD-based compression of a given tensor and the required memory to store the approximation scales linearly in the dimension. Furthermore operations in the TT-format (e.g. matrix-vector multiplication, addition) can also be performed with linear dependence on the dimension of the tensor. Advanced algorithms like cross approximation schemes ([29]) and solver for linear systems in the TT-format ([11, 12]) are available and implemented in the TT-Toolbox (MATLAB) by I. Oseledets that will be used for the numerical experiments in Section 5. There we test the performance and accuracy of our algorithm and compute the ground-state energy and ground-state wavefunction for dierent separations distances of the hydrogen nuclei. We will use atomic units so that = 1, e = 1, m e = 1 and 4π 0 = 1, where is the reduced Planck constant, m e the mass of the electron, e the elementary charge, and 0 the dielectric permittivity of the vacuum. In this system of units, the length unit is the Bohr (about 0.529 Ångstroms) and the energy unit is the Hartree (about 4.36 × 10 −18 Joules). 2 Interaction between two hydrogen atoms 2.1 Problem statement In the following we are interested in the ecient numerical solution of the Born-Oppenheimer approximation of the Schrödinger equation for a pair of hydrogen atoms separated by a distance 2R. We assume that the nucleus of the rst hydrogen atom is positioned at −Re and the nucleus of the second atom is xed at Re, where e ( e = 1) is the unit vector pointing from one hydrogen nucleus to the other. Using atomic units throughout
doi:10.1137/15m102808x fatcat:erg2dqtv6rfonhiywzgzmcfqpu