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USING THE TENSOR-TRAIN APPROACH TO SOLVE THE GROUND-STATE EIGENPROBLEM FOR HYDROGEN MOLECULES
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DOI：10.1137/15M102808X.png)
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Abstract
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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-Schurperturbation method can be used to solve the problem for the difference between a given separation and in finite 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 performance and accuracy of this method.
Keywords:
Schrodinger equation
Born-Oppenheimer approximation
low-rank tensor approximation
tensor-train format
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