Computing vibrational eigenstates with tree tensor network states (TTNS)

Abstract

We present how to compute vibrational eigenstates with tree tensor network states (TTNS), the underlying ansatz behind the multilayer multi-configuration time-dependent Hartree (ML-MCTDH) method. The eigenstates are computed with an algorithm that is based on the density matrix renormalization group (DMRG). We apply this to compute the vibrational spectrum of acetonitrile (CH 3 CN) to high accuracy and compare TTNS with matrix product states (MPS), the ansatz behind the DMRG. The presented optimization scheme converges much faster than ML-MCTDH-based optimization. For this particular system, we found no major advantage of the more general TTNS over MPS. We highlight that for both TTNS and MPS, the usage of an adaptive bond dimension significantly reduces the amount of required parameters. We furthermore propose a procedure to find good trees.

Comparison of the multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalization group (DMRG) for ground state properties of linear rotor chains

“…An MPS can be viewed as a very special tree, where the only flexibility within an MPS-tree is the ordering of the DOFs. An ML-tree, on the other hand, is very flexible 36,56,57 and its topology can be adjusted to the system. Note however that DMRG is not limited to MPSs 36,[58][59][60] .…”

“…An ML-tree, on the other hand, is very flexible 36,56,57 and its topology can be adjusted to the system. Note however that DMRG is not limited to MPSs 36,[58][59][60] .…”

“…The recursive tree structure of the ML-MCTDH ansatz can be viewed as a hierarchical Tucker decomposition, a kind of tensor decomposition (see Ref. 36 and references therein).…”

“…DMRG has been successfully applied to systems with more than one spatial dimension 44 and to real-time evolution [45][46][47] , but it is most efficient for computing ground states of long, one-dimensional systems. Note that the matrix product ansatz used in the DMRG method can be considered as a special case of the ansatz used in the ML-MCTDH method 36 . Despite the methods' shared tensor-based nature, there has been very little overlap between the communities working on ML-MCTDH and DMRG.…”

“…Despite the methods' shared tensor-based nature, there has been very little overlap between the communities working on ML-MCTDH and DMRG. A more systematic comparison of the two methods is appealing and could lead to important improvements for the simulations of molecular quantum systems 36 . The aim of the present paper is twofold: to compare the viability of these two methods for the calculation of rotational ground states; and to promote cross-pollination of the theory and implementation between ML-MCTDH and DMRG in the context of quantum molecular dynamics.…”