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arXiv:1909.13831 · The Journal of Chemical Physics

Computing vibrational eigenstates with tree tensor network states (TTNS)

Henrik R. Larsson

Published Nov 25, 2019• 77 citations
arXivHTMLPDF

Abstract

We present how to compute vibrational eigenstates with tree tensor network states (TTNSs), the underlying ansatz behind the multilayer multiconfiguration 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 (CH3CN) to high accuracy and compare TTNSs with matrix product states (MPSs), 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.

DOI
10.1063/1.5130390

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We present how to compute vibrational eigenstates with tree tensor network states (TTNSs), the underlying ansatz behind the multilayer multiconfiguration 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 (CH3CN) to high accuracy and compare TTNSs with matrix product states (MPSs), 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.
We develop and employ general Tree Tensor Networks (TTNs) to compute the vibrational spectra for two model systems: a set of 64-dimensional coupled oscillators and acetonitrile. We explore various tree architectures, ranging from the simple linear structure of Matrix Product States (MPS), to trees where only the leaf nodes carry a physical leg -- as commonly seen in the underlying ansatz of the Multilayer Multiconfiguration Time-Dependent Hartree (ML-MCTDH) method -- and further to more general trees in which all nodes are allowed to possess a physical leg. In addition, we implement Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) methods and Inverse Iteration methods as eigensolvers. Benchmarking runtime and accuracy shows that all tested topologies can reach high accuracy. For acetonitrile, inverse-iteration refinement brings all 84 computed states below 1~cm−1^{-1}−1 error, while the fork-4 tree, a comb-like tree with four backbone nodes, provides the best overall balance between accuracy and cost. MPS remains computationally attractive, whereas more connected trees generally improve accuracy at fixed bond dimension. All numerical simulations were performed using PyTreeNet, a Python package designed for flexible tensor network computations.
Tree tensor network states (TTNSs) combined with the density matrix renormalization group (DMRG) are emerging as powerful tools for vibrational and vibronic structure simulations in molecules with strong coupling and fluxionality. In this Perspective, we discuss how TTNS methods enable accurate, full-dimensional computations of thousands of eigenstates for molecular systems ranging from quartic-force-field benchmarks to molecules with strong vibronic coupling and protonated water clusters as large as the 33-dimensional Eigen ion, H3_33​O+^++⋅\cdot⋅(H2_22​O)3_33​. We emphasize the close connection and interoperability between DMRG-based TTNS methods and the multilayer multiconfiguration time-dependent Hartree method (ML-MCTDH), which share the same underlying ansatz. We also highlight practical challenges of predictive simulations, including robust error estimation, convergence of observables such as infrared intensities, and optimization of tensor network tree structures. Finally, we outline recent advances toward direct targeting of excited states and discuss opportunities for broader applications in molecular spectroscopy and quantum dynamics.
High Resolution Image Download MS PowerPoint Slide Accurate vibrational spectra are essential for understanding how molecules behave, yet their computation remains challenging, and benchmark data to reliably compare different methods are sparse. Here, we present high-accuracy eigenstate computations for the six-atom, 12-dimensional acetonitrile molecule, a prototypical, strongly coupled anharmonic system. Using a density matrix renormalization group (DMRG) algorithm with a tree-tensor-network-state (TTNS) ansatz, a refinement using TTNSs as basis set, and reliable procedures to estimate energy errors, we compute up to 5,000 vibrational states with error estimates below 0.0007 cm –1 . Our analysis reveals that previous works underestimated the energy error by up to 2 orders of magnitude. Our data serve as a benchmark for future vibrational spectroscopy methods, and our new method offers a path toward similarly precise computations of large, complex molecular systems.
The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalization group (DMRG) are powerful workhorses applied mostly in different scientific fields. Although both methods are based on tensor network states, very different mathematical languages are used for describing them. This severely limits knowledge transfer and sometimes leads to re-inventions of ideas well known in the other field. Here, we review ML-MCTDH and DMRG theory using both MCTDH expressions and tensor network diagrams. We derive the ML-MCTDH equations of motions using diagrams and compare them with time-dependent and time-independent DMRG algorithms. We further review two selected recent advancements. The first advancement is related to optimizing unoccupied single-particle functions in MCTDH, which corresponds to subspace enrichment in the DMRG. The second advancement is related to finding optimal tree structures and on highlighting similarities and differences of tensor networks used in MCTDH and DMRG theories. We hope that this contribution will foster more fruitful cross-fertilization of ideas between ML-MCTDH and DMRG.
We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.
Abstract We present a new variational tree tensor network state (TTNS) ansatz, the three-legged tree tensor network state (T3NS). Physical tensors are interspersed with branching tensors. Physical tensors have one physical index and at most two virtual indices, as in the matrix product state (MPS) ansatz of the density matrix renormalization group (DMRG). Branching tensors have no physical index, but up to three virtual indices. In this way, advantages of DMRG, in particular a low computational cost and a simple implementation of symmetries, are combined with advantages of TTNS, namely incorporating more entanglement. Our code is capable of simulating quantum chemical Hamiltonians, and we present several proof-of-principle calculations on LiF, N2, and the bis(μ-oxo) and μ–η2:η2 peroxo isomers of [Cu2O2]2+.