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arXiv:1309.5060 · The Journal of Chemical Physics

Multi-layer Potfit: An accurate potential representation for efficient high-dimensional quantum dynamics

Frank Otto

Published Jan 6, 2014• 80 citations
arXivHTMLPDF

Abstract

The multi-layer multi-configuration time-dependent Hartree method (ML-MCTDH) is a highly efficient scheme for studying the dynamics of high-dimensional quantum systems. Its use is greatly facilitated if the Hamiltonian of the system possesses a particular structure through which the multi-dimensional matrix elements can be computed efficiently. In the field of quantum molecular dynamics, the effective interaction between the atoms is often described by potential energy surfaces (PES), and it is necessary to fit such PES into the desired structure. For high-dimensional systems, the current approaches for this fitting process either lead to fits that are too large to be practical, or their accuracy is difficult to predict and control. This article introduces multi-layer Potfit (MLPF), a novel fitting scheme that results in a PES representation in the hierarchical tensor (HT) format. The scheme is based on the hierarchical singular value decomposition, which can yield a near-optimal fit and give strict bounds for the obtained accuracy. Here, a recursive scheme for using the HT-format PES within ML-MCTDH is derived, and theoretical estimates as well as a computational example show that the use of MLPF can reduce the numerical effort for ML-MCTDH by orders of magnitude, compared to the traditionally used POTFIT representation of the PES. Moreover, it is shown that MLPF is especially beneficial for high-accuracy PES representations, and it turns out that MLPF leads to computational savings already for comparatively small systems with just four modes.

DOI
10.1063/1.4856135

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The multi-layer multi-configuration time-dependent Hartree method (ML-MCTDH) is a highly efficient scheme for studying the dynamics of high-dimensional quantum systems. Its use is greatly facilitated if the Hamiltonian of the system possesses a particular structure through which the multi-dimensional matrix elements can be computed efficiently. In the field of quantum molecular dynamics, the effective interaction between the atoms is often described by potential energy surfaces (PES), and it is necessary to fit such PES into the desired structure. For high-dimensional systems, the current approaches for this fitting process either lead to fits that are too large to be practical, or their accuracy is difficult to predict and control. This article introduces multi-layer Potfit (MLPF), a novel fitting scheme that results in a PES representation in the hierarchical tensor (HT) format. The scheme is based on the hierarchical singular value decomposition, which can yield a near-optimal fit and give strict bounds for the obtained accuracy. Here, a recursive scheme for using the HT-format PES within ML-MCTDH is derived, and theoretical estimates as well as a computational example show that the use of MLPF can reduce the numerical effort for ML-MCTDH by orders of magnitude, compared to the traditionally used POTFIT representation of the PES. Moreover, it is shown that MLPF is especially beneficial for high-accuracy PES representations, and it turns out that MLPF leads to computational savings already for comparatively small systems with just four modes.
The multi-layer multi-configuration time-dependent Hartree method (ML-MCTDH) is a highly efficient scheme for studying the dynamics of high-dimensional quantum systems. Its use is greatly facilitated if the Hamiltonian of the system possesses a particular structure through which the multi-dimensional matrix elements can be computed efficiently. In the field of quantum molecular dynamics, the effective interaction between the atoms is often described by potential energy surfaces (PES), and it is necessary to fit such PES into the desired structure. For high-dimensional systems, the current approaches for this fitting process either lead to fits that are too large to be practical, or their accuracy is difficult to predict and control. This article introduces multi-layer Potfit (MLPF), a novel fitting scheme that results in a PES representation in the hierarchical tensor (HT) format. The scheme is based on the hierarchical singular value decomposition, which can yield a near-optimal fit and give strict bounds for the obtained accuracy. Here, a recursive scheme for using the HT-format PES within ML-MCTDH is derived, and theoretical estimates as well as a computational example show that the use of MLPF can reduce the numerical effort for ML-MCTDH by orders of magnitude, compared to the traditionally used Potfit representation of the PES. Moreover, it is shown that MLPF is especially beneficial for high-accuracy PES representations, and it turns out that MLPF leads to computational savings already for comparatively small systems with just four modes.
The multiconfiguration time-dependent Hartree (MCTDH) method and its multilayer extension (ML-MCTDH) are powerful algorithms for the efficient computation of nuclear quantum dynamics in high-dimensional systems. By providing time-dependent variational orbitals and an optimal choice of layered effective degrees of freedom, one is able to reduce the computational cost to an amenable number of configurations. However, choices related to selecting properly the mode grouping and tensor tree are strongly system dependent and, thus far, subjectively based on intuition and/or experience. Therefore, herein we detail a new protocol based on multivariate statistics─more specifically, factor analysis and hierarchical clustering─for a reliable and convenient guiding in the optimal design of such complex "system-of-systems" tensor-network decompositions. The advantages of employing the new algorithm and its applicability are tested on water and two floppy protonated water clusters with large amplitude motions.
The multi-layer multi-configurational time-dependent Hartree (MCTDH) approach is an efficient method to study quantum dynamics in real and imaginary time. The present work explores its potential to describe quantum fluids. The multi-layer MCTDH approach in second quantization representation is used to study lattice models beyond one dimension at finite temperatures. A scheme to map the lattice sites onto the MCTDH tree representation for multi-dimensional lattice models is proposed. A statistical sampling scheme previously used in MCTDH calculations is adapted to facilitate an efficient description of the thermal ensemble. As example, a two-dimensional hard-core Bose-Hubbard model is studied considering up to 64 × 64 lattice sites. The single particle function basis set size required to obtain converged results is found to not increase with the lattice size. The numerical results properly simulate the finite temperature Berezinskii-Kosterlitz-Thouless phase transition.
The multiconfiguration time-dependent Hartree (MCTDH) method and its generalization, the multilayer MCTDH (ML-MCTDH), result in equations of motion (EOMs) that are singular when there are virtual orbitals-the unoccupied single-particle functions-in the wave function expansion. For decades this singularity had been numerically removed by regularizing the reduced density matrix. In this Perspective we discuss our recent proposal to regularize the coefficient tensor instead, which has significant impact on both the efficiency and correctness of the EOMs in MCTDH and ML-MCTDH for challenging problems. We further demonstrate that when the system becomes large such that it is necessary to use ML-MCTDH with many layers, it is much more important to employ this new regularization scheme. We illustrate this point by studying a spin-boson model with a large bath that contains up to 100 000 modes. We show that even in the weak coupling regime the new regularization scheme is required to quickly rotate the virtual orbitals into the correct directions in Hilbert space. We argue that this situation can be common for applying a time-dependent tensor network approach to any large enough system.
We present the new Singular-Value Decomposition Multigrid POTFIT (SVD-MGPF), a grid-based tensor decomposition algorithm for large dimensional systems with particular focus on Quantum Dynamical problems. SVD-MGPF constitutes a generalisation of our previous (eigenvalue- decomposition based) EVD-MGPF method [1]. Both MGPF methods provide the Tucker decomposition of a target tensor (e.g. a potential energy surface). For such, a fine grid, the one needed to accurately represent the system, and a coarse one, a subset of the former, are defined. Tucker factor matrices are obtained from a series of POTFIT [2] decompositions carried out on grids which are fine for some degrees of freedom and coarse for the rest. The core tensor is obtained by overlapping these factor matrices with the values of the original tensor on the coarse grid. The difference between SVD-MGPF and EVD-MGPF (or POTFIT) lies in the fact that the Tucker factor matrices are obtained through singular value decompositions of one-particle potential density matrices instead of eigenvalue ones. This is shown to remove the numerical instabilities present in MGPF. Additionally, a black-box method for the choice of the coarse grid choice is proposed. The novelty of the latter is that it leads to a fully-relaxed non-product coarse grid. [1] D. Pel\'aez, H.-D. Meyer, J. Chem. Phys. {\bf 138} 014108, (2013). [2] A. Jaeckle, H.-D. Meyer, J. Chem. Phys. {\bf 104} 7974 (1996).
The multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) approach can suffer from numerical instabilities whenever the wavefunction is weakly entangled. These instabilities arise from singularities in the equations of motion (EOMs) and necessitate the use of regularization of the EOMs. The Projector Splitting Integrator (PSI) has previously been presented as an approach for evolving ML-MCTDH wavefunctions that is free of singularities. Here, we will discuss the implementation of the multi-layer PSI with a particular focus on how the steps required relate to those required to implement standard ML-MCTDH. We demonstrate the efficiency and stability of the PSI for large ML-MCTDH wavefunctions containing up to hundreds of thousands of nodes by considering a series of spin-boson models with up to 106 bath modes and find that for these problems, the PSI requires roughly 3–4 orders of magnitude fewer Hamiltonian evaluations and 2–3 orders of magnitude fewer Hamiltonian applications than standard ML-MCTDH and 2–3/1–2 orders of magnitude fewer evaluations/applications than approaches that use improved regularization schemes. Finally, we consider a series of significantly more challenging multi-spin-boson models that require much larger numbers of single-particle functions with wavefunctions containing up to ∼1.3×109 parameters to obtain accurate dynamics.