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The Journal of Chemical Physics

Implementation of a novel projector-splitting integrator for the multi-configurational time-dependent Hartree approach

Benedikt Kloss, Irène Burghardt, Christian Lubich

Published May 2, 2017• 42 citations
Publisher

Abstract

The variational equations of motion of the multi-configuration time-dependent Hartree (MCTDH) approach contain the inverse of reduced density matrices which are typically ill-conditioned and therefore lead to small stepsizes for numerical time integration. This problem is usually dealt with via regularization of the density matrices, which works well in many cases but still calls for systematic improvement schemes. Recently this problem, its implications and possible solutions have become the subject of increased interest. Notably, a projector splitting integrator for the MCTDH approach that does not require the inversion of reduced density matrices has been proposed [C. Lubich, Appl. Math. Res. Express 2015, 311]. Here, we present the first implementation of this integration scheme. Results for low-dimensional benchmark systems are presented, and the case of initially unoccupied single-particle functions is discussed.

DOI
10.1063/1.4982065

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The variational equations of motion of the multi-configuration time-dependent Hartree (MCTDH) approach contain the inverse of reduced density matrices which are typically ill-conditioned and therefore lead to small stepsizes for numerical time integration. This problem is usually dealt with via regularization of the density matrices, which works well in many cases but still calls for systematic improvement schemes. Recently this problem, its implications and possible solutions have become the subject of increased interest. Notably, a projector splitting integrator for the MCTDH approach that does not require the inversion of reduced density matrices has been proposed [C. Lubich, Appl. Math. Res. Express 2015, 311]. Here, we present the first implementation of this integration scheme. Results for low-dimensional benchmark systems are presented, and the case of initially unoccupied single-particle functions is discussed.
A numerical integrator is proposed for solving the multiconfiguration time-dependent Hartree (MCTDH) equations of motion, which are widely used in computations of molecular quantum dynamics. In contrast to existing integrators, the proposed algorithm does not require inverses of ill-conditioned density matrices and obviates the need for their regularization, allowing for large stepsizes also in the case of near-singular density matrices. The nonlinear MCTDH equations are split into a chain of linear differential equations that can all be efficiently solved by Lanczos approximations to the action of Hermitian-matrix exponentials, alternating with orthogonal matrix decompositions. The integrator is an extension to the Tucker tensor format of recently proposed projector-splitting integrators for the dynamical low-rank approximation by matrices and tensor trains (or matrix product states). The integrator is time-reversible and preserves both the norm and the total energy.
In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.
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.
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.
A multi-layer multi-configurational time-dependent Hartree (MCTDH) approach using a second quantization representation (SQR) based on optimized time-dependent orbitals is introduced. The approach combines elements of the multi-layer MCTDH-SQR approach of Wang and Thoss, which employs a preselected time-independent orbital basis, and the MCTDH for bosons and multi-configuration time-dependent Hartree-Fock approaches, which do not use multi-layering but employ time-dependent orbital bases. In contrast to existing MCTDH-type approaches, the results of the present approach for a given number of configurations are not invariant with respect to unitary transformations of the time-dependent orbital basis. Thus a natural orbital representation is chosen to achieve fast convergence with respect to the number of configurations employed. Equations of motion for the present ansatz, called (multi-layer) MCTDH in optimized second quantization representation, are derived. Furthermore, a scheme for the calculation of optimized unoccupied single-particle functions is given which can be used to avoid singularities in the equations of motion.
A matrix product state formulation of the multiconfiguration time-dependent Hartree (MPS-MCTDH) theory is presented. The Hilbert space that is spanned by the direct products of the phonon degree of freedoms, which is linearly parameterized in the MCTDH ansatz and thus results in an exponential increase in the computational cost, is parametrized by the MPS form. Equations of motion based on the Dirac-Frenkel time-dependent variational principle is derived by using the tangent space projection and the projector-splitting technique for the MPS, which have been recently developed. The mean-field operators, which appear in the equation of motion of the MCTDH single particle functions, are written in terms of the MPS form and efficiently evaluated by a sweep algorithm that is similar to the density-matrix renormalized group sweep. The efficiency and convergence of the MPS approximation to the MCTDH are demonstrated by quantum dynamics simulations of extended excitonic molecular systems.