Importance of Appropriately Regularizing the ML-MCTDH Equations of Motion

Finite temperature dynamics of the Holstein-Tavis-Cummings model

By employing the numerically accurate multiple Davydov Ansatz (mDA) formalism in combination with the thermo-field dynamics (TFD) representation of quantum mechanics, we systematically explore the influence of three parameters-temperature, photonic-mode detuning, and qubit-phonon coupling-on population dynamics and absorption spectra of the Holstein-Tavis-Cummings (HTC) model. It is found that elevated qubit-phonon couplings and/or temperatures have a similar impact on all dynamic observables: they suppress the amplitudes of Rabi oscillations in photonic populations as well as broaden the peaks and decrease their intensities in the absorption spectra. Our results unequivocally demonstrate that the HTC dynamics is very sensitive to the concerted variation of the three aforementioned parameters, and this finding can be used for fine-tuning polaritonic transport. The developed mDA-TFD methodology can be efficiently applied for modeling, predicting, optimizing, and comprehensively understanding dynamic and spectroscopic responses of actual molecular systems in microcavities.

Propagating multi-dimensional density operators using the multi-layer-ρ multi-configurational time-dependent Hartree method

Solving the Liouville-von-Neumann equation using a density operator provides a more complete picture of dynamical quantum phenomena than by using a wavepacket and solving the Schrödinger equation. As density operators are not restricted to the description of pure states, they can treat both thermalized and open systems. In practice, however, they are rarely used to study molecular systems as the computational resources required are even more prohibitive than those needed for wavepacket dynamics. In this paper, we demonstrate the potential utility of a scheme based on the powerful multi-layer multi-configurational time-dependent Hartree algorithm for propagating multi-dimensional density operators. Studies of two systems using this method are presented at a range of temperatures and including up to 13 degrees of freedom. The first case is single proton transfer in salicylaldimine, while the second is double proton transfer in porphycene. A comparison is also made with the approach of using stochastic wavepackets.

Optimal Mode Combination in the Multiconfiguration Time-Dependent Hartree Method through Multivariate Statistics: Factor Analysis and Hierarchical Clustering

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.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Time evolution of ML-MCTDH wavefunctions. II. Application of the projector splitting integrator

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.

Time evolution of ML-MCTDH wavefunctions. I. Gauge conditions, basis functions, and singularities

We derive a family of equations-of-motion (EOMs) for evolving multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH) wavefunctions that, unlike the standard ML-MCTDH EOMs, never require the evaluation of the inverse of singular matrices. All members of this family of EOMs make use of alternative static gauge conditions than those used for standard ML-MCTDH. These alternative conditions result in an expansion of the wavefunction in terms of a set of potentially arbitrary orthonormal functions, rather than in terms of a set of non-orthonormal and potentially linearly dependent functions, as is the case for standard ML-MCTDH. We show that the EOMs used in the projector splitting integrator (PSI) and the invariant EOM approaches are two special cases of this family obtained from different choices for the dynamic gauge condition, with the invariant EOMs making use of a choice that introduces potentially unbounded operators into the EOMs. As a consequence, all arguments for the existence of parallelizable integration schemes for the invariant EOMs can also be applied to the PSI EOMs.

Multi-configuration electron-nuclear dynamics: An open-shell approach

The multi-configuration electron-nuclear dynamics for open shell systems with a spin-unrestricted formalism is described. The mean fields are evaluated using second-order reduced density matrices for electronic and nuclear degrees of freedom. Applications to light-element diatomics including equilibrium geometries, electronic energies, dipole moments, and absorption spectra are presented. The von Neumann entropies for different spin states of a LiH molecule are compared.

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation

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.