Numerical Implementation and Test of the Modified Variational Multiconfigurational Gaussian Method for High-Dimensional Quantum Dynamics

Full wave function cloning for improving convergence of the multiconfigurational Ehrenfest method: Tests in the zero-temperature spin-boson model regime

In this paper, we report a new algorithm for creating an adaptive basis set in the Multiconfigurational Ehrenfest (MCE) method, which is termed Full Cloning (FC), and test it together with the existing Multiple Cloning (MC) using the spin-boson model at zero-temperature as a benchmark. The zero-temperature spin-boson regime is a common hurdle in the development of methods that seek to model quantum dynamics. Two versions of MCE exist. We demonstrate that MC is vital for the convergence of MCE version 2 (MCEv2). The first version (MCEv1) converges much better than MCEv2, but FC improves its convergence in a few cases where it is hard to converge it with the help of a reasonably small size of the basis set.

Variational Approach for Linearly Dependent Moving Bases in Quantum Dynamics: Application to Gaussian Functions

In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schrödinger equation. The method is based on (i) the definition of a linearly independent working space and (ii) a variational construction of the propagator over finite time steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are carried out on a quartic double-well potential with Gaussian basis functions whose centers evolve according to classical equations of motion. We show that the resulting dynamics converges to the exact one and is unitary by construction.

Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods

We benchmark a set of quantum-chemistry methods, including multitrajectory Ehrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfest dynamics, against exact quantum-many-body techniques by studying real-time dynamics in the Holstein model. This is a paradigmatic model in condensed matter theory incorporating a local coupling of electrons to Einstein phonons. For the two-site and three-site Holstein model, we discuss the exact and quantum-chemistry methods in terms of the Born-Huang formalism, covering different initial states, which either start on a single Born-Oppenheimer surface, or with the electron localized to a single site. For extended systems with up to 51 sites, we address both the physics of single Holstein polarons and the dynamics of charge-density waves at finite electron densities. For these extended systems, we compare the quantum-chemistry methods to exact dynamics obtained from time-dependent density matrix renormalization group calculations with local basis optimization (DMRG-LBO). We observe that the multitrajectory Ehrenfest method, in general, only captures the ultrashort time dynamics accurately. In contrast, the surface-hopping method with suitable corrections provides a much better description of the long-time behavior but struggles with the short-time description of coherences between different Born-Oppenheimer states. We show that the multiconfigurational Ehrenfest method yields a significant improvement over the multitrajectory Ehrenfest method and can be converged to the exact results in small systems with moderate computational efforts. We further observe that for extended systems, this convergence is slower with respect to the number of configurations. Our benchmark study demonstrates that DMRG-LBO is a useful tool for assessing the quality of the quantum-chemistry methods.

Controlling energy conservation in quantum dynamics with independently moving basis functions: Application to multi-configuration Ehrenfest

Application of the time-dependent variational principle to a linear combination of frozen-width Gaussians describing the nuclear wavefunction provides a formalism where the total energy is conserved. The computational downside of this formalism is that trajectories of individual Gaussians are solutions of a coupled system of differential equations, limiting implementation to serial propagation algorithms. To allow for parallelization and acceleration of the computation, independent trajectories based on simplified equations of motion were suggested. Unfortunately, within practical realizations involving finite Gaussian bases, this simplification leads to breaking the energy conservation. We offer a solution for this problem by using Lagrange multipliers to ensure the energy and norm conservation regardless of basis function trajectories or basis completeness. We illustrate our approach within the multi-configurational Ehrenfest method considering a linear vibronic coupling model.

Local Measure of Quantum Effects in Quantum Dynamics

The Madelung-de Broglie-Bohm formulation of the Schrödinger equation casts the time-evolution of a wave function as dynamics of an ensemble of quantum, or Bohmian, trajectories, interacting via the nonlocal quantum potential. This trajectory perspective gives insight into the quantumness (or classicality) of a given system due to clear partitioning of the energy into classical and quantum components. Here, we propose a system-independent measure of the quantumness of dynamics, based on the energy time-change, referred to as "quantum power". This measure is local in the coordinate space. Based on applications to model chemical systems, we argue that during the transition from the quantum to classical regime, defined as compression of quantization, the quantum features in dynamics do not "disappear" but are pushed forward in time. This feature may be used to gauge the validity of the semiclassical and other approximate dynamics approaches in applications to anharmonic systems.

Modeling irreversible molecular internal conversion using the time-dependent variational approach with sD2 ansatz

A model of irreversible molecular internal conversion dynamics due to molecular thermal energy dissipation to the bath is presented.

Zombie states for description of structure and dynamics of multi-electron systems

Canonical Coherent States (CSs) of Harmonic Oscillator have been extensively used as a basis in a number of computational methods of quantum dynamics. However, generalising such techniques for fermionic systems is difficult because Fermionic Coherent States (FCSs) require complicated algebra of Grassmann numbers not well suited for numerical calculations. This paper introduces a coherent antisymmetrised superposition of "dead" and "alive" electronic states called here Zombie State (ZS), which can be used in a manner of FCSs but without Grassmann algebra. Instead, for Zombie States, a very simple sign-changing rule is used in the definition of creation and annihilation operators. Then, calculation of electronic structure Hamiltonian matrix elements between two ZSs becomes very simple and a straightforward technique for time propagation of fermionic wave functions can be developed. By analogy with the existing methods based on Canonical Coherent States of Harmonic Oscillator, fermionic wave functions can be propagated using a set of randomly selected Zombie States as a basis. As a proof of principles, the proposed Coupled Zombie States approach is tested on a simple example showing that the technique is exact.

Nonadiabatic photodynamics of phenol on a realistic potential energy surface by a novel multilayer Gaussian MCTDH program

We report the main features of a new implementation of the Gaussian Multi-Configuration Time-Dependent Hartree (G-MCTDH) model. The code allows effective computations of time-dependent phenomena, including calculation of vibronic spectra (in one or more electronic states), relative state populations etc., with the possibility of a multilayer formulation. We have validated the code on the diabatic surfaces recently published by Truhlar and coworkers to study the nonadiabatic photodynamics of phenol. Using an Ehrenfest-like, single-nuclear-configuration (but in a fully quantum formalism) model we calculate the optical spectrum and relative state populations of the system as a function of time.

A new Gaussian MCTDH program: implementation and validation on the levels of the water and glycine molecules

We report the main features of a new general implementation of the Gaussian Multi-Configuration Time-Dependent Hartree model. The code allows effective computations of time-dependent phenomena, including calculation of vibronic spectra (in one or more electronic states), relative state populations, etc. Moreover, by expressing the Dirac-Frenkel variational principle in terms of an effective Hamiltonian, we are able to provide a new reliable estimate of the representation error. After validating the code on simple one-dimensional systems, we analyze the harmonic and anharmonic vibrational spectra of water and glycine showing that reliable and converged energy levels can be obtained with reasonable computing resources. The data obtained on water and glycine are compared with results of previous calculations using the vibrational second-order perturbation theory method. Additional features and perspectives are also shortly discussed.