Perturbative wave-packet spawning procedure for non-adiabatic dynamics in diabatic representation

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.

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-in-Time Error in Variational Quantum Dynamics

The McLachlan "minimum-distance" principle for optimizing approximate solutions of the time-dependent Schrödinger equation is revisited, with a focus on the local-in-time error accompanying the variational solutions. Simple, exact expressions are provided for this error, which are then evaluated in illustrative cases, notably the widely used mean-field approach and the adiabatic quantum molecular dynamics. Based on these findings, we demonstrate the rigorous formulation of an adaptive scheme that resizes on the fly the underlying variational manifold and, hence, optimizes the overall computational cost of a quantum dynamical simulation. Such adaptive schemes are a crucial requirement for devising and applying direct quantum dynamical methods to molecular and condensed-phase problems.

Accurate Neural Network Representation of the Ab Initio Determined Spin-Orbit Interaction in the Diabatic Representation Including the Effects of Conical Intersections

A method for fitting ab initio determined spin-orbit coupling interactions, in the Breit-Pauli approximation, based on quasidiabatic representations using neural network fits is reported. The algorithm generalizes our recently reported neural network approach for representing the dipole interaction. The S₀, S₁, and T₁ states of formaldehyde are used as an example. First, the two singlet states S₀ and S₁ are diabatized with a modified Boys Localization diabatization method. Second, the spin-orbit coupling between singlet and triplet states is transformed to the diabatic representation. This removes the discontinuities in the adiabatic representation. The diabatized spin-orbit couplings are then fit with smooth neural network functions. The analytic representation of spin-orbit coupling interactions in a diabatic basis by neural networks will make accurate full-dimensional quantum dynamical treatment of both internal conversion and intersystem crossing possible, which will help us to gain better understanding of both processes.

Variational nonadiabatic dynamics in the moving crude adiabatic representation: Further merging of nuclear dynamics and electronic structure

A new methodology of simulating nonadiabatic dynamics using frozen-width Gaussian wavepackets within the moving crude adiabatic representation with the on-the-fly evaluation of electronic structure is presented. The main feature of the new approach is the elimination of any global or local model representation of electronic potential energy surfaces; instead, the electron-nuclear interaction is treated explicitly using the Gaussian integration. As a result, the new scheme does not introduce any uncontrolled approximations. The employed variational principle ensures the energy conservation and leaves the number of electronic and nuclear basis functions as the only parameter determining the accuracy. To assess performance of the approach, a model with two electronic and two nuclear spacial degrees of freedom containing conical intersections between potential energy surfaces has been considered. Dynamical features associated with nonadiabatic transitions and nontrivial geometric (or Berry) phases were successfully reproduced within a limited basis expansion.

Geometric Phase Effects in Nonadiabatic Dynamics near Conical Intersections

Dynamical consideration that goes beyond the common Born-Oppenheimer approximation (BOA) becomes necessary when energy differences between electronic potential energy surfaces become small or vanish. One of the typical scenarios of the BOA breakdown in molecules beyond diatomics is a conical intersection (CI) of electronic potential energy surfaces. CIs provide an efficient mechanism for radiationless electronic transitions: acting as "funnels" for the nuclear wave function, they enable rapid conversion of the excessive electronic energy into the nuclear motion. In addition, CIs introduce nontrivial geometric phases (GPs) for both electronic and nuclear wave functions. These phases manifest themselves in change of the wave function signs if one considers an evolution of the system around the CI. This sign change is independent of the shape of the encircling contour and thus has a topological character. How these extra phases affect nonadiabatic dynamics is the main question that is addressed in this Account. We start by considering the simplest model providing the CI topology: two-dimensional two-state linear vibronic coupling model. Selecting this model instead of a real molecule has the advantage that various dynamical regimes can be easily modeled in the model by varying parameters, whereas any fixed molecule provides the system specific behavior that may not be very illustrative. After demonstrating when GP effects are important and how they modify the dynamics for two sets of initial conditions (starting from the ground and excited electronic states), we give examples of molecular systems where the described GP effects are crucial for adequate description of nonadiabatic dynamics. Interestingly, although the GP has a topological character, the extent to which accounting for GPs affect nuclear dynamics profoundly depends on topography of potential energy surfaces. Understanding an extent of changes introduced by the GP in chemical dynamics poses a problem of capturing GP effects by approximate methods of simulating nonadiabatic dynamics that can go beyond simple models. We assess the performance of both fully quantum (wave packet dynamics) and quantum-classical (surface-hopping, Ehrenfest, and quantum-classical Liouville equation) approaches in various cases where GP effects are important. It has been identified that the key to success in approximate methods is a method organization that prevents the quantum nuclear kinetic energy operator to act directly on adiabatic electronic wave functions.

Quantum Nonadiabatic Cloning of Entangled Coherent States

We propose a systematic approach to the basis set extension for nonadiabatic dynamics of entangled combination of nuclear coherent states (CSs) evolving according to the time-dependent variational principle (TDVP). The TDVP provides a rigorous framework for fully quantum nonadiabatic dynamics of closed systems; however, the quality of results strongly depends on available basis functions. Starting with a single nuclear CS replicated vertically on all electronic states, our approach clones this function when replicas of the CS on different electronic states experience increasingly different forces. Created clones move away from each other (decohere), extending the basis set. To determine a moment for cloning, we introduce generalized forces based on derivatives that maximally contribute to a variation of the total quantum action and thus account for entanglement of all basis functions.

Topologically Correct Quantum Nonadiabatic Formalism for On-the-Fly Dynamics

On-the-fly quantum nonadiabatic dynamics for large systems greatly benefits from the adiabatic representation readily available from electronic structure programs. However, conical intersections frequently occurring in this representation introduce nontrivial geometric or Berry phases which require a special treatment for adequate modeling of the nuclear dynamics. We analyze two approaches for nonadiabatic dynamics using the time-dependent variational principle and the adiabatic representation. The first approach employs adiabatic electronic functions with global parametric dependence on the nuclear coordinates. The second approach uses adiabatic electronic functions obtained only at the centers of moving localized nuclear basis functions (e.g., frozen-width Gaussians). Unless a gauge transformation is used to enforce single-valued boundary conditions, the first approach fails to capture the geometric phase. In contrast, the second approach accounts for the geometric phase naturally because of the absence of the global nuclear coordinate dependence in the electronic functions.

Problem-free time-dependent variational principle for open quantum systems

Methods of quantum nuclear wave-function dynamics have become very efficient in simulating large isolated systems using the time-dependent variational principle (TDVP). However, a straightforward extension of the TDVP to the density matrix framework gives rise to methods that do not conserve the energy in the isolated system limit and the total system population for open systems where only energy exchange with environment is allowed. These problems arise when the system density is in a mixed state and is simulated using an incomplete basis. Thus, the basis set incompleteness, which is inevitable in practical calculations, creates artificial channels for energy and population dissipation. To overcome this unphysical behavior, we have introduced a constrained Lagrangian formulation of TDVP applied to a non-stochastic open system Schrödinger equation [L. Joubert-Doriol, I. G. Ryabinkin, and A. F. Izmaylov, J. Chem. Phys. 141, 234112 (2014)]. While our formulation can be applied to any variational ansatz for the system density matrix, derivation of working equations and numerical assessment is done within the variational multiconfiguration Gaussian approach for a two-dimensional linear vibronic coupling model system interacting with a harmonic bath.

Non-stochastic matrix Schrödinger equation for open systems

We propose an extension of the Schrödinger equation for a quantum system interacting with environment. This extension describes dynamics of a collection of auxiliary wavefunctions organized as a matrix m, from which the system density matrix can be reconstructed as ρ̂=mm(†). We formulate a compatibility condition, which ensures that the reconstructed density satisfies a given quantum master equation for the system density. The resulting non-stochastic evolution equation preserves positive-definiteness of the system density and is applicable to both Markovian and non-Markovian system-bath treatments. Our formalism also resolves a long-standing problem of energy loss in the time-dependent variational principle applied to mixed states of closed systems.