Sampling the thermal Wigner density via a generalized Langevin dynamics

Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths

This paper reviews methods that aim at simulating nuclear quantum effects (NQEs) using generalized thermal baths. Generalized (or quantum) baths simulate statistical quantum features, and in particular zero-point energy effects, through non-Markovian stochastic dynamics. They make use of generalized Langevin Equations (GLEs), in which the quantum Bose–Einstein energy distribution is enforced by tuning the random and friction forces, while the system degrees of freedom remain classical. Although these baths have been formally justified only for harmonic oscillators, they perform well for several systems, while keeping the cost of the simulations comparable to the classical ones. We review the formal properties and main characteristics of classical and quantum GLEs, in relation with the fluctuation–dissipation theorems. Then, we describe the quantum thermostat and quantum thermal bath, the two generalized baths currently most used, providing several examples of applications for condensed matter systems, including the calculation of vibrational spectra. The most important drawback of these methods, zero-point energy leakage, is discussed in detail with the help of model systems, and a recently proposed scheme to monitor and mitigate or eliminate it—the adaptive quantum thermal bath—is summarised. This approach considerably extends the domain of application of generalized baths, leading, for instance, to the successful simulation of liquid water, where a subtle interplay of NQEs is at play. The paper concludes by overviewing further development opportunities and open challenges of generalized baths.

A Divergence-Free Wigner Transform of the Boltzmann Operator Based on an Effective Frequency Theory

The centroid effective frequency representation of path integrals as developed by Feynman and Kleinert was originally aimed at calculating partition functions and related quantities in the canonical ensemble. In its path integral formulation, only closed paths were relevant. This formulation has been used by the present authors in order to calculate the many-body Wigner function of the Boltzmann operator, which includes also open paths. This usage of the theory outside of the original intention can lead to mathematical divergence issues for potentials with barriers, particularly at low temperature. In the present paper, we modify the effective frequency theory of Feynman and Kleinert by also including open paths in its variational equations. In this way, a divergence-free approximation to the Boltzmann operator matrix elements is derived. This generalized version of Feynman and Kleinert’s formulation is thus more robust and can be applied to all types of barriers at all temperatures. This new version is used to calculate the Wigner functions of the Boltzmann operator for a quartic oscillator and for a double well potential and both static and dynamic properties are studied at several temperatures. The new theory is found to be essentially as precise as the original one. Its advantage is that it will always deliver a well-defined, even if approximate, Wigner function, which can, for instance, be used for sampling initial conditions for molecular dynamics simulations. As will be discussed, the theory can be systematically improved by including higher-order Fourier modes into the nonquadratic part of the trial action.

Anharmonic spectral features via trajectory-based quantum dynamics: A perturbative analysis of the interplay between dynamics and sampling

The performance of different approximate algorithms for computing anharmonic features in vibrational spectra is analyzed and compared on model and more realistic systems that present relevant nuclear quantum effects. The methods considered combine approximate sampling of the quantum thermal distribution with classical time propagation and include Matsubara dynamics, path integral dynamics approaches, linearized initial value representation, and the recently introduced adaptive quantum thermal bath. A perturbative analysis of these different methods enables us to account for the observed numerical performance on prototypes for overtones and combination bands and to draw qualitatively correct trends for the numerical results obtained for Fermi resonances. Our results prove that the unequal performances of these approaches often derive from the method employed to sample initial conditions and not, as usually assumed, from the lack of coherence in the time propagation. Furthermore, as confirmed by the analysis reported in Benson and Althorpe, J. Chem. Phys. 130, 194510 (2021), we demonstrate, both via the perturbative approach and numerically, that path integral dynamics methods fail to reproduce the intensities of these anharmonic features and follow purely classical trends with respect to their temperature behavior. Finally, the remarkably accurate performance of the adaptive quantum thermal bath approach is documented and motivated.

Nuclear Quantum Effects in Liquid Water at Near Classical Computational Cost Using the Adaptive Quantum Thermal Bath

We demonstrate the accuracy and efficiency of a recently introduced approach to account for nuclear quantum effects (NQEs) in molecular simulations: the adaptive quantum thermal bath (adQTB). In this method, zero-point energy is introduced through a generalized Langevin thermostat designed to precisely enforce the quantum fluctuation-dissipation theorem. We propose a refined adQTB algorithm with improved accuracy and report adQTB simulations of liquid water. Through extensive comparison with reference path integral calculations, we demonstrate that it provides excellent accuracy for a broad range of structural and thermodynamic observables as well as infrared vibrational spectra. The adQTB has a computational cost comparable to that of classical molecular dynamics, enabling simulations of up to millions of degrees of freedom.

Relaxation dynamics through a conical intersection: Quantum and quantum-classical studies

We study the relaxation process through a conical intersection of a photo-excited retinal chromophore model. The analysis is based on a two-electronic-state two-dimensional Hamiltonian developed by Hahn and Stock [J. Phys. Chem. B 104 1146 (2000)] to reproduce, with a minimal model, the main features of the 11-cis to all-trans isomerization of the retinal of rhodopsin. In particular, we focus on the performance of various trajectory-based schemes to nonadiabatic dynamics, and we compare quantum-classical results to the numerically exact quantum vibronic wavepacket dynamics. The purpose of this work is to investigate, by analyzing electronic and nuclear observables, how the sampling of initial conditions for the trajectories affects the subsequent dynamics.