Path Integral Dynamics Research Articles

Path Integral Simulations of Condensed-Phase Vibrational Spectroscopy.

Recent theoretical and algorithmic developments have improved the accuracy with which path integral dynamics methods can include nuclear quantum effects in simulations of condensed-phase vibrational spectra. Such methods are now understood to be approximations to the delocalized classical Matsubara dynamics of smooth Feynman paths, which dominate the dynamics of systems such as liquid water at room temperature. Focusing mainly on simulations of liquid water and hexagonal ice, we explain how the recently developed quasicentroid molecular dynamics (QCMD), fast-QCMD, and temperature-elevated path integral coarse-graining simulations (Te PIGS) methods generate classical dynamics on potentials of mean force obtained by averaging over quantum thermal fluctuations. These new methods give very close agreement with one another, and the Te PIGS method has recently yielded excellent agreement with experimentally measured vibrational spectra for liquid water, ice, and the liquid-air interface. We also discuss the limitations of such methods.

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.

On the “Matsubara heating” of overtone intensities and Fermi splittings

Classical molecular dynamics (MD) and imaginary-time path-integral dynamics methods underestimate the infrared absorption intensities of overtone and combination bands by typically an order of magnitude. Plé et al. [J. Chem. Phys. 155, 2863 (2021)] have shown that this is because such methods fail to describe the coupling of the centroid to the Matsubara dynamics of the fluctuation modes; classical first-order perturbation theory (PT) applied to the Matsubara dynamics is sufficient to recover most of the lost intensity in simple models and gives identical results to quantum (Rayleigh-Schrödinger) PT. Here, we show numerically that the results of this analysis can be used as post-processing correction factors, which can be applied to realistic (classical MD or path-integral dynamics) simulations of infrared spectra. We find that the correction factors recover most of the lost intensity in the overtone and combination bands of gas-phase water and ammonia and much of it for liquid water. We then re-derive and confirm the earlier PT analysis by applying canonical PT to Matsubara dynamics, which has the advantage of avoiding secular terms and gives a simple picture of the perturbed Matsubara dynamics in terms of action-angle variables. Collectively, these variables "Matsubara heat" the amplitudes of the overtone and combination vibrations of the centroid to what they would be in a classical system with the oscillators (of frequency Ωi) held at their quantum effective temperatures [of ℏΩi coth(βℏΩi/2)/2kB]. Numerical calculations show that a similar neglect of "Matsubara heating" causes path-integral methods to underestimate Fermi resonance splittings.

Persistent Homology Metrics Reveal Quantum Fluctuations and Reactive Atoms in Path Integral Dynamics.

Nuclear quantum effects (NQEs) are known to impact a number of features associated with chemical reactivity and physicochemical properties, particularly for light atoms and at low temperatures. In the imaginary time path integral formalism, each atom is mapped onto a “ring polymer” whose spread is related to the quantum mechanical uncertainty in the particle’s position, i.e., its thermal wavelength. A number of metrics have previously been used to investigate and characterize this spread and explain effects arising from quantum delocalization, zero-point energy, and tunneling. Many of these shape metrics consider just the instantaneous structure of the ring polymers. However, given the significant interest in methods such as centroid molecular dynamics and ring polymer molecular dynamics that link the molecular dynamics of these ring polymers to real time properties, there exists significant opportunity to exploit metrics that also allow for the study of the fluctuations of the atom delocalization in time. Here we consider the ring polymer delocalization from the perspective of computational topology, specifically persistent homology, which describes the 3-dimensional arrangement of point cloud data, (i.e. atomic positions). We employ the Betti sequence probability distribution to define the ensemble of shapes adopted by the ring polymer. The Wasserstein distances of Betti sequences adjacent in time are used to characterize fluctuations in shape, where the Fourier transform and associated principal components provides added information differentiating atoms with different NQEs based on their dynamic properties. We demonstrate this methodology on two representative systems, a glassy system consisting of two atom types with dramatically different de Broglie thermal wavelengths, and ab initio molecular dynamics simulation of an aqueous 4 M HCl solution where the H-atoms are differentiated based on their participation in proton transfer reactions.

Path-integral dynamics of water using curvilinear centroids

We develop a path-integral dynamics method for water that resembles centroid molecular dynamics (CMD), except that the centroids are averages of curvilinear, rather than Cartesian, bead coordinates. The curvilinear coordinates are used explicitly only when computing the potential of mean force, the components of which are re-expressed in terms of Cartesian “quasicentroids” (so-called because they are close to the Cartesian centroids). Cartesian equations of motion are obtained by making small approximations to the quantum Boltzmann distribution. Simulations of the infrared spectra of various water models over 150–600 K show these approximations to be justified: for a two-dimensional OH-bond model, the quasicentroid molecular dynamics (QCMD) spectra lie close to the exact quantum spectra, and almost on top of the Matsubara dynamics spectra; for gas-phase water, the QCMD spectra are close to the exact quantum spectra; for liquid water and ice (using the q-TIP4P/F surface), the QCMD spectra are close to the CMD spectra at 600 K and line up with the results of thermostated ring-polymer molecular dynamics and approximate quantum calculations at 300 and 150 K. The QCMD spectra show no sign of the CMD “curvature problem” (of erroneous red shifts and broadening). In the liquid and ice simulations, the potential of mean force was evaluated on-the-fly by generalizing an adiabatic CMD algorithm to curvilinear coordinates; the full limit of adiabatic separation needed to be taken, which made the QCMD calculations 8 times more expensive than partially adiabatic CMD at 300 K, and 32 times at 150 K (and the intensities may still not be converged at this temperature). The QCMD method is probably generalizable to many other systems, provided that collective bead-coordinates can be identified that yield compact mean-field ring-polymer distributions.

Semiclassical Path Integral Dynamics: Photosynthetic Energy Transfer with Realistic Environment Interactions.

This article reviews recent progress in the theoretical modeling of excitation energy transfer (EET) processes in natural light harvesting complexes. The iterative partial linearized density matrix path-integral propagation approach, which involves both forward and backward propagation of electronic degrees of freedom together with a linearized, short-time approximation for the nuclear degrees of freedom, provides an accurate and efficient way to model the nonadiabatic quantum dynamics at the heart of these EET processes. Combined with a recently developed chromophore-protein interaction model that incorporates both accurate ab initio descriptions of intracomplex vibrations and chromophore-protein interactions treated with atomistic detail, these simulation tools are beginning to unravel the detailed EET pathways and relaxation dynamics in light harvesting complexes.

Theoretical spectroscopy using molecular dynamics: theory and application to CH5+ and its isotopologues

Infrared spectroscopy is a powerful technique to unravel the structure and dynamics of molecular systems of ever increasing complexity. For isolated molecules in the gas phase theoretical approaches that directly rely on solving the Schrödinger equation, either approximately or quasi-exactly, are well established. A distinctly different approach to compute infrared spectra can be based on advanced molecular dynamics, itself being based on classical Newtonian dynamics, in conjunction with concurrent first principles electronic structure calculations. At variance with traditional methods, which are formulated in terms of the Schrödinger representation of quantum mechanics, the molecular dynamics approach stems from Heisenberg's representation and thus relies on computing thermal expectation values of time-correlation functions. Crucial in addition to generating the spectra themselves is their decomposition in terms of modes, which can be assigned to correlated atomic motion. This ab initio molecular dynamics route to compute infrared spectra, and its recent extension to quasiclassical techniques relying on approximate path integral dynamics, is covered in the review part of this Perspective. The usefulness of this unconventional approach, which can be generalized beyond infrared spectroscopy, is demonstrated in detail by applying the full machinery in computing and assigning the infrared spectra of protonated methane and its isotopologues. This particular molecule is often considered to be the most prominent member of the class of floppy or fluxional molecules. CH5(+) has been a longstanding challenge for theoretical infrared spectroscopy because it undergoes intricate large-amplitude motion, which is also reviewed. Molecular dynamics based infrared spectroscopy is general and can be applied to diverse systems such as molecular complexes in the gas phase, chromophores in biomolecular environments, and solute-solvent systems in the liquid phase.

Efficient First-Principles Calculation of the Quantum Kinetic Energy and Momentum Distribution of Nuclei

Light nuclei at room temperature and below exhibit a kinetic energy which significantly deviates from the predictions of classical statistical mechanics. This quantum kinetic energy is responsible for a wide variety of isotope effects of interest in fields ranging from chemistry to climatology. It also furnishes the second moment of the nuclear momentum distribution, which contains subtle information about the chemical environment and has recently become accessible to deep inelastic neutron scattering experiments. Here, we show how, by combining imaginary time path integral dynamics with a carefully designed generalized Langevin equation, it is possible to dramatically reduce the expense of computing the quantum kinetic energy. We also introduce a transient anisotropic Gaussian approximation to the nuclear momentum distribution which can be calculated with negligible additional effort. As an example, we evaluate the structural properties, the quantum kinetic energy, and the nuclear momentum distribution for a first-principles simulation of liquid water.

Quantum Molecular Dynamics Calculations of Ultrafast Time Scales and Infrared Spectra of Protonated Methane: Quantifying Isotope-Specific Lifetimes

Understanding the quantum dynamics of molecules undergoing large-amplitude vibrational motion is challenging because of their flat and anharmonic energy landscapes. Using quasiclassical ab initio path integral dynamics, time scales of large-amplitude motion, lifetimes, and IR spectra are computed for protonated methane, CH5+, and its deuterated species, CH4D+, CH3D2+, CH2D3+, CHD4+, and CD5+. The time scales of quantum pseudorotations are shown to depend sensitively on the particular isotopic substitution pattern, which characterizes the measured infrared spectra.

Accelerating the convergence of path integral dynamics with a generalized Langevin equation

The quantum nature of nuclei plays an important role in the accurate modelling of light atoms such as hydrogen, but it is often neglected in simulations due to the high computational overhead involved. It has recently been shown that zero-point energy effects can be included comparatively cheaply in simulations of harmonic and quasiharmonic systems by augmenting classical molecular dynamics with a generalized Langevin equation (GLE). Here we describe how a similar approach can be used to accelerate the convergence of path integral (PI) molecular dynamics to the exact quantum mechanical result in more strongly anharmonic systems exhibiting both zero point energy and tunnelling effects. The resulting PI-GLE method is illustrated with applications to a double-well tunnelling problem and to liquid water.