A Born–Oppenheimer approximation for path integrals with an application to electron solvation in polarizable fluids

Electronic Free Energy Surface of the Nitrogen Dimer Using First-Principles Finite Temperature Electronic Structure Methods

We use full configuration interaction and density matrix quantum Monte Carlo methods to calculate the electronic free energy surface of the nitrogen dimer within the free-energy Born-Oppenheimer approximation. As the temperature is raised from T = 0, we find a temperature regime in which the internal energy causes bond strengthening. At these temperatures, adding in the entropy contributions is required to cause the bond to gradually weaken with increasing temperature. We predict a thermally driven dissociation for the nitrogen dimer between 22,000 to 63,200 K depending on symmetries and basis set. Inclusion of more spatial and spin symmetries reduces the temperature required. The origin of these observations is explored using the structure of the density matrix at various temperatures and bond lengths.

Importance of Nuclear Quantum Effects on Aqueous Electrolyte Transport under Confinement in Ti3C2 MXenes

Protons display a high chemical activity and strongly affect the charge storage capability in confined interlayer spaces of two-dimensional (2D) materials. As such, an accurate representation of proton dynamics under confinement is important for understanding and predicting charge storage dynamics in these materials. While often ignored in atomistic-scale simulations, nuclear quantum effects (NQEs), e.g., tunneling, can be significant under confinement even at room temperature. Using the thermostatted ring polymer molecular dynamics implementation of path integral molecular dynamics (PIMD) in conjunction with the ReaxFF force field, density functional tight binding (DFTB), and NequIP neural network potential simulations, we investigate the role of NQEs on proton and water transport in bulk water and aqueous electrolytes under confinement in Ti₃C₂ MXenes. Although overall NQEs are relatively small, especially in bulk, we find that they can alter both quantitative values and qualitative trends on both proton transport and water self-diffusion under confinement relative to classical MD predictions. Therefore, our results suggest the need for NQEs to be considered to simulate aqueous systems under confinement for both qualitative and quantitative accuracy.

Quantum fluctuations of a fullerene cage modulate its internal magnetic environment

To investigate the effect of quantum fluctuations on the magnetic environment inside a C₆₀ fullerene cage, we have calculated the nuclear magnetic shielding constant of protons in H₂@C₆₀ and HD@C₆₀ systems by on-the-fly ab initio path integral simulation, including both thermal and nuclear quantum effects. The most dominant upfield from an isolated hydrogen molecule occurs due to the diamagnetic current of the C₆₀ cage, which is partly cancelled by the paramagnetic current, where the paramagnetic contribution is enlarged by the zero-point vibrational fluctuation of the C₆₀ carbon backbone structure via a widely distributed HOMO-LUMO gap. This quantum modulation mechanism of the nuclear magnetic shielding constant is newly proposed. Because this quantum effect is independent of the difference between H₂ and HD, the H₂/HD isotope shift occurs in spite of the C₆₀ cage. The nuclear magnetic constants computed for H₂@C₆₀ and HD@C₆₀ are 32.047 and 32.081 ppm, respectively, which are in reasonable agreement with the corresponding values of 32.19 and 32.23 ppm estimated from the experimental values of the chemical shifts.

A molecular dynamics study of nuclear quantum effect on diffusivity of hydrogen molecule

In this paper, the nuclear quantum effect of the hydrogen molecule on its diffusivity was analyzed using the molecular dynamics (MD) method. The centroid MD (CMD) method was applied to reproduce the time evolution of the molecules. The diffusion coefficient of hydrogen was calculated using the Green-Kubo method over a wide temperature region, and the temperature dependence of the quantum effect of the hydrogen molecule on its diffusivity was addressed. The calculated results were compared with classical MD results based on the principle of corresponding state (PCS). It was confirmed that the difference in the diffusion coefficient calculated in the CMD and classical MD methods was small, and the PCS appears to be satisfied on the temperature dependence of the diffusion coefficient, even though the quantum effect of the hydrogen molecules was taken into account. It was clarified that this result did not suggest that the quantum effect on the diffusivity of the hydrogen molecule was small but that the two changes in the intermolecular interaction of hydrogen due to the quantum effect offset each other. Moreover, it was found that this tendency was related to the temperature dependence of the ratio of the kinetic energy of the quantum fluctuational motion to the classical kinetic energy.

Generalized balance equations for charged particle transport via localized and delocalized states: Mobility, generalized Einstein relations, and fractional transport

A generalized phase-space kinetic Boltzmann equation for highly nonequilibrium charged particle transport via localized and delocalized states is used to develop continuity, momentum, and energy balance equations, accounting explicitly for scattering, trapping and detrapping, and recombination loss processes. Analytic expressions detail the effect of these microscopic processes on mobility and diffusivity. Generalized Einstein relations (GER) are developed that enable the anisotropic nature of diffusion to be determined in terms of the measured field dependence of the mobility. Interesting phenomena such as negative differential conductivity and recombination heating and cooling are shown to arise from recombination loss processes and the localized and delocalized nature of transport. Fractional transport emerges naturally within this framework through the appropriate choice of divergent mean waiting time distributions for localized states, and fractional generalizations of the GER and mobility are presented. Signature impacts on time-of-flight current transients of recombination loss processes via both localized and delocalized states are presented.

Solution of a generalized Boltzmann's equation for nonequilibrium charged-particle transport via localized and delocalized states

We present a general phase-space kinetic model for charged-particle transport through combined localized and delocalized states, capable of describing scattering collisions, trapping, detrapping, and losses. The model is described by a generalized Boltzmann equation, for which an analytical solution is found in Fourier-Laplace space. The velocity of the center of mass and the diffusivity about it are determined analytically, together with the flux transport coefficients. Transient negative values of the free particle center-of-mass transport coefficients can be observed due to the trapping to, and detrapping from, localized states. A Chapman-Enskog-type perturbative solution technique is applied, confirming the analytical results and highlighting the emergence of a density gradient representation in the weak-gradient hydrodynamic regime. A generalized diffusion equation with a unique global time operator is shown to arise, reducing to the standard diffusion equation and a Caputo fractional diffusion equation in the normal and dispersive limits. A subordination transformation is used to solve the generalized diffusion equation by mapping from the solution of a corresponding standard diffusion equation.

Path integral Monte Carlo on a lattice: extended states

The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and atoms are restricted to the sites of a one-dimensional lattice. A path integral formalism is developed within the context of the canonical ensemble in which the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. For the case of a free particle, analytical expressions for the energy, its fluctuations, and the qp-qp correlation function are derived and compared with the Monte Carlo simulations. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp for a periodic interaction potential, forcing the qp to occupy extended states. We consider a striped potential in one dimension, where every other lattice site is occupied by an atom with potential ε, and every other lattice site is empty. This potential serves as a stress test for the path integral formalism because of its rapid site-to-site variation. An analytical solution was determined in this case by utilizing Bloch's theorem due to the periodicity of the potential. Comparisons of the potential energy, the total energy, the energy fluctuations, and the correlation function are made between the results of the Monte Carlo simulations and the analytical calculations.

FIRST PRINCIPLES MOLECULAR DYNAMICS INVOLVING EXCITED STATES AND NONADIABATIC TRANSITIONS

Extensions of traditional molecular dynamics to excited electronic states and non-Born–Oppenheimer dynamics are reviewed focusing on applicability to chemical reactions of large molecules, possibly in condensed phases. The latter imposes restrictions on both the level of accuracy of the underlying electronic structure theory and the treatment of nonadiabaticity. This review, therefore, exclusively deals with ab initio "on the fly" molecular dynamics methods. For the same reason, mainly mixed quantum-classical approaches to nonadiabatic dynamics are considered.

Dispersion forces between solvated electrons

Using the path integral centroid approach, we investigate dispersion interactions between electrons solvated in metal-ammonia solutions. We have argued that at finite metal concentrations, the behavior of the solvated electrons is controlled by these interactions. The latter result in a peculiar nonmetal-metal transition, which appears as a sharp dielectric enhancement and a mechanical instability of the system. Our results indicate also that the solvated electrons are to be considered as a two-component mixture consisting of localized and delocalized electrons beyond the critical density corresponding to this mechanical instability.

Path-integral simulations beyond the adiabatic approximation

Within the adiabatic approximation, it is trivial to generalize existing imaginary time path-integral techniques to the case of multiple electronic surfaces. However, there are many times where nonadiabatic effects can play an important role. To this end, we reformulate the well-known path-integral expressions to incorporate multiple potential surfaces, without necessitating the adiabatic approximation. We show that the resulting expression, like its adiabatic counterpart, can be interpreted in terms of a simple classical isomorphic system and thus is amenable to simulation through Monte Carlo techniques. We derive simple expressions to compute expectation values of a general operator in both the nuclear coordinate and electronic state, and demonstrate the existence of a simple internal diagnostic that can be used to evaluate the magnitude of equilibrium nonadiabatic effects.

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.

A comparative study of imaginary time path integral based methods for quantum dynamics

The recently introduced approximate many-body quantum simulation method, ring polymer molecular dynamics (RPMD), is compared to the centroid molecular dynamics method (CMD). Comparisons of simulation results for liquid para-hydrogen at two state points and liquid ortho-deuterium at one state point are presented. The calculated quantum correlation functions for the two methods are shown to be in good agreement with one another for a large portion of the time spectrum. However, as the quantum mechanical nature of the system increases, RPMD is less accurate in predicting the kinetic energy of the system than is CMD. A simplified and highly efficient algorithm is proposed which largely corrects this deficiency.

Solvent reorganization energy of electron-transfer reactions in polar solvents

A microscopic theory of solvent reorganization energy in polar molecular solvents is developed. The theory represents the solvent response as a combination of the density and polarization fluctuations of the solvent given in terms of the density and polarization structure factors. A fully analytical formulation of the theory is provided for a solute of arbitrary shape with an arbitrary distribution of charge. A good agreement between the analytical procedure and the results of Monte Carlo simulations of model systems is achieved. The reorganization energy splits into the contributions from density fluctuations and polarization fluctuations. The polarization part is dominated by longitudinal polarization response. The density part is inversely proportional to temperature. The dependence of the solvent reorganization energy on the solvent dipole moment and refractive index is discussed.