Stereographic projection path integral simulations of (HCl)n clusters (n=2-5): evidence of quantum induced melting in small hydrogen bonded networks

We investigate the quantum thermodynamic properties of small (HCl)(n) clusters using stereographic projection path integral simulations. The HCl stretches are rigid, the orientations are mapped with stereographic projection coordinates, and we make use of the reweighted random series techniques to obtain cubic convergence with respect to the number of path coefficients. Path integral simulations are converged at and above 10 K for the pentamer and above 15 K for the dimer and the trimer. None of the systems display a melting feature in the classical limit. We find an evidence of quantum induced melting between 15 and 45 K.     

Rigid quantum Monte Carlo simulations of condensed molecular matter: water clusters in the n=2-->8 range

The numerical advantage of quantum Monte Carlo simulations of rigid bodies relative to the flexible simulations is investigated for some simple systems. The results show that if high frequency modes in molecular condensed matter are predominantly in the ground state, the convergence of path integral simulations becomes nonuniform. Rigid body quantum parallel tempering simulations are necessary to accurately capture thermodynamic phenomena in the temperature range where the dynamics are influenced by intermolecular degrees of freedom; the stereographic projection path integral adapted for quantum simulations of asymmetric tops is a significantly more efficient strategy compared with Cartesian coordinate simulations for molecular condensed matter under these conditions. The reweighted random series approach for stereographic path integral Monte Carlo is refined and implemented for the quantum simulation of water clusters treated as an assembly of rigid asymmetric tops.     

Stereographic projections path integral for inertia ellipsoids: applications to Arn-HF clusters

The DeWitt formula for inertia ellipsoids mapped by stereographic projection coordinates is developed. We discover that by remapping the quaternion parameter space with stereographic projections, considerable simplification of the differential geometry for the inertia ellipsoid with spherical symmetry takes place. The metric tensor is diagonal and contains only one independent element in that case. We find no difficulties testing and implementing the DeWitt formula for the inertia ellipsoids of asymmetric tops mapped by stereographic projections. The path integral algorithm for the treatment of Rm x S2 manifolds based on a mixture of Cartesian and stereographic projection coordinates is tested for small Arn-HF clusters in the n = 2 to n = 5 range. In particular, we determine the quantum effects of the red shift and the isomerization patterns at finite temperatures. Our findings are consistent with previously reported computations and experimental data for small Arn-HF clusters.     

Molecular dynamics with quantum transitions study of the vibrational relaxation of the HOD bend fundamental in liquid D2O

The molecular dynamics with quantum transitions method is used to study the vibrational relaxation of the HOD bend fundamental in liquid D(2)O. All of the vibrational bending degrees of freedom of the HOD and D(2)O molecules are described by quantum mechanics, while the remaining translational and rotational degrees of freedom are described classically. The effect of the coupling between the rotational and vibrational degrees of freedom of the deuterated water molecules is analyzed. A kinetic mechanism based on three steps is proposed in order to interpret the dynamics of the system. It is shown that intermolecular vibrational energy transfer plays an important role in the relaxation process and also that the transfer of energy into the rotational degrees of freedom is favored over the transfer of energy into the translational motions. The thermalization of the system after the relaxation is reached in a shorter time scale than that of the recovery of the hydrogen bond network. The relaxation and equilibration times obtained compare well with experimental and previous theoretical results.     

Non-Markovian theory of open systems in classical limit

A fully classical limit of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys. 119, 2488 (2003)] is obtained and analyzed. The resulting kinetic equations are capable of describing the evolution of an open system on the entire time axis, including the short-time non-Markovian stage, and are valid beyond linear response regime. We have shown, that proceeding to the classical mechanics limit we restrict the class of allowed correlations between an open system and a canonical bath, so that the initial conditions and the relaxation operator has to be appropriately modified (projected). Disregard of the projection may lead to unphysical behavior, since mechanism of the decay of some correlations is essentially of quantum-mechanical nature, and is not correctly described by classical mechanics. The projection (quantum correction to the kinetics) is particularly important for the non-Markovian regime of relaxation towards canonical equilibrium. The conformity of the developed method to the conventional approaches is demonstrated using a model of Brownian motion (heavy particle in the bath of light ones), for which the obtained non-Markovian equations are reduced to the standard Fokker-Planck equation in phase space.     

Adaptive resolution simulation of liquid para-hydrogen: testing the robustness of the quantum-classical adaptive coupling

Adaptive resolution simulations for classical systems are currently made within a reasonably consistent theoretical framework. Recently we have extended this approach to the quantum-classical coupling by mapping the quantum nature of an atom onto a classical polymer ring representation within the path integral approach [Poma & Delle Site, Phys. Rev. Lett., 2010, 104, 250201]. In this way the process of interfacing adaptively a quantum representation to a classical one corresponds to the problem of interfacing two regions with a different number of effective "classical" degrees of freedom; thus the classical formulation of the adaptive algorithm applies straightforwardly to the quantum-classical problem. In this work we show the robustness of such an approach for a liquid of para-hydrogen at low temperature. This system represents a highly challenging conceptual and technical test for the adaptive approach due to the extreme thermodynamical conditions where quantum effects play a central role.     

Equilibrium properties of quantum water clusters by the variational Gaussian wavepacket method

The variational Gaussian wavepacket (VGW) method in combination with the replica-exchange Monte Carlo is applied to calculations of the heat capacities of quantum water clusters, (H(2)O)(8) and (H(2)O)(10). The VGW method is most conveniently formulated in Cartesian coordinates. These in turn require the use of a flexible (i.e., unconstrained) water potential. When the latter is fitted as a linear combination of Gaussians, all the terms involved in the numerical solution of the VGW equations of motion are analytic. When a flexible water model is used, a large difference in the timescales of the inter- and intramolecular degrees of freedom generally makes the system very difficult to simulate numerically. Yet, given this difficulty, we demonstrate that our methodology is still practical. We compare the computed heat capacities to those for the corresponding classical systems. As expected, the quantum effects shift the melting temperatures toward the lower values.     

On a semiclassical approach to energy transfer in polyatomic molecules

A semiclassical method for energy transfer in polyatomic molecules is suggested. The method is based on a partial quantization of the vibrational degrees of freedom. The remaining degrees of freedom are treated classically. As an example a system with three quantum coordinates is considered. The Coriolis coupling terms are taken into account in the quantum mechanical part of the system and discussed for the N₂ + CO₂ system.     

Quantum monte carlo methods for constrained systems

The torsional ground state for ethane, the torsional, rotational, and mixed torsional and rotational ground state of propane are computed with a version of diffusion Monte Carlo adapted to handle the geometric complexity of curved spaces such as the Ramachandra space. The quantum NVT ensemble average for the mixed torsional and rotational degrees of freedom of propane is computed, using a version of Monte Carlo path integral, also adapted to handle curved spaces. These three problems are selected to demonstrate the generality and the applicability of the approaches described. The spaces of coordinates can be best constructed from the parameters of continuous Lie groups, and alternative methods based on vector spaces, where extended Lagrangian terms would be too cumbersome to implement. We note that the geometric coupling between the torsions and the rotations of propane produces a substantial effect on the ground state energy of propane, and that the quantum effects on the energy of propane are quite large even well above room temperature. © 2014 Wiley Periodicals, Inc.     

Exploring the effects of intramolecular vibrational energy redistribution on the operation of the proton wire in green fluorescent protein

The operation of the proton wire in Green Fluorescent Protein has been simulated by quantum dynamics and considering the coupling to the protein environment by means of a bath of harmonic oscillators. The simulation consists of 36 explicit and fully quantum degrees of freedom: 6 degrees of freedom represent the configuration of the proton wire, which are coupled to 30 bath coordinates. Regimes of weak and strong coupling have been studied. It is found that presence of the bath induces a fast energy transfer from the proton wire to the bath, with characteristic times under 400 fs. This internal vibrational redistribution happens at the expense of the potential energy content of the proton wire, deformed through the interaction to the bath from its uncoupled state. Strong coupling induces a slowing-down of the operation of the wire because it hinders to some extent the approaching of donor and acceptor atoms to distances in which proton transfer can occur. Internal vibrational energy redistribution affects the dynamics, but from our simulations we conclude that it cannot be the only cause responsible for the experimentally reported fluorescence rise times.     

The infrared-driven cis–trans isomerization of nitrous acid HONO III: A mixed quantum–classical simulation

A mixed quantum–classical simulation of the IR-driven cis–trans isomerization of HONO in a Kr matrix at 30 K is presented, treating the hydrogen atom as quantum particle and the Kr matrix as well as intermolecular degrees of freedom of the ONO-body as classical. A new method is presented to time-propagate the coupled set of equations in a DVR basis in internal spherical coordinates, rather than in laboratory frame fixed cartesian coordinates. In spherical coordinates, a much more precise computation of the weak vibrational couplings is possible using a still manageable basis size. Good qualitative agreement between simulation and experiment is obtained, underestimating relaxation and isomerization rates by a modest factor ≈5. Upon matrix fluctuations, frequent curve crossings occur between the initially excited OH-stretch vibration and a closely lying combination mode of torsional and bending coordinate that lead to a transfer of population. The subsequent pathway of energy flow is deduced and discussed within a tier model, where trans-states, that belong to the second tier, are populated through a first tier of states that is composed of combinations of bending and torsional excitations. No specific energy pathway is revealed that would funnel the hydrogen atom directly towards the trans-side, hence the experimentally observed high cis → trans quantum yield of close to one probably has to be explained in a statistical scenario on a timescale much longer than that of the present simulation.     

Quantum mechanical contributions to the structure of liquid water

Path integral Monte Carlo methods are used to study the effect of quantization of the orientational degrees of freedom of water (H₂O), using the ST2 model. A comparison of the classical and quantum atom—atom intermolecular correlation functions show that significant quantitative effects are manifest in the results.     

Incorporation of quantum effects for selected degrees of freedom into the trajectory-based dynamics using spatial domains

The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the "quantum" degrees of freedom describing light particles such as protons, while neglecting with the quantum force for the heavy, nearly classical nuclei. The entire system comprised of light and heavy particles is described by a single wavefunction of full dimensionality. The coordinate space of heavy particles is divided into spatial domains or subspaces. The quantum force acting on the light particles is determined for each domain of similar configurations of the heavy nuclei. This approach effectively introduces parametric dependence of the reduced dimensionality quantum force, on classical degrees of freedom. This strategy improves accuracy of the quantum force and does not restrict interaction between the domains. The concept is illustrated for two-dimensional scattering systems, where the quantum force is required to reproduce vibrational energy of the quantum degree of freedom.     

Quantum dynamics of dissociative chemisorption of CH(4) on Ni(111): Influence of the bending vibration

Two-dimensional, three-dimensional, and four-dimensional quantum dynamic calculations are performed on the dissociative chemisorption of CH(4) on Ni(111) using the multiconfiguration time-dependent Hartree (MCTDH) method. The potential energy surface used for these calculations is 15-dimensional (15D) and was obtained with density functional theory for points which are concentrated in the region that is dynamically relevant to reaction. Many reduced dimensionality calculations were already performed on this system, but the molecule was generally treated as pseudodiatomic. The main improvement of our model is that we try to describe CH(4) as a polyatomic molecule by including a degree of freedom describing a bending vibration in our three-dimensional and four-dimensional models. Using a polyspherical coordinate system, a general expression for the 15D kinetic energy operator is derived, which discards all the singularities in the operator and includes rotational and Coriolis coupling. We use seven rigid constraints to fix the CH(3) umbrella of the molecule to its gas phase equilibrium geometry and to derive two-dimensional, three-dimensional, and four-dimensional Hamiltonians, which were used in the MCTDH method. Only four degrees of freedom evolve strongly along the 15D minimum energy path: the distance of the center of mass of the molecule to the surface, the dissociative C[Single Bond]H bond distance, the polar orientation of the molecule, and the bending angle between the dissociative C[Single Bond]H bond and the umbrella. A selection of these coordinates is included in each of our models. The polar rotation is found to be important in determining the mode selective behavior of the reaction. Furthermore, our calculations are in good agreement with the finding of Xiang et al. [J. Chem. Phys. 117, 7698 (2002)] in their reduced dimensional calculation that the helicopter motion of the umbrella symmetry axis is less efficient than its cartwheel motion for promoting the reaction. The effect of pre-exciting the bend modes is qualitatively incorrect at higher energies, suggesting the necessity of including additional rotational and vibrational degrees of freedom in the model.     

A proton between two waters: insight from full-dimensional quantum-dynamics simulations of the [H2O-H-OH2]+ cluster

The dynamics of a proton between two water molecules is studied by full-dimensional (15 dimensional) quantum dynamics using the multiconfigurational time-dependent Hartree (MCTDH) method. The collision of H(3)O(+) and H(2)O fragments is followed by an ultrafast and nearly irreversible energy transfer from the degrees of freedom that define the hydrogen bond (oxygen-oxygen distance and central proton position) to the rest of the degrees of freedom. The vibrations of the oxygen-oxygen distance are damped within the first 300 fs while the vibrations of the shared proton along the hydrogen bond are damped within the first 150 to 200 fs. Collisions in which the fragments arrive with a high momentum to the interaction distance lead to more recrossing of the transferring proton than collisions with a lower momentum. Slow coordinates, e.g. pyramidalization of the water monomers, have less time to adapt to the incoming or outgoing proton in the case of a high momentum, which leads to an enhanced recrossing effect with respect to slower collisions. In order to understand the energy flow dynamics between the vibration of the shared proton and other degrees of freedom a 5-state model is constructed and exactly solved. The energies and couplings of the states of the model are obtained from the analysis of the infrared spectroscopy of the H(5)O(2)(+) cation, namely from splittings and shifts of the most important spectral lines. The model qualitatively reproduces the key aspects of the full dynamics related to the vibrations of the shared proton, indicating that the proposed coupling scheme is correct.     

Study of vibrational energy localization and redistribution in hydrogen peroxide H2O2 at low energy

Vibrational energy localization and/or redistribution in hydrogen peroxide H2O2 is studied at about 4000 cm(-1) above the quantum mechanical ground state using the ab initio potential energy surface of Koput, Carter, and Handy [J. Phys. Chem. A 102, 6325 (1998)]. In this work, the recently derived canonical perturbation procedure for floppy molecules serves two purposes. First, from the quantum mechanical point of view, it is shown that the energies of the lowest 130 states are reproduced with an average error smaller than 1.5 cm(-1) by a two-dimensional Hamiltonian, which is a function of the torsion and OO-stretch coordinates and momenta, while the other four degrees of freedom contribute only through powers of good quantum numbers. Moreover, the canonical perturbation procedure is also used in classical mechanics calculations, in order to define meaningful local modes, for which the ingoing and outgoing energy flows are monitored. Almost all the individual trajectories launched on the ab initio surface display the same behavior, that is, the superposition of (a) rapid (few hundreds of femtoseconds) and quasiperiodic energy exchanges between the two OH stretches and between the torsion and OO-stretch, and (b) slower (few to several picoseconds) but erratic-looking energy flows between all degrees of freedom. When averaging over large numbers of trajectories with the same local mode energies at time t=0, one observes instead a smooth and irreversible energy flow between all degrees of freedom, which usually thermalize in the range of several tens of picoseconds, that is, on time scales larger than the 5 ps period associated with the quantum density of states.     

Simulating quantum dynamics in classical environments

Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum–classical systems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approximation. The dynamics of these composite systems are governed by a quantum–classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of trajectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many-body environments.