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Register a new userInexpensive modelling of quantum dynamics using path integral generalized Langevin equation thermostats
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The properties of molecules and materials containing light nuclei are affected by their quantum mechanical nature. Modelling these quantum nuclear effects accurately requires computationally demanding path integral techniques. Considerable success has been achieved in reducing the cost of such simulations by using generalized Langevin dynamics to induce frequency-dependent fluctuations. Path integral generalized Langevin equation methods, however, have this far been limited to the study of static, thermodynamic properties due to the large perturbation to the systems dynamics induced by the aggressive thermostatting. Here we introduce a post-processing scheme, based on analytical estimates of the dynamical perturbation induced by the generalized Langevin dynamics, that makes it possible to recover meaningful time correlation properties from a thermostatted trajectory. We show that this approach yields spectroscopic observables for model and realistic systems which have an accuracy comparable to much more demanding approximate quantum dynamics techniques based on full path integral simulations.
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We report an improved method for the calculation of tunneling splittings between degenerate configurations in molecules and clusters using path-integral molecular dynamics (PIMD). Starting from an expression involving a ratio of thermodynamic density matrices at the bottom of the symmetric wells, we use thermodynamic integration with molecular dynamics simulations and a Langevin thermostat to compute the splittings stochastically. The thermodynamic integration is performed by sampling along the semiclassical instanton path, which provides an efficient reaction coordinate as well as being physically well-motivated. This approach allows us to carry out PIMD calculations of the multi-well tunnelling splitting pattern in water dimer, and to refine previous PIMD calculations for one-dimensional models and malonaldehyde. The large (acceptor) splitting in water dimer agrees to within 20% of benchmark variational results, and the smaller splittings are within 10%.
We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. We propose two preconditioned Langevin sampling dynamics, which are shown to have improved stability and sampling accuracy. We present a careful mode analysis of the preconditioned dynamics and show their connections to the normal mode, the staging coordinate and the Matsubara mode representation for ring polymers. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite-dimensional counterpart has a dimension-independent convergence rate. In addition, the preconditioning techniques can be naturally applied to the multi-level quantum systems in the nonadiabatic regime, which are compatible with various numerical approaches.
We introduce a novel approach for a fully quantum description of coupled electron-ion systems from first principles. It combines the variational quantum Monte Carlo (QMC) solution of the electronic part with the path integral (PI) formalism for the quantum nuclear dynamics. On the one hand, the PI molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational QMC can provide Born-Oppenheimer (BO) potential energy surfaces with a precision comparable to the most advanced post Hartree-Fock approaches, and with a favorable scaling with the system size. To deal with the intrinsic QMC noise, we generalize the PI molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of QMC nuclear forces. The variational parameters of the QMC wave function are evolved during the nuclear dynamics, such that the BO potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated even in presence of the Langevin thermostat, thanks to the mapping onto an Ornstein-Uhlenbeck process. This framework turns out to be very efficient also in the case of deterministic ionic forces. The new implementation is validated on the Zundel ion by direct comparison with standard PI Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond room temperature giving the excess proton an increased mobility by quantum tunneling.
Path reweighting is a principally exact method to estimate dynamic properties from biased simulations - provided that the path probability ratio matches the stochastic integrator used in the simulation. Previously reported path probability ratios match the Euler-Maruyama scheme for overdamped Langevin dynamics. Since MD simulations use Langevin dynamics rather than overdamped Langevin dynamics, this severely impedes the application of path reweighting methods. Here, we derive the path probability ratio $M_L$ for Langevin dynamics propagated by a variant of the Langevin Leapfrog integrator. This new path probability ratio allows for exact reweighting of Langevin dynamics propagated by this integrator. We also show that a previously derived approximate path probability ratio $M_{mathrm{approx}}$ differs from the exact $M_L$ only by $mathcal{O}(xi^4Delta t^4)$, and thus yields highly accurate dynamic reweighting results. ($Delta t$ is the integration time step, $xi$ is the collision rate.) The results are tested and the efficiency of path-reweighting is explored using butane as an example.
In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path integral description of the CME, and show how applying Gillespies two conditions to it directly leads to a path integral equivalent to the CLE. We compare this approach to the path integral equivalent of a large system size derivation, and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems, and are readily generalizable.
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