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.     

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.     

Path integral evaluation of the quantum instanton rate constant for proton transfer in a polar solvent

The quantum instanton approximation for thermal rate constants, a type of quantum transition state theory (QTST), is applied to a model proton transfer reaction in liquid methyl chloride developed by Azzouz and Borgis. Monte Carlo path integral methods are used to carry out the calculations, and two other closely related QTST's, namely, the centroid-density and Hansen-Andersen QTST, are also evaluated for comparison using the present path integral approach. A technique is then introduced that calculates the kinetic isotope effect directly via thermodynamic integration of the rate with respect to hydrogen mass, which has the practical advantage of avoiding costly evaluation of the activation free energy. The present application to the Azzouz-Borgis problem shows that the above three types of QTST provide very similar results for the rate, within 30% of each other, which is nontrivial considering the totally different derivations of these QTSTs; the latter rates are also in reasonable agreement with some other previous results (e.g., obtained via molecular dynamics with quantum transitions), within a factor of approximately 2(7) for the H(D) transfer, thus significantly diminishing the possible range of the exact rates. In addition, it is revealed that a small but nonnegligible inconsistency exists in the parametrization of the Azzouz-Borgis model employed in previous studies, which resulted in the large apparent discrepancy in the calculated rates.     

Path integral calculation of thermal rate constants within the quantum instanton approximation: application to the H + CH4 --> H2 + CH3 hydrogen abstraction reaction in full Cartesian space

The quantum instanton approximation for thermal rate constants of chemical reactions [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)], which is modeled after the earlier semiclassical instanton approach, is applied to the hydrogen abstraction reaction from methane by a hydrogen atom, H + CH4 --> H2 + CH3, using a modified and recalibrated version of the Jordan-Gilbert potential surface. The quantum instanton rate is evaluated using path integral Monte Carlo approaches based on the recently proposed implementation schemes [Yamamoto and Miller, J. Chem. Phys. 120, 3086 (2004)]. The calculations were carried out using the Cartesian coordinates of all the atoms (thus involving 18 degrees of freedom), thereby taking explicit account of rotational effects of the whole system and also allowing the equivalent treatment of the four methane hydrogens. To achieve such a treatment, we present extended forms of the path integral estimators for relevant quantities that may be used for general N-atom systems with any generalized reaction coordinates. The quantum instanton rates thus obtained for the temperature range T = 200-2000 K show good agreement with available experimental data, which gives support to the accuracy of the underlying potential surface used.     

Thermodynamic properties of small rare gas clusters by path integral Monte Carlo simulations: a preliminary study

A strategy for reducing the risk of non-ergodic simulations in Monte Carlo calculations of the thermodynamic properties of clusters is discussed with the support of some examples. The results obtained attest the significance of the approach for the low-temperature regime, as non-ergodic sampling of potential energy surfaces is a particularly insidious occurrence. Fourier path integral Monte Carlo techniques for taking into account quantum effects are adopted, in conjunction with suitable tricks for improving the procedure reliability. Applications are restricted to Lennard-Jones clusters of rare-gas systems.     

Improving the convergence of closed and open path integral molecular dynamics via higher order Trotter factorization schemes

Higher order factorization schemes are developed for path integral molecular dynamics in order to improve the convergence of estimators for physical observables as a function of the Trotter number. The methods are based on the Takahashi-Imada and Susuki decompositions of the Boltzmann operator. The methods introduced improve the averages of the estimators by using the classical forces needed to carry out the dynamics to construct a posteriori weighting factors for standard path integral molecular dynamics. The new approaches are straightforward to implement in existing path integral codes and carry no significant overhead. The Suzuki higher order factorization was also used to improve the end-to-end distance estimator in open path integral molecular dynamics. The new schemes are tested in various model systems, including an ab initio path integral molecular dynamics calculation on the hydrogen molecule and a quantum water model. The proposed algorithms have potential utility for reducing the cost of path integral molecular dynamics calculations of bulk systems.     

Hybrid Monte Carlo implementation of the Fourier path integral algorithm

This paper formulates a hybrid Monte Carlo implementation of the Fourier path integral (FPI-HMC) approach with partial averaging. Such a hybrid Monte Carlo approach allows one to generate collective moves through configuration space using molecular dynamics while retaining the computational advantages associated with the Fourier path integral Monte Carlo method. In comparison with the earlier Metropolis Monte Carlo implementations of the FPI algorithm, the present HMC method is shown to be significantly more efficient for quantum Lennard-Jones solids and suggests that such algorithms may prove useful for efficient simulations of a range of atomic and molecular systems.     

Path integral Monte Carlo method for ab initio calculation

The Feynman path integral method is applied to the many-electron problem. We first give new closure relations in terms of ordinary complex and real numbers, which could be derived from an arbitrary complete set of state vectors. Then, in the path integral form, the partition function of the system and the ensemble average of energy are explicitly expressed in terms of these closure relations. It is impossible to evaluate the path integral by direct numerical integrations because of its huge amount of integration variables. Therefore, we develop an algorithm by the Monte Carlo method with constraints corresponding to the normalization condition of states to calculate the required integral. Finally, the ensemble average of energy for the hydrogen molecule is explicitly evaluated by the quantum Monte Carlo method and results are compared with the result obtained by the ordinary full configuration interaction (CI) method. © 1996 John Wiley & Sons, Inc.     

Path-integral virial estimator for reaction-rate calculation based on the quantum instanton approximation

The quantum instanton approximation is a type of quantum transition-state theory that calculates the chemical reaction rate using the reactive flux correlation function and its low-order derivatives at time zero. Here we present several path-integral estimators for the latter quantities, which characterize the initial decay profile of the flux correlation function. As with the internal energy or heat-capacity calculation, different estimators yield different variances (and therefore different convergence properties) in a Monte Carlo calculation. Here we obtain a virial (-type) estimator by using a coordinate scaling procedure rather than integration by parts, which allows more computational benefits. We also consider two different methods for treating the flux operator, i.e., local-path and global-path approaches, in which the latter achieves a smaller variance at the cost of using second-order potential derivatives. Numerical tests are performed for a one-dimensional Eckart barrier and a model proton transfer reaction in a polar solvent, which illustrates the reduced variance of the virial estimator over the corresponding thermodynamic estimator.     

Extrapolation of the time-step bias in diffusion quantum Monte Carlo by a differential sampling technique

A new method of eliminating the finite-time-step error inherent in diffusion quantum Monte Carlo is presented, utilizing an improved version of the existing differential techniques. An implementation is described and results of several small but representative calculations are discussed. The pertinent computation requirements on these systems were reduced by up to a factor of five by the new algorithm. It is speculated that this method may be easily applied to other quantum Monte Carlo and discretized path integral Monte Carlo techniques having related finite step-size errors with a possibility of obtaining similar good results.     

Nuclear quantum effect on the hydrogen-bonded structure of guanine�Ccytosine pair

The structure of Watson?CCrick type guanine?Ccytosine (G?CC) base pair has been studied by classical hybrid Monte Carlo (HMC) and quantum path integral hybrid Monte Carlo (PIHMC) simulations on the semiempirical PM6 potential energy surface. For the three NH?X hydrogen-bonded moieties, the intramolecular NH bonds are found systematically longer while the H?X distance shorter in the PIHMC simulation than in the HMC simulation. We found that the hydrogen bonded length N?X correlates with the H?X distance, but not with the NH distance. A correlation is also between the neighboring hydrogen bonds in the G?CC base pair.     

Phase changes in selected Lennard-Jones X13-nYn clusters

Detailed studies of the thermodynamic properties of selected binary Lennard-Jones clusters of the type X13-nYn (where n=1, 2, 3) are presented. The total energy, heat capacity, and first derivative of the heat capacity as a function of temperature are calculated by using the classical and path integral Monte Carlo methods combined with the parallel tempering technique. A modification in the phase change phenomena from the presence of impurity atoms and quantum effects is investigated.     

A path integral influence functional for excess electron in fluids: Density-functional formulation

In this paper, we propose a path integral influence functional from a solvent to determine a self-correlation function of a quantum particle in classical simple fluid. It is shown that the influence functional is related to a grand potential functional of the pure solvent under a three-dimensional external field arising from a classical isomorphic polymer, on which the quantum particle is mapped. The influence functional can be calculated from the self-correlation function, the solute-solvent and the solvent-solvent pair correlation function. The obtained equation of the self-correlation function is applied to an excess electron problem in fluid helium. The Fourier path-integral Monte Carlo method is employed to perform the path integral of the electron. The solute-solvent pair correlation function is estimated from a reference interaction site model integral equation. These results obtained form our proposed influence functional and from that proposed by Chandler, Singh, and Richardson are compared with those provided by a path integral Monte Carlo simulation with the explicit helium solvent.     

Rotational fluctuation of molecules in quantum clusters. I. Path integral hybrid Monte Carlo algorithm

In this paper, we present a path integral hybrid Monte Carlo (PIHMC) method for rotating molecules in quantum fluids. This is an extension of our PIHMC for correlated Bose fluids [S. Miura and J. Tanaka, J. Chem. Phys. 120, 2160 (2004)] to handle the molecular rotation quantum mechanically. A novel technique referred to be an effective potential of quantum rotation is introduced to incorporate the rotational degree of freedom in the path integral molecular dynamics or hybrid Monte Carlo algorithm. For a permutation move to satisfy Bose statistics, we devise a multilevel Metropolis method combined with a configurational-bias technique for efficiently sampling the permutation and the associated atomic coordinates. Then, we have applied the PIHMC to a helium-4 cluster doped with a carbonyl sulfide molecule. The effects of the quantum rotation on the solvation structure and energetics were examined. Translational and rotational fluctuations of the dopant in the superfluid cluster were also analyzed.     

Nuclear quantum effect and temperature dependency on the hydrogen‐bonded structure of base pairs

The structure of Watson–Crick‐type adenine‐thymine and guanine‐cytosine pairs has been studied by hybrid Monte Carlo (HMC) and path integral hybrid Monte Carlo (PIHMC) simulations with the use of semiempirical PM6‐DH+ method in the gas phase. We elucidated the nuclear quantum effect and temperature dependency on the hydrogen‐bonded moiety of base pairs. It was shown that the contribution of nuclear quantum effect on the hydrogen‐bonded structure is significant not only at low temperature 150 K but also at temperature as high as 450 K. The relative position of hydrogen‐bonded proton between two heavy atoms and the nuclear quantum nature of the proton are also shown. Furthermore, we have applied principal component analysis to HMC and PIHMC simulations to analyze the nuclear quantum effect on intermolecular motions. We found that the ratio of Buckle mode (lowest vibrational mode from normal mode analysis) decreases due to the nuclear quantum effect, whereas that of Propeller mode (second lowest vibrational mode) increases. In addition, nonplanar structures of base pairs were found to become stable due to the nuclear quantum effect from two‐dimensional free energy landscape along Buckle and Propeller modes. © 2013 Wiley Periodicals, Inc.     

Algorithmic differentiation and the calculation of forces by quantum Monte Carlo

We describe an efficient algorithm to compute forces in quantum Monte Carlo using adjoint algorithmic differentiation. This allows us to apply the space warp coordinate transformation in differential form, and compute all the 3M force components of a system with M atoms with a computational effort comparable with the one to obtain the total energy. Few examples illustrating the method for an electronic system containing several water molecules are presented. With the present technique, the calculation of finite-temperature thermodynamic properties of materials with quantum Monte Carlo will be feasible in the near future.     

The phase diagram of water from quantum simulations

The phase diagram of water has been calculated from the TIP4PQ/2005 model, an empirical rigid non-polarisable model. The path integral Monte Carlo technique was used, permitting the incorporation of nuclear quantum effects. The coexistence lines were traced out using the Gibbs-Duhem integration method, once having calculated the free energies of the liquid and solid phases in the quantum limit, which were obtained via thermodynamic integration from the classical value by scaling the mass of the water molecule. The resulting phase diagram is qualitatively correct, being displaced to lower temperatures by 15-20 K. It is found that the influence of nuclear quantum effects is correlated to the tetrahedral order parameter.     

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

We construct improved quantum Monte Carlo estimators for the spherically and system-averaged electron pair density (i.e., the probability density of finding two electrons separated by a relative distance u), also known as the spherically averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth nonlocal operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.     

A fast variational Gaussian wavepacket method: size-induced structural transitions in large neon clusters

The variational Gaussian wavepacket (VGW) approximation provides an alternative to path integral Monte Carlo for the computation of thermodynamic properties of many-body systems at thermal equilibrium. It provides a direct access to the thermal density matrix and is particularly efficient for Monte Carlo approaches, as for an N-body system it operates in a non-inflated 3N-dimensional configuration space. Here, we greatly accelerate the VGW method by retaining only the relevant short-range correlations in the (otherwise full) 3N × 3N Gaussian width matrix without sacrificing the accuracy of the fully coupled VGW method. This results in the reduction of the original O(N(3)) scaling to O(N(2)). The fast-VGW method is then applied to quantum Lennard-Jones clusters with sizes up to N = 6500 atoms. Following Doye and Calvo [JCP 116, 8307 (2002)] we study the competition between the icosahedral and decahedral structural motifs in Ne(N) clusters as a function of N.     

Quantum Monte Carlo simulations of selected ammonia clusters (n = 2-5): isotope effects on the ground state of typical hydrogen bonded systems

Variational Monte Carlo, diffusion Monte Carlo, and stereographic projection path integral simulations are performed on eight selected species from the (NH(3))(n), (ND(3))(n), (NH(2)D)(n), and (NH(3))(n-1)(ND(3)) clusters. Each monomer is treated as a rigid body with the rotation spaces mapped by the stereographic projection coordinates. We compare the energy obtained from path integral simulations at several low temperatures with those obtained by diffusion Monte Carlo, for two dimers, and we find that at 4 K, the fully deuterated dimer energy is in excellent agreement with the ground state energy of the same. The ground state wavefunction for the (NH(3))(2-5) clusters is predominantly localized in the global minimum of the potential energy. In all simulations of mixed isotopic substitutions, we find that the heavier isotope is almost exclusively the participant in the hydrogen bond.