Efficient estimators for quantum instanton evaluation of the kinetic isotope effects: application to the intramolecular hydrogen transfer in pentadiene

The quantum instanton approximation is used to compute kinetic isotope effects for intramolecular hydrogen transfer in cis-1,3-pentadiene. Due to the importance of skeleton motions, this system with 13 atoms is a simple prototype for hydrogen transfer in enzymatic reactions. The calculation is carried out using thermodynamic integration with respect to the mass of the isotopes and a path integral Monte Carlo evaluation of relevant thermodynamic quantities. Efficient "virial" estimators are derived for the logarithmic derivatives of the partition function and the delta-delta correlation functions. These estimators require significantly fewer Monte Carlo samples since their statistical error does not increase with the number of discrete time slices in the path integral. The calculation treats all 39 degrees of freedom quantum mechanically and uses an empirical valence bond potential based on a molecular mechanics force field.     

用变分Monte Carlo方法处理分子

对变分量子Ｍｏｎｔｅ Ｃａｒｔｏ方法提出了一种种算法：将传统的Ｈａｒｔｒｅｅ－Ｆｏｅｋ方法与量子Ｍｏｎｔｅ Ｃａｒｌｏ方法有机结合在一起；导出了“局部能”的解析式；使用了一种新的相关函数和随机数发生器。我们用这个新算法计算了Ｈ２、ＬｉＨ、Ｌｉ２、Ｈ２Ｏ、Ｆ２分子的基态和ＣＨ２分子的＾３Ｂ１、＾１Ａ１态的能量。计算结果表明，这个新算法在精度和统计误差两个方面比一般ＶＭＣ过程都要好得多。     

Monte Carlo computation of ground‐state energy derivatives

Monte Carlo is a simple technique, which uses random numbers to compute ground‐state energies of small molecules (and quantum systems in general). The results always have a small statistical error, which poses a major obstacle when estimating properties defined as ground‐state‐energy derivatives (such as the molecule's geometry, its vibrational frequencies, polarizabilities, etc.). In this article, we present and demonstrate an approach that makes an accurate Monte–Carlo estimation of such derivatives possible. This is achieved by realizing that the simulation constitutes an autocorrelated stochastic process, whose proper analysis then enables us to estimate various energy derivatives as a combination of total correlation between readily computable quantities. The resulting procedure is a natural extension of the usual Monte Carlo algorithm for computing the ground‐state energy, with relatively small computational overhead. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Prediction of sound velocity and compressibility via molecular simulation at fixed entropy

A molecular-level simulation route is proposed to compute the isentropic thermodynamic properties in a fluid system by Monte Carlo simulation at fixed entropy. The method involves computation of the pressure response of a system to an infinitesimal change in system density by introduction of a single molecule, while retaining the system volume as well as the absolute molar entropy. The probability for accepting a change in temperature during the Monte Carlo moves was weighted against the argument proposed by Smith et al. [W.R. Smith, M. Lísal, I. Nezbeda, Chem. Phys. Lett. 426 (2006) 436–440]. Application to fluid argon has confirmed superior accuracy for the technique within the gas state to yield results within 1.2% of the measured values for the range of thermodynamic conditions investigated.     

Calculating expectations with time-dependent perturbations in quantum Monte Carlo

We show that a small perturbation periodic in imaginary time can be used to compute expectation values of nondifferential operators that do not commute with the Hamiltonian within the framework of quantum diffusion Monte Carlo. Some results for the harmonic oscillator and the helium atom are presented showing the validity of the proposed method.     

Monte Carlo determination of the oscillator strength and excited state lifetime for the Li 22S → 22P transition

A recently developed Monte Carlo method is used to compute the 2²S → 2²P transition dipole moment of Li. This approach employs a guided Metropolis random walk with quantum Monte Carlo “side” walks to sample the required probability distributions. The transition dipole moment is employed to obtain the oscillator strength and excited-state lifetime. Our most accurately converged calculations yield an oscillator strength of 0.742(7) and excited-state lifetime of 27.41(35) ns. These results are in excellent agreement with precise experimental measurements of 0.742(1) and 27.29(4) ns, respectively. In addition, single-state expectation values are computed for both states. Monte Carlo parameters, such as the time step size and the convergence time, are varied in order to study their effect on computed results.     

Quantum Monte Carlo calculations of the potential energy curve of the helium dimer

We report results of two quantum Monte Carlo methods -- variational Monte Carlo and diffusion Monte Carlo -- on the potential energy curve of the helium dimer. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all.     

A method to compute probability current in generic coordinates

A method to compute probability current and its surface integral, the total flux, for systems of many particles of different masses is presented, based on transforming the wave function and its gradient onto a mass-weighted coordinate system. As a test for this methodology, it has been applied to a nontrivial 6-dimensional quantum dynamics study of a model of the operation of the proton-wire in Green Fluorescent Protein [O. Vendrell, R. Gelabert, M. Moreno, and J. M. Lluch, J. Phys. Chem. B, 112, 5500-5511 (2008)]. An adaptive Monte Carlo method is proposed, with favorable scaling properties for future applications, to solve the flux integral. Comparison of total reactive flux with the time derivative of the survival probability is satisfactory, corroborating the adequacy of the derivation. Using the new method the flux can quantitatively be divided into its positive and negative contributions, or more relevantly, into tunneling and classical parts.     

精确固定节面量子Monte：Carlo差值法

本文提出了精确固定节面量子Monte Carlo差值法，这个新算法能够在精确固定节面量子Monte Carlo方法的基础上直接计算两个体系之间的能量差，且使计算结果的统计误差达到10-2 kJ/mol 数量级，获得电子相关能90%以上。我们把这个新算法应用于分子势能面的研究中，使用一个“刚性移动”模型，利用Jacobi变换使分子两个几何构型的能量计算具有很好的正相关性，因而能得到准确的能量差值，于是精确的分子势能面就可以得到。这个新算法已经被使用到BH分子基态势能曲线和H₃分子势能面的研究。这个算法还可应用于分子光谱、化学反应能量变化值等领域的研究。     

剩余函数量子Monte Carlo方法

为量子Ｍｏｎｔｅ Ｃａｒｌｏ方法提出一条新途径－剩余函数法,引入了Ｓｃｈｒｏｅｄｉｎｇｅｒ方程剩余函数的概念,利用剩余函数将一种新的有明显物理意义的试探函数应用到量子Ｍｏｎｔｅ Ｃａｒｌｏ过程中,这种试探函数是通过一种迭进式的方式确定的,它不需要在Ｍｏｎｔｅ Ｃａｒｌｏ过程中优化参数。文中我们将给出这种试探函数的具体形式,证明由这种试探函数求出的能量期望值收敛于体系真实的能量值;文中还给出这种试探     

Electron pair localization function: a practical tool to visualize electron localization in molecules from quantum Monte Carlo data

In this work we introduce an electron localization function describing the pairing of electrons in a molecular system. This function, called "electron pair localization function," is constructed to be particularly simple to evaluate within a quantum Monte Carlo framework. Two major advantages of this function are the following: (i) the simplicity and generality of its definition; and (ii) the possibility of calculating it with quantum Monte Carlo at various levels of accuracy (Hartree-Fock, multiconfigurational wave functions, valence bond, density functional theory, variational Monte Carlo with explicitly correlated trial wave functions, fixed-node diffusion Monte Carlo, etc). A number of applications of the electron pair localization function to simple atomic and molecular systems are presented and systematic comparisons with the more standard electron localization function of Becke and Edgecombe are done. Results illustrate that the electron pair localization function is a simple and practical tool for visualizing electronic localization in molecular systems.     

Gamma distribution model to provide a direct assessment of the overall quality of quantum Monte Carlo-generated electron distributions

Our objective is to assess the accuracy of simulated quantum Monte Carlo electron distributions of atoms and molecules. Our approach is first to model the exact electron distribution by a linear combination of gamma distribution functions, with parameters chosen to exactly reproduce highly accurate literature values for a number of selected moments for the system of interest. In application to the ground-state electron distributions of helium and dihydrogen, a high level of accuracy of the model was confirmed upon comparing its predicted moments, not used in the model's parametrization, to those calculated from high-level theory. Next, we generated electron-electron and electron-nucleus distributions for dihydrogen from electron positions outputted from a variety of quantum Monte Carlo algorithms. Upon juxtaposition of the simulated distributions with the putatively exact one that we derived from the model, we quantified the error in simulated distributions. The most accurate distributions were obtained from no-compromise reptation quantum Monte Carlo, a recently developed algorithm designed to ameliorate the distributions' time-step bias. Marginally less accurate distributions were generated from fixed-node diffusion Monte Carlo with descendant counting and detailed balance.     

A diffusion quantum Monte Carlo study of geometries and harmonic frequencies of molecules

This article describes an approach in determination of equilibrium geometries and harmonic frequencies of molecules by the Ornstein-Uhlenbeck diffusion quantum Monte Carlo method based on the floating spherical Gaussians. In conjunction with a projected and renormalized Hellmann-Feynman gradient and an electronic energy at variational Monte Carlo and diffusion quantum Monte Carlo, respectively, the quasi-Newton algorithm implemented with the Broyden-Fletcher-Goldfarb-Shanno updated Hessian was used to find the optimized molecular geometry. We applied this approach to N2 and H2O molecules. The geometry and harmonic frequencies calculated were consistent with some sophisticated ab initio calculated values within reasonable statistical uncertainty.     

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.     

Computational studies on thermodynamic properties, effective diameters, and free volume of argon using an ab initio potential

A quantum mechanical derived ab initio interaction potential for the argon dimer was tested in molecular simulations to reproduce the thermophysical properties of the vapor-liquid phase equilibria using the Gibbs ensemble Monte Carlo simulations as well as the liquid and supercritical equation of state using the NVT Monte Carlo simulations. The ab initio interaction potential was taken from the literature. A recently developed theory [R. Laghaei et al., J. Chem. Phys. 124, 154502 (2006)] was used to compute the effective diameters of argon in fluid phases and the results were subsequently applied in the generic van der Waals theory to compute the free volume of argon. The calculated densities of the coexisting phases, the vapor pressure, and the equation of state show excellent agreement with experimental values. The effective diameters and free volumes of argon are given over a wide range of densities and temperatures. An empirical formula was used to fit the effective diameters as a function of density and temperature. The computed free volume will be used in future investigations to calculate the transport properties of argon.     

Quantum Monte Carlo for electronic excitations of free-base porphyrin

Accurate calculations of allowed and nonallowed transitions in porphyrin are reported. Using the quantum Monte Carlo method in the diffusion Monte Carlo variant, the vertical transition between the ground state singlet and the second excited state singlet as well as the adiabatic transition between the ground state and the lowest triplet state have been computed for this 162-electron system. The present theoretical results are compared to experiment and to results of other theoretical methods. The diffusion Monte Carlo energy differences are found to be in excellent agreement with experiment.     

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.     

Computational methods in coupled electron-ion Monte Carlo simulations.

In the last few years, we have been developing a Monte Carlo simulation method to cope with systems of many electrons and ions in the Born-Oppenheimer approximation: the coupled electron-ion Monte Carlo method (CEIMC). Electronic properties in CEIMC are computed by quantum Monte Carlo rather than by density functional theory (DFT) based techniques. CEIMC can, in principle, overcome some of the limitations of the present DFT-based ab initio dynamical methods. The new method has recently been applied to high-pressure metallic hydrogen. Herein, we present a new sampling algorithm that we have developed in the framework of the reptation quantum Monte Carlo method chosen to sample the electronic degrees of freedom, thereby improving its efficiency. Moreover, we show herein that, at least for the case of metallic hydrogen, variational estimates of the electronic energies lead to an accurate sampling of the proton degrees of freedom.