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.     

A version of diffusion Monte Carlo method based on random grids of coherent states. II. Six-dimensional simulation of electronic states of H2

We report a new version of the diffusion Monte Carlo (DMC) method, based on coherent-state quantum mechanics. Randomly selected grids of coherent states in phase space are used to obtain numerical imaginary time solutions of the Schrodinger equation, with an iterative refinement technique to improve the quality of the Monte Carlo grid. Accurate results were obtained, for the appropriately symmetrized two lowest states of the hydrogen molecule, by Monte Carlo sampling and six-dimensional propagation in the full phase space.     

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.     

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。     

Diffusion quantum Monte Carlo for equilibrium structures and harmonic frequencies of ethane and ozone molecules

Application of the Ornstein-Uhlenbeck diffusion quantum Monte Carlo method in combination with a trial wave function constructed from the floating spherical Gaussian orbitals and spherical Gaussian geminals to studies on the equilibrium structures and harmonic frequencies of ethane and ozone is presented. These Monte Carlo computed results are compared with those of experiments as well as the coupled cluster methods with the correlation consistent basis sets for the two molecules. For ozone, we also compare the Monte Carlo results with the results from multireference calculations.     

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.     

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.     

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 rare event sampling method for diffusion Monte Carlo using smart darting

We identify a set of multidimensional potential energy surfaces sufficiently complex to cause both the classical parallel tempering and the guided or unguided diffusion Monte Carlo methods to converge too inefficiently for practical applications. The mathematical model is constructed as a linear combination of decoupled Double Wells [(DDW)(n)]. We show that the set (DDW)(n) provides a serious test for new methods aimed at addressing rare event sampling in stochastic simulations. Unlike the typical numerical tests used in these cases, the thermodynamics and the quantum dynamics for (DDW)(n) can be solved deterministically. We use the potential energy set (DDW)(n) to explore and identify methods that can enhance the diffusion Monte Carlo algorithm. We demonstrate that the smart darting method succeeds at reducing quasiergodicity for n ? 100 using just 1 × 10(6) moves in classical simulations (DDW)(n). Finally, we prove that smart darting, when incorporated into the regular or the guided diffusion Monte Carlo algorithm, drastically improves its convergence. The new method promises to significantly extend the range of systems computationally tractable by the diffusion Monte Carlo algorithm.     

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.     

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.     

应用Monte Carlo方法计算He原子包含电子相关作用的基态能量

应用Monte Carlo方法计算He原子包含电子相关波函数的基态能量,获得了与精确值非常接近的结果．实践表明,应用Monte Carlo方法有可能在多电子体系中直接采用包含任意2个电子间距离ry的函数作为变分函数来考虑电子相关作用．     

Optimum and efficient sampling for variational quantum Monte Carlo

Quantum mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial wave functions, that is to variational quantum Monte Carlo. Almost all previous implementations employ samples distributed as the physical probability density of the trial wave function, and assume the central limit theorem to be valid. In this paper we provide an analysis of random error in estimation and optimization that leads naturally to new sampling strategies with improved computational and statistical properties. A rigorous lower limit to the random error is derived, and an efficient sampling strategy presented that significantly increases computational efficiency. In addition the infinite variance heavy tailed random errors of optimum parameters in conventional methods are replaced with a Normal random error, strengthening the theoretical basis of optimization. The method is applied to a number of first row systems and compared with previously published results.     

Rydberg states with quantum Monte Carlo

Calculations on Rydberg states are performed using quantum Monte Carlo methods. Excitation energies and singlet-triplet splittings are calculated for two model systems, the carbon atom (3P and 1P) and carbon monoxide ((1Sigma and 3Sigma). Kohn-Sham wave functions constructed from open-shell localized Hartree-Fock orbitals are used as trial and guide functions. The fixed-node diffusion quantum Monte Carlo (FN-DMC) method depends strongly on the wave function's nodal hypersurface. Nodal artefacts are investigated for the ground state of the carbon atom. Their effect on the FN-DMC results can be analyzed quantitatively. FN-DMC leads to accurate excitation energies but to less accurate singlet-triplet splittings. Variational Monte Carlo calculations are able to reproduce the experimental results for both the excitation energies and the singlet-triplet splittings.     

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.     

HF in clusters of molecular hydrogen: II. Quantum solvation by H2 isotopomers, cluster rigidity, and comparison with CO-doped parahydrogen clusters

We present a comprehensive theoretical study of the quantum solvation of the HF molecule by small clusters of the H2 isotopomers, p-H2, HD, and o-D2, with up to 13 hydrogen solvent molecules. This complements our earlier work on the HF-doped parahydrogen clusters [H. Jiang and Z. Bacic, J. Chem. Phys. 122, 244306 (2005)]. The ground-state properties of the clusters are calculated exactly using the diffusion Monte Carlo method. Detailed information is obtained regarding the size and isotopomer dependences of the energetics, vibrationally averaged structures, and their rigidity. The rigidity of these clusters is investigated further by analyzing the distributions of their principal moments of inertia from the diffusion Monte Carlo simulations. The clusters are found to be rather rigid, especially when compared with the pure parahydrogen clusters of the same size. Extensive comparison is made with the quantum Monte Carlo results for the CO-doped parahydrogen clusters and significant differences are observed in the size evolution of certain properties, notably the chemical potential.     

Electron affinities with diffusion quantum Monte Carlo for C2 and BO molecules

To verify the performance of the fixed-node diffusion quantum Monte Carlo method in electron affinities calculations, the adiabatic electron affinities of C(2) and BO molecules calculated by the fixed-node Ornstein-Uhlenbeck diffusion quantum Monte Carlo simulations guiding by trial wave functions constructed from the floating spherical Gaussian orbitals and spherical Gaussian geminals are presented in this work. The random walk based results, 3.264(43) and 2.507(32) eV for C(2) and BO, respectively, are compared with the available best experimental determined values, 3.269(6) and 2.508(8) eV for C(2) and BO, respectively, and the results of other theoretical calculations.     

Path integral ground state study of finite-size systems: application to small (parahydrogen)N (N=2-20) clusters

We use the path integral ground state method to study the energetic and structural properties of small para-H2 clusters of sizes ranging from 2 to 20 molecules. A fourth order formula is used to approximate the short imaginary-time propagator and two interaction potentials are considered. Our results are compared to those of exact basis set calculations and other quantum Monte Carlo methods when available. We find that for all cluster sizes considered, our results show a lower ground state energy than literature values obtained by diffusion Monte Carlo and variational Monte Carlo. For the dimer and trimer, ground state energies are in good agreement with exact results obtained using the discrete variable representation. Structural properties are found to be insensitive to the choice of interaction potential. We explore the use of Pekeris coordinates to analyze the importance of linear arrangement in trimers and for trimers within clusters of larger size.     

Self-optimizing Diffusion Quantum Monte Carlo Calculation: the Potential Energy Curve of C_2

11NTRODUCTIONDiffusionquantumMonteCarlo(DMC)isoneofthesimplestofthevariousMonteCarlotechniquesavailabletosolvetheSchrodingerequation,forarecentre-viewofDMC(seeRef[1i).Foravarietyofsmallatomsandmolecules,DMChasbeenshowntobecapableofprovidinganestimateoftheground-state(nonrelativistic)energywithanaccuracycomparabletogoodqualityClcalculations,evenwhenarelativelysimpletrialwavefunctionisemployed.However,todate,thereareseveralobstaclesinaDMCcalculation:(1)BeforetheDMCcomputation,thepar…