An optimized initialization algorithm to ensure accuracy in quantum Monte Carlo calculations

Quantum Monte Carlo (QMC) calculations require the generation of random electronic configurations with respect to a desired probability density, usually the square of the magnitude of the wavefunction. In most cases, the Metropolis algorithm is used to generate a sequence of configurations in a Markov chain. This method has an inherent equilibration phase, during which the configurations are not representative of the desired density and must be discarded. If statistics are gathered before the walkers have equilibrated, contamination by nonequilibrated configurations can greatly reduce the accuracy of the results. Because separate Markov chains must be equilibrated for the walkers on each processor, the use of a long equilibration phase has a profoundly detrimental effect on the efficiency of large parallel calculations. The stratified atomic walker initialization (STRAW) shortens the equilibration phase of QMC calculations by generating statistically independent electronic configurations in regions of high probability density. This ensures the accuracy of calculations by avoiding contamination by nonequilibrated configurations. Shortening the length of the equilibration phase also results in significant improvements in the efficiency of parallel calculations, which reduces the total computational run time. For example, using STRAW rather than a standard initialization method in 512 processor calculations reduces the amount of time needed to calculate the energy expectation value of a trial function for a molecule of the energetic material RDX to within 0.01 au by 33%.     

Random number generators tested on quantum Monte Carlo simulations

We have tested and compared several (pseudo) random number generators (RNGs) applied to a practical application, ground state energy calculations of molecules using variational and diffusion Monte Carlo metheds. A new multiple recursive generator with 8th‐order recursion (MRG8) and the Mersenne twister generator (MT19937) are tested and compared with the RANLUX generator with five luxury levels (RANLUX‐[0–4]). Both MRG8 and MT19937 are proven to give the same total energy as that evaluated with RANLUX‐4 (highest luxury level) within the statistical error bars with less computational cost to generate the sequence. We also tested the notorious implementation of linear congruential generator (LCG), RANDU, for comparison. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

A sparse algorithm for the evaluation of the local energy in quantum Monte Carlo

A new algorithm is presented for the sparse representation and evaluation of Slater determinants in the quantum Monte Carlo (QMC) method. The approach, combined with the use of localized orbitals in a Slater-type orbital basis set, significantly extends the size molecule that can be treated with the QMC method. Application of the algorithm to systems containing up to 390 electrons confirms that the cost of evaluating the Slater determinant scales linearly with system size.     

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。     

A new algorithm for the fixed-node quantum Monte Carlo method

A novel algorithm is proposed for the fixed-node quantum Monte Carlo (FNQMC) method.In contrast to previous procedures,its "guiding function" is not optimized prior to diffusion quantum Monte Carlo (DMC) computation but synchronistically in the diffusion process The new algorithm can not only save CPU time,but also make both of the optimization and diffusion carried out according to the same sampling fashion,reaching the goal to improve each other This new optimizing procedure converges super-linearly,and thus can accelerate the particle diffusion During the diffusion process,the node of the "guiding function" changes incessantly,which is conducible to reducing the "fixed-node error" The new algorithm has been used to calculate the total energies of states X3B1 and a1A1 of CH2 as well as π-X2B1 and λ-2A1 of NH2 The singlet-triplet energy splitting (λEsT) in CH2 and π energy splitting in NH2 obtained with this present method are (45 542±1.840) and (141.644±1.589) kJ/mol,respectively The calculated     

量子Monte Carlo处理激发态

提出了用于电子激发态的剩余函数变分量子ＭｏｎｔｅＣａｒｌｏ（ＳＦＶＭＣ）方法,已经证明：若激发态的初始波函数与基态的初始波函数属于对称性不同的不可约表示时,该激发态的ＳＦＶＭＣ方法与基态的ＳＦＶＭＣ方法完全相同;若激发态的初始波函数与基态的初始波函数有相同的对称性时,只要对激发态的初始波函数作正交性修正,则其态的ＳＦＶＭＣ方法亦可推到该激发态的情况。文章导出了这第二类激发态的ＳＦＶＭＣ方法的详细计     

Quantum Monte Carlo Calculation of Correlation Effects on Bond Orders

We present a computational approach, using quantum Monte Carlo, that provides some insight into the effect of electron correlation on chemical bonding between individual pairs of atoms. Our approach rests upon a recently suggested relation between the bond order and charge fluctuations with respect to atomic domains. Within the present implementation we have taken a compromise between conceptual rigour and computational simplicity. In a first step atomic domains were obtained from Hartree-Fock (HF) densities, using Bader’s definition of atoms in molecules. These domains were used in a second step in quantum Monte Carlo calculations to determine bond orders for pairs of atoms. Correlation effects have been studied by comparison of HF bond orders with those obtained from pure diffusion quantum Monte Carlo calculations. We illustrate this concept for C–O and C–S bonds in different molecular environments. Our results suggest an approximate linear relation between bond order and bond length for these kinds of bonds.     

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     

一种高效的优化量子Monte Carlo波函数的新方法

提出了一个优化量子MonteCarlo波函数的新方法，与前人的方法相比，它不使用变分原理而是极小化Schrodinger方程的剩余量；不按Ψ2取样而是按Ψ2（EL－E）2取样；不用差分而是用分析导数；不使用传统的速降法而是使用一个步长自动调节的下降法，它具有拟牛顿性质，因而是超线性收敛的．由于这四项新技术的使用，使得波函数的优化快速、准确，收敛速度比前人提出的方法快3～5倍．     

Efficiency of nested Markov chain Monte Carlo for polarizable potentials and perturbed Hamiltonians

Nested Markov chain Monte Carlo is a rigorous way to enhance sampling of a given energy landscape using an auxiliary, approximate potential energy surface. Its practical efficiency mainly depends on how cheap and how different are the auxiliary potential with respect to the reference system. In this article, a combined efficiency index is proposed and assessed for two important families of energy surfaces. As illustrated for water clusters, many‐body polarizable potentials can be approximated by simplifying the polarization contribution and keeping only the two‐body terms. In small systems, neglecting polarization entirely is also acceptable. When the reference potential energy is obtained from diagonalization of a quantum mechanical Hamiltonian, a first‐order perturbation scheme can be used to estimate the energy difference occuring on a Monte Carlo move. Our results indicate that this perturbation approximation performs well provided that the number of steps between successive diagonalization is adjusted beforehand. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem 110:2342–2346, 2010     

A benchmark quantum Monte Carlo study of the ground state chromium dimer

We report calculations of the ground state energy and binding curve of the chromium dimer using the variational and diffusion quantum Monte Carlo (VMC and DMC) methods. We examined various single‐determinant and multideterminant wavefunctions multiplied by a Jastrow factor as a trial/guiding wavefunction for VMC/DMC. The molecular orbitals in the single determinants were calculated using restricted or unrestricted Hartree–Fock or density functional theory (DFT) calculations where five commonly used local (SVWN5), semilocal (PW91 and BLYP), and hybrid (B1LYP and B3LYP) functionals were examined. The multideterminant expansions were obtained from the generalized valence bond and (truncated) unrestricted configuration interaction with single and double excitations (UCISD) methods. We also examined a UCISD wavefunction in which UCISD expansions were added to the UB3LYP single‐determinant reference, and their coefficients were optimized at the VMC level. In addition to the wavefunction dependence, the effects of pseudopotentials and backflow transformation were also investigated. The UB3LYP single‐determinant and multideterminant wavefunctions were found to give the variationally best DMC energies within the framework of single‐determinant and multideterminants, respectively, though both the DMC energies were higher than twice the DMC atomic energy. Some of the VMC binding curves show a flat or quite shallow well bottom, which gets recovered deeper by DMC. All the DMC binding curves have a minimum indicating a bound state, but the unrestricted ones overestimate the equilibrium bond length. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Monte Carlo correction programme for PC

A Monte Carlo correction program for quantitative microanalysis on PC computer is introduced in this paper. The elastic scattering is described by the screened Rutherford cross section. Instead of computing the energy loss according to the actual path between two scatterings we have defined the Bethe inelastic cross section determined by the Bethe-slowing-down approximation. It is assumed that it causes no angular departure of the scattered electron. In the second model we took into account the angular dependence of inelastic scattering assuming that the primary electron interacts with quasi-free atom electrons. On the basis of these two models analytical Monte Carlo programmes were developed and experimentally tested on some oxide glass. Our results are fully comparable to those obtained by ten world microprobe laboratories using classical ZAF correction or Bence-Albee methods. We have found that introducing angular part of the inelastic cross section analytical results did not significantly change. All of our results were carried out for bulk specimens but extending it to layers is under the development.     

Monte Carlo simulations for predicting the precision of results and for optimizing data acquisition schedules

Monte Carlo simulations can be used to determine the precision of an analytical method if the standard deviations of the component unit operations are estimated accurately. Alternative methods for estimating the standard deviation have been compared by evaluating the success of Monte Carlo simulations to predict the precision of a second-order rate constant determined by spectrophotometry and of an equivalent weight and acid dissociation constant determined by potentiometry. Monte Carlo simulation has also been used with simplex optimization to predict a data acquisition schedule which gives high precision in the equivalent weight determination. By comparison with a naive design, a 22-fold improvement was predicted. A 15-fold improvement was observed experimentally.     

Molecule‐adapted basis sets optimized with a quantum Monte Carlo method

The Monte Carlo simulated annealing method is adapted to optimize correlated Gaussian‐type functions in nonrelativistic molecular environments. Starting from an atom‐centered atomic Gaussian basis set, the uncontracted functions are reoptimized in the molecular environments corresponding to the H₂O, CN^?, N₂, CO, BF, NO⁺, CO₂, and CS systems. These new molecular adapted basis sets are used to calculate total energies, harmonic vibrational frequencies, and equilibrium geometries at a correlated level of theory. The present methodology is a simple and effective way to improve molecular correlated wave functions, without the need to enlarge the molecular basis set. Additionally, this methodology can be used to generate hierarchical sequences of molecular basis sets with increasing size, which are relevant to establish complete basis set limits. © 2014 Wiley Periodicals, Inc.     

A Parallelised High Performance Monte Carlo Simulation Approach for Complex Polymerisation Kinetics

A novel, parallelised approach to Monte Carlo simulations for the computation of full molecular weight distributions (MWDs) arising from complex polymerisation reactions is presented. The parallel Monte Carlo method constitutes perhaps the most comprehensive route to the simulation of full MWDs of multiple chain length polymer entities and can also provide detailed microstructural information. New fundamental insights have been developed with regard to the Monte Carlo process in at least three key areas: (i) an insufficient system size is demonstrated to create inaccuracies via poor representation of the most improbable events and least numerous species; (ii) advanced algorithmic principles and compiler technology known to computer science have been used to provide speed improvements and (iii) the parallelisability of the algorithm has been explored and excellent scalability demonstrated. At present, the parallel Monte Carlo method presented herein compares very favourably in speed with the latest developments in the h‐p Galerkin method‐based PREDICI software package while providing significantly more detailed microstructural information. It seems viable to fuse parallel Monte Carlo methods with those based on the h‐p Galerkin methods to achieve an optimum of information depths for the modelling of complex macromolecular kinetics and the resulting microstructural information.

     

Algorithm and application of Monte Carlo simulation for multi-dispersive copolymerization system

A Monte Carlo algorithm has been established for multi-dispersive copolymerization system, based on the experimental data of copolymer molecular weight and dispersion via GPC measurement. The program simulates the insertion of every monomer unit and records the structure and microscopical sequence of every chain in various lengths. It has been applied successfully for the ring-opening copolymerization of 2,2-dimethyltrimethylene carbonate (DTC) with δ-caprolactone (δ-CL). The simulation coincides with the experimental results and provides microscopical data of triad fractions, lengths of homopolymer segments, etc., which are difficult to obtain by experiments. The algorithm presents also a uniform frame for copolymerization studies under other complicated mechanisms.     

A scalable parallel Monte Carlo method for free energy simulations of molecular systems

We present a method of parallelizing flat histogram Monte Carlo simulations, which give the free energy of a molecular system as an output. In the serial version, a constant probability distribution, as a function of any system parameter, is calculated by updating an external potential that is added to the system Hamiltonian. This external potential is related to the free energy. In the parallel implementation, the simulation is distributed on to different processors. With regular intervals the modifying potential is summed over all processors and distributed back to every processor, thus spreading the information of which parts of parameter space have been explored. This implementation is shown to decrease the execution time linearly with added number of processors.     

Monte Carlo simulation for the micellar behavior of amphiphilic comb-like copolymers

Micellar behaviors in 2D and 3D lattice models for amphiphilic comb-like copolymers in water phase and in water/oil mixtures were simulated. A dynamical algorithm together with chain reptation movements was used in the simulation. Three-dimension displaying program was pro-grammed and free energy was estimated by Monte Carlo technigue. The results demonstrate that reduced interaction energy influences morphological structures of micelle and emulsion stems greatly; 3D simulation showing can display more direct images of morphological structures; the amphiphilic comb-like polymers with a hydrophobic main chain and hydrophilic side chains have lower energy in water than in oil.