Comparison of a quantum random number generator with pseudorandom number generators for their use in molecular Monte Carlo simulations

Four pseudorandom number generators were compared with a physical, quantum‐based random number generator using the NIST suite of statistical tests, which only the quantum‐based random number generator could successfully pass. We then measured the effect of the five random number generators on various calculated properties in different Markov‐chain Monte Carlo simulations. Two types of systems were tested: conformational sampling of a small molecule in aqueous solution and liquid methanol under constant temperature and pressure. The results show that poor quality pseudorandom number generators produce results that deviate significantly from those obtained with the quantum‐based random number generator, particularly in the case of the small molecule in aqueous solution setup. In contrast, the widely used Mersenne Twister pseudorandom generator and a 64‐bit Linear Congruential Generator with a scrambler produce results that are statistically indistinguishable from those obtained with the quantum‐based random number generator. © 2017 Wiley Periodicals, Inc.     

Zori 1.0: a parallel quantum Monte Carlo electronic structure package

The Zori 1.0 package for electronic structure computations is described. Zori performs variational and diffusion Monte Carlo computations as well as correlated wave function optimization. This article presents an overview of the implemented methods and code capabilities.     

Quality of random number generators significantly affects results of Monte Carlo simulations for organic and biological systems

We have simulated pure liquid butane, methanol, and hydrated alanine polypeptide with the Monte Carlo technique using three kinds of random number generators (RNG's)—the standard Linear Congruential Generator (LCG), a modification of the LCG with additional randomization used in the BOSS software, and the “Mersenne Twister” generator by Matsumoto and Nishimura. While using the latter two RNG's leads to reasonably similar physical features, the LCG produces significant different results. For the pure fluids, a noticeable expansion occurs. Using the original LCG on butane yields, a molecular volume of 171.4 ų per molecule compared to about 163.6–163.9 ų for the other two generators, a deviation of about 5%. For methanol, the LCG produces an average volume of 86.3 ų per molecule, which is about 24% higher than the 68.8–70.2 ų obtained with the RNG's in BOSS and the generator by Matsumoto and Nishimura. In case of the hydrated tridecaalanine peptide, the volume and energy tend to be noticeably greater with the LCG than with the BOSS (modified LCG) RNG's. For the simulated hydrated extended conformation of tridecaalanine, the difference in volume reached about 87%. The uniformity and periodicity of the generators do not seem to play the crucial role in these phenomena. We conclude that, it is important to test a RNG's by modeling a system such as the pure liquid methanol with a well‐established force field before routinely employing it in Monte Carlo simulations. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

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 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     

Manager-worker-based model for the parallelization of quantum Monte Carlo on heterogeneous and homogeneous networks

A manager-worker-based parallelization algorithm for Quantum Monte Carlo (QMC-MW) is presented and compared with the pure iterative parallelization algorithm, which is in common use. The new manager-worker algorithm performs automatic load balancing, allowing it to perform near the theoretical maximal speed even on heterogeneous parallel computers. Furthermore, the new algorithm performs as well as the pure iterative algorithm on homogeneous parallel computers. When combined with the dynamic distributable decorrelation algorithm (DDDA) [Feldmann et al., J Comput Chem 28, 2309 (2007)], the new manager-worker algorithm allows QMC calculations to be terminated at a prespecified level of convergence rather than upon a prespecified number of steps (the common practice). This allows a guaranteed level of precision at the least cost. Additionally, we show (by both analytic derivation and experimental verification) that standard QMC implementations are not "perfectly parallel" as is often claimed.     

Reliable radical stabilization energies from diffusion Monte Carlo calculations

We assess the performance of variational (VMC) and diffusion (DMC) quantum Monte Carlo methods for calculating the radical stabilization energies of a set of 43 carbon-centered radical species. Even using simple single-determinant trial wavefunctions, both methods perform exceptionally well, with mean absolute deviations from reference values well under the chemical accuracy standard of 1 kcal/mol. In addition, the use of DMC results in a highly concentrated spread of errors, with all 43 results within chemical accuracy at the 95% confidence level. These results indicate that DMC is an extremely reliable method for calculating radical stabilization energies and could be used as a benchmark method for larger systems in future.     

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     

Monte Carlo simulations of biomolecules: The MC module in CHARMM

We describe the implementation of a general and flexible Monte Carlo (MC) module for the program CHARMM, which is used widely for modeling biomolecular systems with empirical energy functions. Construction and use of an almost arbitrary move set with only a few commands is made possible by providing several predefined types of moves that can be combined. Sampling can be enhanced by noncanonical acceptance criteria, automatic optimization of step sizes, and energy minimization. A systematic procedure for improving MC move sets is introduced and applied to simulations of two peptides. The resulting move sets allow MC to sample the configuration spaces of these systems much more rapidly than Langevin dynamics. The rate of convergence of the difference in free energy between ethane and methanol in explicit solvent is also examined, and comparable performances are observed for MC and the Nosé-Hoover algorithm. Its ease of use combined with its sampling efficiency make the MC module in CHARMM an attractive alternative for exploring the behavior of biomolecular systems.     

Field-programmable gate arrays and quantum Monte Carlo: Power efficient coprocessing for scalable high-performance computing

Massively parallel architectures offer the potential to significantly accelerate an application relative to their serial counterparts. However, not all applications exhibit an adequate level of data and/or task parallelism to exploit such platforms. Furthermore, the power consumption associated with these forms of computation renders “scaling out” for exascale levels of performance incompatible with modern sustainable energy policies. In this work, we investigate the potential for field-programmable gate arrays (FPGAs) to feature in future exascale platforms, and their capacity to improve performance per unit power measurements for the purposes of scientific computing. We have focused our efforts on variational Monte Carlo, and report on the benefits of coprocessing with a FPGA relative to a purely multicore system.     

Dynamics of polyethylene melts studied by Monte Carlo simulations on a high coordination lattice

A detailed comparison is made between the experiment, prior simulations by other groups, and our simulation based on a newly designed dynamic Monte Carlo algorithm, on the dynamics of polyethylene (PE) melts. The new algorithm, namely, noncross random two-bead move has been developed on a high coordination lattice (the 2nnd lattice) for studying the dynamics of realistic polymers. The chain length (molecular weight) in our simulation ranges from C40 (562 Da) to C324 (4538 Da). The effects of finite chain length have been confirmed and significant non-Gaussian statistics evidently results in nonstandard static and dynamic properties of short PE chains. The diffusion coefficients scale with molecular weight (M) to the −1.7 power for short chains and −2.2 for longer chains, which coincides very well with experimental results. No pure Rouse scaling in diffusion has been observed. The transitional molecular weight to the entanglement regime is around 1500 Da. The detailed mean square displacements of middle bead (g1) are presented for several chain lengths. The reptation-like slowdown can be clearly observed only above M ∼ 2400 Da. The slope 0.25 predicted by the theory for the intermediate regime is missing; instead a slope close to 0.4 appears, indicating that additional relaxation mechanism exists in this transitional region. The relaxation times extracted by fitting the autocorrelation function of end-to-end vectors with reptation model scale with M to 2.5 for long chains, which seemingly conflicts with the scaling of diffusion. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2556–2571, 2006     

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     

自优化扩散量子Monte Carlo差值法

提出了自优化扩散量子MonteCarlo差值法,这是一个集优化、扩散和相关取样三项技术于一身的MonteCarlo新算法.这个算法能够在扩散过程中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^-5hartree数量级,获得相关能达80%以上.应用该方法研究分子势能面,使用"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和分子势能面.另外,我们还首创了"平衡后留样"技术,可节省50%以上的计算量.该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.     

Fixed—node Quantum Monte Carlo：A Novel Approach

The fixed-node quantum Monte Carlo (FNQMC)1,2 method has made it possible to calculate the electronic structure of relatively large molecular systems. These large systems range from positron complexes [NH2, Ps] with ～10 electrons to C20 isomers with 120 electrons, silicon crystal structures of 250 atoms and 1000 valence electrons. In the practical calculation for FNQMC method, in general, a minimal basis set of Slater-type atomic orbital (STO) and Jastrow functions are taken to cons…     

Predicting helical hairpins from sequences by Monte Carlo simulations

The key problem in polypeptide‐structure prediction is with regard to thermodynamics. Two factors limit prediction in ab initio computer simulations. First, the thermodynamically dominant conformations must be found from an extremely large number of possible conformations. Second, these low‐energy forms must deviate little from the experimental structures. Here, we report on the application of the diffusion‐controlled Monte Carlo approach to predict four α‐helical hairpins with 34–38 residues by global optimization, using an energy optimized on other supersecondary structures. A total of seven simulations is carried out for each protein starting from fully extended conformations. Three proteins are correctly folded (within 3.0 Å rms from the experimental structures), but the fourth protein cannot distinguish between several equienergetic conformations. Possible improvement of the energy model is suggested. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 582–589, 2000     

Correlation method for variance reduction of Monte Carlo integration in RS-HDMR

The High Dimensional Model Representation (HDMR) technique is a procedure for efficiently representing high-dimensional functions. A practical form of the technique, RS-HDMR, is based on randomly sampling the overall function and utilizing orthonormal polynomial expansions. The determination of expansion coefficients employs Monte Carlo integration, which controls the accuracy of RS-HDMR expansions. In this article, a correlation method is used to reduce the Monte Carlo integration error. The determination of the expansion coefficients becomes an iteration procedure, and the resultant RS-HDMR expansion has much better accuracy than that achieved by direct Monte Carlo integration. For an illustration in four dimensions a few hundred random samples are sufficient to construct an RS-HDMR expansion by the correlation method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

Dynamic Monte Carlo simulations of competition in crystallization of mixed polymers grafted on a substrate

Nowadays, preparation of poly(lactide) (PLA) with high content of stereocomplex crystallites (SCs) is receiving more and more attentions. The stereocomplex formation between enantiomeric poly(l ‐lactide) and poly(d ‐lactide) can efficiently improve the corresponding mechanical properties and thermal stability. In the current work, using dynamic Monte Carlo simulations, the microscopic mechanism of SC formation in grafted polymers was investigated. The increase in grafting density can lead to the enhancement of SC formation. On one hand, the miscibility between the chains of different types can be improved due to the grafting. On the other hand, the increase in grafting density can result in the higher degree of chain extension and the change in crystal nucleation and growth modes. The changes facilitate the stereocomplex formation. These findings not only provide an effective way to prepare PLAs with high content of SCs, but also reveal the underlying mechanisms controlling the SC formation. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 89–97     

Monte Carlo simulations of polymer dynamics: Recent advances

A brief review is given of applications of Monte Carlo simulations to study the dynamical properties of coarse-grained models of polymer melts, emphasizing the crossover from the Rouse model toward reptation, and the glass transition. The extent to which Monte Carlo algorithms can mimic the actual chain dynamics is critically examined, and the need for the use of coarse-grained rather than fully atomistic models for such simulations is explained. It is shown that various lattice and continuum models yield qualitatively similar results, and the behavior agrees with the findings of corresponding molecular dynamics simulations and experiments, where available. It is argued that these simulations significantly enhance our understanding of the theoretical concepts on the dynamics of dense macromolecular systems. © 1997 John Wiley & Sons, Inc.