Path integral quantum Monte Carlo method

Carlo method (VQMC), the diffusion quantum Monte Carlo method (DQMC), the Green s function quantum Monte Carlo method (GFQMC), and the path integral quantum Monte Carlo method (PIQMC). These methods are by their nature strongly related and each has its own peculiar advantages and disadvantages relative to the others. 

Condensed-phase Electronic Systems Path Integral Simulations Monte Carlo Quantum Methods for Electronic Structure Rates of Chemical Reactions Wave Packets. 

Barker, J.A., A quantum-statistical Monte Carlo method Path integrals with boundary conditions, J. Chem. Phys. 1979, 70, 2914-2918... 

Miller, T. F., Ill Clary, D. C., Torsional path integral Monte Carlo method for calculating the absolute quantum free energy of large molecules, J. Chem. Phys. 2003,119, 68-76... 

By increasing pressure and/or decreasing temperature, ionic quantum effects can become relevant. Those effects are important for hydrogen at high pressure [7, 48]. Static properties of quantum systems at finite temperature can be obtained with the Path Integral Monte Carlo method (PIMC) [19]. We need to consider the ionic thermal density matrix rather than the classical Boltzmann distribution ... 

The quantum mechanical generalization of the anisotropic-planar-rotor Hamiltonian (2.5) is devised and investigated by quasiharmonic, quasiclas-sical, and path-integral Monte Carlo methods in Refs. 213 and 218. Here, thermal fluctuations compete with quantum fluctuations which adds a qualitatively new dimension to the scarce one-dimensional phase diagram of the classical anisotropic-planar-rotor model (2.5). The phase behavior of this model is much richer, and the phenomenon of reentrant orientational quantum melting is observed in a certain regime of the phase diagram. 

This review is a brief update of the recent progress in the attempt to calculate properties of atoms and molecules by stochastic methods which go under the general name of quantum Monte Carlo (QMC). Below we distinguish between basic variants of QMC variational Monte Carlo (VMC), diffusion Monte Carlo (DMC), Green s function Monte Carlo (GFMC), and path-integral Monte Carlo (PIMC). 

In this short review we have pointed out only very few of the basic issues involving the simulation of chemical systems with Quantum Monte Carlo. What has been achieved in the last few years is remarkable very precise calculations of small molecules, the most accurate calculations of the electron gas, silicon and carbon clusters, solids, and simulations of hydrogen at temperatures when bonds are forming. New methods have been developed as well high-accuracy trial wavefunctions for atoms, molecules, and solids, treatment of atomic cores, and the generalization of path-integral Monte Carlo to treat many-electron systems at positive temperatures. 

The physical adsorption isotherms on carbon materials have been studied theoretically using Grand Canonical Monte Carlo simulations and an effective classical potential [8], or using Feynmaim path formalism in conjunction with the Monte Carlo method to take into account the quantum effects [9]. To simulate hydrogen adsorption accurately at low temperature, these quantum effects have to be included. In this last case hydrogen is considered as a quantum fiuid. The basic idea of Feynman path integral formalism is to look at the possible paths that a particle can take to move from one point to another. 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

This review article is divided into two major sections, the first of which details the theoretical basis of RQMC (Sect. 18.2). Initially we describe quantum Monte Carlo sampling from the pure distribution and mixed distribution F, showing that the RQMC approach to sample from the pure distribution rests on Metropolis-Hastings (MH [25,26]) sampling, as does the variational path integral (VPI [27]) method. As already mentioned, RQMC proposes reptafion-type moves while... 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

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