Monte Carlo, quantum

In the variational quantum Monte Carlo (VQMC) method, the expectation value of the energy ( ) and/or another average property of a system is determined by Monte Carlo integrations. The expectation value of the energy is typically determined for a trial function j/q using Metropolis sampling based on tj/g. It is given by 

The term Hyg/Vo S local energy Ei. In determining ( ), it is not necessary to carry out analytic integrations and, since only differentiation of the trial wavefunction is required to evaluate the local energy, the trial wave-function may take any desired functional form. It may even include interelectron distances explicitly. Thus, relatively simple trial functions may incorporate electron correlation effects rather accurately and produce expectation values of the energy well below those of the Hartree-Fock limit. Except in the limit of a large number of terms the VQMC method is not an exact method. 

The Metropolis sampling procedure provides a means of sampling points in configuration space with specified probabilities in this case with probabilities proportional to the square of the wavefunction. Starting from an arbi- 

The step sizes for a typical Metropolis walk are usually chosen to give an acceptance ratio of about one-half in order to maximize the rate of diffusion and improve the sampling speed. Serial correlation of points is usually high. In many-dimensional (or many-electron) systems, the steps may be taken one dimension (or one electron) at a time or all at once. The optimum step sizes and/or combinations of steps depend strongly on the nature of the system treated. 

Another alternative, likely to be more efficient than Metropolis sampling, is the use of probability density functions P. These relatively simple functions, which approximate and mimic the density of the more complex function V / , can be sampled directly without a Metropolis walk and the associated serial correlation. Sample points of unit weight are obtained with probabilities proportional to the probability density P, and their weights are multiplied by the factor if /P to give overall v /o weighting. The expectation value of the energy ) is then given by 

Replacing t by -ir yields the imaginary-time Schrodinger equation 

47) can be viewed as a diffusion equation in the spatial coordinates of the electrons with a diffusion coefficient D equal to j. The source and sink term S is related to the potential energy V. In regions of space where V is attractive (negative) the concentration of diffusing material (here the wavefunction) will accumulate and it will decrease where V is positive. It turns out that if we start from an initial trial wavefunction and propagate it forward in time using eq. (11.47), 

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Anderson J B, Traynor C A and Boghosian B M 1993 An exact quantum Monte-Carlo calculation of the helium-helium intermolecular potential J. Chem. Phys. 99 345... 

Hollenstein H, Marquardt R, Quack M and Suhm M A 1994 Dipole moment function and equilibrium structure of methane In an analytical, anharmonic nine-dimenslonal potential surface related to experimental rotational constants and transition moments by quantum Monte Carlo calculations J. Chem. Phys. 101 3588-602... 

Ra]agopal G, Needs R J, James A, Kenney S D and Foulkes W M C 1995 Variational and diffusion quantum Monte Carlo calculations at nonzero wave vectors theory and application to diamond-structure germanium Phys. Rev. B 51 10 591-600... 

Umrigar C J, Wilson K G and Wilkins J W 1988 Optimized trial wavefunctions for quantum Monte Carlo calculations Phys. Rev. Lett. 60 1719-22... 

Eckstein FI, Schattke W, Reigrotzki M and Redmer R 1996 Variational quantum Monte Carlo ground state of GaAs Phys. Rev. B 54 5512-15... 

Towler M D, Flood R Q and Needs R J 2000 Minimum principles and level splitting in quantum Monte Carlo excitation energies application to diamond Phys. Rev. B 62 2330-7... 

Filippi C and Umrigar C J 2000 Correlated sampling in quantum Monte Carlo a route to forces Phys. Rev. B 61 R16 291... 

Grossman J C and Mitas L 1995 Quantum Monte Carlo determination of electronic and structural properties of Si clusters (n 20) Phys. Rev. Lett. 74 1323... 

A method that avoids making the HF mistakes in the first place is called quantum Monte Carlo (QMC). There are several types of QMC variational, dilfusion, and Greens function Monte Carlo calculations. These methods work with an explicitly correlated wave function. This is a wave function that has a function of the electron-electron distance (a generalization of the original work by Hylleraas). 

Quantum Monte Carlo (QMC) methods are computations that use a statistical integration to calculate integrals which could not be evaluated analytically. These calculations can be extremely accurate, but often at the expense of enormous CPU times. There are a number of methods for obtaining excited-state energies from QMC calculations. These methods will only be mentioned here and are explained more fully in the text by Hammond, Lester, and Reynolds. 

PW91 (Perdew, Wang 1991) a gradient corrected DFT method QCI (quadratic conhguration interaction) a correlated ah initio method QMC (quantum Monte Carlo) an explicitly correlated ah initio method QM/MM a technique in which orbital-based calculations and molecular mechanics calculations are combined into one calculation QSAR (quantitative structure-activity relationship) a technique for computing chemical properties, particularly as applied to biological activity QSPR (quantitative structure-property relationship) a technique for computing chemical properties... 

M. Suzuki, ed. Quantum Monte Carlo Methods. Springer Series in Solid State Sciences. Vol. 74. New York Springer, 1987. 

Accurate values of the correlation functional are available thanks to the quantum Monte Carlo calculations of Ceperley and Alder (1980). These values have been interpolated in order to give an analytic form to the correlation potential (Vosko, Wilk and Nusair, 1980). 

The answer to this question as well as the question of the precise meaning of the term ab initio itself in the context of quantum chemistry seems to differ considerably according to the particular researcher that one might consult.3 Some authors I have questioned claim that the two terms are used interchangeably to mean calculations performed without recourse to any experimental measurement. This would include Hartree-Fock, and many of the DFT functionals, along with quantum Monte Carlo and Cl methods. 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

Inserting equation (6-14) into equation (6-12) retrieves the p4/3 dependence of the exchange energy indicated in equation (3-5). This exchange functional is frequently called Slater exchange and is abbreviated by S. No such explicit expression is known for the correlation part, ec. However, highly accurate numerical quantum Monte-Carlo simulations of the homogeneous electron gas are available from the work of Ceperly and Alder, 1980. 

Marchi, M. Sprik, M. Klein, M. L., Calculation of the free energy of electron solvation in liquid ammonia using a path integral quantum Monte Carlo simulation, J. Phys. Chem. 1988, 92, 3625-3629... 

Sese, L. M., A quantum Monte Carlo study of liquid Lennard-Jones methane, path-integral and effective potentials, Mol. Phys. 1992, 76, 1335-1346... 

James B. Anderson, Quantum Monte Carlo Atoms, Molecules, Clusters, Liquids, and Solids. 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

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