Advances in Chemical Physics: Monte Carlo Methods in by David M. Ferguson, J. Ilja Siepmann, Donald G. Truhlar, Ilya

By David M. Ferguson, J. Ilja Siepmann, Donald G. Truhlar, Ilya Prigogine, Stuart A. Rice

In Monte Carlo tools in Chemical Physics: An creation to the Monte Carlo approach for Particle Simulations J. Ilja Siepmann Random quantity turbines for Parallel functions Ashok Srinivasan, David M. Ceperley and Michael Mascagni among Classical and Quantum Monte Carlo tools: "Variational" QMC Dario Bressanini and Peter J. Reynolds Monte Carlo Eigenvalue equipment in Quantum Mechanics and Statistical Mechanics M. P. Nightingale and C.J. Umrigar Adaptive Path-Integral Monte Carlo equipment for exact Computation of Molecular Thermodynamic houses Robert Q. Topper Monte Carlo Sampling for Classical Trajectory Simulations Gilles H. Peslherbe Haobin Wang and William L. Hase Monte Carlo ways to the Protein Folding challenge Jeffrey Skolnick and Andrzej Kolinski Entropy Sampling Monte Carlo for Polypeptides and Proteins Harold A. Scheraga and Minh-Hong Hao Macrostate Dissection of Thermodynamic Monte Carlo Integrals Bruce W. Church, Alex Ulitsky, and David Shalloway Simulated Annealing-Optimal Histogram equipment David M. Ferguson and David G. Garrett Monte Carlo equipment for Polymeric structures Juan J. de Pablo and Fernando A. Escobedo Thermodynamic-Scaling tools in Monte Carlo and Their program to section Equilibria John Valleau Semigrand Canonical Monte Carlo Simulation: Integration alongside Coexistence traces David A. Kofke Monte Carlo tools for Simulating section Equilibria of advanced Fluids J. Ilja Siepmann Reactive Canonical Monte Carlo J. Karl Johnson New Monte Carlo Algorithms for Classical Spin platforms G. T. Barkema and M.E.J. NewmanContent:

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111. METHODS FOR RANDOM NUMBER GENERATION In this section we describe some popular basic generators and their parallelization. We also mention about “combined generators,” which are obtained from the basic ones. A. Linear Congruential Generators The most commonly used generator for pseudo-random numbers is the linear congruential generator (LCG) [20] : x, = ax,- + b(mod m) (34 where rn is the modulus, a the multiplier, and c the additive constant or addend. The size of the modulus constrains the period, and it is usually chosen to be either prime or a power of 2.

There are two broad categories of methods that go beyond HartreeFock in constructing wavefunctions : configuration interaction (CI), and many-body perturbation theory. In CI one begins by noting that the exact 44 DARIO BRESSANINI AND PETER J. REYNOLDS M-electron wavefunction can be expanded as a linear combination of an infinite set of Slater determinants that span the Hilbert space of electrons. These can be any complete set of M-electron antisymmetric functions. One such choice is obtained from the Hartree-Fock method by substituting all excited states for each MO in the determinant.

REYNOLDS Expectation values of nondifferential operators may be obtained simply as Differential operators are only slightly more difficult to sample, since we can write 11. MONTE C A R L 0 SAMPLING OF A TRIAL WAVEFUNCTION A. Metropolis Sampling The key problem is how to create and sample the distribution Y:(R) (from now on, for simplicity, we consider only real trial wavefunctions). This is readily done in a number of ways, possibly familiar from statistical mechanics. Probably the most common method is simple Metropolis sampling [ 6 ] .

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