Monte Carlo path integral

Ceperley D M 1996 Path integral Monte Carlo for fermions Monte Carlo and Molecular Dynamics of Condensed Matter Systems vol 49, ed K Binder and E G Ciccotti (Bologna Italian Physical Society) pp 443-82... 

Herman M F, Bruskin E J and Berne B J 1982 On path integral Monte Carlo simulations J. Chem. Phys. 76 5150-5... 

V. Path Integral Monte Carlo — Analysis of Quantum Effects in... 

Phase transitions in two-dimensional layers often have very interesting and surprising features. The phase diagram of the multicomponent Widom-Rowhnson model with purely repulsive interactions contains a nontrivial phase where only one of the sublattices is preferentially occupied. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are molecular layers of H2, D2, N2 and CO molecules on graphite substrates. We review the path integral Monte Carlo (PIMC) approach to such phenomena, clarify certain experimentally observed anomalies in H2 and D2 layers, and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are analyzed via PIMC as well. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions where quantum effects play a role. 

In Sec. II we briefly review the experimental situation in surface adsorption phenomena with particular emphasis on quantum effects. In Section III models for the computation of interaction potentials and examples are considered. In Section IV we summarize the basic formulae for path integral Monte Carlo and finite size scahng for critical phenomena. In Section V we consider in detail examples for phase transitions and quantum effects in adsorbed layers. In Section VI we summarize. 

V. PATH INTEGRAL MONTE CARLO — ANALYSIS OF QUANTUM EFFECTS IN ADSORBED LAYERS... 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

Path integral Monte Carlo simulations were performed [175] for the system with Hamiltonian (Eq. (25)) for uj = ujq/J = A (where / = 1) with N = 256 particles and a Trotter dimension P = 64 chosen to achieve good computer performance. It turned out that only data with noise of less than 0.1% led to statistically reliable results, which were only possible to obtain with about 10 MC steps. The whole study took approximately 5000 CPU hours on a CRAY YMP. 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

Thirumalai, D. Wallqvist, A. Berne, B.J., Path-integral Monte Carlo simulations of electron localization in water clusters, J. Stat. Phys. 1986, 43, 973-984... 

Thirumalai, D. Hall, R. W. Berne, B. J., A path integral Monte Carlo study of liquid neon and the quantum effective pair potential, J. Chem. Phys. 1984, 81, 2523-2527... 

Ortiz, V. Lopez, G. E., Fourier path integral Monte Carlo study of a two-dimensional model quantum monolayer, Mol. Phys. 2002,100, 1003-1009... 

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... 

Srinivisan, J. Volobuev, Y. L. Mielke, S. L. Truhlar, D. G., Parallel Fourier path-integral Monte Carlo calculations of absolute free energies and chemical equilibria, Comput. Phys. Commun. 2000, 128, 446-464... 

Barrat, J.-L. Loubeyre, P Klein, M. L., Isotopic shift in the melting curve of helium a path integral Monte Carlo study, J. Chem. Phys. 1989, 90, 5644—5650... 

Chandler D, Wolynes PG (1981) J Chem Phys 74 4078. Quantum Path Integral Monte Carlo can also provide a way to evaluhte the rate in non-adiabatic reactive flux correlation theory Wolynes PG (1987) J Chem Phys 87 6559... 

WaUqvist A, Thirumalai D, Berne BJ. (1987) Path integral Monte Carlo smdy of the hydrated electron. J Chem Phys 86 6404-6418. 

As a final example in this subsection we mention the recent study of Zillich and Whaley who examined LiH solvated in He clusters with up to 100 atoms. The authors used a path integral Monte Carlo simulation approach, whose details shall not be discussed further here. The LiH-He interaction potential was found to be highly anisotropic with attractions for He approaching the molecule in a direction parallel to the molecular axis, but with strong repulsions for He approaching the molecule in a direction perpendicular to the molecular bond. Despite these repulsions, the authors found that LiH prefers to occupy central regions of the LiH He clusters for n larger than 10-15. 

Weht, R., Kohanoff, J., Estrin, D., Chakravarty, C. (1998). An ab initio path integral Monte Carlo simulation method for molecules and clusters application to Li4 and Lij. . Chem. Phys. 108, 8848-8858. 

IV. Fourier Path Integral Monte Carlo Methods... 

In Eq. (3.2) m is the mass of particle i at coordinate r,-, M is the total mass, and Rc is the constraining radius. We have chosen a continuous constraining potential because it is well suited to the Fourier path integral Monte Carlo calculations to be described in the next section. 

IV. FOURIER PATH INTEGRAL MONTE CARLO METHODS... 

P. H. Zong, D. M. Ceperley (1998) Path integral Monte Carlo calculation of electronic forces. Phys. Rev. E 58, p. 5123 C. Rickwardt, P. Nielaba, M. H. Muser and K. Binder (2001) Path integral Monte Carlo simulations of silicates. Phys. Rev. B 63, 045204... 

A more accurate, but also more complex form, will be discussed later in the section on Path Integral Monte Carlo. Note that we have S3mimetrized the primitive form in order to reduce the systematic error of the factorization [19]. The explicit form of the kinetic propagator is the Green s function of the Bloch equation of a system of free particles [19,21], i.e. a diffusion equation in configurational space... 

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