Path integral Monte Carlo quantum dynamics

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. 

The formulas just developed are clearly relevant to quantum dynamics, but their relevance to the Monte Carlo computation of molecular thermodynamic properties has not yet been developed. It turns out that we can develop a theory of quantum statistical mechanics [33] that is completely analogous to the Feynman path-integral version of quantum dynamics. 

Much work has been recently done in extending path integral and centroid molecular dynamics (CMD) methods to include nuclear quantum effects in classical MD simulations [230-233]. Tachikawa etal [230] studiedp-CH O) (n= 1-3) by ab initio hybrid Monte Carlo and ab initio path integral simulations. Their simulation showed that, due to quantum effects, the average hydrogen-bonded... 

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

In the Fourier method each path contributing to Eq. (4.13) is expanded in a Fourier series and the sum over all contributing paths is replaced by an equivalent Riemann integration over all Fourier coefficients. This method was first introduced by Feynman and Hibbs to determine analytic expressions for the harmonic oscillator propagator and has been used by Miller in the context of chemical reaction dynamics. We have further developed the approach for use in finite-temjjerature Monte Carlo studies of quantum sys-tems, and we have found the method to be very useful in the cluster studies discussed in this chapter. 

Compared with other areas such as ab initio electronic structure theory and molecular dynamics and Monte Carlo calculations, path integral simulation is a relative latecomer to the field of computational chemistry. While the analytical advantages of formulating quantum mechanics in terms of path integrals have influenced modem physics profoundly for the past 40 years, its computational advantages in areas of chemistry were not appreciated until rather late. 

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