Feynman-Hibbs model

FIG. 13 Herringbone order parameter and total energy for N2 (X model with Steele s corrugation). Quantum simulation, full line classical simulation, dotted line quasiharmonic theory, dashed line Feynman-Hibbs simulation, triangles. The lines are linear connections of the data. (Reprinted with permission from Ref. 95, Fig. 4. 1993, American Physical Society.)... 

Fig. 11.1. The Helmholtz free energy as a function of /3 for the three free energy models of the harmonic oscillator. Here we have set h = uj = 1. The exact result is the solid line, the Feynman-Hibbs free energy is the upper dashed line, and the classical free energy is the lower dashed line. The classical and Feynman-Hibbs potentials bound the exact free energy, and the Feynman-Hibbs free energy becomes inaccurate as the quantum system drops into the ground state at low temperature...

The Feynman-Hibbs and QFH models perform quite well in free energy calculations as long as the quantum corrections are modest. The conditions for validity of the approximations are given above. 

Here U N) is the interaction potential energy for the complete system at a specific configuration, uniformly the same quantity that has been discussed above. U(TV) is an ejfective potential designed to be used in classical-limit partition function calculations, e.g. Eq. (3.17), p. 40, in order to include quantum mechanical effects approximately. We will call this /(TV) the quadratic Feynman-Hibbs (QFH) model. In Eq. (3.67), Mj is the mass of atom j, and V is the Laplacian of the... 

A more ambitious model than QFH, and one that is expected to be more accurate, is the Feynman-Hibbs (FH) model (Feynman and Hibbs, 1965) for which... 

We make this approximation for the inner gaussian kinetic energy average. This inequality shows that our approximation will yield an upper bound to the exact chemical potential. The exercises to follow derive the Feynman-Hibbs and QFH models from this approximation. 

In their simulations, L6vesque et used a standard Leimard-Jones interaction potential between hydrogen molecules, and included the effect of quadrupolar interactions by adding a Coulomb interaction term in which each hydrogen molecule is represented by three effective charges q (q = 0.4829e at the position of the protons and q = -e at the centre of mass of the molecule). The adsorbate-adsorbent interaction was modeled with a standard Lennard-Jones potential. In order to partially account for quantum effects at 77 K, a semi-classical approach based on the Feynman-Hibbs effective potential was used ... 

Classical statistical mechanical theory is, for the most part, adequate for the solutions treated in this book (Benmore et al, 2001 Tomberli et al, 2001), as has been discussed more specifically elsewhere (Feynman and Hibbs, 1965). It is important to distinguish that issue of statistical mechanical theory from the theory, computation, and modeling involved in the interaction potential energy U N). The potential distribution theorem doesn t require specifically simplified forms for /(2V) on grounds of statistical mechanical principal simplifications can make calculations more practical, of course, but those are issues to be addressed for specific cases. 

We discuss initially the model (Feynman and Hibbs, 1965, Section 10-3)... 

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