Quantum mechanics path integral Monte Carlo

The quantum mechanical generalization of the anisotropic-planar-rotor Hamiltonian (2.5) is devised and investigated by quasiharmonic, quasiclas-sical, and path-integral Monte Carlo methods in Refs. 213 and 218. Here, thermal fluctuations compete with quantum fluctuations which adds a qualitatively new dimension to the scarce one-dimensional phase diagram of the classical anisotropic-planar-rotor model (2.5). The phase behavior of this model is much richer, and the phenomenon of reentrant orientational quantum melting is observed in a certain regime of the phase diagram. 

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. 

The PIQMC method is the result of coupling of Feynman s path integral formulation of quantum mechanics with Monte Carlo sampling techniques to produce a method for finite temperature quantum systems. The calculations are not much more complicated than DQMC and produce a sum over all possible states occupied as for a Boltzmann distribution. In the limit of zero temperature... 

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. 

Figure 3. Real and imaginary parts of the position correlation function for the quartic oscillator described in this section at a high temperature, h(op-ypil Q, Solid lines exact quantum mechanical results. Markers FBSD-path integral results with iV = 1 and 10,000 Monte Carlo points per integration variable. Dashed lines classical results.

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