Path integral Monte Carlo configurations

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... 

The probability PXqr q ) fo move the reaction coordinate centroid variable from the reactant configuration to the transition state is readily calculated [108] by PIMC or PIMD techniques [17-19] combined with umbrella sampling [77,108,123] of the reaction coordinate centroid variable. In the latter computational technique, a number of windows are set up which confine the path centroid variable of the reaction coordinate to different regions. These windows connect in a piecewise fashion the possible centroid positions in going from the reactant state to the transition state. A series of Monte Carlo calculations are then performed, one for each window, and the centroid probability distribution in each window is determined. These individual window distributions are then smoothly joined to calculate the overall probability function in Eq. (4.11). An equivalent approach is to calculate the centroid mean force and integrate it from the reactant well to barrier top (i.e., a reversible work approach for the calculation of the quantum activation free energy [109,124]). 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

Figure 5.6 Illustration of the discretization of the integration path into Mr = 7 overlapping intervals. A configuration (replica) is asscxriated with each interval. Each interval, in turn, is comprised of four states of an expanded ensemble. Expanded-ensemble Monte Carlo moves that change the values of XN and % N within one interval are indicated by horizontal...

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