Quantum Monte Carlo technique sampling techniques

Another technique which has gained prominence in recent years is the Quantum Monte Carlo (QMC) technique. This technique maps a d-dimensional quantum model onto a d - - 1 dimensional classical model via a Trotter decomposition of the partition function or the ground state projection operator [56, 57]. The quantum model is then studied by performing a Monte Carlo sampling procedure on the classical model in higher dimension. For fermions, the mapping of the interacting quantum model system to the classical system could... 

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]). 

Geometry optimization of the proposed mimetic is included as part of the design analysis to ensure the feasibility of the desired molecular conformation. MM and semiempirical quantum mechanical methods have been used most extensively for these purposes. Conformational analysis of the proposed mimetic allows the determination of an energy profile for the molecule under consideration. This has been used by researchers to assess where the desired conformation for the mimetic resides on the molecular potential energy surface. Monte Carlo, MD, and distance geometry-based conformational search techniques have been employed extensively to sample conformational space. Computational methods that attempt to approximate the efifects of aqueous solvation on the conformational profile of the mimetic are being used more frequently as part of these efforts. 

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... 

---