Multiscale reference function

In addition to the Hankel-Hadamard moment quantization method for the bosonic ground state energy (as well as other states, provided their nodes axe known), two other, powerful, moment quantization methods have been recently developed and used to generate the energy and wavefunction for arbitrary states. Both use the same basis representation, that corresponding to the Multiscale Reference Function (MRF) formalism. 

The Multiscale Reference Function (MRF) Representation We can recover the wavefunction at point b by taking... 

Handy, C. R., Khan, D. and Wang, Xiao-Qian, (2000) Multiscale Reference Function Quantization of the —(ix) Non-Hermitian Polynomial Potentials , preprint Clark Atlanta University. 

To account for the effect of a sufficiently broad, statistical distribution of heterogeneities on the overall transport, we can consider a probabilistic approach that will generate a probability density function in space (5) and time (t), /(i, t), describing key features of the transport. The effects of multiscale heterogeneities on contaminant transport patterns are significant, and consideration only of the mean transport behavior, such as the spatial moments of the concentration distribution, is not sufficient. The continuous time random walk (CTRW) approach is a physically based method that has been advanced recently as an effective means to quantify contaminant transport. The interested reader is referred to a detailed review of this approach (Berkowitz et al. 2006). 

Under the circumstances, a number of theoretical methods have been already developed to improve the QM/MM-MD method, e.g., the modification of the semi-empirical QM Hamiltonians [7, 52-54], the optimization of the QM/MM empirical parameters [10] and the replacement of the empirical repulsion potential functions [55]. However, these methods need the numerical values of some reasonable reference quantities to optimize the parameters for some specific molecular systems. Moreover, it is usually hard to obtain the reference experimental or computational ones in solution. It is, therefore, reasonable and plausible as a second best strategy that the closer MM solvent molecules around the QM solute should be included into the QM region to avoid the serious problems in the boundary between QM and MM regions. This is because the most serious problem is originating in the quantum-mechanical behaviors. On the basis of such strategy, we have developed the number-adaptive multiscale (NAM) QM/MM-MD [56, 57] and the QM/MM-MD method combined with the fragment molecular orbital (FMO) one, i.e., FMO-QM/MM-MD method [20]. 

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