The formal definition of the generalized discrete variable representation (DVR) for quantum mechanics and its connection to the usual variational basis representation (VBR) is given. Using the one dimensional Morse oscillator example, we compare the ‘‘Gaussian quadrature’’ DVR, more general DVR’s, and other ‘‘pointwise’’ representations such as the finite difference method and a Simpson’s rule quadrature. The DVR is shown to be accurate in itself, and an efficient representation for optimizing basis set parameters. Extensions to multidimensional problems are discussed.
ReaxFF (van Duin, A.C.T.; Dasgupta, S.; Lorant, F.; Goddard, W.A. J. Phys. Chem. A, 2001, 105, 9396-9409) reactive potentials are parametrized for cyclotrimethylene trinitramine (RDX) and 1,1-diamino-2,2-dinitroethene (FOX-7) in a novel application combining data envelopment analysis and a modern self-adaptive evolutionary algorithm to optimize multiple objectives simultaneously and map the entire family of solutions. In order to correct the poor crystallographic parameters predicted by ReaxFF using its base parametrization (Strachan, A.; van Duin, A. C. T.; Chakraborty, D.; Dasgupta S.; Goddard, W. A. Phys. Rev. Lett., 2003, 91, 098301), we augmented the existing training set data used for parametrization with additional (SAPT)DFT calculations of RDX and FOX-7 dimer interactions. By adjusting a small subset of the ReaxFF parameters that govern long-range interactions, the evolutionary algorithm approach converges on a family of solutions that best describe crystallographic parameters through simultaneous optimization of the objective functions. Molecular dynamics calculations of RDX and FOX-7 are conducted to assess the quality of the force fields, resulting in parametrizations that improve the overall prediction of the crystal structures.
A discrete variable representation for scattering problems is developed. In this representation the potential matrix is diagonal, with elements being the potential evaluated at the proper quadrature points. The angular momentum operators may be treated exactly up to truncation of the basis set and provide the coupling in the coordinate-labeled discrete variable representation. The definition of the inner product over the internal coordinates as quadratures rather than integrations allows a discrete matrix transformation to be used to diagonalize any potential matrix. This framework allows one to obtain approximate solutions in a particularly simple and efficient manner and is presented in detail for atom–diatom collisions. At large values of the scattering distance the coupled equations may be solved to a high degree of accuracy using the distorted wave approximation in the finite basis representation. At small values of the scattering distance an exactly analogous technique may be used to obtain an approximate solution in the discrete variable representation. The solutions may be matched exactly and the exact scattering boundary conditions applied. Numerical results from atom–rigid rotor collisions are presented.
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