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.

Various methods have been proposed for smoothing and denoising data sets, but a distinction is seldom made between the two procedures. Here, we distinguish between them in the signal domain and its transformed domain. Smoothing removes components (of the transformed signal) occurring in the high end of the transformed domain regardless of amplitude. Denoising removes small-amplitude components occurring in the transformed domain regardless of position. Methods for smoothing and denoising are presented which depend on the recently developed discrete wavelet (DW) transform technique. The DW smoothing and denoising methods are filtering techniques that are applied to the transformed data set, prior to back-transforming it to the signal domain. These DW techniques are compared with other familiar methods of smoothing (Savitzky-Golay) and denoising through filtering (Fourier transform). Preparatory to improving experimental data sets, synthetic data sets, comprised of ideal functions to which a known amount of noise has been added, are examined. The filter cutoffs are systematically explored, and DW techniques are shown to be highly successful for data sets with great dynamic range. In the minority of cases, smoothing and denoising are nearly interchangeable. It is shown that DW smoothing compresses with the most predictable results, whereas DW denoising compresses with minimal distortion of the signal.Experimental spectra are often complicated by noise, which may be due to interfering physical or chemical processes, imperfections in the experimental apparatus, or any of a number of causes which result in random, spurious fluctuations of the signal received at the detector. By definition, noise is instantaneously irreproducible although often it may be characterizable on average if it obeys Gaussian or Poisson distribution statistics, for example. In the present work, an interfering signal of another compound is not regarded as noise, nor are drift, baseline, signal pickup, or "cyclic noise" such as that used in a broader definition by Erickson et al. 1 included.

Distributed Gaussian bases (DGB) are defined and used to calculate eigenvalues for one and multidimensional potentials. Comparisons are made with calculations using other bases. The DGB is shown to be accurate, flexible, and efficient. In addition, the localized nature of the basis requires only very low order numerical quadrature for the evaluation of potential matrix elements.

The many-body expansion of the interaction potential between atoms and molecules is analyzed in detail for different types of interactions involving up to seven atoms. Elementary clusters of Ar, Na, Si, and, in particular, Au are studied, using first-principles wave-function-and density-functional-based methods to obtain the individual n-body contributions to the interaction energies. With increasing atom number the many-body expansion converges rapidly only for long-range weak interactions. Large oscillatory behavior is observed for other types of interactions. This is consistent with the fact that Au clusters up to a certain size prefer planar structures over the more compact three-dimensional Lennard-Jones-type structures. Several Au model potentials and semiempirical PM6 theory are investigated for their ability to reproduce the quantum results. We further investigate small water clusters as prototypes of hydrogen-bonded systems. Here, the many-body expansion converges rapidly, reflecting the localized nature of the hydrogen bond and justifying the use of two-body potentials to describe water-water interactions. The question of whether electron correlation contributions can be successfully modeled by a many-body interaction potential is also addressed.

Reactions of ozone with ethene and propene leading to primary ozonide ͑concerted and stepwise ozonolysis͒ or epoxide and singlet molecular oxygen ͑partial ozonolysis͒ are studied theoretically. The mechanism of concerted ozonolysis proceeds via a single transition structure which is a partial diradical. The transition structures and intermediates in the stepwise ozonolysis and partial ozonolysis mechanisms are singlet diradicals. Spin-restricted and unrestricted density functional methods are employed to calculate the structures of the closed-shell and diradical species. Although the partial diradicals exhibit moderate to pronounced instability in their RDFT and RHF solutions, RDFT is required to locate the transition structure for concerted ozonolysis. Spin projected fourth-order Møller-Plesset theory ͑PMP4͒ was used to correct the DFT energies. The calculated pre-exponential factors and activation energies for the concerted ozonolysis of ethene and propene are in good agreement with experimental values. However, the PMP4//DFT procedure incorrectly predicts the stepwise mechanism as the favored channel. UCCSD͑T͒ predicts the concerted mechanism as the favored channel but significantly overestimates the activation energies. RCCSD͑T͒ is found to be more accurate than UCCSD͑T͒ for the calculation of the concerted mechanism but is not applicable to the diradical intermediates. The major difficulty in accurate prediction of the rate constant data for these reactions is the wide range of spin contamination for the reference UHF wave functions and UDFT solutions across the potential energy surface. The possibility of the partial ozonolysis mechanism being the source of epoxide observed in some experiments is discussed.