Shepard point

Using a suitable quantum mechanical electronic structure method, the energy, gradient, and Hessian can be calculated at an arbitrary geometry, q, which is called an interpolation point or a Shepard point. Near q Vii(q), V(q), and V22(q) may be expanded as a Taylor series, yielding... 

Banerjee A, Adams N, Simons J and Shepard R 1985 Search for stationary points on surfaces J. Phys. Chem. 89 52... 

Equation (42) provides an accurate PES only in the vicinity of (i). Many different reference configurations, which we call data points , are needed. Suppose we have A iata such data points. Then, a modified Shepard interpolation58 gives the PES as a weighted average of these Taylor expansions ... 

The reviews of the MCSCF method by Shepard [129] and Werner [130] are too technical to summarize briefly. We shall content ourselves with pointing out that the structure of our MCDHF(B) equations is consistent with the nonrela-tivistic theory, so that much of the nonrelativistic formalism can be adopted with little change. The nonrelativistic energy expression is of the form of (231) and can be regarded as a function of the CSF coefficients Cj and the MO expansion coefficients where... 

Energy derivatives are essential for the computation of dynamics properties. There are several dynamics-related methods available in gamess. The intrinsic reaction coordinate (IRC) or minimum energy path (MEP) follows the infinitely damped path from a first-order saddle point (transition state) to the minima connected to that transition state. In addition to providing an analysis of the process by which a chemical reaction occurs (e.g. evolution of geometric structure and wavefunction), the IRC is a common starting point for the study of dynamics. Example are variational transition state theory (VTST [55]) and the modified Shepard interpolation method developed by Collins and co-workers... 

Proposed temporal variations in the CCD for the major oceans. (After van Andel, 1975 van Andel, T.J.H. 1960. Sources and dispersion of Holocene sediments, northern Gulf of Mexico. In Recent Sediments, Northwest Gulf of Mexico, Shepard, F.P, Phleger, F.B., and van Andel, T.J.H., eds., American Association of Petroleum Geologists, Tulsa, OK, 44-55. Data points from Atlantic Deep Sea Drilling Sites produce the diffuse Cretaceous and Eocene zone shown in the figure. Reprinted with permission of the AAPG whose permission is required for further use.)... 

Gradient methods discussed above use a quadratic function (energy, gradient and approximate Hessian) to model the energy surface near the transition state. Distance-weighted interpolants provide a more flexible functional form that can interpolate arbitrarily spaced points with a smooth differentiable function. For a gradient-based optimization, the Shepard interpolation functions seem appropriate... 

Recently, Jasien k Shepard (11) proposed a new approach for representing molecular potential energy surfaces. They expand the energy (and derivatives, if available) about a line which connects the reactants with the products through the saddle point. This line can be a simple interpolation between the geometries of the above points or it can be an approximate reaction... 

The Shepard diagram is a scatter plot in which the dissimilarities between the molecules, measured as the distances in the fuU n-dimensional space, are compared with the corresponding distances in the projected 2-dimensional space. A large spread is an indicator of a poor MDS projection and vice versa up to the unreachable extreme where all points fall on one line which indicates a perfect MDS projection. 

Banerjee, A., Adams, N., Simons, J., 8c Shepard, R. (1985). Search for stationary points on surfaces. Journal of Physical Chemistry, 89, 52. 

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