B-spline method

A comprehensive theoretical investigation of PECD in a series of substituted oxiranes has been presented by Stener et al. [53] using the B-spline method. Variations in the predicted dichroism were found as both a function of initial orbital, and of the chemical substitution about the epoxy ring. These substitutions do not induce very significant geometry changes in the optimized ring structure. 

Two wider ranging, more systematic investigations of conformational dependence have since been performed to establish whether the conformational sensitivity noted in the above PECD smdies may generally provide a means for identifying and distinguishing gas-phase structure of suitable chiral species. The B-spline method has been applied to the model system (l/f,2f )-l,2-dibromo-l,2-dichloro-l,2-difluoroethane [60]. Rotation around the C C bond creates three stable conformational possibilities for this molecule to adopt. The results for both core and valence shell ionizations reaffirm an earlier conclusion a and p are almost unaffected by the rotational conformation adopted, whereas the PECD varies significantly. Eor the C Ij ionization to show any sensitivity at aU to the relative disposition of the halogen atoms further reinforces the point made previously in connection with the core level PECD phenomenon. 

The two computational methods, CMS-Xa and LCAO B-spline DPT, for now provide consistent, comparable results [57] with little to choose between them in comparison with experiment in those cases presented here (Sections I.D. 1. a and I. D.a.2). The B-spline method holds the upper hand aesthetically by its avoidance of a model potential semiempirically partitioned into spherical atomic regions. More importantly it olfers greater scope for future development, particularly as the inevitable increases in available computing power open new doors. 

A common problem for both methods lies in the use of potentials that do not possess the correct net attractiveness. This can have the consequence that continuum feamres appear shifted in energy. In particular, there is evidence that the LB94 exchange-correlation potential currently used for the B-spline calculations, although possessing the correct asymptotic behavior for ion plus electron, is too attractive, and near threshold features can then disappear below the ionization threshold. An empirical correction can be made, offsetting the energy scale, but this can mean that dynamics within a few electronvolts of threshold get an inadequate description or are lost. There is limited scope to tune the Xa potential, principally by adjustment of the assumed a parameter, but for the B-spline method a preferable alternative for the future may well be use of the SAOP functional that also has correct asymptotic behavior, but appears to be better calibrated for such problems [79]. 

Ye, T., Mittal, R., Udaykumar, H. S., and Shyy, W. J. Computat. Phys. 156, 209-240 (1999). Yusof, J. M. Combined immersed boundaries/B-splines methods for simulations of flows in complex geometries. CTR Annual Research Briefs, NASA Ames/Stanford University (1997). 

Oberlin D, Scheraga HA (1998) B-spline method for energy minimization in grid-based molecular mechanics calculations. J Comp Chem 19 71-85... 

Nuclear polarization corrections for hydrogenlike Pb and ions have been reconsidered recently [66]. Recalculations were performed within two different numerical approaches the direct numerical integration [62 - 64] and the B-spline method [67]. Besides a comparison with analytical evaluations [65], an enlarged set of nuclear excitations has been taken into account in these calculations. 

The fitted curve is a close approximation to the redemption yield curve, and is also very smooth. However, the fit is inaccurate at the very short end, indicating an underpriced 6-month bond, and also does not approximate the long end of the curve. For this reason, B-spline methods are more commonly used. 

We derive daily zero-coupon yield curves from five countries of the Eurozone (France, Germany, Italy, Spain, and the Netherlands) during the period from 2 January 2001 to 21 August 2002, using zero-coupon rates with 26 different maturities ranging from one month to 30 years. The yield curves are extracted from daily Treasury bond market prices by using a standard cubic B-splines method. Our input baskets are composed of... 

In this section we will analyse the various applications of the DFT B-spline method, starting from the beginning to the present start of the art. In order to give a clear exposition, we have chosen to proceed accordingly to the size of the system studied. So we will start with atoms and will finish with large fullerenes, passing through hydrides and medium sized molecules. 

A generalized partial differential equation solver which handles simultaneous parabolic, one dimensional elliptic, ordinary and integral equations and uses B-splines with an adaptive grid was written to solve the model. Further details on the model and solution method can be found in Reference 14. 

Stener and co-workers [59] used an alternative B-spline LCAO density functional theory (DFT) method in their PECD investigations [53, 57, 60-63]. In this approach a normal LCAO basis set is adapted for the continuum by the addition of B-spline radial functions. A large single center expansion of such... 

Vedam, H., Venkatasubramanian, V., and Bhalodia, M A B-spline base method for data compression, process monitoring and diagnosis. Comput. Chem. Eng. 22(13), S827-S830 (1998). 

Table IV. Comparison of Overall and Response Dispersion portions of Amount Bandwidths of the Transformed Method (Kurtz, et al.) with Bandwidths of the Spline Method (Wegscheider) for Dataset B.

Table V includes the same comparison as Table IV between Kurtz transformed method and Wegscheider s spline method for Dataset B and shows the actual amount units. It gives the reader a clear idea of the actual sizes of estimated amount ranges resulting from the uncertainty in response data.

For most applications the most stable method of calculating a spline is to express it in terms of the set of Basis- or B-splines (12) associated... 

The spline surface S(x,y) consists of a set of bicubic polynomials, one in each panel, joined together with continuity up to the second derivative across the panel boundaries. Because each B-spline only extends over four adjacent knot intervals, the functions B.(x)C.(y) are each non-zero only over a rectangle composed of 16 adjacent panels in a 4 x 4 arrangement. The amount of calculation required to evaluate the coefficients y may be reduced by making use of this property. As before, least-squares methods may be used if the number of data exceeds (h+4)(jJ+4), which is usually the case. 

Fano [96]. The K-matrix method has been successfully applied to a number of problems in atomic physics [167,193], particularly in conjunction with 13-splines [75,194-200]. B-splines are a convenient tool to represent accurately in finite boxes the radial component of continuum atomic orbitals [82], their major feature being a property known as effective completeness [201]. 

The B-spline K-matrix method follows the close-coupling prescription a complete set of stationary eigenfunctions of the Hamiltonian in the continuum is approximated with a linear combination of partial wave channels (PWCs) [Pg.286]

T.K. Fang, T.N. Chang, B-spline-based multichannel K-matrix method for atomic photo ionization, Phys. Rev. A 61 (2000) 062704. 

Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-splines for the spatial direction and various GEAR methods in time (Ref 247). PDEPACK and DSS (Ref 247) use finite differences in the spatial direction and GEARB in time (Ref 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction both codes are restricted to modeling chemical reactors. 

Figure 1. Titration curves for monovalent cation-hydrogen exchange. (A) Na at 0.11 (B) Na at 0.3 I (C)Na at 0.5 I (D) K" at 0.11. Every third data point is shown. Interpolation is by a cubic spline method.

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