B. Spline

Cubic B-Splin es Cubic B-splines can also be used to solve differential equations (Refs. 105 and 266). 

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... 

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. 

In Fig. 5, a similar selection of results for the tran -difluoro-oxrrane molecule is presented, again comparing B-spline and CMS-Xa calculations as before. In this case, the outer two orbitals have a reversed ordering the O lone parr (orbital 9b) is the HOMO-1 while the HOMO (11a) has C—C—O a-bonding characteristics. The 5b level is analogous to the 9a in methyl oxirane. 

Figure 5. Calculated angular parameters (= P) and for rmnr-(2S,3S)-difluoro-oxirane solid curves CMS-Xa broken curves B-spline (Ref. [53]). Results are given for ionization from four different orbitals (a) 11a (HOMO) (b) 9b (HOMO-1) (c) 5b (d) 3a/2b C H.

Another detailed comparative study of B-spline and CMS-Xa calculations that has been presented [57] addresses core and valence-shell PECD in camphor. For this molecule, a substantial amount of relevant experimental PECD data for the core [56] and valence-shell ionization [36, 56, 64, 65] is now available, allowing a full three-way comparison to be performed. Detailed discussion of the interpretation of the experimental results achieved with these calculations is deferred until Section VI.B, but it is helpful here to summarize the conclusions regarding the computational approaches. 

A similar convergence study has been reported for the B-spline calculation for trans-2,3 dimethyloxirane molecule, et al. [53] The authors in fact comment that the parameter is no more demanding on basis set size than the p parameter, but their data ([53] Fig. 1) show that when the asymptotic is increased from 10 to 15 there is a significantly greater improvement in the former angular parameter. A value of. (max = 15 was chosen for all subsequent B-spfine calculations for oxiranes, and the same limit tends to be applied in the other reported B-spline calculations of chiral molecule PECD [60, 61]. 

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]. 

Figure 16. Carbonyl C PECD from enantiomers of camphor. The experimentally derived data (Ref. [56]) for the (iS)-enantiomer have been negated prior to plotting on expectation that they will then fall on the same trend line as the (/ )-enantiomer data. The CMS-Xa and B-spline calculations (Ref. [57]) for the (R)-camphor enantiomer are included for comparison. The inset shows the (R)-camphor structure.

Unfortunately, experimental difficulties precluded measurements closer to threshold, and the B-spline calculation also does not properly span this near threshold region down to the onset [57]. However, the general trend rising above 5 eV is for the dichroism to become attenuated, easily rationalized as the ejected photoelectron displaying less sensitivity to the chiral molecular potential as it acquires more energy. 

Figure 17. Camphor PECD from the HOMO (carbonyl oxygen lone-pair) ionization. Experimental data are (S)-camphor, (R)-camphor (both from Ref. [36]) (R)- and (S)-enantiomer data from Ref [64] O data from Ref [65]. Also shown as curves are CMS-Xa calculations (Ref. [36]) and B-spline calculations (Ref [57]).

We employ a B-spline basis representation for the distribution function ... 

B-spline coefficients that globally minimize the performance index is determined. 

The unknown permeability is represented using tensor product B-splines, which are given by the product of univariate B-splines ... 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

In order to ensure successful minimization of the performance index and to enhance our ability to determine the global optimum, we select the corresponding finite-dimensional representation in a different manner than before. We again use B-splines to represent the unknown functions ... 

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). 

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). 

Other commonly employed and related sets of approximating polynomials are Hermite polynomials and B splines. Particularly in the latter case, the functions possess the desired properties of smoothness across patch boundary intersections, strong locality leading to simplification of the A coefficient matrix, and efficiency of computation. In the following discussion the B functions may be viewed, up to specific values, as any of the aforementioned types. 

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

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