B-spline basis sets

Taking the radial wave functions and energies for states n from the B-spline basis set, we may easily carry out the double sums in (128). The partial-wave contributions to from terms in square bracket are listed in Table 2. These terms fall-off approximately as L for large L and may easily be extrapolated. We find E = —0.0373736 a.u., leading to a binding energy of -0.8990800 a.u., differing from experiment by 0.5%. 

To solve Eqs. (176-181), an angular momentum decomposition is first carried out and the equations are then reduced to coupled equations involving single-body radial wave functions only. The radial wave functions for states v, m, n, a, b, are taken from a B-spline basis set [36] and the resulting coupled radial equations are solved iteratively. The core equations (176-177) are solved first and the valence equations (179-181) are then solved for valence states of interest using the converged core amplitudes. 

The set % is further symmetrized accordingly to the point group symmetry, and real harmonics are employed in order to avoid complex arithmetic. In this implementation the electron density is generated from a previous LCAO conventional DPT calculation with the program ADF [20], and further expanded using a numerical integration scheme with the same basis set in order to build the hamiltonian matrix. The Poisson equation is solved to get the Hartree term, as in the atomic calculation, and the B-spline basis set has proven very flexible so that the particular multipolar boundary conditions are easily satisfied. 

Keywords Electronic correlation Relativistic effects Isotopic effects Multiply excited states Hollow atoms B-spline basis set Charge transfer Photodissociation Radiative association... 

Using a truncated diagonalization method with these B-spline basis sets, we have calculated the decay properties of all doubly- excited AlAl and A151 states of and... 

The idea of using B-spline basis sets for the representation of vibrational molecular wave functions emerged rapidly. For a Morse potential and a two-dimensional Henon-Heiles potential, we have assessed the efficiency of the B-splines over the conventional DVR (discrete variable representation) with a sine or a Laguerre basis sets [50]. In addition, the discretization of the vibrational continuum of energy when using the Galerkin method allows the calculation of photodissociation cross-sections in a time-independent approach. 

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. 

N.(u) are the normalised B-spline basis functions defined on the knot set ... 

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 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]. 

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... 

W.R. Johnson, S.A. Blundell, J. Sapirstein, Finite basis sets for the Dirac equation constructed from B splines, Phys. Rev. A 37 (1988) 307. 

Next, we expand the solutions to the radial Dirac equation in a finite basis. This basis is chosen to be a set of n B-splines of order k. Following deBoor [33], we divide the interval [0, R] into segments. The end points of these segments are given by the knot sequence tj, i = 1,2, , n + k. The B-splincs of order k, Bi k r), on this knot sequence are defined recursively by the relations. 

The set of B-splines of order k on the knot sequence f< forms a complete basis for piecewise polynomials of degree fc — 1 on the interval spanned by the knot sequence. We represent the solution to the radial Dirac equation as a linear combination of these B-splines and work with the B-spline representation of the wave functions rather than the wave functions themselves. 

Ho also computed series of high-lying Feshbach-type resonances converging to the TV = 3,..,9 hydrogen threshols (133). Several tens of resonances were reported. Chen (134) applied a basis set of B-spline functions within the saddle-point complex rotation method to compute parameters of twelve and low lying Feshbach resonances. 

Over the many decades of the development and the implementation of mefh-ods for the computation of bound elecfronic structures and/or of properties associated with the continuous spectrum (e.g., resonances), the standard approach has been the use of formalisms that in practice utilize a single set of orthonormal basis sets for the calculation of relevanf mafrices, wave-functions, etc. (e.g., Gaussian or Slater functions, or B-splines, etc). However, regardless of the level of sophistication of fhe formalism, when it comes to the solution of MEPs or of difficult problems such as the ones discussed in the previous sections, this approach has serious practical limitations regarding both efficiency and interpretation. 

Accordingly, instead of expecting to obtain ho say from a somehow identifiable root of a diagonalized H matrix constructed from a large basis set of orthonormal functions or B-splines, ho for an isolafed sfafe was computed directly in the form... 

The description of the photoionization process by means of a method based on the Density Functional Theory (DFT) is reviewed. The present approach is based on a basis set expansion in B-spline functions, which are particularly suited to deal with the boundary conditions of the continuum states. Both Kohn-Sham (KS) and its extension to the Time Dependent (TD-DFT) formalism are considered. The computational aspects of the method are described the implementations for atoms, for molecules in One Centre Expansion (OCE) and for molecules with the Linear Combination of Atomic Orbital (LCAO) scheme. The applications of the method are discussed, from atoms to large fullerenes, with comparison with available experimental data. 

Up to now all the computational procedures have been precisely defined, with the exception of the basis set. In the atomic case all the equations are of course simplified to only the radial coordinate, the angular part being analytical. The radial functions are expanded in a basis set of B-splines over a selected interval [0, RmaxI- These are piecewise polynomial functions, completely defined in terms of... 

In practice, the spline is expressed as a set of basis functions, with the general spline being a combination of these. This may be arrived at using B-splines. The B-spline for a specified number of knot points Xq,.,7Q is (5.8). 

Remarks (i) For conventional FS-TARMA models, the functional subspaces include linearly independent basis functions selected from an ordered set, such as Qiebyshev, trigonometric, b-splines, wavelets, and other functions. For simplicity a functional subspace is often selected to include consecutive basis functions up to a maximum index. Yet, for purposes of model parsimony (economy) and effective estimation, some functions may not be necessary and may be dropped, (ii) An FS-TARMA ( a> c) p ... 

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