Partial wave expansions

Problems in chemical physics which involve the collision of a particle with a surface do not have rotational synnnetry that leads to partial wave expansions. Instead they have two dimensional translational symmetry for motions parallel to the surface. This leads to expansion of solutions in terms of diffraction eigenfiinctions. 

We call this a partial M/ave expansion. To detennine tire coefficients one matches asymptotic solutions to the radial Scln-ddinger equation with the corresponding partial wave expansion of equation (A3.11.106). It is customary to write the asymptotic radial Scln-ddinger equation solution as... 

The asymptotic partial wave expansion of equation (A3.11.106) can be developed using the identity... 

B2.2.6.7 PARTIAL WAVE EXPANSION FOR TRANSPORT CROSS SECTIONS... 

PECD in Camphor Convergence of the Partial Wave Expansion 1. Data Acquisition and Analysis Molecular Conformation and Substitution Effects 1. Data Acquisition and Treatment... 

Substitution of Eq. (18) into the partial wave expansion formula [Eq. (A.3)] gives for the continuum state... 

A common finding of computational PECD studies is that a relatively large partial wave expansion, typically running to niax > 15 is required. Chiral molecules necessarily are of very low, or no, symmetry, and hence are quite... 

Following normal practice, it is convenient to replace the continuum fimction in Eq. (A.l) or (A.2) with an incoming wave normalized partial wave expansion [39, 40, 118] ... 

Let us now introduce the partial waves expansion of [Pg.29]

Then tpj and ipE are obtained by mean of the partial wave expansion of the gaussian functions ... 

This empirical result is consistent with the theoretical analysis of the partial-wave expansion (where the truncation of the FCI expansion is based on the angular-momentum quantum number rather than on the principal quantum number n), for which it has been proved that the truncation error is proportional to L-3 when all STOs up to l = L are included in the FCI wavefunction [49, 50],... 

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

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