Dimensional singularities

Eq. (38) consists of the well known two dimensional singular stress field, / F(v), multiplied by the factor, J Cj" cos fi z, which makes the field "three dimensional". Thus the... 

Figure 13. Cumulative contributions (% of total) to electronic energy of ground-state H2 arising from dimensional singularities. For scaled distance Ru = 1. Lowest curve (dashed) shows contribution from second-order pole at U = 1 middle curve (dot-dashed) sum of first- and second-order poles at > = 1 uppermost curve (solid) adds contributions from singularties at 00 limit.

With appropriate dimensional scalings, theD — 1 limit of Schrodinger equations for coulombic systems provides important information about the dimension dependence of energy eigenvalues. The energy typically has a second-order pole at D = 1, the residue of which can often be exactly determined. We demonstrate this with some simple examples and then review a systematic procedure for characterizing a class of dimensional singularities found in coulombic problems. 

A characteristic feature of ground state energies of coulombic systems is the presence of a second-order pole at = 1. The origin of these poles has been explained by Doren and Herschbach [14,16] in terms of an analysis of the Schrodinger equation at particle coalescences. Their method of analysis allows one to predict the locations and types of a certain class of dimensional singularities without actually having to solve for the function E S). We will illustrate this first for central-potential problems and then for many-particle systems. 

Eq. (27) implies that the wavefunction will have dimensional singularities at those values of P such that (P -f- e — 2) can be zero that is, P=0, -1, -2,..., in addition to the case P = 1 just discussed. However, when P < 1 these singularities will appear in both the numerator and the denominator of Eq. (24) for (ff) they cancel out, leaving E regular. As long as cq 0, the only such dimensional singularity in E will be at P = 1. 

The partial sums 5 of Eq. (11) are not suitable approximants for summing dimensional expansions for atoms and molecules since they cannot model the dimensional singularities. More effective approximants can be developed by incorporating what is known in advance about the singffiarity structure of the energy function E(S) into the form of the approximants. 

Finally, we mention one other method, the shifted expansion, which was discovered empirically [44] before the dimensional singularity structure was understood. This method consists of reexpanding the expansion parameter 8 = 1/Din terms of (D — where a is an arbitrary shift parameter. For the ground state of He the optimal shift parameter was found to be [Pg.303]

For n = 0 the shifted expansion is identical to the rescaled expansion, Eq. (41), with a second-order pole at = 1, but they differ at higher order since the shifted expansion adds additional higher-order poles at <5 = 1. We know from dimensional singularity analysis [18] that there are no poles at = 1 of order higher than two, so we can expect that the shifted expansion will be less and less accurate than the rescaled expansion as n increases. This is indeed what happens in practice [11]. The shifted expansion procedure has been rather popular [31], but now that we have information about the dimensional singularity structure of the problem this method should probably be considered obsolete. 

Finally, we note that excited states can have a dimensional singularity structure that appears to be qualitatively different from that found for the ground state. In Chapter 8.1 the 6 expansions of three excited states of helium axe analyzed. The ls2s state appears to have the same structure as the groimd state, while the ls2s 5 and 2p states seem to have a rather different structure. 

A. F. Hegarty, Uniformly accurate methods for two-dimensional singular perturbation problems, in BAIL Proceedings of the 4th Intern. Conference on Boundary and Interior Layers, S. K. Godunov, J. J. H. Miller, and V. A. Novikov, Eds., Boole Press, Dublin, 1986, pp. 314-319. 

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