Addition theorems as three-dimensional Taylor expansions

Abstract

A principal tool for the construction of the addition theorem of a r Ји١Ž . function f is the translation operator e , which generates f r q rЈ by doing a threedimensional Taylor expansion of f around r. In atomic and molecular quantum mechanics, one is usually interested in irreducible spherical tensors. In such a case, the application of the translation operator in its Cartesian form e xЈѨ rѨ x e yЈѨ rѨ y e zЈѨ rѨ z leads to serious technical problems. A much more promising approach consists of the use of an operator expansion for e r Ји١ which contains exclusively irreducible spherical tensors. This expansion contains as differential operators the Laplacian ٌ 2 and the spherical tensor m Ž . gradient operator Y Y ٌ , which is an irreducible spherical tensor of rank l. Thus, if l m Ž . Y Y ٌ is applied to another spherical tensor, the angular part of the resulting expression l is completely determined by angular momentum coupling, whereas the radial part is obtained by differentiating the radial part of the spherical tensor with respect to r. Consequently, the systematic use of the spherical tensor gradient operator leads to a considerable technical simplification. In this way, the originally three-dimensional expansion problem in x, y, and z is reduced to an essentially one-dimensional expansion problem in r. The practical usefulness of this approach is demonstrated by constructing the Laplace expansion of the Coulomb potential as well as the addition theorems of the regular and irregular solid harmonics and of the Yukawa potential.

The Spherical Tensor Gradient Operator

“…Alternative derivations of this result are discussed in [120,Section 7]. In the vast majority of all LCAO-MO electronic structure calculations, Gaussian functions are used as basis functions.…”

“…Consequently, any attempt to provide a reasonably complete bibliography would clearly be beyond the scope of this article. The interested reader is referred to the long, but nevertheless incomplete lists of references in [120,122].…”

Combining Slater-type orbitals and effective core potentials

“…(8). General theory of addition theorems was given in a pedagogical way by Weniger [38,39]. Note that mathematical correctness (and usefulness) of some formulations of the addition theorems is still subject to a debate [40].…”

“…We first note that the rate of convergence of Eq. (38) is not significantly affected by a change of values l, α, and γ, the only important variables being β, n. The sum (38) converges faster when β decreases or n increases. Therefore, we can consider the worst-case scenario of I 00 as a function of β.…”

“…Interestingly enough, there has been a significant progress on this issue in recent years. In fact, looking at only the past 15 years, there are many notable works of Bouferguene et al [10][11][12][13], Rico et al [14][15][16][17][18][19][20][21][22][23][24], Hoggan et al [25][26][27][28][29][30], Pachucki [31][32][33][34][35], and others [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. In particular, for the diatomic systems STOs can be now used routinely [51].…”