Helium atom partial-wave expansion

One of the most commonly used expansions in the theory of the electronic structure of atoms and molecules is the partial wave expansion, in which individual atomic orbitals are expressed as products of radial functions and spherical harmonics. Appropriately symmetrized sums of products of the spherical harmonics for the coordinates of each particle can be formed to yield eigenfunctions of total Lz, S, and Sz. To prevent lengthy expressions involving 3-j and 6-j symbols from obscuring the essential physics, I shall focus on the partial wave expansion for an 5-state of the helium atom ... 

To see how electron localisation for large D would effect the partial wave expansions for many-electron atoms, we may for simplicity consider the helium atom, which is described by three internal coordinates, ri, T2, and. The localisation in the variables ri and V2 can easily be accommodated by using basis functions with flexible length scales, but the localisation in 6 would be very slowly described by truncated partial wave expansions, which contain no internal angular scaling parameter. Thus as D increases, the partial wave expansion has less trouble describing the electron-electron cusp, but more and more trouble describing the localisation of electrons. 

The partial-wave expansion of the ground-state helium atom... 

Table 7.1 The conveigence of the ground-state energy (in E, ) in the partial-wave expansion of the helium atom. is the Cl energy calculated in a basis of AOs of angular momentum I

Fig. 7.8. The energy increments logio(—tt) of the partial-wave expansion of the ground-state helium atom plotted against logio (L -I-1/2). The thick grey line represents the function C4 (L -H l/2) with C4 fitted to the calculated numbers. Atomic units are used.

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