Orbital collapse computation

The limit cycle found in the previous section holds only for 103 — 031 small. Obviously, once the limit cycle exists, it can be continued, either globally or until certain bad things happen such as the period tending to infinity or the orbit collapsing to a point. It is very difficult to show analytically that these events do not occur. Moreover, the computations necessary to actually prove the asymptotic stability of the bifurcating orbit are very difficult. We discuss briefly some numerical computations, shown in Figure 8.1, which suggest answers to both these problems. 

The understanding of orbital collapse remained in a rather unsatisfactory state until around 1969 when, as a result of considerable progress in numerical methods and the advent of fast computers, it became a relatively simple matter to solve the radial Schrodinger equation in the Hartree-Fock and related schemes, which take better account of the shell structure of the atom. 

This method is probably as accurate as some other simple pseudopotential approaches. However, there appear to be some difficulties in improving it to the standard of some of the recent pseudopotential calculations. Attempts to use larger than minimal basis sets required the inclusion of a Phillips-Kleinman term in addition to the orthogonality procedure in order to prevent collapse of the valence orbitals into the core space. Thus in calculations on AlaQ, Vincait had to include not only the A1 3s and Cl 3j and 3p shells but also the A12p and Cl 2s and 2p shells explicitly in the valence-electron basis in order to obtain good results. Consequently this calculation was not substantially less expensive in computing time than an equivalent all-electron calculation. 

In the case H211, the vectors Rq, Aq, and Pq (which describe projections perpendicular to the direction of periodicity) vanish for all choices of orbitals and nuclei. This causes all the incomplete Bessel functions appearing in the computations to reduce to exponential integrals, with all the singular terms collapsing to the form given in Eq. (16) of the present communication. A more comprehensive test is provided by the case H2X, because the... 

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