Multi-centre expansion

Below we show that as an alternative to the multi-centre expansion (5.28), the wave function may be written in terms of a one-centre expansion of the form... 

Of those methods that aim to mimic ab initio results some are very accurate indeed. The disadvantage is that they may be only marginally more economical than the full ah initio calculation. Such methods are PDDO (Projectors of Diatomic Differential Overlap), developed by Newton and co-workers,56-58 and LEDO (Limited Expansion of Diatomic Overlap), devised by Billingsley and Bloor.59 Both methods are essentially ways of accurately approximating some of the multi-centre integrals. Both approximate a two-centre distribution by one-centre distribution according to... 

In the nonrelativistic limit, only the (LL LL) class of multi-centre integrals is needed this class is still the most important in relativistic calculations. In terms of the conventional expansion in powers of a, a DHF calculation at this level includes all the one-centre relativistic effects, screening by the Coulomb interaction, and some of the spin-other-orbit effects. The (LL 55) and (55 55)... 

The vast majority of contemporary molecular calculations overcome the convergence problems of the one-centre expansion method, which are associated with the description of off-centre nuclei, by employing basis functions located on a number of centres usually coinciding with the nuclei in the molecule. Rather accurate calculations for molecules containing more than one non-hydrogenic nucleus can be performed by using a multi-centre basis set. The use of such basis sets does, however, give rise to new problems. 

In independent-electron models of atomic and molecular electronic structure, such as the Hartree-Fock approximation, only fimctions corresponding to the first few values of the angular momentum quantum number / contribute significantly to the energy, or to other expectation values, when multi-centre basis sets are employed. However, in treatments which take account of electron correlation effects, the higher harmonics are known to be important. A considerable amount of data is available on the convergence properties of the harmonic expansion for atoms and the importance of higher-order terms in the harmonic expansion for molecular systems has also been demonstrated. ... 

In this Section I review our work on force fleld design, which is based on the transferabihty of atomic multipole moments. We adopt Stone s spherical tensor formahsm and the idea of a multi-centre multipole expansion, one site for each nucleus. However, the moments are obtained via integration over an atomic basin of the appropriate tensor times the electron density, rather than via the DMA route. Starting with the electrostatic potential I move on... 

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