Primitive Gaussian distributions

So far we have only derived the multipole expansion for primitive Gaussian distributions. As pointed out in the introductory example, one of the main strengths of the multipole expansion is that it can be used to treat the interactions of several primitive charge distributions simultaneously by combining them into one single, albeit more complicated, distribution. It will turn out to be useful to translate the centers of multipole expansions to different points in space e.g., if (A) is an expansion about A, we must find a way to convert it to a series about A — t, where t is the translation vector. 

The explicit nature of p, (for example, point charges or products of primitive Gaussians) is irrelevant for most of what follows. In the present context, N is not the number of electrons, but the number of distributions, which is assumed to be proportional to the number of atoms in the molecule. 

We first consider a simple case the transition from primitive to contracted Gaussian distributions. To this end, the multipole expressions for primitive charge distributions flap = and flyg = have to be contracted with the contraction coefficients and as... 

The method that is used in most of the work described in this chapter is the distributed multipole analysis (DMA) of Stone,which is implemented in the CADPAC ab initio suite. DMA is based on the density matrix p,y of the ab initio wavefunction of the molecule, expressed in terms of the Gaussian primitives q that comprise the atomic orbital basis set ... 

Au(l) has a 5d ( S) electronic state configuration. The RCEPs used here were generated from Dirac-Fock (DF) aU-electron (AE) relativistic atomic orbitals, and therefore implicitly include the major radial scahng effects of the core electron on the valence electrons. In the closed-sheU metal-ion system these indirect relativistic effects are dominant. In the smaU-core RCEP used here the 5s 5p subsheUs are included in the valence orbital space together with 5d, 6s and 6p atomic orbitals, and aU must be adequately represented by basis functions. The gaussian function basis set on each metal atom consists of the pubUshed [4 P3 ] distribution which is double-zeta each for the 5sp and 6sp orbitals, and triple-zeta for the 5d electrons. The standard notation for this basis set is RCEP-4111/31 IG to show exphcitly the gaussian primitive distribution in each basis function. 

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