By Ramon Carbo, Joseph M. Riera

We stay in a molecular global, virtually closed shell in nature, and therefore Chemistry has been a technological know-how facing closed shell mol ecules. even though, the excessive measure of experimental sophistication reached long ago decade has made extra obvious the position of open shell constructions in chemical study. A parallel phenomenon will be saw within the improvement of SCF idea, the place closed shell molecular calculations at any point of complexity compose the most physique of references which are bought in Quantum Chemistry this present day. along with the linkage among experimental and theoretical behaviour, there are, evidently, different purposes which might be hooked up to an absence of molecular open shell calculations. between others, there has been no connec tionbetween closed or open shell theoretical remedies. during this demeanour, many computational beneficial properties utilized by closed shell connoisseurs haven't been prolonged to different computational parts. because the paintings of Roothaan in 1960, the open shell molecular panorama has been, the oretically, a really closed one. extra improvement of SCF idea, which has resulted in an outburst of multiconfigurational strategies, has paid no, or very faint, cognizance to the interconnection among those SCF thought complex good points, the open shell framework and closed shell universal perform. an excellent theoretical aim, in most cases talking, and particularly within SCF thought, may well include a strategy that are used to resolve a given chemical challenge, in the actual and approx imate limits of the theory.

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Theorem: (Structure of the PE hamiltonian). The off-diagonal hamiltonian matrix elements with respect of the PEF basis set have the following algorithm (6. V) Proof: <~ pC7l I ~~ > I q This follows since between any pair of SD with K construction, more than three differences. t L there exist, by Then, between two SD with K = L there are only two differences, and a maximum coincidence. • At the same time, the hamiltonian diagonal elements can be written as (7. V) + + where 95 I I r~s~ t t I fi Vk I kEU (2 J r~ rs - Krs) + 2 Ie£?

V) A .. L o(iES K; 2 1J jES K) K ~ B .. 1J L Aij - 2 K,L (L>K) a (n C~ - 1 The sums in the energy run over all the orbitals entering in the set ~= ~ S~. }, {A .. } and {B .. 11 will be entirely applicable in this case. 3. Two Electron Systems As an application example of the framework stud"ied in the previous section, a two electron problem will be studied here. In this case, each configuration will be associated with an index set of cardinal number unity and each configuration index will be made equal to a MO index.

1. 1 , 1 F II 1,) will only multiply the F eigenvalues by the {ail factors. Since the differences between Fock operators of each shell vanish, in this case: 11. Corollary From a general energy expression, a unique pseudo secular equation attached to a unique coupling operator can be obtained, upon variation. This last form can be applied without further manipulations to any simpler case. The Coupling Operator is constructed in such a manner that it can be used to obtain the initial energy expression.