By Ramon Carbo, Joseph M. Riera

We dwell in a molecular global, nearly closed shell in nature, and for that reason Chemistry has been a technology facing closed shell mol ecules. although, the excessive measure of experimental sophistication reached long ago decade has made extra obvious the position of open shell buildings in chemical learn. A parallel phenomenon could 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 that are received in Quantum Chemistry this day. in addition to 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 good points utilized by closed shell connoisseurs haven't been prolonged to different computational components. because the paintings of Roothaan in 1960, the open shell molecular panorama has been, the oretically, a really closed one. additional improvement of SCF conception, which has ended in an outburst of multiconfigurational approaches, has paid no, or very faint, consciousness to the interconnection among those SCF thought complicated positive factors, the open shell framework and closed shell universal perform. an outstanding theoretical aim, in general conversing, and specifically within SCF thought, may well encompass a strategy that are used to unravel a given chemical challenge, in the actual and approx imate limits of the theory.

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5{C l ;C 2 ; ••• ) = 2. Complete MCPE Energy Expression Let's suppose that a closed shell ground state wavefunction ~ is built up with all the Nc configurations available in a given problem, then if 1, one can wr i te (2. V) E = L C~ <~K~I ~K> K + L K,L (KIL) <5 {n - 1 = # (SK () SL» CKC L <~K 1c:W'1 ~L> From here it is moderately easy to show that E . L w. ~tf))J 1. ~- h.. 1. + L (A, . J.. ) - B .. K .. ) . 43 where (3. V) (4-. V) (5. 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~.

SjO;i>; l. iEP. The implications become more apparent if we consider the case where all eigenvalues have a common shifting parameter, say, A o;v has now a manyfold degeneracy and the eiThen, the eigenvalue A o,v genvectors have an accompanying arbitrariness. NOW, going back to the general case, where the matrix M is not necessarily diagonal with respect to the vectors {Io;p>}, PEP, and considering a new matrix T, with elements defined according to the expression 26 (10. III) t .. = O•• A ~J ~J .

In order to develop a useful theoretical background with immediate application to a computational algorithm, the most convenient procedure is to start with a generalized secular equation in matrix form (1. III) . Oij> The hermitean matrix RO can be modified by a new hermitean matrix to form (2. III) v, R = RO + V. In order to observe the action of V in the eigenspace of RO' it is necessary to consider V as the projection of an hermitean arbitrary matrix, say M, over some subspace of the eigenspace of RO' that is (3.