Brillouins’ theorem, electron correlation

Because the Cl wavefunction is a much more flexible wavefunction than is o, Eci < Esc we lower the energy in line with the effect of correlating electron motions. [If we add only single excitations, there can be no improvement of the RHF or UHF energy, since singles do not directly mix with these solutions (Brillouin s theorem). Double excitations have to be included to introduce electron correlation.]... 

The importance of a given double excitation depends on the energy connected with the electron relocation and the arrangement of points A,B,C,D. Yet this simplistic reasoning suggests single citations do not carry any correlation (this is confirmed by the Brillouin theorem) and this is why their role is very small. Moreover, it also suggests that double citations should be very important. 

An important implication of Brillouin s theorem is the guarantee of no first-order errors for all one-electron operators provided the exact Hartree-Fock wavefunctions are used.426 This follows because the first-order correction to the correlation effect is derived from two-electron excitations. 

Methods based on such procedures may be ranked by the level of excitations admitted in the construction of from Pq- only one electron at a time is excited, one speaks of single excitations or, more simply, singles. However, as discussed in the CASSCF section, Brillouin s theorem proves that singly excited cannot mix with a variationally optimized Pq in closed-shell molecules, and that only special types of singly excited Pj can mix with Pq in open-shell molecules. In contrast, simultaneous promotions of two electrons [i.e., double excitations ( doubles )] are always important, because Coulomb repulsions involve two electrons and doubles provide correlation between pairs of electrons. Triple excitations ( triples ), which affect the wavefunction for three electrons, are usually less important, whereas inclusion of quadruples accounts for the effects of different double excitations on each other. These different excitation levels are denoted by letters S (singles), D (doubles), T (triples), and Q (quadruples). 

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