Brillouin’s theorem

The Microstate Cl Method lowers the energy of the uncorrelated ground state as well as excited states. The Sngly Btcited Cl Method is particularly appropriate for calculating UV visible spectra, and does not affect the energy of the ground state (Brillouin s Theorem). 

In addition, the numerator will be nonzero only for double substitutions. Single substitutions are known to make this expression zero by Brillouin s theorem. Triple and higher substitutions also result in zero value since the Hamiltonian contains only one and two-electron terms (physically, this means that all interactions between electrons occur pairwise). 

Brillouin s theorem (Brillouin, 1933) tells us that the singly excited states do not interact with the HF ground state. This theorem is true for all HF wavefunc-tions, and does not depend on the ZDO or LCAO approximations. This means that... 

If we used perturbation theory to estimate the expansion coefficients c etc., then all the singly excited coefficients would be zero by Brillouin s theorem. This led authors to make statements that HF calculations of primary properties are correct to second order of perturbation theory , because substitution of the perturbed wavefunction into... 

The Brillouin s theorem has been shown to hold in the case of the HPHF function [2], As a result, any variations of the orbitals which minimizes the HPHF total energy, can be expressed as ... 

Figure 7.4 Structure of the CI matrix as blocked by classes of determinants. The HF block is the (1,1) position, the matrix elements between the HF and singly excited determinants are zero by Brillouin s theorem, and between the HF and triply excited determinants are zero by the Condon-Slater rules. In a system of reasonable size, remaining regions of the matrix become increasingly sparse, but the number of determinants in each block grows to be extremely large. Thus, the (1,1) eigenvalue is most affected by the doubles, then by the singles, then by the triples, etc...

Let us return, however, to singly excited determinants. While, like triples, they fail to interact with the ground state (although in this case because of Brillouin s theorem), they too mix with doubles and thus can have some influence on the lowest eigenvalue. In this instance, there are sufficiently few singles compared to doubles that it does not make the problem significantly more difficult to include them, and this level of theory is known as CISD. 

Born-Oppenheimer approximation, 22, 219 Bound state, 209-210 Bid, with alkenes, 260 Brillouin s theorem, 241 Bromide ion (Br ) effect on Ao, 181 trails effect, 181 as X ligand, 176 Bromine (Br2) sigma bond, 77 Bromochloromethane, 13 ll-Bromo- ii/o-9-chloro-7-... 

The second problem is the much more realistic one of the effect of a limited basis set expansion. This is clearly a more serious problem because only for linear molecules or those with a few first-row atoms can the Hartree-Fock limit be reached at present. For many of the molecules with which theoretical chemists deal, wave-functions of such accuracy are not available but it may be some comfort to know that even if they were they need not give very good answers It should be mentioned that Brillouin s theorem applies to any SCF wavefunction, but unless the wavefunction is near the Hartree-Fock limit the electron distribution cannot be expected to be a close representation of the true one. No general treatment of this problem has been given neither does one seem possible since it would depend on the ways in which the basis set under consideration was weak, and these may be many. 

The HF term includes intemuclear repulsions, and the perturbation correction E(2) is a purely electronic term. E1 2 is a sum of terms each of which models the promotion of pairs of electrons. So-called double excitations from occupied to formally unoccupied MOs (virtual MOs) are required by Brillouin s theorem [89], which says, essentially, that a wavefunction based on the HF determinant Z> plus a determinant corresponding to exciting just one electron from Dl cannot improve the energy. 

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