Two-electron terms

A more severe challenge is the efficient treatment of the DKH transformation of the two-electron terms. Here, the problem is actually two fold. First, the electron density or the orbital densities which enter the electron-electron interaction integrals are affected by the picture-change effect on the density. Then, these integrals contribute to the potential into which the Fock-type operator is expanded. Moreover, the interaction integrals are not necessarily even, block-diagonal operators, which has led to the restrictions made in section 11.1. 

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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). 

Each of the MNDO, AMI and PM3 methods involves at least 12 parameters per atom orbital exponents, Cj/pi one-electron terms, II /p and j3s/p two-electron terms, Gss, Gsp, Gpp, Gp2, Hsp, parameters used in the core-core repulsion, a and for the AMI and PM3 methods also a, b and c constants, as described below. 

The aja, operator tests whether orbital i exists in the wave function, if that is the case, a one-electron orbital matrix element is generated, and similarly for the two-electron terms. Using the Hamiltonian in eq. (C.6) with the wave function in eq. (C.4) generates the first quantized operator in eq. (C.3). 

The second, two-electron term in Eq. (2.18) is much harder to evaluate, as it describes the energy of an electron as it moves through the average field created by the remaining (N— 1) electrons. After a considerable amount of mathematical magic, however, the energetic contribution of the two-electron interactions can be written as... 

By grouping one-electron and two-electron terms together the Hamiltonian may be written as... 

The energy matrices contain one-electron terms (which can be written down in terms of the AOM e parameters) and two-electron terms, expressed as multiples of the Racah parameters B and C. Values of the and Racah parameters which provide the best fit to the experimental data are then found. Most work has been done on the tetragonal (D4h) chromophores M X where the N atoms (equatorial) are provided by amine ligands. Only three AOM parameters can be determined since there are only three independent orbital splitting parameters eff(N), e0(X) and en.(X) can be found if ew(N) is taken to be zero, saturated amines having no orbitals available for jr-overlap. 

It should be noted that what has been demonstrated for the p-CSE does not hold for the 1-CSE because the Hamiltonian includes two-electron terms. In fact. 

Note this resembles the (iV-particle) Fock operator that appears in Hartree-Fock theory, but the contribution of the two-electron term is only half the normal contribution in the Fock operator. However, if we consider making the energy stationary w.r.t. variations in the reference, we must also consider the second term in Eq. (34), where we find... 

Notice the appearance both here and above of the vital parameter U/4/9 which is the measure of the importance of the two-electron terms to the one-electron ones. This discussion is exactly analogous to the Hubbard treatment of solids which we will study below. 

Figure 5 Areas of validity of Mulliken-Hund (M-H) and Heitler London (H-L) viewpoints, k is the ratio U/4/J, the ratio of one of the two-electron terms to a multiple of one of the one-electron terms.

The Breit-Pauli Hamiltonian with an external field contains all standard one- and two-electron contributions as well as the magnetic interaction of the electrons and their interactions with an external electromagnetic field. We may group the various contributions in the Breit-Pauli Hamiltonian according to one-and two-electron terms,... 

In contrast to the one-electron terms, the reduction of the 4x4 Dirac-Breit Hamiltonian to the 2x2 Breit-Pauli Hamiltonian is very tedious for the two-electron terms as each interaction term has to be transformed according to the Foldy-Wouthuysen protocol. As the derivation can be found for example in Refs. (56-58) and in detail in Ref. (21), we only present here the transformed terms and discuss their dimension. The two-electron Breit-Pauli operator gBP (i, j) reads... 

For an excited tttt configuration, we have the possibilities 5 = 0 and 5=1 for the total spin quantum number hence we get two electronic terms, a singlet and a triplet. (If one or both of the pi energy levels involved are degenerate, as in C6H6, we get more than two electronic terms from the 7777 configuration.) Lots of different notations are in use in... 

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