One-electron terms

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. 

In this latter formula, the two electron repulsion integral is written following Mulliken convention and the one electron integrals are grouped in the matrix e. In this way, the one-electron terms of the Hamiltonian are grouped together with the two electron ones into a two electron matrix. Here, the matrix is used only in order to render a more compact formalism. 

The first, one-electron term is readily simplified by realizing that all of the N electrons in the molecule are indistinguishable. This integral describes the motion of each electron about the fixed atomic nuclei in the absence of all other electrons, and can therefore 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. 

We can now consider explicitly how configurations interact to produce electronic states. Our first task is to define the Hamiltonian operator. In order to simplify our analysis, we adopt a Hamiltonian which consists of only one electron terms and we set out to develop electronic states which arise from one electron configuration mixing. 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

The left-hand side (LHS) of Eq. (4.26) will teU us what to use in Eq. (4.25) instead of Concerning the one-electron terms, the integrals carried out solely over the valence space yield the nuclear-electronic potential energy of the N ... 

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.

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... 

Here, S denotes the inverse overlap matrix element between basis functions and v. With Eqs. (102) and (103) the one-electron terms in Eq. (100) reduce to... 

The only terms remaining to be defined in the assembly of the HF secular determinant are the one-electron terms for off-diagonal matrix elements. These are defined as... 

Now we shall have to express operators for physical quantities in terms of irreducible tensors in the spaces of total angular momentum and quasispin. One-electron terms of relativistic energy operator (2.1) (formulas (2.2)-(2.4)) are expressed in terms of operators (23.69), (23.71)-(23.73) in a trivial way. With two-electron operators the procedure of deriving the pertinent relations is more complex. The relativistic counterpart of (18.50)... 

For olefins, the first term has been studied in terms of the symmetry D2 independently by two groups913. The second term, which was also called the one-electron term, is usually neglected in cases of electronically allowed transitions because it is small compared with the other two terms. It is even smaller in the olefinic case, where the molecules contain mainly nonpolar bonds. Scott and Yeh73 have therefore concentrated on the third term, which arises from the coupling of a considered transition a with all the transitions of different chromophores within the molecule. If transition a falls energetically below all the transitions of the other chromophores, the latter can be viewed as polarizable groups and their anisotropies fi determine the dynamic coupling74. The final expression obtained was... 

The term E%(eq) is the potential energy of the clumped nuclei configuration at the equilibrium geometry R0 (crude level), hpQ is one -electron term ( core Hamiltonian ) for equilibrium nuclear configuration Ro, and upq (Q) represents matrix element of electron - vibration (phonon) coupling. 

It leaves intact the fermion operators related to the /1-th group itself. By virtue of this the two-electron operators WBA result in a renormalization of one-electron terms in the Hamiltonians for each group. <4 = 1,..., M. The expectation values ((b+b ))B are the one-electron densities. The Schrodinger equation eq. (1.193) can be driven close to the standard HFR form. This can be done if one defines generalized Coulomb and exchange operators for group A by their matrix elements in the carrier space of group A ... 

Hamiltonians thus defined contain large one-electron terms describing the attraction of electrons to the unscreened atomic cores in an alien subsystem ... 

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