Two-electron interaction

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

The main difficulty in solving the Schrodinger equation (Eq. II. 1) for a many-electron system comes from the two-electron interaction terms... 

Secondary orbital interaction had been proposed to explain predominant formation of endo attack prodncts in Diels Alder reaction of cyclopentadiene and dienophiles by Hoffmann and Woodward [22]. According to this rnle, the major stereoisomer in Diels-Alder reactions is that it is formed through a maximum accumulation of double bonds. In the Diels-Alder reactions, secondary orbital interaction consists of a stabilizing two-electron interaction between the atoms not involved in the formation or cleavage of o bonds (Scheme 19). 

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... 

The antiaromatic region is not important for the reactivity of the parent enediyne because the activation energy is determined only by the energy difference between the reactant and the TS. However, for the cyclic enediynes in Fig. 7 in which the C1-C6 distances are 3.39 and 2.92 A, respectively, antiaromaticity of the reactant should be relevant to the reaction kinetics. In addition, the role of repulsion between the in-plane filled orbitals is accentuated by a parallel decrease in the attractive two-electron interaction between the re and re orbitals which vanishes at the 3.2 A distance between the terminal carbon atoms. 

The terms etc. represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (3) in addition to the electron... 

Hamiltonians involving more than two electron interactions. I shall use this to illustrate the general case of arbitrary p. The second-order reduced density matrix (2-RDM) of a pure state ij/, a function of four particles, is defined as follows ... 

The following important conclusions can be drawn from the above results [88JST(163)173]. First, the values of [2A ]n are nearly equal for furan and pyrrole hence the correct aromaticity trend can be ascertained only if the [XAE], contributions are also taken into account. Thus, the relative aromatic character of the compounds under discussion is determined by the sum of the stabilizing effects of the two electron interactions. These are the stabilization energy AE, referring to the interaction be-... 

Let us consider approximations in accounting for the Breit interaction, that we made when outer core and valence electrons are included in GRECP calculations with Coulomb two-electron interactions, but inner core electrons are absorbed into the GRECP. When both electrons belong to the inner core shells, the Breit effect is of the same order as the Coulomb interaction between them. Though Bff does not contribute to differential (valence) properties directly, it can lead to essential relaxation of both core and valence shells. This relaxation is taken into account when the Breit interaction is treated by self-consistent way in the framework of the HEDB method [33, 34]. 

Figure 5.1 The CNDO formalism for estimating repulsive two-electron interactions fails to distinguish in one-center cases between different orbitals (top example for the case of methylene) and in two-center cases either between different orbitals or different orbital orientations (bottom example for the case of hydrazine)...

Since all the two-electron interaction operators are invariant under rotations in the space of total momentum J of two particles, we shall consider only the scalar two-electron operators... 

In a relativistic case let us consider for the same purposes two subshells of equivalent electrons ni /1 2j 2 2 - General expressions do not depend on the explicit form of a two-electron interaction, therefore we shall not specify the operator type. Here, non-diagonal matrix elements of two sorts occur N1N2 — Ni — 1 IV2 + 1 and N1N2 — Ni — 2 N2 + 2. In the first case we have... 

Figure 35 Schematic representation of the two-center, two-electron interaction between transition metals and Z-type ligands.

As an approximation to equation (11) it is useful to consider a valence hamiltonian for which only the one-electron operators are transformed and the two-electron interactions remain unchanged. The hamiltonian now simplifies to ... 

---