Bielectronic operators

In addition, F contains two bielectronic operators. They describe the interaction between the electron occupying spin orbital i and the other electrons found in the atom. So, for the interaction between electrons 1 and 2 at a distance ri2, we have the Coulomb operator Jj and the exchange operator Kj defined by... 

The choice of the basis set. One may perform a new S transform for the H conformation and identify the new orthogonalized AOs with that of the bicentric problem. This is the most direct solution but the orthogonaliz-ation tails will be different and the use in this new basis of the effective bielectronic Hamiltonian given in Eq. (136) for instance may result in uncontrolled effects. One may also express the bielectronic operator of Eq. (136) in the non-orthogonal basis set and calculate the Hamiltonian matrix of H in the basis of non-orthogonal determinants, antisymmetrized products of Is AOs. The problem to solve is then of H-ES) type and it faces a typical non-orthogon ity problem of VB methods, which has been a major drawback of these approaches. 

Now we turn to the bielectronic integrals contained in the left-hand side of Eq. (3.5). They require a little attention. Consider the Coulomb operator Jj from Eq. (2.5). The integral J dri is over all space, but can be split into two contributions, namely, in short-hand notation, as... 

The dispersion contribution to the interaction energy in small molecular clusters has been extensively studied in the past decades. The expression used in PCM is based on the formulation of the theory expressed in terms of dynamical polarizabilities. The Qdis(r, r ) operator is reworked as the sum of two operators, mono- and bielectronic, both based on the solvent electronic charge distribution averaged over the whole body of the solvent. For the two-electron term there is the need for two properties of the solvent (its refractive index ns, and the first ionization potential) and for a property of the solute, the average transition energy toM. The two operators are inserted in the Hamiltonian (1.2) in the form of a discretized surface integral, with a finite number of elements [15]. 

In these expressions, h, J and K are the usual monoelectronic coulombic, bielectronic coulombic and exchange operators, respectively, and R are the one-particle density operators expressed in the and 6 orbital basis ... 

But it must be clear that this reduction of information and this focus on some low part of the spectrum proceed differently and lead to completely different tools. The effective Hamiltonians appear as N-electron operators acting in well defined finite bases of iV-electron functions. The effective Hamiltonians obtained from the exact bielectronic Hamiltonian introduce three- and four-body interactions. They may essentially be expressed as numbers multiplied by products of creation and annihilation operators. In contrast, the pseudo-Hamiltonians keep an a priori defined analytic form, sometimes simpler than the exact Hamiltonian to mimic. For instance, the... 

Figure 2. The various types of CW interactions in a four-electron-three-orbital system. Bielectronic charge transfer and mixed" bielectronic interactions are omitted. It Is Important to note that monoelectronic polarization can operate only within an H- but not within a D- or U- bound system, by symmetry.

Furthermore, note that the U-bound 4e-4o system can be thought of as a composite of two subsystems and large polyelectronic molecules are made up of many different subsystems because of operative symmetry constraints. This means, that, in general, a large part of bielectronic correlation in molecules is reproduced only at the SCF-MO-CI (VB or MOVB) level, a situation fundamentally different from the one encountered before in the case of monoelectronic polarization. 

Since bielectronic correlation couples subsystems while pair or hole monoelectronic polarization operates within each subsystem, it follows immediately that the gauche isomer will be favored by W and the 1,1 isomer by polarization since these mechanisms come strongly into play only if fragment orbitals spanning the same AO s are available. This is the first quantum chemical derivation of coulomb polarization selection in molecules. 

---