Coulombic repulsion, spin-paired electrons

When r-i = r2 the triplet product functions, Eqs. (3.7.4)-(3.7.6), vanish The two electrons cannot be localized in the same spot this is called a "Fermi hole." When r1 = rz, then the singlet spatial wavefunction, Eq. (3.7.3), is finite and nonzero. This consequence of Fermi-Dirac statistics is called "spin pairing" Two electrons with opposite spins attract each other (despite the classical Coulomb repulsion), while two electrons with the same spin repel each other. 

Electron correlation in the triplet is a consequence of the Pauli exclusion principle, which is embodied mathematically in the antisymmetrization of the electronic wave function. Anti-symmetrization results in zero probability for two electrons of the same spin simultaneously being in the same AO in an LCAO-MO wave function. This obviously reduces the Coulomb repulsion between these electrons from the value, Jxy, that is calculated by computing the Coulomb repulsion energy between a pair of uncorrelated electrons, one in f/x and the other in ffy. 

This difference between these two carbenes can be attributed to the presence of the lone pair of electrons on nitrogen in 4b. Two-center, three-electron bonding in the molecular plane of 4b, involving the lone pair, produces some unpaired a spin density on nitrogen. Coulombic repulsion between the a and tt electrons of... 

The filled (Is)2 cores lead to additional steric repulsions with the incoming donor hybrid. These combine with nuclear Coulomb repulsions to oppose high electronic overlap. Figure 3.6 displays the calculated increase in steric repulsion as each filled 2s spin-orbital of Li2 collides with the filled core Is of the opposite atom. The paired one-electron steric repulsions shown in Fig. 3.6 are similar to the ionic two-electron steric repulsions of Fig. 2.11, and the wave-mechanical origin of the steric pressure would be analogous to that described in the discussion surrounding Fig. 2.12. 

The operator 2Jj computes the Coulombic repulsion energy between an electron in v[/, and the pair of electrons in /,-, assuming that the electrons in v[/, and in /,- move independently of each other. The operator ICj corrects the Coulombic repulsion energy, computed from 2Jj, for the fact that antisymmetrization of results in correlation between an electron in /,- and the electron of the same spin in v[/,. 

The motion of each electron in the Hartree-Fock approximation is solved for in the presence of the average potential of all the remaining electrons in the system. Because of this, the Hartree-Fock approximation, as discussed earlier, does not provide an adequate description of the repulsion between pairs of electrons. If the electrons have parallel spin, they are effectively kept apart in the Hartree-Fock method by the antisymmetric nature of the wavefunction, producing what is commonly known as the Fermi hole. Electrons of opposite spin, on the other hand, should also avoid each other, but this is not adequately allowed for in the Hartree-Fock method. The avoidance in this latter case is called the Coulomb hole. 

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