Spin pairing of electrons

Despite the quantitative victory of molecular orbital (MO) theory, much of our qualitative understanding of electronic structure is still couched in terms of local bonds and lone pairs, that are key conceptual elements of the valence bond (VB) picture. VB theory is essentially the quantum chemical formulation of the Lewis concept of the chemical bond [1,2]. Thus, a chemical bond involves spin-pairing of electrons which occupy valence atomic orbitals or hybrids of adjacent atoms that are bonded in the Lewis structure. In this manner, each term of a VB wave function corresponds to a specific chemical structure, and the isomorphism of the theoretical elements with the chemical elements creates an intimate relationship between the abstract theory and the nature of the... 

The first ionization energy of sulfur (Is 2s 2p 3s 3p 3p 3p ) is less than that of phosphorus (Is 2s 2p 3s 3p 3p 3p ) because less energy is required to remove an electron from the dp orbitals of sulfur than from the half-filled 3p orbitals of pbospborus. The presence of a spin pair of electrons results in greater electron repulsion compared to two unpaired electrons in separate orbitals. 

The names of bond types o and n bonds are formed by spin pairing of electrons on adjacent atoms. 

The spin-pairing scheme of the product, B)j., is different from that of the reactant. This happens if at least two pairs of electrons have exchanged partners. In other words, at least three electi ons need to be involved. 

The key to the correct answer is the fact that the conversion of one type-V (or VI) structures to another is a phase-inverting reaction, with a 62 species transition state. This follows from the obseiwation that the two type-V (or VI) stiucture differ by the spin pairing of four electrons. Inspection shows (Fig. 28), that the out-of-phase combination of two A[ structmes is in fact a one,... 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

In Pauli s model, the sea of electrons, known as the conduction electrons are taken to be non-interacting and so the total wavefunction is just a product of individual one-electron wavefuncdons. The Pauli model takes account of the exclusion principle each conduction electron has spin and so each available spatial quantum state can accommodate a pair of electrons, one of either spin. 

Naively it may be expected that the correlation between pairs of electrons belonging to the same spatial MO would be the major part of the electron correlation. However, as the size of the molecule increases, the number of electron pairs belonging to different spatial MOs grows faster than those belonging to the same MO. Consider for example the valence orbitals for CH4. There are four intraorbital electron pairs of opposite spin, but there are 12 interorbital pairs of opposite spin, and 12 interorbital pairs of the same spin. A typical value for the intraorbital pair correlation of a single bond is 20kcal/ mol, while that of an interorbital pair (where the two MO are spatially close, as in CH4) is 1 kcal/mol. The interpair correlation is therefore often comparable to the intrapair contribution. 

To show how orbital diagrams are obtained from electron configurations, consider the boron atom (Z = 5). Its electron configuration is ls22s22p1. The pair of electrons in the Is orbital must have opposed spins (+j, or f j). The same is true of the two electrons in the 2s orbital. There are three orbitals in the 2p sublevel. The single 2p electron in boron could be in any one of these orbitals. Its spin could be either up or down. The orbital diagram is ordinarily written... 

According to this model, a covalent bond consists of a pair of electrons of opposed spin within an orbital. For example, a hydrogen atom forms a covalent bond by accepting an electron from another atom to complete its Is orbital. Using orbital diagrams, we could write... 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

The kinetic energy in the Hartree-Fock scheme is evidently too low, owing to the fact that we have assumed the existence of a simplified uncorrelated motion, whereas the particles in reality have much more complicated movements because of their tendency to avoid each other. The potential energy, on the other hand, comes out much too high in the HF scheme essentially due to the fact that we have compelled a pair of electrons with opposite spins together in the same orbital in space. 

C atom has one unpaired electron in each of its four sp hybrid orbitals and can therefore form four cr-bonds that point toward the corners of a regular tetrahedron. The C—C bond is formed by spin-pairing of the electrons in one sp hybrid orbital of each C atom. We label this bond hybrid orbital composed of 2s- and 2/t-orbitals on a carbon atom, and the parentheses show which orbitals on each atom overlap (Fig. 3.15). Each C—H bond is formed by spin-pairing of an electron in one of the remaining sp hybrid orbitals with an electron in a 1 s-orbital of an H atom (denoted His). These bonds are denoted cr(C2s/ Hls). 

Introduction of the half-integral spin of the electrons (values h/2 and —fe/2) alters the above discussion only in that a spin coordinate must now be added to the wavefunctions which would then have both space and spin components. This creates four vectors (three space and one spin component). Application of the Pauli exclusion principle, which states that all wavefunctions must be antisymmetric in space and spin coordinates for all pairs of electrons, again results in the T-state being of lower energy [equations (9) and (10)]. 

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