Most spin-alternated determinants

FIGURE 8.1 Energies of the singlet ( Fo) and triplet excited ( P ) states of ethylene relative to the most spin-alternant determinant (MSAD). 

Apart from these simplifying assumptions, a fundamental difference between qualitative VB theory and spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO determinant, without any a priori bias for a given electronic coupling into bond pairs. Instead of an interplay between VB structures, a molecule is viewed then as a collective spinordering The electrons tend to occupy the molecular space (i.e., the various atomic centers) in such a way that an electron of a spin will be surrounded by as many p spin electrons as possible, and vice versa. Determinants having this property, called the most spin-alternated determinants (MSAD) have the lowest energies (by virtue of the VB rules, in Qualitative VB Theory) and play the major role in electronic structure. As a reminder, the reader should recall from our discussion above that the unique spin-alternant determinant, which we called the quasiclassical state, is used as a reference for the interaction energy. 

Spin densities determine many properties of radical species, and have an important effect on the chemical reactivity within the family of the most reactive substances containing free radicals. Momentum densities represent an alternative description of a microscopic many-particle system with emphasis placed on aspects different from those in the more conventional position space particle density model. In particular, momentum densities provide a description of molecules that, in some sense, turns the usual position space electron density model inside out , by reversing the relative emphasis of the peripheral and core regions of atomic neighborhoods. 

Exercise 8.3 Combining the MSAD with a minor second most alternant determinant leads to the qualitative spin multiplicities displayed in Scheme 8.Ans.2... 

Synthesis of the closely related acyclic (19) and macrocyclic (20) polyradicals has recently been reported (Figure 5.1).1231 The -conjugated carbanions (e.g., the calix[4]-arene-based tetraanion and the related calix[3]arene-based trianion) were synthesized and studied.1241 Oxidation of these tetra- and tri-anions gave the corresponding tetra- and tri-radicals, respectively. It has been shown in closely related systems that it is not the shape or overall geometric symmetry of the molecules, but rather it is the juxtaposition of the carbenic centers within the jt-cross-conjugated structure, that is most important in determining the spin multiplicity of the alternant hydrocarbon molecule.1251... 

This sort of model can easily be generalised to deal with more than two electrons and other assignments of the spins. The case of most interest in molecular studies is that in which a set of molecular orbitals are all occupied by two electrons. Thus if there were two electrons, one of either spin, in both orbitals and y>2, the total wave function would be a 4 x 4 determinant. But most of the features of the two-electron model are retained. The system could be alternatively described as consisting of two electrons in each of the equivalent orbitals. The effect of the antisymmetry principle is then to keep electrons of the same spin apart, the motion of the two opposite spin-types being uncorrelated. 

Some of the most efficient algorithms currently available for spin-coupled calculations involve the expansion of the SC wavefunction in terms of Slater determinants, leading to a summation over (N-p) -dimensional cofactors for the p-particle density matrix [23]. For calculations involving the full spin space, a further saving can be achieved by the use of projected spin functions [24], An alternative strategy is provided by CASVB (see Sect. 5). 

The three rows in Table II refer to properties of the spin density. In the first one it vanishes, which implies that we must have doubly filled orbitals in the corresponding determinants. The two classes TICS and CCW differ by having real and complex orbitals, respectively. In the second row of Table II there are three classes with a nonvanishing spin density with a fixed direction—hence the term axial. Of these the class ASDW is most well known—one example being furnished by the alternant molec-... 

---