Symmetric Spin Exchange

The esr spectrum of the diradical -PTM—CC1=CC1—PTM (p. 395) in solution, at room temperature, consists of a broad main line and three satellite pairs. However, the hcc values are half those of the PTMs (a, 13.9 aromatic, j 5.0G PTM- 29.5, 12.5-10.7 G) (Ballester et al, 1980b) showing the existence of rapid spin spin exchange between the two odd electrons and coupling with one magnetically active nucleus with spin 1 /2, as predicted for symmetrical diradicals (Reitz and Weissman, 1960). Its abnormal half-width in solution (2.15 G) at room temperature suggests that it is a... 

To gain insight into the meaning of the symmetric (or Heisenberg) spin exchange interaction, we consider a spin dimer consisting of two equivalent spin-1/2 spin sites, I and 2, with one electron at each spin site (24.7) [1,26]. The energy of the spin... 

Finally, let us consider molecules with identical nuclei that are subject to C (n > 2) rotations. For C2v molecules in which the C2 rotation exchanges two nuclei of half-integer spin, the nuclear statistical weights of the symmetric and antisymmetric rotational levels will be one and three, respectively. For molecules where C2 exchanges two spinless nuclei, one-half of the rotational levels (odd or even J values, depending on the vibrational and electronic states)... 

If / = 1 for each nucleus, as in H2 and N2, the total wave function must be symmetric to nuclear exchange. There are nine nuclear spin wave functions of which six are symmetric and three antisymmetric to exchange. Figure 5. f 8 illustrates the fact that ortho- ll2 (or N2)... 

Just as for diatomics, for a polyatomic molecule rotational levels are symmetric (5 ) or antisymmetric (a) to nuclear exchange which, when nuclear spins are taken into account, may result in an intensity alternation with J. These labels are given in Figure 6.24. 

The spin part pl can be derived by labelling the electrons 1 and 2 and remembering that, in general, each can have an a or /i spin wave function giving four possible combinations a(l)P(2), P(l)a(2), a(l)a(2) and P(l)P(2). Because the first two are neither symmetric nor antisymmetric to the exchange of electrons, which is equivalent to the exchange of the labels 1 and 2, they must be replaced by linear combinations giving... 

Equation (7.23) expresses the total electronic wave function as the product of the orbital and spin parts. Since J/g must be antisymmetric to electron exchange the Ig and Ag orbital wave functions of oxygen combine only with the antisymmetric (singlet) spin wave function which is the same as that in Equation (7.24) for helium. Similarly, the Ig orbital wave function combines only with the three symmetric (triplet) spin wave functions which are the same as those in Equation (7.25) for helium. 

Further, if the wave function depends also on the electron spins, spin variables over all electrons should also be integrated we will see this below, in the calculation of exchange hole. The expression in the curly brackets above is exactly the XC hole PxCM(r, r ) defined in Equation 7.17. A comparison with Equation 7.19a shows that adding the hole to the density is similar to subtracting the density of one electron p(r )/N from it. The hole thus represents a deficit of one electron from the density. This is easily verified by integrating p tM(V, r ) over the volume dr, which gives a value of — 1. However, the structure of the hole is not simple and this is because of the motion of different electrons correlated due to the Pauli exclusion principle and the Coulomb interaction between them. Finally we note that the product p(r)p cM(r, r ) is symmetric with respect to an exchange in the variables... 

We must now combine the nuclear wave functions with the rest of the molecular wave function to generate a total wave function which is antisymmetric with respect to exchange of Fermions. For Bosons the total wave function must be symmetric. To do so we write r]r = i rans r]rviB rot r Nuc-spiN and recognize that both the vibrational and translational wave functions are symmetric. Rotational wave functions... 

The spin-free two-particle excitation operators and density matrices are symmetric with respect to simultaneous exchange of the upper and lower indices, but neither symmetric nor antisymmetric with respect to exchange of either upper or lower indices separately ... 

Aq is the spectral peak volume of a single proton and n is the number of protons at the spin site i. Obviously, when different spin sites have different populations, i.e., rii rij, neither the product matrix A(rm) A(0) nor the exchange matrix L is symmetric, eq. (11). This also follows from the principle of detailed balance [28, 48],... 

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