Van Vleck’s equation

Turning from the more practical side of recent advances, let us consider the routine aspect of the calculation of molecular susceptibilities from the eigenvalues and eigenvectors produced by some ligand field model. The usual procedure, using Van Vleck s equation... 

The principles of calculation of the size of the spin-orbit interaction are straightforward in the case of a free atom or for truly localized atomic orbitals of atoms within molecules, e.g. the 4f orbitals in rare earth compounds. Thus Hund accurately predicted values of the relevant moments L, J and S for rare earth ions in solid salts and aqueous solutions and showed that their magnetic moments were given by Van Vleck s equation (equation 15) ... 

These definitions can be easily incorporated into a computer program which will generate the appropriate coupled spin states and energies, and then used to calculate the magnetic susceptibility under Van Vleck s equation (11). The more complete thermodynamic equation for X is given in Eq. (12). 

Low-temperature turnover in the susceptibility curves required incorporation of a (T - d) term in Van Vleck s equation (11), as well as a small component of paramagnetic impurity. Use of variable field experimental data in this low-temperature region, combined with the thermodynamic form of x, Eq. (12), would have been desirable. Exploration of wide parameter space in the fitting process and use of best-fit contour maps of e/ M vs JcrM yielded the following parameter values ... 

C=370cm-, QJQ,=U0, g=2.1, =+110cm-. Temperature dependence of 1/Xm- Curves fitted according to van Vleck s equation... 

An instructive illustration of the effect of molecular motion in solids is the proton resonance from solid cyclohexane, studied by Andrew and Eades 101). Figure 10 illustrates their results on the variation of the second moment of the resonance with temperature. The second moment below 150°K is consistent with a Dsi molecular symmetry, tetrahedral bond angles, a C—C bond distance of 1.54 A and C—H bond distance of 1.10 A. This is ascertained by application of Van Vleck s formula, Equation (17), to calculate the inter- and intramolecular contribution to the second moment. Calculation of the intermolecular contribution was made on the basis of the x-ray determined structure of the solid. 

The readers will undoubtedly agree that the elucidations and derivations of the equations 13—15 are beyond the scope of this book, but chemists with some physical inclination may find some satisfaction in reading through Van Vleck s [207] Theory of Electric and Magnetic Susceptibilities to quench their mathemetical thirst. Instead of tabulating the measured magnetic moments of all europium compounds with variations we will mention the magnetic behaviour of the individual compounds as we deal with them. 

Various methods have been developed for dealing with the anomalous commutation relationships in molecular quantum mechanics, chief among them being Van Vleck s reversed angular momentum method [10]. Most of these methods are rather complicated and require the introduction of an array of new symbols. Brown and Howard [15], however, have pointed out that it is quite possible to handle these difficulties within the standard framework of spherical tensor algebra. If matrix elements are evaluated directly in laboratory-fixed coordinates and components are referred to axes mounted on the molecule only when necessary, it is possible to avoid the anomalous commutation relationships completely. Only the standard equations given earlier in this chapter are used to derive the required results it is just necessary to keep a cool head in the process ... 

In writing this equation, we have made use of Van Vleck s pure precession hypothesis [12], in which the molecular orbital /.) is approximated by an atomic orbital with well-defined values for the quantum numbers n, l and /.. Such an orbital implies a spherically symmetric potential and its use is most appropriate when the electronic distribution is nearly spherical. Examples of this situation occur quite often in the description of Rydberg states. It is also appropriate for hydrides like OH where the molecule is essentially an oxygen atom with a small pimple, the hydrogen atom, on its side. Accepting the pure precession hypothesis allows the matrix elements of the orbital operators to be evaluated since... 

Eq. (1.36) is called the Van Vleck equation. In that form the Zeeman operator operates only to first order. Indeed, we should include the second order Zeeman term, which allows the interaction between the ground S multiplet (j functions) with all the excited ones ([Pg.18]

Derivation perturbation theory for eigenvalues (except S = 1, where the variation method is applied), van Vleck equation (linear magnetics)... 

While the singlet ground state will be unaffected by an external magnetic field, the S 1 state will become Zeeman split into the Ms =—1,0, 1 sublevels. Thus, the energies of the four possible 15, Ms) states are known (10, 0)) = Jex (ll, + 1)) = —g/rB77 (ll, 0)) = 0 (ll, —1))= I /ab//. When these four expressions are introduced into the Van Vleck equation, the Bleaney-Bowers equation7 is obtained. This equation describes the temperature dependence of the susceptibility (for the zero-field limit), independent of the sign of Jcx ... 

For the purposes of illustrating the Van Vleck equation, the effects of zero-field splitting for S = 1 can be examined as shown in Figure 3.10. In this case, it is no longer true that all of the terms are equal to 0 and so the Curie law will not be obeyed. With the applied field applied along the z direction, the = 0 state will be unaffected, but the 1 states will be Zeeman split and will vary linearly with applied field. 

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