Van Vleck method

Diffusional behavior of sorbed species is studied by NMR using one of three approaches the van Vleck method of moments, relaxation measurements, and the pulsed-field-gradient method. An example of the use of the method of moments is the work of Stevenson (194) on H resonances in zeolite H-Y (see Section III,K). Another is the study by Lechert and Wittem (284) of C6H6 and C6H3D3 adsorbed on zeolite Na-X. Analysis of second moments of H resonances allowed the intra- and intermolecular contributions to the spectra to be extracted. Similarly, second moments of H and 19F spectra of cyclohexane, benzene, fluorobenzene, and dioxane on Na-X provided information about orientation of molecules within zeolitic cavities (284-287). 

We need not spend much time with the Dirac-Van Vleck method here. Its successes occur for very simple configurations such as p and for configurations involving s electrons. For us the prime example of the latter is f which can be readily handled by alternative techniques. Although the method excited some attention in the 1930 s, it is now little more than a curiosity. It had little impact on the development of atomic theory. 

We must now discuss the way in which the transformation can be determined. We shall limit ourselves to a brief discussion of two methods the Van Vleck method and the Bloch equations as formulated by Durand. The so-called Van Vleck method (see Ref. 44 for a very readable account) is conceptually very simple and we begin with this technique. 

In the Van Vleck method, we take the transformation (/ to be a unitary transformation. The transformation U can be written in terms of a parameter matrix G as... 

Nuclear magnetic resonance spectroscopy of the solutes in clathrates and low temperature specific heat measurements are thought to be particularly promising methods for providing more detailed information on the rotational freedom of the solute molecules and their interaction with the host lattice. The absence of electron paramagnetic resonance of the oxygen molecule in a hydroquinone clathrate has already been explained on the basis of weak orientational effects by Meyer, O Brien, and van Vleck.18... 

This book has been written in an attempt to provide students with the mathematical basis of chemistry and physics. Many of the subjects chosen are those that I wish that I had known when I was a student It was just at that time that the no-mans-land between these two domains - chemistry and physics - was established by the Harvard School , certainly attributable to E. Bright Wilson, Jr., J. H. van Vleck and the others of that epoch. I was most honored to have been a product, at least indirectly, of that group as a graduate student of J. C. Decius. Later, in my post-doc years. I profited from the Harvard-MIT seminars. During this experience I listened to, and tried to understand, the presentations by those most prestigious persons, who played a very important role in my development in chemistry and physics. The essential books at that time were most certainly the many publications by John C. Slater and the Bible on mathematical methods, by Margeneau and Murphy. They were my inspirations. 

Serber[12] and Van Vleck and Sherman[13] continued the analysis and introduced symmetric group arguments to aid in dealing with spin. About the same time the Japanese school involving Yamanouchi and Kotani[14] published analyses of the problem using symmetric group methods. 

For discussion of experimental methods and more detailed discussion of the theory see C. P. Smyth, Dielectric Constant and Molecular Structure, McGraw-Hill Book Co., New York, 1955 or J. W. Smith, Electric Dipole Moments, Butterworths, London, 1955. A more detailed theoretical treatment is given by J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University PresB, 1932. 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

The earliest work employed the method of moments, and Fraissard et al (192) were the first to consider in detail the second moment of the proton NMR resonance in nonspinning solids. In general, when the line shape is determined solely by the I-I and I-S magnetic dipole-dipole interaction, the Van Vleck second moment, M2, is given by (193)... 

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

Huzinaga was the recipient of the 1994 John C. Polanyi Award of the Canadian Society for Chemistry. In his award lecture he described his model potential method, which deals only with the active electrons in molecular and solid state calculations. An invited review article,59 based on his 1994 Polanyi Award lecture, chronicles his efforts to develop a sound theoretical framework for the core-valence separation of electrons, a problem Van Vleck and Sherman60 once referred to as the nightmare of the inner core. ... 

One must agree that the precise recipe implied by Van Vleck s and Sherman s language is daunting. The use of characters of the irreducible representations in dealing with spin state-antisymmetrization problems does not appear to lead to any very useful results. Prom today s perspective, however, it is known that some irreducible representation matrix elements (not just the characters) are fairly simple, and when applications are written for large computers, the systematization provided by the group methods is useful. 

Serber[15] has contributed to the analysis of symmetric group methods as an aid in dealing with the twin problems of antisymmetrization and spin state. In addition, Van Vleck espoused the use of the Dirac vector model[16] to deal with permutations. [17] Unfortunately, this becomes more difficult rapidly if permutations past binary interchanges are incorporated into the theory. Somewhat later the Japanese school involving Yamanouchi[18] and Kotani et al.[19] also published analyses of this problem using symmetric group methods. 

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 ... 

It took several decades for the effective Hamiltonian to evolve to its modem form. It will come as no surprise to learn that Van Vleck played an important part in this development for example, he was the first to describe the form of the operator for a polyatomic molecule with quantised orbital angular momentum [2], The present formulation owes much to the derivation of the effective spin Hamiltonian by Pryce [3] and Griffith [4], Miller published a pivotal paper in 1969 [5] in which he built on these ideas to show how a general effective Hamiltonian for a diatomic molecule can be constructed. He has applied his approach in a number of specific situations, for example, to the description of N2 in its A 3 + state [6], described in chapter 8. In this book, we follow the treatment of Brown, Colbourn, Watson and Wayne [7], except that we incorporate spherical tensor methods where advantageous. It is a strange fact that the standard form of the effective Hamiltonian for a polyatomic molecule [2] was established many years before that for a diatomic molecule [7]. 

If we compare these equations with the projection operator expansion given in equation (7.43), we find that the expressions are identical up to and including the X2 contribution but that the 7.3 term derived here corresponds not to the X3 term in the expansion (7.43) but to its symmetrised (Hermitian) form discussed at the end of section 7.2. Since the discrepancies that arise from these two different forms are of order Xs or higher, the effective Hamiltonians derived by the two methods are identical to order X3. In the literature the Van Vleck transformation is normally implemented by use of equations (7.67) to (7.70) although the X3 contribution (7.70) has often been ignored. 

Very recently, an interesting modification of HETCOR correlation was published by Chan and co-workers.69 In this experiment, a DQ filter is incorporated into the pulse sequence of HETCOR spectroscopy so that a DQ excitation profile can be obtained by measuring a series of 2D spectra (Figure 14). This method offers a simple experimental approach to extract the van Vleck second moment of a multiple-spin system under high-resolution condition. Hydroxyapatite (Ca10(PO4)6(OH)2) and brushite (CaHP04 2H20) were used as reference samples. 

Crystal field levels in rare earth metals have been determined by measurements of properties such as (i) specific heat, (ii) Van Vleck susceptibility, (iii) magnetization in high magnetic fields, (iv) paramagnetic resonance, (v) Mossbauer effect, (vi) inelastic neutron scattering and (vii) miscellaneous methods. 

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