Van Vleck contribution

For Sm3+ and Eu3+ the /T values (given in emu kelvin per mole) obtained by including the Van Vleck contribution are reported in brackets. The experimental values refer to the Ln2(S04)3 8H20 series, obtained as an average of the different measurements reported by Van Vleck [9]. 

Here Ar( ) isothermal susceptibility of eq. (36) and xA ) is the isolated susceptibility that is identical to the Van Vleck contribution Ar (T) in eq. (36). They differ by the Curie contributions of the degenerate CEF states. The latter are due to the thermal repopulation of those states when they are split by a static external strain. Eq. (A9) seems to suggest that even an arbitrary small sound frequency, ft) 0, is too high to allow for repopulation of states. Thus Curie terms would be absent and the isolated quadrupolar susceptibility would determine the T-dependence of the sound velocities at finite frequencies. However, this singular behaviour of t 7 (ft)) in eq. (A9) is an artifact of the RPA approximation used,... 

The so-called mixed- or intermediate-valent compounds have been of considerable interest since their discovery in the early 1970s (see Varma (1976) for a review). SmS is a classic example (Jayaraman et al. 1970), which transforms under pressure from divalent (Sm ) to a valence of 2.8. In the divalent state the susceptibility, which is small, arises from van Vleck contributions mixing neighboring J-levels. Moon et al. (1978) have calculated an effective C2 — 1 for these contributions. Reference to table 2 shows that the Sm state has C2 = 6.0, so that the form factor would seem to be an excellent method by which to test the concept of mixed valence. 

The diamagnetic susceptibility x of LaSe2 was measured at 88 to 440 K. Due to -0.3% paramagnetic impurities, x varies linearly with ITT. The extrapolation to zero gives x(0) = -24x10" cm /mol. According to the additivity rule, x(0) Xdia should be -122 x 10" cm /mol if available data for La " and Se " are used. The difference corresponds to a paramagnetic Van Vleck contribution. This arises from deformation of the ions due to mixed ionic-covalent bond character, Lashkarev, Savitskii [14]. 

The first attempts to rationalize the magnetic properties of rare earth compounds date back to Hund [10], who analysed the magnetic moment observed at room temperature in the framework of the old quantum theory, finding a remarkable agreement with predictions, except for Eu3+ and Sm3+ compounds. The inclusion by Laporte [11] of the contribution of excited multiplets for these ions did not provide the correct estimate of the magnetic properties at room temperature, and it was not until Van Vleck [12] introduced second-order effects that agreement could be obtained also for these two ions. 

For the latter two ions, the presence of low-lying excited states makes the inclusion of both the first-order contribution of the excited states and the second-order effects due to coupling of the ground / state with the excited states crucial in a correct estimation of the room-temperature values of xT. Indeed, second-order contribution in Van Vleck [9] expansion of the susceptibility is inversely proportional to the energy difference between the ground and the excited states ... 

In his 1933 paper on "tautomerism," Ingold began discussion with references to the physics of the electron, citing a 1923 paper by J. J. Thomson and a recent book by John H. Van Vleck.62 He noted Lewis s contributions (1923) to the notion of inductive effect ( ) in which electrons remain bound by their original atomic nuclei Lowry (1923) to the notion of electromeric effect, in which there is a displacement of a duplet, shifting from one pair of atoms to... 

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

An important fact underlying this approach to analysis of molecular spectra based on a formalism arising from pioneering work of Dunham and van Vleck, and advances contributed by their successors, or any other approach, is that not merely coefficients A],], or their equivalents but also various... 

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. 

Another additive term in the magnetic susceptibility arises from the temperature-independent core diamagnetism of all the ions in a solid. For YBa2Cu307 the core diamagnetism is approximately -2 x 10 7 based on a calculation using Pascal s constants (9). This small negative contribution serves to reduce the total susceptibility. A third possible contribution arises from Van Vleck paramagnetism (10) caused by excited states in the atoms of the... 

Van Vleck (80) illustrated how Eq. (73) can be identified in the energy expression of the two interacting electrons in a non-relativistic field-free framework. For such a system the contribution to the electron-electron interaction energy Ey comprises the Coulomb energy Jy and the exchange energy Ky,... 

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,... 

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). 

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