Anharmonic thermal motion formalism

In low-spin transition metal complexes, the preferential occupancy of the d orbitals in the crystal field tends to create excess density in the voids between the bonds, which means that anharmonicity tends to reinforce the electron density asphericity. We will discuss, in the following sections, to what extent the two effects can be separated by combined use of aspherical atom and anharmonic thermal motion formalisms. 

More quantitatively, the effect of the thermal motion follows from the anharmonic thermal motion formalisms discussed in chapter 2. In the bcc structure, the relevant nonzero anharmonic term in the one-particle potential is the anisotropic, cubic site-symmetry allowed, part of uJuku um in expression (2.39). The modified potential for the cubic sites is given by (Willis 1969, Willis and Pryor 1975)... 

Zucker, U. H., and Schulz, H. Statistical approaches for the treatment of anhar-monic motion in crystals. 1. A comparison of the most frequently used formalisms of anharmonic thermal vibrations. Acta Cryst. A38, 563-568 (1982). 

There are certain subtleties associated with the order to which the expansion of eqn (5.67) is carried and for a discussion of which we refer the perspicacious reader to chap. 25 of Ashcroft and Mermin (1976). If we maintain the use of normal coordinates, but now including the anharmonic terms which are represented via these coordinates, it is seen that the various modes are no longer independent. For the purposes of the present discussion, we see that these anharmonic terms have the effect of insuring that the equation of motion for the n mode is altered by coupling to the rest of the mode variables. This result bears formal resemblance to the Navier-Stokes equations which when written in terms of the Fourier variables yield a series of mode coupled equations. We have already noted that the physics of both thermal expansion and thermal conductivity demand the inclusion of these higher-order terms. 

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