Space-fixed quantisation scheme

For complexes with very small anisotropies and small reduced masses, such as those containing He or H2, a space-fixed quantisation scheme is appropriate. However, for most other Van der Waals complexes, the anisotropy is strong enough for the diatom angular momentum to be quantised along the R vector, and the total wavefunction is most conveniently expanded in body-fixed functions. 

With the introduction of electronic angular momentum, we have to consider how the spin might be coupled to the rotational motion of the molecule. This question becomes even more important when electronic orbital angular momentum is involved. The various coupling schemes give rise to what are known as Hund s coupling cases they are discussed in detail in chapter 6, and many practical examples will be encountered elsewhere in this book. If only electron spin is involved, the important question is whether it is quantised in a space-fixed axis system, or molecule-fixed. In this section we confine ourselves to space quantisation, which corresponds to Hund s case (b). 

We have derived the total Hamiltonian expressed in a space-fixed (i.e. non-rotating) coordinate system in (2.36), (2.37) and (2.75). We can now simplify the electronic Hamiltonian 3Q,i by transforming the electronic coordinates to the molecule-fixed axis system defined by (2.40) because the Coulombic potential term, when expressed as a function of these new coordinates, is independent of 0, ip and x From a physical standpoint it is obviously sensible to transform the electronic coordinates in this way because under the influence of the electrostatic interactions, the electrons rotate in space with the nuclei. We shall take the opportunity to refer the electron spins to the molecule-fixed axis system in this section also, and leave discussion of the alternative scheme of space quantisation to a later section. Since we assume the electron spin wave function to be completely separable from the spatial (i.e. orbital) wave function,... 

Finally, in section 2.10 we considered the alternative transformation scheme in which the electron spin remained quantised in the space-fixed coordinate system. The Hamiltonian for this situation is easily derived from (3.267), (3.268) and (3.269) by making the substitutions... 

In the Born-Oppenheimer approximation the basis set for 3Q,i would consist of products of electronic space and spin functions. Transformation to the gyrating axis system may involve transformation of both space and spin variables, leading to a Hamiltonian in which the spin is quantised in the molecule-fixed axis system (as, for example, in a Hund s case (a) coupling scheme) or transformation of spatial variables only, in which case spatially quantised spin is implied (for example, Hund s case (b)). We will deal in detail with the former transformation and subsequently summarise the results appropriate to spatially quantised spin. 

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See also in sourсe #XX -- [ ]

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