Transformation of the electronic coordinates to molecule-fixed axes

Note that the two terms in (2.73) do not commute since is a function of the Euler angles. We can substitute either (2.69) or (2.73) into (2.3 8) to obtain the modified form of (2.53), 

Along with the introduction of the redundant coordinate x, we define the normalisation condition as 

9 sinrf cosec0 3 cosrf cos0 d sind cot0 d 

Transformation of the electronic coordinates to molecule-fixed axes 

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, 

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Transformation of the electronic coordinates to molecule-fixed axes [1 51... 

Next we consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia. To achieve inversion of all particles with respect to space-fixed axes, we first rotate all the electrons and nuclei by 180° about the c axis (which is perpendicular to the molecular plane) we then reflect all the electrons in the molecular ab plane. The net effect of these two transformations is the desired space-fixed inversion of all particles. (Compare the corresponding discussion for diatomic molecules in Section 4.7.) The first step rotates the electrons and nuclei together and therefore has no effect on the molecule-fixed coordinates of either the electrons or the nuclei. (The abc axes rotate with the nuclei.) Thus the first step has no effect on tpel. The second step is a reflection of electronic spatial coordinates in the molecular plane this is a symmetry plane and the corresponding operator Oa has the possible eigenvalues +1 and — 1 (since its square is the unit operator). The electronic wave functions of a planar molecule can thus be classified as having... 

We wish to divide XT into a part describing the nuclear motion and a part describing the electronic motion in a fixed nuclear configuration, as far as possible. Equations (2.36) and (2.37) do not themselves represent such a separation because 3 is still a function of R,

partial differential operators with respect to these coordinates. The obvious way to remove the effects of nuclear motion from. >iel is by transforming from space-fixed axes to molecule-fixed axes gyrating with the nuclei. 

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