Space-fixed inversion

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

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... 

P and S. Note that states with different values of /. or. S have different energies, partly because of the electron-electron interaction term in the electronic Hamiltonian. A further symmetry classification that should be mentioned is the parity of an atomic state which depends on the behaviour of the total wave function under space-fixed inversion. This is either even (g) or odd (u) and is determined by, summed over all the electrons in the atom. 

It is important to distinguish between the space-fixed inversion operator E defined here and the molecule-fixed inversion operator, denoted i. The latter defines the g,u character of functions of molecule-fixed coordinates in appropriate systems (i.e. those with a centre of symmetry) but says nothing about the overall parity of the state. It is therefore a less powerful operator than E. ... 

The effect of space-fixed inversion on the Euler angles and on molecule-fixed coordinates... 

Substituting these results in equation (6.207), we obtain the transformation properties of the molecule-fixed coordinates of a point i under space-fixed inversion ... 

The transformation of general Hund s case (a) and case (b) functions under space-fixed inversion... 

Figure 6.24. The effectofthe space-fixed inversion operator E on the molecule-fixed coordinate system (x, y, z). The molecule-fixed coordinate system is always taken to be right-handed. After the inversion of the electronic and nuclear coordinates in laboratory-fixed space, the (x, y, z) coordinate system is fixed back onto the molecule so that the z axis points from nucleus 1 to nucleus 2 and the y axis is arbitrarily chosen to point in the same direction as before the inversion. As a result, the new values of the Euler angles (ip 6, x ) are related to the original values , 9, x)by

In a case (a) basis set, the electron spin angular momentum is quantised along the linear axis, the quantum number E labelling the allowed components along this axis. Because we have chosen this axis of quantisation, the wave function is an implicit function of the three Euler angles and so is affected by the space-fixed inversion operator E. An electron spin wave function which is quantised in an arbitrary space-fixed axis system,. V. Ms), is not affected by E, however. This is because E operates on functions of coordinates in ordinary three-dimensional space, not on functions in spin space. The analogous operator to E in spin space is the time reversal operator. 

The two component states of orbital degeneracy in a diatomic molecule have opposite parity. As we described in chapter 6, parity is the symmetry label associated with the behaviour of a wave function under the space-fixed inversion operator E ... 

We follow the conventions described by Brown, Kaise, Kerr and Milton [115] in order to form parity-conserved functions, as discussed in detail in, section 6.9. Parity is related to the behaviour of a state or function under the space-fixed inversion operator... 

We should not leave this discussion of the intensity of rotational transitions without some mention of the parity selection rule. Electric dipole transitions involve the interaction between the oscillating electric field and the oscillating electric dipole moment of the molecule. The latter is represented in quantum mechanics by the transition moment fjLx b,a) given in equation (6.300). For this transition moment to be non-zero, the integrand ijry i ust be totally symmetric with respect to all appropriate symmetry operations, which includes the space-fixed inversion operator E. Now the electric dipole moment operator,... 

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