The space-fixed inversion operator

The inversion operator E is defined as the operator which transforms a function f(X , Yi, Zi) into a new function which has the same value as f(—Xh — Yi,—Zi) 

Clearly, if this operator is applied twice to a wave function, the system reverts to its original configuration  

The following behaviour is consistent with this result  

If a quantum system transforms according to the upper sign, the state has a positive parity and, if according to the lower sign, a negative parity. 

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 

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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... 

Figure 6.24. The effect of the space-fixed inversion operator 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 (0, 0, / ) are related to the original values 0, x)hy

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,... 

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... 

The nature of molecular inversion can be understood if we consider an operation E whose effect on the position vectors of all the particles of a molecule (atomic nuclei and electrons) in the space-fixed system of coordinates is defined as... 

The PI group operations are defined by their effect on the space-fixed coordinates of the atomic nuclei and electrons. Since our molecular wavefunctions are written in terms of the vibrational coordinates, the Euler angles and the angle p, we must first determine the effect of the PI group operations on these variables. In the case of inversion this can lead to certain problems both in the understanding of the concepts of molecular symmetry and in the proper use of group theoretical methods in the classification of the states of ammonia. 

Prior to digressing on the subject of nuclear exchange symmetry, we mentioned that a new symmetry element besides (molecule-fixed) was required to classify electronic-rotational states in homonuclear diatomics. A logical choice is i (molecule-fixed), an operation which belongs to but not It may be shown that z (molecule-fixed) is equivalent to Xj (space-fixed), and so the procedures worked out in the foregoing discussion may be used to classify ji/ZeiZrot) as either (s) or (a) under in lieu of determining their behavior under molecule-fixed inversion. The dipole moment operator ft in homonuclear molecules is (s) under Xj [11]. This leads to the conclusion that only states ij/e Xrot > with like symmetry under Xff can be connected by El transitions in electronic band spectra ... 

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