Methane Symmetry Operations - Triply Degenerate Vibrational Wave Functions

13. Application of the Continuous Three-Dimensional Rotation-Reflection Group to Triply Degenerate Vibrational Wave Functions

13.1 Spherical polar vibrational coordinates and angular momentum operators

In order to apply the full continuous pure rotation group to triply degenerate vibrational coordinates, it is convenient to express the latter in spherical polar form

where Q_s² = Q_sx² + Q_sy² + Q_sz². The vibrational angular momentum operators given in (eq. 36) become, after a tedious but well known calculation,

where P_θs = −iℏ(∂/∂θ_s), etc., and s = 3 or 4. Note that entries in the first column of the 3 × 3 matrix on the right of (eq. 39) are arbitrary, since the first component of the column vector on the right is zero.

Comparison of (eq. 25) and (eq. 39) shows that the form of the operators L_sx , L_sy , L_sz  is identical to the form of the operators J_X, J_Y, J_Z provided that we replace the momentum p_χ in (eq. 25) by zero. [Attempts to relabel φ_s as χ_s , and to write (eq. 39) in a fashion analogous to (eq. 23) fail because of difficulties with the minus signs.] We thus conclude, or prove by direct substitution, that eigenfunctions |L k_L⟩ of the form

belong to the eigenvalues L(L + 1)ℏ² and k_Lℏ, respectively, of L² and L_z.  

13.2   Comparison of vibrational and rotational angular momentum eigenfunctions

The projection of the vibrational angular momentum along the molecule-fixed z axis is indicated by the second subscript in the D function in (eq. 40), whereas the projection of the total angular momentum along the molecule-fixed z axis is indicated by the first subscript in (eq. 26). This difference between (eq. 26) and (eq. 40) is related to the fact that the molecule-fixed components of J commute with the anomalous [30] sign of i, while the laboratory-fixed components of J and both the molecule-fixed and laboratory-fixed components of other angular momentum operators commute with the normal [29] sign of i, and to the fact that the Hamiltonian operator involves terms in the difference J − ζL of the two angular momenta, not the sum J + ζL. Further discussion of these two points will not be given, though further manifestations of these differences occur elsewhere in this section.

The normalization factors in (eq. 26) and (eq. 40) differ by (2π)^1/2 because vibrational configuration space for a triply degenerate vibration is characterized by two angular variables (θ_sφ_s), whereas rotational configuration space is characterized by three (χθφ). The analog χ_s of the third Eulerian angle has been set to zero in (eq. 40). It can be given any value without altering D^(j)_µ′ µ functions having µ′ = 0.  

13.3   Transformation properties of the vibrational angular momentum eigenfunctions

The triply degenerate vibrational coordinates Q_sx , Q_sy , Q_sz  transform under the pure rotation operators $$R_{\alpha\beta\gamma}$$  of the point group T_d as

By extension, we require Q_sx , Q_sy , Q_sz  to transform according to this equation under any pure rotation.

The question now arises of how the eigenfunctions |L k_L⟩ transform. This can be determined rather easily by making use of some slightly formalistic arguments, designed to make the transformation algebra of this section analogous to that used in previous sections.

Equation (eq. 41) can be rewritten in terms of the spherical polar forms of the vibrational variables, and recast somewhat to read

In (eq. 42) we have introduced an extraneous variable χ_s, and generated with its help a complete 3 × 3 matrix of the form S(χ_sθ_sφ_s) from the row vector [sinθ_s cosφ_s, sinθ_s sinφ_s, cosθ_s]. Actually, however, because of the presence of the row vector [0 0 1], (eq. 42) makes a statement only about how the third row of the matrix S(χ_sθ_sφ_s) transforms and this third row does not involve the extraneous variable χ_s.

We now generalize (eq. 42) by omitting the row vector [0 0 1] and the scalar Q_s, to obtain

It can be seen that this generalized equation introduces no inconsistencies with respect to the physically meaningful (eq. 42). Subjecting (eq. 43) to the similarity transformation defined by (eq. 30) and (eq. 31), making use of a property [22] of the D matrices

and extending, as in Sec. 8, validity of the transformed (eq. 43) from L = 1 to all L, we find that

Specializing this to k″ = 0, and making use of (eq. 40) we obtain

 

13.4 Van Vleck's reversed angular momenta

The rather awkward looking transformation (eq. 46) can be made analogous to (eq. 33) by utilizing the concept of "reversed angular momenta," first introduced by Van Vleck [30] for this purpose. Quite simply, if one considers not [L_x , L_y , L_z], but instead

$$ \left[\tilde{L}_x, \tilde{L}_y, \tilde{L}_z \right] ≡ \left[-L_x,-L_y,-L_z \right] ~ , $$

then the

$$\tilde{L}$$ 

operators commute with the anomalous sign of i just as J_x , J_y , J_z  do. A function belonging to the eigenvalue k_Lℏ of L_z and L(L + 1)ℏ² of L² clearly belongs to the eigenvalue

$$\tilde{k}_L \hbar = -k_L \hbar ~of ~ \tilde{L}_z$$

and

$$\tilde{L}(\tilde{L} + 1) \hbar^2 = L(L+1) \hbar^2 ~ of ~ \tilde{L}^2 ~. $$

Defining the eigenfunctions

$$|\tilde{L} ~ \tilde{k}_L \rangle ~ of ~ \tilde{L}^2 ~, ~ \tilde{L}_z$$

to be

where

$$L ~ = ~ \tilde{L}$$

and

$$k_L = -\tilde{L}_L$$

insures that matrix elements of the "reversed" ladder operators

$$\tilde{L}_x \mp i\tilde{L}_y$$

are real and positive. Equation (46) can then be rewritten, using the

$$|\tilde{L} - k \rangle ~, ~$$

functions rather than the |L +k⟩, as

an expression analogous to (eq. 33).  

13.5 The molecule-fixed inversion operation i

Since the vibrational coordinates Q_sx , Q_sy , Q_sz transform according to the representation F₂ of T_d, and since F₂ correlates with the representation D_u⁽¹⁾ of the full three-dimensional rotation-reflection group, it is convenient and consistent to require Q_sx , Q_sy , Q_sz to transform into their negatives under the molecule-fixed inversion operation i, even though i does not occur in the T_d point group. This transformation of Q_sx , Q_sy , Q_sz can be achieved by replacing the vibrational spherical polar coordinates θ_sφ_s in (eq. 38) by

Direct substitution of (θ_s)_new and (φ_s)_new from (eq. 49) into Wigner's [22] D functions as used in (eq. 40) shows that the vibrational wave functions

$$| \tilde{L} \tilde{k}_L \rangle $$

transform into (−1)^L times themselves under the molecule-fixed inversion operation i, i.e., that the vibrational wave functions

$$| \tilde{L} \tilde{k}_L \rangle $$

for a triply-degenerate mode transform [14] as D_g^(L) for even L and as D_u^(L) for odd L.