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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

$$ \left[ \begin{array}{c} Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] \equiv Q_s \left[ \begin{array}{l} \sin\theta_s \, \cos\phi_s\\ \sin\theta_s \, \sin\phi_s\\ \cos\theta_s \end{array}\right] ~ , $$

(eq. 38)

where Qs2 = Qsx2 + Qsy2 + Qsz2. The vibrational angular momentum operators given in (eq. 36) become, after a tedious but well known calculation,

$$ \left[ \begin{array}{c} L_{sx} \\ L_{sy} \\ L_{sz} \end{array}\right] = \left[ \begin{array}{ccc} +\csc\theta_s \, \cos\phi_s & -\sin\phi_s & -\cot\theta_s \, \cos\phi_s \\ +\csc\theta_s \, \sin\phi_s & +\cos\phi_s & -\cot\theta_s \, \sin\phi_s \\ 0 & 0 & +1 \end{array}\right] ~ \left[ \begin{array}{c} 0 \\ P_{\theta_s} \\ P_{\phi_s} \end{array}\right] ~ , $$

(eq. 39)

where Pθs = −iℏ(∂/∂θs), 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 Lsx , Lsy , Lsz  is identical to the form of the operators JX, JY, JZ provided that we replace the momentum pχ in (eq. 25) by zero. [Attempts to relabel φs as χ, 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 kL⟩ of the form

$$|L \, k_L \rangle = [(2L+1)/4\pi]^{1/2} \, {\cal D}_{0k_L}^{(L)}\, (\{0\, \theta_s \phi_s \}) ~ , $$

(eq. 40)

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

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 (θsφs), whereas rotational configuration space is characterized by three (χθφ). The analog χs 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 Qsx , Qsy , Qsz  transform under the pure rotation operators $$R_{\alpha\beta\gamma}$$  of the point group Td as

$$R_{\alpha\beta\gamma} ~ \left[ \begin{array}{c} Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] = \left[ \begin{array}{c} ~\\ S(\alpha\beta\gamma) \\~ \end{array}\right] ~ \left[ \begin{array}{c} Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] ~ . $$

(eq. 41)

By extension, we require Qsx , Qsy , Qsz  to transform according to this equation under any pure rotation.

The question now arises of how the eigenfunctions |L kL⟩ 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

$$\begin{eqnarray*} R_{\alpha\beta\gamma}[Q_{sx}, Q_{sy}, Q_{sz}] &=& R_{\alpha\beta\gamma}[\sin\theta_s\cos\phi_s,\sin\theta_s\sin\phi_s,\cos\theta_s]\,Q_s\\ &=& R_{\alpha\beta\gamma}[0 \, 0 \, 1 ] \cdot S(\chi_s \theta_s \phi_s) \cdot Q_s\\ &=& [0 \, 0 \, 1 ] \cdot S(\chi_s \theta_s \phi_s) \cdot S^{\rm tr}(\alpha\beta\gamma) \cdot Q_s ~ . \end{eqnarray*}$$

(eq. 42)

In (eq. 42) we have introduced an extraneous variable χs, and generated with its help a complete 3 × 3 matrix of the form Ssθsφs) from the row vector [sinθs cosφs, sinθs sinφs, cosθs]. 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 Ssθsφs) transforms and this third row does not involve the extraneous variable χs.

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

$$ R_{\alpha\beta\gamma}\,S(\chi_s\theta_s\phi_s) = S(\chi_s\theta_s\phi_s) \cdot S^{\rm tr}(\alpha\beta\gamma) ~ ,$$

(eq. 43)

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

$${\cal D}_{\mu^\prime\mu}^{(j)} (\{\alpha\beta\gamma\}) = (-1)^{\mu-\mu^\prime} ~ {\cal D}_{-\mu^\prime-\mu}^{(j)*} (\{\alpha\beta\gamma\}) ~ ,$$

(eq. 44)

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

$$R_{\alpha\beta\gamma}~ {\cal D}_{k^{\prime\prime}k_L}^{(L)} (\{\chi_s\theta_s\phi_s\})= \sum_{k_L^\prime}~{\cal D}_{k^{\prime\prime}k_L^\prime}^{(L)} (\{\chi_s\theta_s\phi_s\}) \, (-1)^{k_L^\prime-k_L} ~ {\cal D}_{-k_L-k_L^\prime}^{(L)} (\{\alpha\beta\gamma\}) ~ . $$

(eq. 45)

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

$$R_{\alpha\beta\gamma}~ |L\, k_L\rangle = \sum_{k_L^\prime} \, (-1)^{k_L^\prime-k_L} ~ {\cal D}_{-k_L-k_L^\prime}^{(L)} (\{\alpha\beta\gamma\}) ~ |L\, k_L^\prime\rangle ~ . $$

(eq. 46)


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, L, Lz], but instead

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

then the


operators commute with the anomalous sign of i just as J, J, J do. A function belonging to the eigenvalue kLℏ of Lz and L(L + 1)ℏ2 of L2 clearly belongs to the eigenvalue

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


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

Defining the eigenfunctions

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

to be

$$|\tilde{L} \, \tilde{k}_L\rangle \equiv (-1)^{k_L} \, [(2L+1)/4\pi ]^{1/2} ~ {\cal D}_{0\,k_L}^{(L)} (\{0\, \theta_s\, \phi_s\}) = (-1)^{k_L} \, |L\, k_L\rangle ~ ,$$

(eq. 47)


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


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

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

$$R_{\alpha\beta\gamma}~ |\tilde{L} \, \tilde{k}_L \rangle = \sum_{\tilde{k}_L^\prime} ~ {\cal D}_{\tilde{k}_L \tilde{k}_L^\prime}^{(L)} \, (\{\alpha\beta\gamma\}) ~ |\tilde{L} \, \tilde{k}_L^\prime \rangle ~ ,$$

(eq. 48)

an expression analogous to (eq. 33).  

13.5 The molecule-fixed inversion operation i

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

$$\begin{eqnarray*} (\theta_s)_{\rm new} &=& \pi - \theta_s\\ (\phi_s)_{\rm new} &=& \pi + \phi_s ~ . \end{eqnarray*}$$

(eq. 49)

Direct substitution of (θs)new and (φs)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 Dg(L) for even L and as Du(L) for odd L.


Created September 28, 2016, Updated February 17, 2022