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Methane Symmetry Operations - Symmetry Properties of Rotational Wave functions and Direction Cosines

7.   Symmetry Properties of Rotational Wave functions and Direction Cosines

It is in the determination of symmetry properties of functions of the Eulerian angles, and in particular in the question of how to apply sense-reversing point-group operations to these functions, that the principal differences arise in group-theoretical discussions of methane. The treatment given here follows from the discussion of Section 4. The two other most commonly followed treatments, due to Jahn [5−8] and Moret-Bailly [9−11], respectively, will be discussed briefly in Section 12.

7.1 Symmetric-top rotational basis functions

The molecule-fixed components Jx,Jy,Jz of the total angular momentum operator can be written as [4]

$$\left[\begin{array}{c} J_x\\ J_y\\ J_z \end{array}\right] = \left[\begin{array}{ccc} +\cos\chi \cot\theta & \sin\chi & -\cos\chi \csc\theta \\ -\sin\chi \cot\theta & \cos\chi & +\sin\chi \csc\theta \\ +1 & 0 & 0 \end{array}\right] \left[\begin{array}{c} p_\chi\\ p_\theta\\ p_\phi \end{array}\right] ~ , $$

(eq. 23)

where pχ = −iħ(∂/∂χ), etc. [4, 28]. Since the transformation from molecule-fixed to laboratory-fixed vector components takes place via the matrix S in (eq. 10), the laboratory-fixed components JX,JY,JZ of the total angular momentum operator can be obtained from (eq. 23) and

$$\left[\begin{array}{c} J_X\\ J_Y\\ J_Z \end{array}\right] = [S^{-1}(\chi\theta\phi)] \left[\begin{array}{c} J_x\\ J_y\\ J_z \end{array}\right] ~ . $$

(eq. 24)


$$\left[\begin{array}{c} J_X\\ J_Y\\ J_Z \end{array}\right] = \left[\begin{array}{ccc} +\csc\theta \cos\phi & -\sin\phi & -\cot\theta \cos\phi \\ +\csc\theta \sin\phi & +\cos\phi & -\cot\theta \sin\phi \\ 0 & 0 & +1 \end{array}\right] \left[\begin{array}{c} p_\chi \\ p_\theta \\ p_\phi \end{array}\right] ~ .$$

(eq. 25)

It can be shown, by direct application of the differential operators, that symmetric top rotational basis functions |kJm⟩ defined in terms of Wigner's $${\cal D}_{\mu^\prime\mu}^{(j)}\, (\{\alpha\beta\gamma\})$$ functions (equation (15.27) of [22]) as

$$|kJm\rangle = [(2J + 1)/8\pi^2]^{1/2} ~ {\cal D}_{km}^{(J)}(\{\chi\theta\phi\}) ~ ,$$

(eq. 26)

are eigenfunctions of the operators Jz and JZ belonging to the eigenvalues kħ and mħ, respectively, and give rise to real positive matrix elements for the laboratory-fixed [29] (JX ± iJY) and molecule-fixed [30] (Jx ‡ iJy) ladder operators. Substitution of the Eulerian angle transformations of Table 5 in the function |kJm⟩ leads to the transformations shown in Table 11.

Table 11. Transformation of the symmetric top function |kJm⟩ under the symmetry operations of the D2d subgroup given in Table 5

E |kJm⟩ = |kJm   C2(x) |kJm⟩ = (−1)J |−kJm
S4(z) |kJm⟩ = (−i)k |kJm C2(y) |kJm⟩ = (−1)J−k |−kJm
S43(z) |kJm⟩ = (+i)k |kJm σd(110) |kJm⟩ = (−1)J (−i)k |−kJm
C2(z) |kJm⟩ = (−1)k |kJm σd(−110) |kJm⟩ = (−1)J (+i)k |−kJm

If we further take sums and differences for K = | k | ≥ 1,

$$| K^\pm Jm\rangle = 2^{-1/2} \, [\,|KJm\rangle \pm | -KJm\rangle] ~ ,$$

(eq. 27)

we find symmetry species for these functions under the D2d symmetry operations as given in Table 12.

Table 12. Symmetry species in D2d of the Wang sum and difference rotational functions specified in (eq. 27)

  J = even J = odd
  |0Jm  |K+Jm i |K+Jm  |0Jm  |K+Jm i |KJm 
K = 0  A1     A2    
K = 1 mod 4 Ex Ey Ey Ex  
K = 2 mod 4 B1 B2 B2 B1
K = 3 mod 4 Ex Ey   Ey + Ex  
K = 4 mod 4 A1 A2 A2 A1

As mentioned earlier, functions belonging to a single symmetry species of the full point group Td are rather less convenient to write down. In fact, it is necessary to introduce linear combinations of the symmetric top functions much more complicated than sums and differences [31]. These will not be given here, though transformation properties of the |kJm⟩ under operations of the full point group Td are given in Section 8.

7.2 Direction cosines

The discussion of transformation properties and symmetry species is rather simple for the direction cosines. These quantities are just the nine elements of the 3 × 3 rotation matrix Sθφ), and transformations of the Eulerian angles were in fact defined originally in Section 4.2 and Section 4.3 to insure certain transformation properties of the direction cosines themselves. Equations (eq. 14) and (eq. 18) prescribe the transformations of the direction cosine matrix when χnew, θnew, φnew are substituted for χ,θ,φ. Since the matrices M in (eq. 14) must be taken from Table 2, we see that the three functions in any column of the direction cosine matrix S transform like functions of species F2x,F2y,F2z as far as proper rotations are concerned. Since the matrices N in (eq. 18) must also be taken from Table 2, and since the transformation (eq. 18) involves −N, not +N, the columns of the direction cosine matrix do not transform like F2x,F2y,F2z as far as improper rotations are concerned.

It happens, however, that the matrices in Table 3 are identical to those in Table 2 for proper rotations and equal to the negatives of those in Table 2 for improper rotations. Thus, the three functions in any column of the direction cosine matrix Sθφ) transform like functions of species F1x,F1y,F1z under the operations of the full molecular symmetry group Td.


Created September 20, 2016, Updated March 1, 2023