Molecular frames for a symmetry-adapted rotational basis set

Figures & data

Figure 1. A visual representation of the Euler angles. (a) The first two Euler angles ϕ and θ which are essentially angles defining the spherical coordinates. The first rotation of ϕ is about the z-axis of both the RF- and BF-frame which initially coincide. The BF-frame is then rotated by θ about the y-axis of the BF-frame or the $y^{\prime }$-axis of the figure. (b) The final Euler angle χ is due to a rotation about the z-axis of the BF-frame or the $z^{\prime }$-axis of the figure.

Figure 2. The bond vector BF-frame for the ABA molecule with the x-axis parallel to the A${}_1$–B bond and the y-axis in the plane of the three atoms. The axes are offset for clarity.

Table 1. The transformation of the t and s vectors.

Object	Equation	Transformation
${\mathit{s}}_{k,\alpha }$	${\hat{\mathit{e}}}_F{\mathrm{\partial }}_{\alpha F}{q}_k$	${\mathit{s}}_{k^{\prime },{\alpha }^{\prime }}$
${\mathit{s}}_{g,\alpha }$	[29]	$N_{hg}{\mathit{s}}_{h,{\alpha }^{\prime }}\mp \dot{\phantom{a}}{\gamma }_k{\mathit{s}}_{k,\alpha }{N}_{zg}$
${\mathit{t}}_{k,\alpha }$	${\mathrm{\partial }}_k{\mathit{R}}_{\alpha }$	${\mathit{t}}_{k^{\prime },{\alpha }^{\prime }}\pm \dot{\phantom{a}}{\gamma }_{k^{\prime }}{\mathit{t}}_{z,{\alpha }^{\prime }}$
${\mathit{t}}_{g,\alpha }$	${\hat{\mathit{e}}}_g\times {\mathit{R}}_{\alpha }$	$N_{hg}{\mathit{t}}_{h,{\alpha }^{\prime }}$
${\hat{J}}_g$	[34]	$N_{hg}{\hat{J}}_h$ [18]
${\hat{\pi }}_k$	$-i\hslash \mathrm{\partial }/\mathrm{\partial }{q}_k$	${\hat{\pi }}_{k^{\prime }}\pm \dot{\phantom{a}}{\gamma }_{k^{\prime }}{\hat{J}}_z$

Note: This assumes that ${R_{\alpha }}^{\prime }=R_{{\alpha }^{\prime }}$ and that ${q_k}^{\prime }=q_{k^{\prime }}$. The matrix is N is that of Equation (13(13) ${r_{\alpha }}^{\prime }=M^T\tilde{M}{\tilde{r}}_{\alpha }\equiv N{\tilde{r}}_{\alpha }$(13) ). The term ${\dot{\gamma }}_k={\mathrm{\partial }}_k\gamma$ where γ is the angle in the χ transformation ${\chi }^{\prime }=\pm \chi +\gamma$ of Table 3. The sign in front ${\dot{\gamma }}_k$ corresponds to the left and right transformations of Table 3. The definition of the rotational s-vector ${\mathit{s}}_{g,\alpha }$ and the BF angular momentum operator components ${\hat{J}}_g$ are given in [29] and [34], respectively. Summation over repeated indices is assumed.

Table 2. The transformation of $q_k$ in the left column and the corresponding transformed vibrational s-vector and vibrational momentum operator.

Coordinate	s-vector	Momentum operator
$q_k\to -q_{k^{\prime }}$	$-{\mathit{s}}_{k^{\prime },{\alpha }^{\prime }}$	$-({\hat{\pi }}_{k^{\prime }}\pm \dot{\phantom{a}}{\gamma }_{k^{\prime }}{\hat{J}}_z)$
$q_k\to a{q}_k+b{q}_{k^{\prime }}$	$a{\mathit{s}}_{k,{\alpha }^{\prime }}+b{\mathit{s}}_{k^{\prime },{\alpha }^{\prime }}$	$a({\hat{\pi }}_k\pm \dot{\phantom{a}}{\gamma }_k{\hat{J}}_z)+b({\hat{\pi }}_{k^{\prime }}\pm \dot{\phantom{a}}{\gamma }_{k^{\prime }}{\hat{J}}_z)$

Note: The transformations of the Cartesian coordinates, the rotational s-vectors and the angular momentum operators remain the same as Table  1.

Table 3. The change in the Euler angles for a given rotation of the BF-frame.

	$R_{\alpha }^{\pi }$	$R_z^{\text{β}}$
θ	$\pi -\theta$	θ
ϕ	$\varphi +\varphi$	ϕ
χ	$2\pi -2\alpha -\chi$	$\chi +\text{β}$

Note: $R_{\alpha }^{\pi }$ and $R_z^{\text{β}}$ are rotations of the BF-frame. $R_{\alpha }^{\pi }$ is a rotation of π radians about an axis in the xy plane making an angle α about with the x-axis (α is measured in the right-handed sense about the z-axis) and $R_z^{\text{β}}$ is a rotation of β radians about the z-axis (β measured in the right handed sense about the z-axis) [18].

Figure 3. The principal axis system for the ABA molecule in vibrational equilibrium with the z-axis pointing out of the plane and the y-axis bisecting the A1–B–A2 bond. The axes are offset for clarity.

Table 4. The irrep of the function $|J,|k|,m,p〉$ for the MS group ${\mathcal{C}}_{2\mathrm{v}}$(M) depending on the values of J, $|k|$ and p.

J	k	$p(J,|k\left|\right)$	Γ
even	even	1	A${}_1$
even	even	$-1$	A${}_2$
odd	even	$-1$	A${}_1$
odd	even	1	A${}_2$
even	odd	$-1$	B${}_1$
even	odd	1	B${}_2$
odd	odd	1	B${}_1$
odd	odd	$-1$	B${}_2$

Figure 4. The structure of the CH₃Cl molecule.

Figure 5. The equilibrium Eckart frame for CH₃Cl. The z-axis is along the C–Cl bond and the x-axis in the plane formed by Cl–C–H${}_1$.

Figure 6. The dihedrals that determine the value of ${\varphi }_1=1/3({\theta }_{21}-{\theta }_{13})$.

Figure 7. The dihedrals that determine the value of ${\varphi }_2=1/3({\theta }_{32}-{\theta }_{21}).$

Figure 8. All the angles relevant angles to determine the angle between $x_1$ and $x_2$.

Figure 9. The structure of the C₂H₆ molecule.

Figure 10. The equilibrium Sayvetz frame for C₂H₆. The z-axis is parallel to the C–C bond and the x-axis bisects the dihedral between the planes formed by C–C–H${}_1$ and C–C–H${}_4$.

Table 5. The generators of the extended group ${\mathcal{G}}_{36}$(EM) of C₂H₆ and their effect on the torsional angle (τ) and the equivalent rotation of the generator [32].

Transformed τ	Equivalent rotation	${\mathcal{G}}_{36}($EM) generator
$\tau -4\pi /3$	E	(123)(456)
τ	$R_z^{2\pi /3}$	(132)(456)
$2\pi -\tau$	E	$\left(14\right)\left(26\right)\left(35\right)(ab{)}^{\ast }$
τ	$R_0^{\pi }$	$\left(14\right)\left(25\right)\left(36\right)\left(ab\right)$
$\tau +2\pi$	$R_z^{\pi }$	$E^{\prime }$

Figure 11. The geometric frame for C₂H₆. The z-axis is parallel to the C–C bond. The x-axis is at an angle $\varphi =(q_{15}+q_{17})/\left(3\sqrt{2}\right)$ from the bisector of the dihedral angle between the planes formed by C–C–H${}_1$ and C–C–H${}_4$ (colour online).

The geometric frame fro C2H6. The z-axis is parallel to the C-C bond. The x-axis is at an angle φ = (q15 + q17)/32) from the bisector of the dihedral angle between the planes formed by C-C-H1 and C-C-H4. The bisector is the line which the φ angle points to in the figure (the red line in the web version of this article).

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