Eckart Conditions - Overall Translation and Rotation

Overall Translation and Rotation

The vibrational (internal) modes are invariant under translation and infinitesimal rotation of the equilibrium (reference) molecule if and only if the Eckart conditions apply. This will be shown in this subsection.

An overall translation of the reference molecule is given by

for any arbitrary 3-vector . An infinitesimal rotation of the molecule is given by

$\vec{R}_A^0 \mapsto \vec{R}_A^0 + \Delta\varphi \; ( \vec{n}\times \vec{R}_A^0)$

where Δφ is an infinitesimal angle, Δφ >> (Δφ)², and is an arbitrary unit vector. From the orthogonality of to the external space follows that the satisfy

$\sum_{A=1}^N \vec{q}^{\,A}_r = \vec{0} \quad\mathrm{and}\quad \sum_{A=1}^N \vec{R}^0_A\times \vec{q}^A_r = \vec{0}.$

Now, under translation

$q_r \mapsto \sum_A\vec{q}^{\,A}_r \cdot(\vec{d}^A - \vec{t}) = q_r - \vec{t}\cdot\sum_A \vec{q}^{\,A}_r = q_r.$

Clearly, is invariant under translation if and only if

$\sum_A \vec{q}^{\,A}_r = 0,$

because the vector is arbitrary. So, the translational Eckart conditions imply the translational invariance of the vectors belonging to internal space and conversely. Under rotation we have,

$q_r \mapsto \sum_A\vec{q}^{\,A}_r \cdot \big(\vec{d}^A - \Delta\varphi \; ( \vec{n}\times \vec{R}_A^0) \big) = q_r - \Delta\varphi \; \vec{n}\cdot\sum_A \vec{R}^0_A\times\vec{q}^{\,A}_r = q_r.$

Rotational invariance follows if and only if

$\sum_A \vec{R}^0_A\times\vec{q}^{\,A}_r = \vec{0}.$

The external modes, on the other hand, are not invariant and it is not difficult to show that they change under translation as follows:

$\begin{align} s_i &\mapsto s_i + M \vec{f}_i \cdot \vec{t} \quad \mathrm{for}\quad i=1,2,3 \\ s_i &\mapsto s_i \quad \mathrm{for}\quad i=4,5,6, \\ \end{align}$

where M is the total mass of the molecule. They change under infinitesimal rotation as follows

$\begin{align} s_i &\mapsto s_i \quad \mathrm{for}\quad i=1,2,3 \\ s_i &\mapsto s_i + \Delta \phi \vec{f}_i \cdot \mathbf{I}^0\cdot \vec{n} \quad \mathrm{for}\quad i=4,5,6, \\ \end{align}$

where I0 is the inertia tensor of the equilibrium molecule. This behavior shows that the first three external modes describe the overall translation of the molecule, while the modes 4, 5, and, 6 describe the overall rotation.

Read more about this topic: Eckart Conditions

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