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
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
Now, under translation
Clearly, is invariant under translation if and only if
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,
Rotational invariance follows if and only if
The external modes, on the other hand, are not invariant and it is not difficult to show that they change under translation as follows:
where M is the total mass of the molecule. They change under infinitesimal rotation as follows
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
Famous quotes containing the words translation and/or rotation:
“The Bible is for the Government of the People, by the People, and for the People.”
—General prologue, Wycliffe translation of the Bible (1384)
“The lazy manage to keep up with the earths rotation just as well as the industrious.”
—Mason Cooley (b. 1927)