Using the previous definitions for atomic positions, one can see that
the system is overdefined. There are three coordinates reflecting rotation
of the system as a whole, three coordinates indicating the translational
motion of the system, and 3N coordinates reflecting the individual
atomic displacements. This gives 3N+6 coordinates which is six more
than is needed. Thus, there are six Eckart-Sayvetz conditions
which can be imposed to reduce the number of degrees of freedom.
In vector form, they are as follows.
Since the origin is at the center of mass, the following must be true:
Likewise,
What these equations mean is that during a vibration, the center of mass
must not move and that no linear momentum is generated. This causes the
roto-translational and vibro-translational energies to go to zero.
So, in the rigid rotor model of the atom, one can see that translational
energy is completely separable from the rotations and vibrations.
Using the previous equations to simplify the expression for the kinetic
energy:
The last term can be manipulated as follows:
Now, by imposing the condition that during a vibration no angular
momentum is generated one can simplify the second term.
The final form of the kinetic energy is now:
The last term is known as the Coriolis energy and is usually very small.
It is commonly grouped in with the rotational energy.