Separation of variables

Separation of internal motion from center-of-mass motion

First we consider two masses m1 and m2 in one dimension, interacting through a potential that depends only upon their relative separation (x1 – x2), so that V(x1, x2) = V(x1 – x2). Here we are considering an arbitrary potential acting along the internuclear vector (taken to be along the x axis for simplicity). Given that the force acting upon the jth particle is Fj = -(dV/dxj), we can express Newton's equations for m1 and m2 as follows:

The definitions of the center of mass coordinate and the relative coordinate x are

where (M = m1 + m2). Using this definition of x, we can show that F1 = -F2. The trick here is to recognize that

Since x = x1 – x2 we have,

 

We then solve for x1 and x2 in terms of X and x. Plug these into Newtons' equations to obtain two coupled equations in terms of X and x alone. Separate these equations to find a center-of-mass equation (X alone) and a relative coordinate equation (x alone). The potential is a relative function of x1 – x2 and therefore only will appear in the x equation. The solutions will have the form.

where m is the reduced mass

 

Now begin with the definitions:

Solve for x1:

We substitute in for x2 to obtain an equation in x1 alone.

We place all of the x1 terms on the right-hand side.

 

Note that 1 + m2/m1 is equal to M/m1. Therefore, we can divide both sides by m1/M to obtain

Now solve for x2.

Substitute in for x1:

Move all terms in x2 to the right-hand side.

Note that 1 + m1/m2 is equal to M/m2. Therefore, we can divide both sides by m2/M to obtain:

Now, we set up the differential equations for the motion of the particles 1 and 2 equations

using the above information

The sum of these two equations gives

To eliminate X we divide the first by m1 and second by m2

  

and then take the difference.

Notice that we can use the definition

to rewrite the equation as

 

The same set approach can be applied to motion along the y axis. Starting with

the motion can be separated into the center-of-mass motion:

and internal motion:

The importance of this set of transformations is that we have shown for any diatomic molecule how motions can be separated into center-of-mass motion (molecular translation) and internal motions (vibration and rotation). In the following we demonstrate the separation of internal motions into vibrational and rotational motions.

Separation of rotation and vibration

The rigid rotator approximation is the starting point for rotational spectroscopy. For a diatomic we can demonstrate this using two dimensions as shown below.

The origin of the coordinate system for the diatomic molecule is shown as the center of mass. In the rigid rotator approximation we assume that the internuclear distance is fixed (we will it R). Then we can transform into polar coordinates:

x = R cosq

y = R sinq

There is no rotational potential. In Cartesian coordinates the rotational kinetic energy is given by

We have included projections onto both the x and y axes. The differential elements can be transformed as:

dx = - Rsinqdq

dy = Rcosq dq

Substituting these into the kinetic energy expression we have:

where we have introduced the moment of inertia for a diatomic molecule I = mR2.

In two-dimensions there are two degrees of freedom for the generic diatomic molecule we discuss here. By fixing R and concentrating on the rotation angle q we have described one of the degrees of freedom. We now consider motion along the internuclear vector as the molecule rotates. We could do this at a fixed angle q or by considering a rotating coordinate system. To make things easy we will consider the motion along the x coordinate when q = 0. However, our result will be general and will apply to any arbitrary angle q.

Treating the diatomic as a classical spring the Hooke's law restoring force along x is F = -kx. The equation of motion is:

The general solution is x = Acos(wt) where

The constant A describes the amplitude of the vibration. The classical equations of motion can be replaced with quantum mechanical equations of motion now that we have separated the variables.

Orientation averaging

In a typical experiment molecules can have any orientation in three dimensions but the electromagnetic radiation is polarized along one axis. Thus, for any interaction of a dipole with a radiation field we must average over the orientations of the molecules at any given time. We have seen that the transition probability between two states i and j is proportional to |mij.E|2. The strength of the interaction is the dot product of mij and E so it is determined by the angle q for x-polarized radiation. This is shown in a two-dimensional coordinate system below.

The average over all angles q is given by

The results derived here are valid for three dimensions as well.

The expression used for the dipole mij depends on the type of spectroscopy we consider.

For rotational spectroscopy mij is the permanent dipole moment of the molecule. As the molecule rotates in space that dipole oscillates at a frequency that can interact with electromagnetic radiation in the microwave region of the spectrum.

mij = mground_state

For vibrational spectroscopy it is the change in dipole moment as the molecule vibrates that can interact with electromagnetic radiation. Considering only motion along the x-direction this means.

mij = dmground_state/dx

for interaction with x-polarized radiation.

Electronic spectroscopy consists of interaction of an electronic transition moment with electromagnetic radiation: