D.51 Born-Oppenheimer nuclear motion

This note gives a derivation of the Born-Oppenheimer Hamiltonian eigenvalue problems (9.13) for the wave functions of the nuclei.

First consider an exact eigenfunction

You can do this for any set of nuclear coordinates that you like, but the coefficients

So, to be really precise, the wave function of

where superscripts n indicate nuclear coordinates. (The nuclear spins are really irrelevant, but it cannot hurt to keep them in.)

Consider what this means physically. By construction, the square
electron eigenfunctions

(D.31) |

In the unsteady case, the

So far, no approximations have been made; the only thing that has been
done is to define the nuclear wave functions

Note first that the eigenfunctions can be taken to be real since the
Hamiltonian is real. If the eigenfunctions were complex, then their
real and imaginary parts separately would be eigenfunctions, and both
of these are real. This argument applies to both the electron
eigenfunctions separately as well as to the full eigenfunction. The
trick is now to take an inner product of the equation above with a
chosen electron eigenfunction

What do you get? Consider the terms in reverse order, from right to
left. In the right hand side, the electron-coordinate inner product

Turning to the latter four terms in the left-hand side, remember that
by definition the electron eigenfunctions satisfy

and if you then take an inner product of

That leaves only the nuclear kinetic term, and that one is a bit
tricky. Recalling the definition (9.4) of the kinetic
energy operator

Remember that not just the nuclear wave functions, but also the
electron wave functions depend on the nuclear coordinates. So, if you
differentiate out the product, you get

Now if you take the inner product with electron eigenfunction

Well, whether you like it or not, the exact equation is, collecting
all terms and rearranging,

The first thing to note is the final sum in (D.32). Unless
you can talk away this sum as negligible, (9.13) is not
valid. The off-diagonal

coefficients, the vibration” and “electronic,

if you are
wondering.)

Let’s have a closer look at (D.33) and
(D.34) to see how big the various terms really are. At
first appearance it might seem that both the nuclear kinetic energy

How about the coefficients

But there are exceptions. If different energy levels get close
together, the electron wave functions become very sensitive to small
effects, including small changes in the nuclear positions. When the
wave functions have become sensitive enough that they vary
significantly under nuclear position changes comparable in size to the
nuclear wave function blobs, you can no longer ignore the

You can be a bit more precise about that claim with a few tricks.
Consider the factors

appearing in the

For

The first equality is just a matter of the definition of the electron eigenfunctions and taking the second

As far as the final term in

The first term should not be large: while the left hand side of the inner product has a large component along

The diagonal part of matrix

Note that in a transient case that starts out as a single nuclear wave
function