# Wave function (orbital) rotation matrix

## Main Question or Discussion Point

Dear friends,

I've come across this questions when studying biatomic molecules. Here's my problem:
You have the following two wave functions:

Psi_1 = px(A) + px(B)
Psi_2 = py(A) + py(B)

here px(A) is the px orbital wave function of the A nucleus, px(B) of the B nucleus and so on.

Now we want to rotate these wave functions in order to test their symmetry. I just can't get the way it is done:

C (Psi_1, Psi_2) = R (Psi_1, Psi_2)

here C is the rotation operator, (Psi_1, Psi_2) is a column vector and R is the rotational matrix of coordinates!
Why does this work? If one told me rotate a function, I would rotate the coordinates with the R matrix, not the function values themselves...
Thanks in advance,
botee

## Answers and Replies

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cgk
Science Advisor
If you're reading theoretical chemistry text, "orbital rotation" usually does not refer to a rotation of orbitals in actual physical 3D space. Rather, it refers to a unitary transformation amongst the orbitals; that is, you build a new set of orbitals via
$$\tilde\phi_i(\vec r) = \sum_j U_{ij} \phi_j(\vec r)$$
where U is a unitary (=orthogonal, in the real case) matrix. This is what is happening here. Note that such a orbital rotation amongst occupied orbitals leaves the Slater determinant build from them invariant.

However, in this particular case this orbital rotation may or may not be equivalent to a rotation in 3D-space (it depends on the orientation of the molecule if it is) .