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We know that an atomic orbital wavefunction may be written in terms of polar coordinates, $$\psi (r, \theta, \phi) = R(r) A(\theta, \phi)$$ where $R(r)$ is the radial component and $A(\theta, \phi)$ is the angular part.

There are many orbitals of interest, but in particular I'm interested in applying a symmetry operation on $d_{xy}$. We know that the angular part of this orbital is $\sin^2 (\theta) \sin(2 \phi)$. Suppose I want to rotate only along one axis, let's say $\theta$, so I fix $\phi$ and rotate $\theta$ by $\pi/2$. This means that the rotated orbital is now written as $A(\theta + \pi/2, \phi)=\sin^2(\theta + \pi/2) \sin(2 \phi)$. We know from trig. that $\sin(a + \pi/2) = \cos(a)$, therefore the angular part of our transformed orbital may now be written as $$A(\theta+\pi/2, \phi)=\cos^2 (\theta) \sin(2\phi)$$

however, graphical approaches to this problem for the real orbital suggest this $\hat C_4$ rotation should give back $-d_{xy}$. What am I missing?

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up vote 3 down vote accepted

You did the rotation incorrectly--- rotating to add 90 degrees to $\theta$ doesn't add a constant to $\theta$, it rotates the x-y plane by 90 degrees. The angle which used to be $\phi$ is now $\theta$-like in that it rotates in a plane including the z axis, and the $\phi$ dependence is completely changed.

The easiest way to rotate this angular wavefunction is to note that it is the angular part of the quadrtic polynomial 2xy. Rotating by 90 degrees around the y axis takes z to x and x to minus z, and so gives the polynomial - 2yz. Subsituting the angular form of z and y, you get

$$ A(\theta,\phi) = - \cos(\theta) \sin(\theta) \sin(\phi) = - \cos(2\theta)\sin(\phi) $$

In general, don't rotate angular wavefunctions in polar coordinates. Write them out as polynomials and rotate their rectangular coordinate form. There is a tabulated way to write the rotation for angular wavefunctions in terms of themselves, but this is usually more trouble than converting in the way I did above.

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Where can I read more about this tabulated way to write them in terms of themselves? The reason I'm interested in this is because I'm trying to discover the transformation properties of symmetry adapted linear combinations for myself, but I've found I cannot reliably attack this via the graphical approach. – Chris Oct 12 '11 at 7:46
Importantly I'm curious about this because I don't see how the rotated angular function shown above is related to $-d_{xy}$. – Chris Oct 12 '11 at 7:51
The rotation of the angular functions into themselves is known as the integer spin representations of the rotation group, and it is covered in most quantum mechanics boos, under "angular momentum". If you ignore the physics, you can read this chapter directly. The formulas you want use a "D" matrix to write the rotated versions of the angular parts in terms of other Y's with the same l and different m, you'll see a D_{mm'}(\theta). – Ron Maimon Oct 12 '11 at 16:49
Cool. I found that I get my answer by rotating about the $z$ axis instead. Thanks for the help! – Chris Oct 12 '11 at 17:42
A nice Java applet is here: You can rotate orbitals with your mouse. – Vladimir Kalitvianski Oct 12 '11 at 18:51

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