Electron angular momentum in diatomic molecules

[kelly0303]

Hello! I just started reading some molecular physics and I am a bit confused about the electron angular momentum in diatomic molecules. Let's say we have just 2 protons and an electron for simplicity and we are in the Born-Oppenheimer approximation, so we assume that the nuclei are fixed in space. Given that we don't have a central potential the angular momentum quantum number is not conserved. However, its projection on the internuclear axis is conserved and this is something that holds for any diatomic molecule (with some subtleties related to Hund cases, but let's assume we are in Hund case a, so this projection is well defined). So the way I visualize this, vectorially, is a vector corresponding to the angular momentum, that rotates at an angle around the internuclear axis (similar to a magnetic moment around a magnetic field). So the magnitude and precession angle seem to be constant (and hence the projection). But this looks to me just like the projection of an electron angular momentum along the z-axis on an atom (the momentum precess around the z-axis, and its projection gives the quantum numbers $m_l$). So I am not sure what exactly it is not conserved about angular momentum in molecules, as to me it seems like the behavior of the angular momentum vector is the same as in atoms, where we know it is conserved. Can someone help me understand? Thank you!

 

[dRic2]

I'm not familiar with this topic, hope I don't say something illy, but here

So the way I visualize this, vectorially, is a vector corresponding to the angular momentum, that rotates at an angle around the internuclear axis (similar to a magnetic moment around a magnetic field). So the magnitude and precession angle seem to be constant (and hence the projection).

aren't you supposing too much ? You previously said that only the projection on the axis is constant, that's it. Why are you supposing the precession angle to be constant also ?

 

[kelly0303]

I'm not familiar with this topic, hope I don't say something illy, but here

aren't you supposing too much ? You previously said that only the projection on the axis is constant, that's it. Why are you supposing the precession angle to be constant also ?

Sorry, we also have that the $L^2$ operator commutes with the Hamiltonian, so I assumed that the length of the vector is constant.

 

[dRic2]

$L^2$, or $J^2$ ? If $L^2$ commutes with H, orbital angular momentum should be conserved. Right ? And that's not the case. If $J^2$ commutes with H then total angular momentum (orbital + spin) is conserved and this is always true for an isolated system.

 

[Twigg]

It's been a good while since I studied this but here goes. I thought $L^2$ was not a good quantum number in Hund's case a? I thought it was just the projection $\Lambda$ that was good.