Legitimacy in visualizing the orbital overlap

[bearcharge]

A chemist is becoming suspicious here:

So, for visual convenience, the so-called dxy, dxz, dyz, dz2, dx2-y2 orbitals are actually linear combination of eigenfunctions. But chemists have been using the geometric feature of these 'manmade' orbitals to make sense the chemical bonding successfully. If they are not eigenfunctions, or 'real' orbitals, how can such success be justified?

 

[DrDu]

This has nothing to do with visual convenience!
The orbitals you are referring to are linear combinations of eigenfunctions of the operators $L^2$ and, at least for the hydrogen atom, of the hamiltonian H. But they are combinations of degenerate eigenfunctions, so they are eigenfunctions themselves and , for the description of an isolated H atom, they are neither superior or inferior to the eigenfunctions of $L_z$. However, in chemistry they have some definite advantages over the orbitals labeled by the magnetic quantum number m. Namely, they are real and not complex functions and are more localized. I.e. the molecular environment will break in many cases the rotational symmetry around the z axis so that the real valued orbitals which are localized so as to form bonds will remain approximate zeroth order eigenfunctions although their degeneracy is lifted.

 

[bearcharge]

Thanks for the answer. Just checking if my understanding is correct or not:

1. These real functions are also eigenfunctions of L², L_z and H (for hydrogen atom) because of the degeneracy.

2. These real functions are good approximations of eigenfunctions when symmetry is lowered/degeneracy is lifted. Is this implying complex functions are not good approximations in this case?

thanks!

 

[DrDu]

1. They aren't eigenfunctions of $L_z$. But there is no good reason why they should, as the hamiltonian does not depend on it.
2. This depends. In linear molecules, it is sometimes better to work with complex functions as the rotational symmetry around the internuclear axis remains (and can be chosen as the z-axis). In most other molecules, the real orbitals have advantages, be it only that you can use float number type instead of complex numbers in computer programs. Generally bonding is better described using e.g. px and py orbitals instead of complex doughnut shaped p+ and p- orbitals.

 

[bearcharge]

Thanks!

1. Doesn't L_z commute with L² for hydrogen atom?

2. Can I explain the whole situation to a chemist who does not know quantum physics very well in this way:

The real orbitals we use for describing bonding are actually linear combinations of solved eigenfunctions. They turned out to be appropriate for describing bonding in most situations. So, most of the times, chemists can live happily in a world where they can think about the bonding on the basis of geometric configuration of these real orbitals without worrying out the legitimacy of doing so.

thanks!

 

[DrDu]

1. Of course Lz commutes with L2. There are other operators which do so (e.g. trivially Lx or Ly) which do not necessarily commute with Lz. So what is your reason that you insist that a set of degenerate orbitals must be eigenfunctions of Lz? There is no problem with legitimacy.
The hydrogen atom has other symmetry operations which do not even commute with L2 (the Runge Lenz vector). However, they are peculiar to H, too, and thus have also little relevance for chemistry.

2. Beginning physicists have a strange tendency to try to teach chemists how they think how chemistry works. Believe me, at least theoretical chemists have a very good level of quantum mechanics.

 

[bearcharge]

Thanks for exposing my ignorance on these points and thanks for helping me with a deeper understanding.