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Derivation of Atomic Orbital Shapes
By Brad Rodgers on Sunday, March 24, 2002 - 03:03 am:

What mathematics is required to derive the shapes of the atomic orbitals (s,p,d, and f)? I've asked one of my teachers why the orbitals even form seperate shapes, and she said to consult a book or find a website, but I'm not even sure where to look. I'm pretty sure it's related to schrodinger's equation, but I must admit I have no real knowledge about this equation. If the mathematics is not too complex (and not too long), could someone prove why certain orbitals take certain shapes?

Also, why does the octet rule hold for some bond scenarios but not others (e.g. why can BF3 exist)? Actually, why does the octet rule hold at all?

Sorry to be asking chemistry questions on this board, but the class everyone must take at our school is most certainly aimed at giving people a basis in industry, not the sciences. It provides statements, but really doesn't tell why they are true, or if it does, the thinking is sloppy ,unmathematical and could be easily argued against. Of course no one really argues but me, to which end I can't get too far, which brings us back to why I'm posting on this board.

Thanks,

Brad

By William Astle on Sunday, March 24, 2002 - 12:12 pm:

I think its's quite complicated for all but the simplest potentials.

The Hydrogen Atom is relatively easy. For this you solve the equation using an electrostatic potential which goes like -1/r. The general solution is an evil product of polynomials named after various mathematicians. The lowest energy state gives a probability density function like f(r) = Ce-cr where c, C are some positive real constants - which is rather unlike the classical idea of an electron orbiting a proton.

As I understand it from rusty chemistry....

An atom takes the lowest possible energy state. An atom has a number of possible 'orbitals' into which electrons can be slotted. The (chemical) energy of the atom is roughly determined by the orbital configuration of the electrons. The octet rule is a rough approximation for describing the configurations of lowest energy. Unfortunately it is a crap one, which is why it doesn't always work. The nasty thing, if I recall correctly, is that the positions of electrons do not independently alter the energy of the atom. (So you can't say 'with five electrons the following configuration is the lowest energy, when we add a couple more we should preserve the original configuration and put the extra pair into empty orbitals to get the new lowest energy configuration').


That may well be a pile of rubbish - if it is hopefully someone will correct it.

By William Astle on Sunday, March 24, 2002 - 12:18 pm:

Incidently - the reason that any compound exists is because the change to another lower energy molecular configuration (if it exist) requires more (activiation) energy than is swimming around in the form of heat, photons etc.

By Philip Ellison on Thursday, April 04, 2002 - 10:54 pm:

You're right about using Schrodinger's equation. From what I've seen of your posts, the maths involved shouldn't be beyond you. However, it is beyond me, so you'd better get a book out of your local library! I believe that it can only be solved algebraically for hydrogen... anything more complicated and you can only get numerical approximations from computer programs.