I Would like to have the structure of "covalent coordinate" molecules explained in terms of either molecular orbitals or hybridizations. Specifically, I am interested in the sulfate ion and carbond monoxide. I am not interested in hearing terms like "filling the valence shell" or "becoming like a noble gas". Thanks

Second, if you want the full explanation in terms of "molecular orbitals or hybridizations," you really ought to read the references given in this MadSci answer. Incidentally, at least one of those references concluded that the Lewis dot picture (with octets and formal charges on sulfur) is a better fit to the electron density distribution than the "expanded valence" picture we get from hybridization theory (for which, of course, the best source is Pauling's The Nature of the Chemical Bond).

But I'll see what I can do for you, in terms of both molecular orbital theory and valence-bond theory - qualitatively, of course. A good general reference for this sort of thing is F.A. Cotton's classic text, Chemical Applications of Group Theory. If you want the full mathematical treatment of what is presented schematically below, you should work your way through a copy. Even then you won't have the mathematical equipment to do other than guess at the orbital energies...

• Valence-bond Theory considers bonds as combinations of overlapping one-electron atomic orbitals, which may or may not be hybridized. This approach is set out in many texts but is usually represented using conventions drawn from Lewis theory, which was first proposed by G.N. Lewis in "The Atom and the Molecule," Journal of the American Chemical Society 38, 762-785 (1916).
• Molecular Orbital Theory, in its simplest form (e.g. Hückel and Extended Hückel Theory), combines atomic orbitals to form molecular orbitals according to the symmetry of the molecule, then fills them two electrons at a time. More sophisticated versions of molecular orbital theory exist, with various ways of taking electron-electron interactions into account, but these are usually add-ons to the basic method.

Contents

1. Carbon monoxide in terms of hybridization and resonance (i.e. valence bond) theory.
2. Carbon monoxide in terms of molecular orbital theory.
3. The sulfate ion in terms of hybridization and resonance theory.
4. The sulfate ion in terms of molecular orbital theory.
5. The bottom line

1. Carbon monoxide in terms of hybridization and resonance (i.e. valence bond) theory.

Oxygen begins with six valence electrons, distributed (for example) thus: one in sp₁, two in sp₂, two in p_x, one in p_y, shown schematically in Figure 2.

Figure 1		Figure 2

However, the one-electron orbitals on oxygen and carbon don't match up well, so we can fix things one of two ways:

1. We "move" (this is pure bookkeeping!) an electron from oxygen's p_x to carbon's sp₁, so that both atoms now have the same electron configuration and can form a triple bond by the overlap of the one-electron orbitals (Figure 3). There is one lone pair each on carbon and on oxygen, and because we have "removed" an electron from oxygen and "given" it to carbon, there are formal charges.
   
   Figure 3
2. We "move" an electron from p_x on carbon to carbon's sp₁. Overlap of the remaining one-electron orbitals results in a double bond, with one lone pair of electrons on carbon and two on oxygen; all this is shown schematically in Figure 4.
   
   Figure 4

Of course, valence-bond theory requires that we mix these two models (which can in fact be stated mathematically); this is called "resonance" and is represented schematically in Figure 5. The true structure is neither of these models, but a combination, and includes contributions from electronegativity which shift electron density from carbon back to oxygen. The calculated charge density surface (HF/6-31G*) is shown in Figure 6.

Figure 5		Figure 6

2. Carbon monoxide in terms of molecular orbital theory.

More strictly speaking, we need to realize that neither the C nor the O are at the molecule's center of symmetry, so that we must take symmetry-adapted linear combinations of the atomic orbitals -- which then turn out to be molecular orbitals right off. In point of fact, we don't usually work through the group-theoretical analysis because simple visual combination gets us to the same place. And so, to work.

There are two sets of orbitals of P symmetry, and they are mutually perpendicular so that they cannot overlap. They combine to form two pairs of degenerate molecular orbitals, one pair bonding (p) and the other antibonding (p*). These are shown in Figure 7.

Figure 7

There are also two sets of orbitals of S symmetry, and they are not mutually perpendicular. We can combine them in two equivalent ways:

1. We can first take independent combinations of the atomic s and p_z orbitals, and then allow the sets to interact (Figure 8).
   
   Figure 8
2. We can first combine the s and p_z orbitals on each atom, producing sp hybrids, and then form combinations suited to the overall molecular symmetry (Figure 9).
   
   Figure 9

Figure 10

3. The sulfate ion in terms of hybridization and resonance theory.

Figure 11

Figure 12

However, this is not the only possible valence-bond picture! We can entirely ignore p-d p-bonding (which is not supported by quantum chemical calculations anyhow) and present a picture which fully satisfies the Lewis octet rule. In this representation, sulfur is still sp³ hybridized, but d-orbitals on sulfur do not participate in bonding. Instead, bonding is semipolar (that is, a combination of covalent and ionic) and can be represented by the Lewis structure of Figure 13. Of course, the +2 formal charge on sulfur is not quite a true representation. An HF/6-31G* calculation calculated the charge on sulfur to be +1.77! Induction, driven by the large separation of charge in Figure 13, pulls electrons back toward sulfur so that the charge density surface for sulfate ion (Figure 14) shows a moderation of the negative charges on the oxygen atoms (the blue color is not positive charge, just "less negative").

Figure 13	Figure 14

4. The sulfate ion in terms of molecular orbital theory.

Interestingly, sulfur d atomic orbitals (except possibly d_z2) do not appear to have the proper symmetry to form molecular orbitals in a tetrahedral molecule. The additional bonds generated by molecular orbital calculations are p bonds between the oxygen atoms, in which sulfur participates very little, if at all (for an example, see Figure 15). There are also several oxygen lone-pair orbitals such as Figure 16.

sulfate	Figure 15	Figure 16

Molecular orbital theory bears out the simple Lewis picture shown above in Figure 13; this is also the conclusion of A. Reed and P.v.R. Schleyer, Journal of the American Chemical Society 109, 7362-73 (1987), ibid. 112, 1434-45 (1990); E. Magnusson, ibid. 7940-51.

5. The bottom line

The operative question, as always, is: does it correctly predict the observed electron density and chemical reactivity? For more on this, see "The Chemical Core of Chemistry" by Joachim Schummer, HYLE, 4, 129-162 (1998).

God never saw an orbital.

-- Walter Kauzmann to Maitland Jones
quoted on p. 67 of M. Jones, Organic Chemistry, Norton, 1997.