The pictorial representations of orbitals, the charges on the atoms, the shapes, bond lengths and bond angles that we determine using a program like HyperChem are merely the visual manifestation of a complex set of calculations (remember we are attempting to solve the Schrodinger equation for the molecule). Some of the numerical results of those calculations are stored in the log file (if you choose to create one).

Of particular interest to us are the eigenvalues (the energies of the molecular orbitals) and the eigenvectors (the wavefunctions for the molecular orbitals). The eigenvalues are self explanatory, and you can compare them to the orbital energies in the energy level diagram resulting from a particular calculation. Remember that our molecular orbital model constructs model orbitals by taking linear combinations of atomic orbitals (think back to making the MO's of homonuclear diatomics). The eigenfunctions are those linear combinations.

The portion of the log file that is of interest to us at this time is the section labeled eigenvalues and eignevectors that is nearest the bottom of the file (depending on how far you were from an optimized structure when you began there may be only a few or a very large number of such sections. One solution to this is to perform the calculation before starting the log file, then with the log file open do a single point calculation (since the molecule is already at the minimized structure). Here is the relevant section from the log file for CO.

We infer the origins of any of the molecular orbitals by reading down the appropriate column. For example, orbital 5 is the highest occupied molecular orbital in CO (why do we know this is the case). We would conclude that the wavefunction for orbital 5 was given by:

In this case HyperChem has chosen the z axis as the internuclear axis. Hence, the negative lobe of pz on one atom is pointing toward the positive lobe of pz on the other atom, so if the pz orbitals have opposite signs, as they do here, the result is a bonding interaction (draw yourself some pictures if this is confusing).

Remember that it is the square of the coefficients that tell us about the fraction of a particular atomic orbital in the molecular orbital. So in this case we would say that the molecular orbital we are looking at comes has about 44% C2s character, 35% C2pz character, less than 1% O2s character and about 22% O2pz character. So this orbital is mostly C in nature (about 80%) and is the reason we consider this highest occupied orbital to be similar to a lone pair of electrons on the C atom.

Here is a spreadsheet that explores these coefficients in more detail. In the first portion of the spreadsheet are the coefficients straight form the log file. In cells A16 through I24 are the squares of those coefficients. In row 26 are the sums of the squares from the columns. These all add to 1, as they must, since we are just summing the fraction of the various AOs in each MO, and this must add to 1. In K16-K24 are the sums of the squares from the rows corresponding to the various AOs. These also must add to 1 as each AO is used completely in forming the MOs.

In A39-L37 I am trying to give a sense of how the charge is determined. The coefficients are shown only for the occupied molecular orbitals (1-5). On the right side of the region I first determine that adding the squares of the carbon orbital coefficients in the occupied MOs gives 1.92 C orbitals used, meaning 3.84 electrons are associated with the C atom in the molecule. But carbon came with 4 electrons meaning that it now "has" 0.16 fewer than previously giving it a charge of 0.16. The same argument for O shows that 6.16 electrons are associated with O in the molecule, but O only brought 6 electrons to the molecule giving it a charge of -0.16. These are the charges (to one more significant figure) that HyperChem reports for a PM3 calculation on CO.