LCAO MO & Overlap

From WikidChem

Lcao_mo_& _overlap Proprietors: Jingying Yang & Alex Kazberouk Molecules are made from overlapping atoms.

Instead of hybridizing orbitals (like sp2 or whatever), here we are hybridizing atomic orbitals from 2 different atoms to find how they overlap and how we can model bonding

Ψ(x1,y1,z1) = (1/√2)( AOa + AOb) = the sum of AOs. Just like we sum up orbitals to get hybridization…and thus the closer we are to one atom, the larger the larger AOa or AOb will be)

When we square this function that we have designated as the sum of AOs, we get: Ψ^2(x1,y1,z1) = (1/2)(AOa^2 + AOb^2 + 2AOa x AOb)=1/2(AOa^2 + AOb^2) + AOa x AOb Square because psi^2 is what we need for electron density

When two molecules are at a great distance, either AOa or AOb (or both) will be negligible at any point in space because the molecules are so far apart. Thus, for molecules at a great distance, Ψ^2(x1,y1,z1)=1/2(AOa^2 + AOb^2)

This model also works really really well right next to a nucleus where AOa for instance is large and AOb is small

For molecules at a bonding distance, however, AOa x AOb will not be negligible between the atoms because the values of both AOa and AOb will be significant.

Thus, for a bonded molecule, the electron probability density (Ψ^2) is as follows: Ψ^2=1/2(AOa^2 + AOb^2) + AOa x AOb. For the same two atoms, if they were unbonded (ie. at a great distance), their Ψ^2 would be as follows: Ψ^2=1/2(AOa^2 + AOb^2).

This gives two equations that can be used to find the electron density difference, which is used to show bonding. This seems very similiar to what we do in X-ray diffraction density difference maps - calculate unbonded electron density using quantum mechanics (individual orbitas of atoms) and then take that away from what we actually measure to find bonding

Electron density difference = (Electron density of a bonded molecule) - (Electron density of those unbonded atoms) = 1/2(AOa^2 + AOb^2) + AOa x AOb? - 1/2(AOa^2 + AOb^2)? = AOa x AOb Thus, AOa x AOb is the electron density overlap that forms when two atoms bond to make a molecule. The one half makes it so that there is no coefficient next to the AOa/AOb ((a+b)^2 = a^2 + b^2 + 2ab)

One small problem with the above, though, is that 1/2(AOa^2 + AOb^2) = 1/2(1+1)=1 so 1/2(AOa^2 + AOb^2) + AOa x AOb > 1. This means that we must renormalize this problem, so the coefficient of 1/2 is actually slightly smaller than 1/2 to make the probability density sum to 1.

This is only needed for quantitative measures and is not necessary to understand what bonding looks like or what the general shape is.