What Variational Parameter?

Ah, the crux of the problem, is it not? Up until now, we've just assumed we have some set of molecular orbitals or , which we can manipulate at will. But how does one come up with even approximate solutions to the many body Schrödinger equation without having to solve it? Start with the celebrated linear combination of atomic orbitals to get molecular orbitals (LCAO-MO) approximation. This allows us to use some set of (approximate) atomic orbitals, the basis functions which we know and love, to expand the MOs in. In the most general terms,

remains a spatial molecular orbital, is a spatial atomic orbital (perhaps symmetry orbital, but no matter), and are the contraction coefficients by which we transform from one basis to another. Armed with only this, we should be able to compose the electronic energy in the atomic orbital basis. Why, you ask? Because we have an expression in terms of integrals over MOs. To variationally minimize that energy, we need to vary the MOs themselves, but have no way to do that, since they remain these amorphous constructs. By defining them a bit more precisely, we should arrive at a point where an obvious set of variational parameters (hint: ) present themselves. Begin with the closed shell HF energy in terms of spatial MOs.

Since and ,
 
is a one electron integral over atomic orbitals. Do we have something we can actually calculate!? Take an aside and examine this quantity superficially.

Typically, basis functions are constructed to mimic true atomic orbitals. The Hydrogen atom can be described rigorously, and the eigenfunctions found. The 1s orbital looks something like . It satisfies all the appropriate boundary conditions, having a cusp at the nucleus and exponentially decaying to zero at infinity. Higher angular momentum functions, like 2p's and 3d's, can be built from this basic framework through adding the angular nodes by multiplying in factors of x, y, and z. Basis functions such as these are called slater-type orbitals. If instead of the exponential we use a gaussian function, , we loose the boundary conditions but generate a more tractable problem when it comes to calculating integrals. Using a linear combination of single cartesian gaussian-type orbitals to approximate a slater-type orbital gives better computational accuracy without too much more effort. Here's the functional forms of all three types:

Just taking a 1s SGTO for illustrative purposes, what is that one electron integral?

Hey! We can do that!

Back to the problem at hand, we now need to expand the two electron integrals in the MO basis. Following a procedure analogus to equation 64, we get

All of these AO integrals can be calculated and stored, to be called up when needed to evaluate the electronic energy. The closed shell energy in the AO basis can be written as

is the density matrix, a product of AO-MO coefficient matrices, or