- This approach of writing the unknown molecular orbitals in terms of a
fixed basis
is
known as the
*linear combination of atomic orbitals*(LCAO) approach. - Classic qualitative approach to molecular orbital theory.
- Originally introduced with the idea of using atomic orbitals as the basis, but
this creates problems.
- To form
**F**, we*must*be able to compute the integrals over the one- and two-electron operators in the Hamiltonian.

- Similar qualitative reasoning as H atom: exponential behavior near
nucleus, and exponential decay at long range.
- So use
a

*Slater-type orbital*(STO) centered on nucleus A. - Works nicely for atoms (one-center integrals); works for diatomic molecules
(two-center integrals) but requires some numerical integration.
- Works for linear polyatomics, but integral expressions are very difficult and
numerical integration is required in a number of cases.
- No algorithm for nonlinear molecules. None.

- Boys suggested Gaussian-type orbitals (GTO)
or the "Cartesian" form

which are closely related.

- Multicenter integrals are all easy (relatively speaking). Require at most a
one-dimensional numerical integration, by quadrature.
- What about the physics.

figure

figure

figure

- Cannot solve H exactly with finite expansion in GTOs (needs only one
STO!).
- Combination of several GTOs much better - need bigger GTO basis sets than STO
sets.
- Some GTO sets combine a few GTOs together into one basis function before we
start (
*contracted*basis functions). - Perform quite well: virtually all electronic structure calculations use
Gaussians.