The LCAO SCF Method

• 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.

  

Basis functions

• 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.

Other choices of basis functions

• 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.

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GTO basis sets

• 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.