Electron Correlation

• Obvious solution: extend the approximate wave function so that instead of just products of one-electron functions, it contains two-electron functions of rij (remember Hylleraas?).

• Fine for two electrons, troublesome for three... .

• Make more use of the basis set. For instance, we have virtual MOs as well as occupied MOs. If denotes the Hartree-Fock determinant

  

  we can start to replace the occupied orbitals with virtual orbitals to build more configurations:

  

  etc.

Configuration Interaction

• The variation principle tells us that if we allow all these configurations to mix - configuration interaction (CI) -- we will get a lower energy than Hartree-Fock: thus we are accounting for electron correlation.

  

• We can then optimize the coefficients c by making

  

  stationary.

• This has obvious similarities with the MCSCF approach we came up with for H2. However, in CI we do not reoptimize any orbitals.

• Typically, MCSCF is used only to deal with the problem of near degeneracies, sometimes termed nondynamical correlation, as opposed to the dynamical correlation mediated by in H.

• For M MOs (N occupied), the number of configurations behaves as

  

  For Ne (10 electrons) in a SZP basis (15 basis functions) this is about 30 million terms. Not generally feasible to include all of them!

• We recall that the Hamiltonian contains only one- and two-electron operators, and that the MOs form an orthonormal set.

• Then all matrix elements

  

  for three or more substitutions. Thus only the single and double excitations interact with .

• Restrict to only single and double excitations (CISD):

  

• This is a popular and longstanding method for calculating correlation energies. Its advantages and disadvantages will be discussed later.

Perturbation Theory

• Mathematical method for dealing with an unknown/intractable system, by building on a known/tractable system.

• For example, Hartree-Fock seems to be a pretty good approximation: treat correlation by looking at the difference between the exact Hamiltonian and the Fock operator.

• Expand the exact wave function and energy in a power series about the Hartree-Fock solution: expect a convergent series of contributions from each order (term).

• Second order (denoted MP2) cheap. MP3 more expensive and less reliable, MP4 is as high as is usually practical.

• Idea works best when exact wave function strongly dominated by Hartree-Fock. Fails for significant nondynamical correlation.

CI or Perturbation Theory?

• CI is variational -- upper bound to the exact energy. Can hangle nondynamical correlation.

• Perturbation theory is size-extensive: scales correctly with the number of particles in the system. Sometimes called "size-consistency".

• Suppose we have n He atoms a long distance apart. The total energy is clearly nEHe. True for the Hartree-Fock model, and true for perturbation theory. Not true for CI: energy goes as for large n.

• Certainly, correct scaling is more desirable than an upper bound: how are we to compare systems of different sizes otherwise?

The Coupled-Cluster Method

• Similar to CI: seeks to optimize coefficients of single and double excitations.

• Nonvariation, but exactly size-extensive.

• Much more reliable than perturbation theory or CI. CCSD (singles and doubles), CCSD(T) (adds estimate of the triples). CCSD(T) is the best Hartree-Fock based method we have.

• Approximate coupled-cluster methods: coupled-pair functional (CPF), quadratic CI (QCI).