- 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
- We can then optimize the coefficients c by making
- 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
- 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
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.
- 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
- Expand the exact wave function and energy in a power series about the
Hartree-Fock solution: expect a convergent series of contributions from each
- 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
- 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
- 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