Friday, May 20, 2011

Measures of (static) electronic correlations in molecules

I particularly like the Figure below taken from a paper The radical character of the acenes: A density matrix renormalization group study by Johannes Hachmann, Jonathan Dorando, Michael Avil├ęs, and Garnet Chan.
HUNO = highest occupied natural orbital          
LUNO = lowest occupied natural orbital

Note how as the acene gets longer the spread in orbital occupation number increases.

In a nice review article Schmidt and Gordon state:
The presence of occupation numbers between 0.1 and 1.9 indicates considerable multireference character. 
(Multi-reference character means how many Slater determinants are required to describe the state.)

It seems to me that the spread in the above distribution gives a measure of the number orbitals M one needs to include in the active space of a Complete Active Space (CAS) calculation. In a Hartree-Fock calculation the occupations are all 2 or 0. The number of natural orbital occupations that deviate significantly from 2 or 0 is a good measure of M. It may be possible to relate this to various quantum entanglement measures.

Readers with backgrounds in solid state physics may note the similarity to the above picture to how  in Fermi liquids one can quantify the strong electronic correlations in terms of the discontinuity in orbital occupation number at the Fermi surface (discussed in an earlier post).


  1. 've decided that the distinction between "static" and "dynamic" correlation is not meaningful for CASSCF wavefunctions. This is clear, because when the active space spans the entire orbital manifold, one has a full configuration interaction - which obviously contains both static and dynamic correlations. The issue is further confused because the usual formulation of "CASPT2" actually does not use the CASSCF Hamiltonian as a zeroth order reference - it uses a one-electron hamiltonian made from CASSCF orbitals. There are other formulations of perturbation theory on a CASSCF reference that DO use the CASSCF Hamiltonian as a zeroth order guess, but this means that there already correlation contributions to the reference.

    I think that the Schmidt/Gordon suggestion for the limits of occupation number (i.e. 0.1 to 0.9) are probably too tight.

    I also suspect that state-averaging may change the interpretation, too. If there are correlations that affect all states identically, then these will result in orbital occupation broadening without changing the amount of information required to index the state space.

  2. I think the real issue here is the possibility for multiple solutions. If the character of the target state changes then so will the orbitals. You may have two solutions with significant near-one occupation but with different orbital spaces.