On 15/03/2012, at 9:23 AM,
Jürgen Gräfenstein jurgen-$-chem.gu.se wrote: Sent to CCL
by: =?iso-8859-1?Q?J=FCrgen_Gr=E4fenstein?= [jurgen:+:chem.gu.se] On 14
Mar, 2012, at 13:48 , Seth Olsen seth.olsen!^!uq.edu.au
wrote:
It's also worth mentioning that I've only ever dealt with
state-averaged problems. It is possible that distinguishing between
"static" and "dynamic" correlations may be operationally useful for ground state
problems, where the idea of a well-defined reference makes somewhat more sense
(although still not very much, because of the orbital invariance of the CAS
expansion).
The latter is actually no issue. You can
rotate the orbitals in a HF or CASSCF wave function but you will still keep its
character, i.e. single-reference or
multi-reference.
Not sure if I understand.
The invariant spaces are different for the two cases. For an
evenly-weighted state average, unitaries on the target space leave the
state-averaged ensemble invariant. This latter point is probably more
important than the orbital invariance, because the idea of a "reference" (i.e. a
special state from which the expansion is built) dissolves; all states in the
target space are on the same footing. I think my point is that the state
that is correlated is not really any state in a state-averaged scheme, but the
average ensemble itself. There are additional complications that
arise then, and its not clear that the concepts used for ground-state models
will carry over. It's pretty clear that the consensus is that "static" correlations
are correlations required by the constraint that the state transform as a
particular irrep of the symmetric group, and that that "dynamic" correlations
are associated with the Coulomb hole.
If there was no
Coulomb repulsion the electrons would not avoid each other and the ground state
for H_2, however stretched, would be (1 \sigma_g)^2. Unfortunately, misleading
statements of the kind "dynamic correlation is driven by Coulomb interaction,
static correlation by the near-degeneracy of two or more configurations" are
quite common in the literature. Both kind of correlations are driven by Coulomb
repulsion; however, a set of quasi-degenerate configurations responds
differently to electron-electron repulsion than a bunch of configurations higher
up in energy.
Hmmm. OK. Probably
the issue is that the symmetric group constraint entangles all degrees of
freedom while the Coulomb operator generates pairwise entanglements.
Probably the effects aren't separable (which would be why we are having
this discussion, I suppose). This is a serious problem for mathematics
(not just chemistry) because while pairwise entanglements are amenable to
analysis with Schmidt decompositions (i.e. the SVD), there is no good analogue
for tensors of rank > 2. This problem seems to underlie a lot of
current issues.
Your point about quasi-deneracy merits
some more thought. So, you're suggesting that when the broadening of the
energies by the correlation is smaller than their splitting, the correlation is
"dynamic"? Maybe a self-energy concept can be leveraged
here. On the other hand, both static
and dynamic correlation has to maintain the symmetry (more strictly, the IRREP)
of the wave function. That is, in both cases only configurations from the right
IRREP contribute to the CI expansion. Also, dynamic correlation is probably
dominated by, but definitely not restricted to, two.electron
interactions. I get all that. But, this
tells you right away that it is not possible to build any operator whose
expectation will give you a measure of pure "static" or "dynamic" correlations.
This is because the requirement of transformation as an irrep of S_n will
entangle all degrees of freedom, while the Coulomb operator generates pairwise
entanglements. The operators act on different Hilbert spaces.
-Seth
The proper definition of static and dynamic
correlation becomes topical in connection with CAS-DFT methods, which a number
of authors (including myself) have struggled and struggle with. In this context,
a physically motivated, "waterproof" definition of the two correlation
contributions would be of great
value.
I don't envy you having to
wrestle with the representability problems there.
--------------------------------------------------- Seth Olsen ARC
Australian Research Fellow 6-431 Physics Annexe School of Mathematics and
Physics The University of Queensland Brisbane QLD 4072 Australia seth.olsen+/-uq.edu.au+61 7 3365
2816
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