From owner-chemistry@ccl.net Thu Mar 15 00:04:00 2012 From: "Seth Olsen seth.olsen*uq.edu.au" To: CCL Subject: CCL: CASSCF does not produce spin densities Message-Id: <-46499-120314235841-5965-neNBpr3GjQ3UURigP32EAw^-^server.ccl.net> X-Original-From: Seth Olsen Content-Type: multipart/alternative; boundary=Apple-Mail-12--858178517 Date: Thu, 15 Mar 2012 13:57:03 +1000 Mime-Version: 1.0 (Apple Message framework v1084) Sent to CCL by: Seth Olsen [seth.olsen{}uq.edu.au] --Apple-Mail-12--858178517 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=iso-8859-1 On 15/03/2012, at 9:23 AM, J=FCrgen Gr=E4fenstein jurgen-$-chem.gu.se = wrote: >=20 > Sent to CCL by: =3D?iso-8859-1?Q?J=3DFCrgen_Gr=3DE4fenstein?=3D = [jurgen:+:chem.gu.se] >=20 > On 14 Mar, 2012, at 13:48 , Seth Olsen seth.olsen!^!uq.edu.au wrote: >=20 >>=20 >> 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). =20 >=20 > 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. >=20 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. >=20 > 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. >=20 > 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. >=20 >> 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 >=20 > 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. =20 >=20 > Best regards, > J=FCrgen >=20 >=20 >=20 > -=3D This is automatically added to each message by the mailing script = =3D- > To recover the email address of the author of the message, please = change>=20>=20>=20 > Subscribe/Unsubscribe:=20>=20>=20 > Job: http://www.ccl.net/jobs=20 > Conferences: = http://server.ccl.net/chemistry/announcements/conferences/ >=20>=20>=20>=20 >=20 --------------------------------------------------- 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 --------------------------------------------------- Unless stated otherwise, this e-mail represents only the views of the = Sender and not the views of The University of Queensland --Apple-Mail-12--858178517 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=iso-8859-1

Sent to CCL by: = =3D?iso-8859-1?Q?J=3DFCrgen_Gr=3DE4fenstein?=3D = [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.  


Best = regards,
J=FCrgen



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