[molpro-user] How to read the CSFs in a MRCI calculation?
José Cortés
zolidus at gmail.com
Fri Aug 28 22:24:41 BST 2015
Again thank you very much for your reply
I have been able to obtain the expansion in terms of determinants. I was
missing multiply each of the internally contracted configurations (ICC) by
the normalizing factor.
In short I summarize the procedure
The part of the double excitations in the wave function is expressed as
\sum_{i>j}\sum_{ab}\sum_{p}N_{ab}^{ijp}C_{ab}^{ijp}\Psi_{ijp}^{ab}
Molpro prints C_{ab}^{ijp} in the output file, and where N_{ab}^{ijp} is
the remaining normalization factor.
1) The first step is to convert \Psi_{0} from CSF to determinants.
2) After, the ICC are obtained in terms of determinants
\Psi_{ijp}^{ab}=\frac{1}{2}(\hat{E}_{ai,bj}+p\hat{E}_{bi,aj})\Psi_{0}
resulting in 8 cases (3 are zero) depending if i=j a=b p=1, i=j a=b p=-1,
etc. In developing the algebra, everything is put in terms of creation and
annihilation, where care must be taken with the phase factors (1 or -1)
when applying operators, depending whether if there are an even or odd
electrons number to the left in the occupation-number vector.
3) the normalization factors are then calculated
N_{ab}^{ijp}=\sqrt{<\Psi_{ijp}^{ab}|\Psi_{klq}^{cd}>}
<\Psi_{ijp}^{ab}|\Psi_{klq}^{cd}>=\frac{1}{2}\delta_{pq}(\delta_{ac}\delta_{bd}+p\delta_{ad}\delta_{bc})S_{ij,kl}^{p}
S_{ij,kl}^{p}=<0|\hat{E}_{ik,jl}+p\hat{E}_{il,jk}|0>
similarly, there are eight different cases. When a!=b, an additional
multiplication is performed of N_{ab}^{ijp} by \sqrt{2}.
4) Finally, the coefficients for each of the determinants are summed.
By this way, I got the CI vector norm that is indicated in the output of
Molpro, and with a little more work, I could calculate the total energy of
the system.
Regards
José
2015-08-22 20:32 GMT-05:00 Gerald Knizia <knizia at theochem.uni-stuttgart.de>:
> Dear José,
> this may be a bit complicated; I am actually not entirely sure how this
> works internally in this program. But there are some things to consider:
>
> - The internally contracted configurations are by themselves not
> necessarily orthogonal in the internal space, and thus they need to be
> orthogonalized. However, if I am not mistaken, the coefficients which
> are printed are actually the coefficients for the "raw",
> non-orthogonalized spaces (i.e., singlet/triplet coupled E^ab_ij
> operators applied on the reference function directly, not the
> orthogonalized counterparts). That means that if you just take those
> coefficients, expand the internally contracted configurations (ICCs) in
> determinants, and then compute the overlap (using the determinant
> overlaps! not just square-summing the coefficients), you should get the
> correct result. I think this is what you are doing.
>
> - One thing to take care of are the reference coefficients used. If I am
> not very mistaken (someone else: please correct me if I am wrong), then
> the CI coefficients of the reference function |Phi> which are used to
> define the ICCs are *NOT* updated during the MRCI iterations (as this
> would require re-orthogonalization and lots of other things to be done,
> and lead to inconsistencies about the external space between iterations,
> which would make things hard to converge). That is: The determinants
> used to define the ICCs are not equal to the determinants which you get
> from the pure internal part of the wave function; rather, they stay
> fixed on the original reference function. This may be easily missed.
>
> Not sure if this helps, but at this moment I have no further ideas on
> this. Btw: great work reproducing the norms in the internal and
> singly-external space. That must have taken a while :).
>
> Best wishes,
> Gerald
>
>
> On Thu, 2015-08-20 at 03:29 -0500, José Cortés wrote:
> > Dear Professor Knizia
> >
> > Thank you very much for your reply. I have a question concerning the
> > coefficients of
> > the doubly external configurations.
> >
> > My aim is obtain the CI vector in terms of determinants. I know this
> > can not be done
> > directly in the mrci module of Molpro, so I wrote a perl script.
> >
> > I have no problems in the case of the reference function and simple
> > external configurations.
> > I can go from CSF to determinants by the genealogical coupling scheme.
> > However, this
> > is more difficult in the doubly external configurations case.
> >
> > First, i transform each contracted configuration in terms of
> > uncontracted CSF´s
> >
> > \Psi_{ijp}^{ab}=\frac{1}{2}(\hat{E}_{ai,bj}+p
> > \hat{E}_{bi,aj})\Psi_{0}
> > =\sum_{P\nu}<\Psi_{ijp}^{ab}|\Psi_{P
> > \nu}^{ab}>\Psi_{P\nu}^{ab}
> >
> > where \Psi_{0} is the reference function (p=1 singlet coupling, p=-1
> > triplet coupling).
> > Then, I multiply each uncontracted CSF by the associated coefficient
> > with \Psi_{ijp}^{ab}
> > (listed in the output). Later I express each uncontracted CSF in
> > terms of determinants.
> > As I understand it, the labels i, j, a, b, p in \Psi_{ijp}^{ab} refer
> > to
> > "I J -> K L NP" in the output of Molpro.
> >
> > However, performing this procedure does not achieve a square norm that
> > matches the output
> >
> > CLASS SQ.NORM ECORR1 ECORR2
> > +++++++++++++++++++++++++++++++++++++++++++++++++++
> > Internals 0.00001717 0.00000000 -0.00000069
> > Singles 0.00065315 -0.00263386 -0.00263924
> > Pairs 0.00787253 -0.03783725 -0.03783119
> >
> > I do not understand that I'm doing wrong.
> > Perhaps, the coefficients associated with \Psi_{ijp}^{ab} are refer to
> > a non-orthonormal basis?
> > I would appreciate any help that you could provide me.
> >
> >
> > regards
> >
> > Jose Jara
> > Universidad Nacional
> > Autónoma de México
> >
> >
> > 2015-08-12 17:01 GMT-05:00 José Cortés <zolidus at gmail.com>:
> > Thank you very much for your reply.
> > I had never used the MRCI module in Molpro and I thought that
> > the CI expansion would be made on terms of uncontracted CSF
> > like in some other programs.
> >
> > I also found the answer in the literature recommended for the
> > MRCI module in MOLPRO (J. Chem. Phys. 89, 5803 (1988))
> >
> > Regards
> > José Jara
> >
> >
> > 2015-08-12 12:15 GMT-05:00 Gerald Knizia
> > <knizia at theochem.uni-stuttgart.de>:
> > Dear José,
> > you are right about \ and / indicating spin couplings
> > in the
> > geneological coupling scheme. Regarding the doubly
> > external
> > configurations: These are not individual CSFs, but
> > they indicate
> > internally contracted configurations, in which
> > spin-free excitation
> > operators are applied to the entire reference
> > function:
> >
> > |Phi^ij_ab> = E^{ab}_{ij} |Phi>
> >
> > with |Phi> being the full MCSCF-type reference
> > function (not individual
> > CSFs), i and j being occupied orbital labels, and a
> > and b being external
> > orbitals.
> >
> > The operator E^{ab}_{ij} denotes explicitly
> >
> > E^{ab}_{ij} = \sum_{\sigma \in A,B} \sum_{\tau \in
> > A,B} c^{a \sigma}
> > c^{b \tau} c_{j \tau} c_{i \sigma}
> >
> > where \sigma and \tau sum over spin labels.
> >
> > Best wishes,
> > Gerald
> >
> >
> > On Sun, 2015-08-09 at 21:51 -0500, José Cortés wrote:
> > > Dear molpro users
> > >
> > > I have a question about the output of the CSFs in a
> > mrci calculation.
> > > As an example suppose the He2 molecule with the
> > 6-311G basis,
> > > where an active space of 4e,4o was chosen in the
> > part of the casscf
> > > calculation.
> > > Such election leaves only two external orbitals. As
> > I understand the
> > > symbols
> > > "/" and "\" concerning to relative spin couplings,
> > which are related
> > > to the t vectors
> > > in the genealogical coupling scheme.
> > >
> > > For example for the following configurations (which
> > are in the output
> > > of the MRCI calculation)
> > >
> > > Reference coefficients greater than
> > 0.0000000
> > > ================================
> > > 2200 0.9959538
> > > /\/\ 0.0600897
> > >
> > > Coefficients of singly external
> > configurations greater than
> > > 0.0000000
> > >
> > ================================================
> > > /\\0 5.1 -0.0062116
> > > 20\0 6.1 -0.0055316
> > >
> > > the t vectors specified in the complete basis set
> > would be like:
> > > 220000
> > > /\/\00
> > > /\\0/0
> > > 20\00/
> > >
> > > However, my question arises in the doubly external
> > configurations
> > >
> > > Coefficients of doubly external
> > configurations greater than
> > > 0.0000
> > > ===========================================
> > > PAIR I J -> K L NP SYM
> > REF
> > > COEFFICIENTS
> > > 4 2.1 2.1 5.1 5.1 1 1
> > 1 -0.02645858
> > > 1 1.1 1.1 5.1 5.1 1 1
> > 1 -0.02035037
> > > 4 2.1 2.1 6.1 6.1 1 1
> > 1 -0.01367284
> > > 9 3.1 3.1 5.1 5.1 1 1
> > 1 -0.01342122
> > >
> > > In this case, how the t-vectors should be specified
> > for each one of
> > > the configurations in the complete basis set?
> > > How do I get in this case each one of the
> > configurations?
> > > I would appreciate any information you could give me
> > about it.
> > >
> > > Regards
> > > José Jara
> > > Universidad Nacional
> > > Autónoma de México
> > >
> >
> > > _______________________________________________
> > > Molpro-user mailing list
> > > Molpro-user at molpro.net
> > > http://www.molpro.net/mailman/listinfo/molpro-user
> >
> >
> >
> >
> >
> >
>
>
>
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