a question of terminology

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Peterson, Kirk

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Mar 1, 2021, 12:38:35 PMMar 1
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Dear Dirac experts,

 

My apologies in advance for the undoubtably naive question, but while recognizing differences in implementation for atomic cases, e.g., finite basis sets vs. numerical radial/angular solutions, is there an actual difference between 4-component MCSCF (as in Dirac) and what is referred to as simply MCDHF (as in Grasp2k) in the literature ?  Likewise, is a 4c-KRCI in Dirac  essentially equivalent to a MCDHF + RCI calculation?  I want to be careful not to mix terms, but on the surface these appear to be equivalent.

 

thanks in advance,

 

-Kirk

Trond SAUE

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Mar 1, 2021, 12:59:02 PMMar 1
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Hi Kirk,

I believe Ken to be the best placed to answer this question, but here is a start: A major difference between basis set and numerical code is the ease at which one can obtain virtual orbitals. At the SCF level codes like GRASP solve differential equations for each OCCUPIED orbital, whereas in finite basis we solve for all orbitals in a single shot and get virtual orbitals in the same go. From that point of view MCSCF is nice for numerical codes, since it allows to optimize also some orbitals with low occupation, or, i other words, by making your MCSCF big enough you get enough virtuals. I leave the rest to Ken...

All the best,

   Trond

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Peterson, Kirk

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Mar 1, 2021, 1:20:59 PMMar 1
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Hi Trond,

 

thanks for the quick response, this is helpful.  From your text below, it seems to me that recovering dynamical correlation would then be more difficult for the numerical codes.

 

best wishes,

 

-Kirk

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Trond SAUE

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Mar 1, 2021, 1:25:35 PMMar 1
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Yes, they have to define big CI-spaces for MCSCF and then have to cope with problematic convergence for orbitals of feeble occupation.

On 3/1/21 7:20 PM, Peterson, Kirk wrote:
it seems to me that recovering dynamical correlation would then be more difficult for the numerical codes.
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Kenneth Dyall

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Mar 1, 2021, 4:34:42 PMMar 1
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Hi Kirk,

The numerical solution of differential equations for MCDHF in GRASP is not equivalent to MCSCF in Dirac (or other basis set codes). There are two issues. 

One is that the numerical integration constructs correlating spinors that are essentially a sum over occupied orbitals and an integration over both continua, positive and negative. This renders the procedure subject to variational collapse, as there is no projection onto a set of positive- and negative-energy states that defines which spinors should be occupied and which should not (the no-pair approximation). I remember that Paul Indelicato found this problem when doing MCDHF (with the Desclaux atomic code, very similar to GRASP), and had to introduce explicit projection to get convergence. Basically, to separate out the positive- and negative-energy states, you have to define a one-particle Fock-type operator, for which the spinors are eigenfunctions. Then it's clear which is which. In numerical MCDHF the eigenvalues of the correlating spinors can become large and negative, and hence have significant negative-energy components. These large contributions mean (in a CI sense) you are putting electrons into the negative-energy states, but in the wrong way. It has to be done according to QED, with renormalization and all that stuff. 

The second issue that the configurations can be selected: the GRASP approach is usually not a complete active space (CAS). It can be, if you define it that way. The same is true for a MCDHF+RCI calculation vs MCSCF/CI in Dirac.

And as Trond points out, the big problem with the numerical approaches is the difficulty of generating virtuals, especially making sure that they are eigenfunctions of a one-particle Fock operator and thus aren't contaminated by negative-energy states. The MCDHF method is intended to recover dynamic as well as static correlation, but it's doing it in the wrong way as it's not consistent with QED.

All the best,
Ken.


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Peterson, Kirk

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Mar 1, 2021, 4:52:51 PMMar 1
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Hi Ken,

 

thanks very much for the detailed explanation.  The papers that use MCDHF and MCDHF+RCI never seem to talk about how those methods compare to finite basis set relativistic methods except maybe to cast doubt on their accuracy.  Do you have a feeling for why the atomic physics community seems to be so entrenched in using MCDHF-based approaches?

 

I need to respond to some referee comments that imply that codes like Molpro or Dirac (note the emphasis on codes rather than methods) aren't often used for atomic calculations and their accuracy somehow is suspect. I thought that was very odd and wanted to make sure I reply with some accurate statements.

 

best wishes,

 

-Kirk

 

From: <dirac...@googlegroups.com> on behalf of Kenneth Dyall <diracso...@gmail.com>
Reply-To: "dirac...@googlegroups.com" <dirac...@googlegroups.com>
Date: Monday, March 1, 2021 at 1:34 PM
To: Dirac Users <dirac...@googlegroups.com>
Subject: Re: [dirac-users] a question of terminology

 

Hi Kirk,

Kenneth Dyall

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Mar 1, 2021, 6:44:33 PMMar 1
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Hi Kirk,

Yeah, that's a difficult issue, and there's a variety of methods that people use in atomic physics; MCDHF/RCI isn't the only one. Some groups use finite-element methods, which generates a complete spectrum much like basis set methods. Others do use coupled-cluster methods for correlation - in fact, this is where the Fock-space coupled cluster has been used a lot, first by Uzi Kaldor, and now by Ephraim Eliav and others, and particularly on heavy element properties. Finite-element methods lend themselves to the calculations needed for accurate calculation of the Lamb shift, for example.

As far as accuracy is concerned, in some areas of atomic physics the relativistic methods are extended far beyond the Dirac-Coulomb method, with use of the full finite-frequency Breit interaction and QED effects to high orders. But these are usually only done on few-electron systems, and not on many-electron systems, where the calculations (QED in particular) are extremely difficult. So it might be this kind of issue that you're running into with the referee. 

I was going to comment before that MDCHF is a bit like coupled-cluster, as the iteration process for the radial functions is a bit like the iteration process for obtaining the amplitudes. If you thought of expanding the radial functions from MCDHF in some suitable one-particle basis, you would get an expansion that probably represents something like coupled cluster. I don't know how widely atomic physicists recognize the problems with MCDHF and its compatibility with QED, so it might be best to avoid that conversation for now!

So maybe the bottom line is to agree that Dirac and Molpro are somewhat limited in their treatment of higher-order relativistic effects for atoms, but have considerable power when it comes to correlation (and cite Eliav's work, perhaps). 

Ken.

Peterson, Kirk

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Mar 1, 2021, 7:22:53 PMMar 1
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Hi Ken,

 

yes, I'm very aware of the work by Eliav since in some ways it is similar to what my group does, particularly when they use CCSD(T) rather than just FS-CCSD (although we tend to not do 4c everywhere).  The referee didn't seem to be too worried about high level relativistic effects, so I will definitely avoid bringing up the QED issues.  It is interesting to think about comparing the MCDHF approach to coupled cluster. Some recent previous work on the electron affinity of Th atom (MCDHF/RCI) ended in amazing agreement with experiment, but we've found that correlation of the 5d electrons (via coupled cluster) can have a fairly large effect but this is something they ignored.

 

best,  -Kirk

Kenneth Dyall

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Mar 1, 2021, 10:38:33 PMMar 1
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Hi Kirk,

Yes the amazing agreement in that case is really a Pauling point (appalling point :) ), where cancellation of errors gives the experimental value. Of course it's not worth anything because there's no variational theorem for the EA, and one should instead show convergence in both 1-particle and n-particle spaces. 

Ken.

Trond SAUE

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Mar 2, 2021, 3:41:50 AMMar 2
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HI Ken,

I am not sure that I follow you here; you are saying that correlating spinors are sum/integrals ?

All the best,

   Trond

On 3/1/21 10:34 PM, Kenneth Dyall wrote:

One is that the numerical integration constructs correlating spinors that are essentially a sum over occupied orbitals and an integration over both continua, positive and negative.
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Kenneth Dyall

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Mar 2, 2021, 10:25:58 AMMar 2
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Hi Trond,

Yes. They are not eigenfunctions of a one-particle operator, not for the reference state at least. This is true for spinors coming from the numerical methods. You can expand the correlating orbitals as a linear combination of eigenfunctions of a Fock operator. They are essentially like natural orbitals, optimized for maximum correlation content.

All the best,
Ken.

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Peterson, Kirk

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Mar 2, 2021, 11:47:08 AMMar 2
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Hi Ken,

 

I was thinking that as well, that's why we always try to do what you suggest below, although showing convergence in the n-particle space is the most difficult (particularly for SO).  Previously we have though gone to CCSDTQ for the EA of Th and for several An-containing small molecules with good effect (scalar rel only for those parts).

 

-Kirk

 

PS - I'll have to start using the term "appalling point"  :)

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