Calculating MRCI size-consistency corrections

[Léo Chaussy]

Dear molpro users,

I am trying to calculate Davidson-Silver correction for a series of systems (E. R. Davidson and D. W. Silver, Chem. Phys. Lett., 1977, 52, 403–406.).

To begin I started by trying (and failing) to reproduce the Davidson, fixed reference cluster corrected energy given in MRCI output. I am doing an a two-state MRCI calculation on top of a two state CASSCF on a metal ion.

If I use naively the "Coefficient of the reference function (fixed)" printed in molpro output, i cannot find molpro Davidson corrected (fixed) energy by simply using the formula directly. It only works in the relaxed case.

If I try to calculate the  c_n coefficients for state n using equation (6) of the MRCI documentation on Cluster corrections for my relatively small expansion (3 CSF), I do not find the same Davidson correction either which probably means that I am taking the wrong C_0 (printed using print,ref=1)  and C_n (printed in output of MRCI as Reference coefficients).

I must be missing something, what is the correct way of printing the coefficient c_n of the (fixed) reference function in the MRCI wave function ?

Regards

Leo Chaussy

[Mariusz Radon]

Dear Leo:

I am not sure if I understood your problem correctly, but please keep in mind that the Davidson correction computed by Molpro is *not* the Davidson-Silver correction described in the 1977 paper which you mentioned:

- standard Davidson: (1-C_0^2)/(C_0^2) [eq. 1' in Davidon-Silver paper]

- Davidson-Silver: (1-C_0^2)/(2*C_0^2-1) [eq. 19 in Davidon-Silver paper]

Maybe this difference contributes to your confusion?

Best wishes,

Mariusz

--
Mariusz Radon, Ph.D., D.Sc.
Associate Professor
Faculty of Chemistry, Jagiellonian University

Address: Gronostajowa 2, 30-387 Krakow, Poland
Room C1-06, Phone: 48-12-686-24-89
E-mail: mra...@chemia.uj.edu.pl (marius...@uj.edu.pl)
Web: https://tungsten.ch.uj.edu.pl/~mradon
ORCID: https://orcid.org/0000-0002-1901-8521

[Léo Chaussy]

Thanks for your reply.

I might not have been very clear in my first message... My idea was to recalculate first the standard Davidson correction on a simple system because I wanted to be sure of which coefficient to select in the output calculation to calculate the Davidson-Silver fixed correction (the notation seems to differ in molpro's output and in the documentation https://www.molpro.net/manual/doku.php?id=the_mrci_program ).

I assumed that the  "Coefficient of reference function:   C(0)" in the CI output is what I need to apply the various correction formula. In the relaxed case, doing a 7 roots SA-CASSCF on Co+ (3F), I manage to reproduce the Davidson corrected energy (relaxed) for every root using the C(0) relaxed coefficient. But in the fixed case, applying the same formula leads to inconsistencies visible in the table joined in this message (except for the first root which is actually a CISD). 

I failed to understand the source of these inconsistencies so I wanted to share it here, before trying to move forward to bigger systems.

I join the input file used to produce these results as well.

Regards

Co+, 3F
State	E_MRCI	E_correlation	C(0) (fixed)	C(0) (relaxed)	E_Davidson_calculated_fixed	E_davidson_molpro_fixed	E_Davidson_calculated_relaxed	E_davidson_molpro_relaxed
1.1	-1392.219671	-0.66677308	0.96969465	0.96969465	-1392.261999	-1392.261999	-1392.261999	-1392.261999
1.2	-1392.219746	-0.66684768	0.92293518	0.96966151	-1392.335758	-1392.262127	-1392.262127	-1392.262127
2.2	-1392.219651	-0.66675303	0.92296189	0.9696896	-1392.335602	-1392.261985	-1392.261985	-1392.261985
1.3	-1392.219746	-0.66684768	0.90777609	0.96966151	-1392.362123	-1392.262127	-1392.262127	-1392.262127
2.3	-1392.219651	-0.66675303	0.90780236	0.96968961	-1392.361961	-1392.261985	-1392.261985	-1392.261985
1.4	-1392.219746	-0.66684768	0.88148857	0.96966149	-1392.411108	-1392.262127	-1392.262127	-1392.262127
2.4	-1392.219651	-0.66675303	0.88151411	0.96968961	-1392.410936	-1392.261985	-1392.261985	-1392.261985

Input

 ***,Co +  3F
  memory,300,M
 
    GTHRESH,energy=1.d-6;gradient=1.d-3;
    GPRINT,orbitals,civector;
    GEXPEC,dm,qm,rel,darwin,massv;
 
    symmetry,x,y   ! enforce C_2v point group
    geometry={ ! geometry block
    Co
    }
 
   basis=aug-cc-pwCVTZ-DK

   set, dkroll=1;

   {hf,old;
   wf, 26, 1, 2;
   }

   text,Cobalt lowest triplet F state


   {multi;
    frozen,3,1,1,0;
    closed,5,2,2,0;
    occ,7,3,3,1;

! State-averaged CASSCF over 7 states
    wf,26,1,2 ! A1 components
    state,1 ! one A1 states
    wf,26,2,2 ! B1 components
    state,2 ! two B1 states
    wf,26,3,2 ! B2 components
    state,2 ! two B2 states
    wf,26,4,2 ! A2 component
    state,2 ! two A2 state
    dm; ! density matrix
    expec,LZ ! get LZ mean values
    TRAN2,LXX,LYY,LZZ   ! get L_x**2, L_y**2, L_z**2, L**2 mean values -> check the proper angular momentum of the atomic states
    } ! end of CASSCF block

 

   text, A1 state
   {ci;
    core,3,1,1,0;
    closed,5,2,2,0;
    occ,7,3,3,1
    wf,26,1,2;
    state,1
   }
 
   text, B1 states
   {ci;
    core,3,1,1,0;
    closed,5,2,2,0;
    occ,7,3,3,1;
    wf,26,2,2;
    state,2
   }
 
   text, B2 states
   {ci;
    core,3,1,1,0
    closed,5,2,2,0
    occ,7,3,3,1
    wf,26,3,2;
    state,2
   }
 
   text, A2 states
   {ci;
    core,3,1,1,0;
    closed,5,2,2,0;
    occ,7,3,3,1;
    wf,26,4,2;
    state,2
   }

--- ! end of file

[Mariusz Radon]

Dear Leo:

I do not know the source code and have no idea what happens here, but the energies marked as "E_Davidson_calculated_fixed" look like weird compared to other values. This is because the "C(0) (fixed)" values are quite low for all roots except the first one. These "C(0) (fixed)" values are very different from the relaxed ones.

If you know the "E_davidson_molpro_fixed" calculated by Molpro, you can obtain the actual C0 (fixed) coefficients which were used in the Molpro's internal computation. I guess they will be very closed to the relaxed C0's and very different from those printed as "C(0) (fixed)".

So, it looks like there is a problem with the values printed as "C0(fixed)", but no problem in the actual computation of the Davidson correction. If this happens with the recent version of Molpro, I think you should consider reporting a possible bug.

Best wishes,

Mariusz

--
Mariusz Radon, Ph.D., D.Sc.
Associate Professor
Faculty of Chemistry, Jagiellonian University

Address: Gronostajowa 2, 30-387 Krakow, Poland
Room C1-06, Phone: 48-12-686-24-89
E-mail: mra...@chemia.uj.edu.pl (marius...@uj.edu.pl)
Web: https://tungsten.ch.uj.edu.pl/~mradon
ORCID: https://orcid.org/0000-0002-1901-8521

[Hans-Joachim Werner]

The corrections are described in detail in the manual, https://www.molpro.net/manual/doku.php?id=the_mrci_program#cluster_corrections_for_multi-state_mrci
The problem with the „fixed“ correction is that the reference states in the 2-state cases are degenerate. This means they can mix arbitrarily and therefore
c_0 is not uniquely defined.
Best regards
Joachim Werner

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[Léo Chaussy]

Thanks for the precision, I did not guess that from the manual but it all makes a lot more sense now.

Regards

Leo Chaussy

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