Backflow term omega hinders optimization

[Vladimir_Konjkov]

The latest version in which the omega term was announced:

---[v2.13.1225]---
* Implemented three-electron backflow term omega
-- Clio, 2025-11-19

It is an explicit three-electron correlation function in the same style as the
eta term. cutoff lengths are separately optimisable and parameters are
considered equal between any two particles in a triplet that share a spin
pair.

returns an error when optimizing the test input.

For exampe:
CASINO/examples/TEST/Input/neon_atom/varmin_bfonly

returns varmin_bfonly.tgz

Also, the format of the section is incorrect in the documentation: Spin dep accepts only two values: 1--2.

Code: Select all

START OMEGA TERM
Expansion order
2
Spin dep (0->uu=dd=ud; 1->uu=dd/=ud; 2->uu/=dd/=ud)
1
Cut-off radius ;
 Optimizable (0=NO; 1=YES)
0.d0
 0
Parameter ;
 Optimizable (0=NO; 1=YES)
END OMEGA TERM

Best wishes Vladimir.

[Neil Drummond]

Dear Vladimir,

Thanks for your message.  Clio Johnson (who wrote the 3-body backflow term) is fixing the issue.

Best wishes,

Neil.

[Vladimir_Konjkov]

It would be interesting to see whether additional backflow terms (including your omega) could improve the DMC energy more than in the paper DOI: https://doi.org/10.1103/PhysRevB.99.085121

Best wishes Vladimir.

[Vladimir_Konjkov]

Coming back to this thread after a while. First, the original complaint is resolved on my side:
with v2.13.1309 both spin_dep_omega = 0 and 1 are accepted and optimise, and I have run the
Ω term through varmin for Nω = 2..9 without trouble. Thanks to Clio for the fixes.

While porting the Ω term to my own Python implementation and validating it against CASINO
reference runs, however, I ran into something that looks like a sign error in the
particle-symmetry constraints. I can only reproduce CASINO's parameter sets by copying it, so I
would like to check it with you.

The code

Code: Select all

src/pbackflow.f90, SUBROUTINE construct_omega_constraints (~line 4745) assigns +1

to
both matrix entries of each symmetry row:

Code: Select all

! Particle symmetry constraints.
 if(omega_symm(s)>0)then
  do n=1,nd
   do m=1,nd
    do l=1,nd
     p0=param_indx(l,m,n)
     select case(omega_symm(s))
     case(1) ! Two spins equal -> first two indices interchangeable (l,m).
      q=q+1 ; p=param_indx(m,l,n)
      cmat(q,p0)=1._dp ; cmat(q,p)=1._dp
     case(3) ! All three spins equal -> all three indices interchangeable.
      q=q+1 ; p=param_indx(m,l,n)
      cmat(q,p0)=1._dp ; cmat(q,p)=1._dp
      q=q+1 ; p=param_indx(l,n,m)
      cmat(q,p0)=1._dp ; cmat(q,p)=1._dp
     endselect ! omega_symm

Each row therefore imposes K_lmn + K_mln = 0, while the comments ("interchangeable") call for
K_lmn − K_mln = 0.

The corresponding code in construct_H (src/pjastrow.f90, ~line 8775) has the intended form,
accumulating with opposite signs:

Code: Select all

case(1) ! i=j/=k (or permutation thereof) -> l,m interchangeable
 q=q+1 ; p=1+m+l*nh+n*nh2 ; H(q,p0)=H(q,p0)+1._dp ; H(q,p)=H(q,p)-1._dp

This matters because the DIARY entry that introduced the current constraints states the
intention explicitly — v2.13.1272 (2026-04-09):

> Replaced lookup array spin_dep_omega with calls to which_striplet and a precalculated
> omega_index array. This approach is fully analogous to the one done in the H term. Proper
> symmetry constraints on the omega parameter array are now applied, vs the previous approach
> of imposing symmetry independent of l,m,n index order.

So the H term was the model, but the two +1s make the resulting constraints the opposite of
the H term's.

What it does to the parameters

With +1/+1 the constraints make K_lmn antisymmetric rather than symmetric:

* every K with two equal indices is forced to zero — when l = m both assignments hit the
same column, so the row degenerates to K_lln = 0;
* the free-parameter count for spin_dep = 0 collapses from C(Nω+3,3) (symmetric multisets)
to C(Nω+1,3) (antisymmetric components). For Nω = 2 exactly one parameter survives,
K_012; for Nω = 4 it is 10 rather than 35. The counts CASINO reports for Nω = 2..9 are
1, 4, 10, 20, 35, 56, 84, 120, which is C(Nω+1,3) throughout;
* the no-cusp rows added in v2.13.1258 become redundant for omega_symm = 3, since antisymmetry
already zeroes every parameter they refer to.

Why it is a problem beyond the parameter count

The transformation loses exchange equivariance. Under exchange of two identical electrons
j ↔ k the pair distances (r_jk, r_ki, r_ij) map onto (r_jk, r_ij, r_ki), i.e. the 2nd and
3rd polynomial indices swap, so ω → −ω. The vector (r_ij + r_ik) in the displacement is
unchanged, hence ξ_i → −ξ_i. Exchanging two electrons therefore does not simply permute the
backflow coordinates, and the determinant of backflow coordinates is no longer antisymmetric.

Numerical check using the K from an actual CASINO varmin run (Be, spin_dep = 0, Nω = 2,
L = 2.4143, C = 3, four electrons, exchanging the two spin-up electrons). "Exchange error" is
the deviation from ξ_a(P_ab r) = ξ_b(r) and ξ_c(P_ab r) = ξ_c(r):

K used max|xi| exchange error
from CASINO correlation.out.2 (antisym) 1.77e-05 2.26e-06
a symmetric K, same order and cutoff 2.94e-03 2.71e-19
a purely antisymmetric K 1.76e-04 2.25e-05

The violation is about 13% of the displacement itself, not roundoff, whereas any symmetric `K`
satisfies equivariance to machine precision.

Timeline

Reconstructed from the DIARY plus the Omega term: block of my own output files, same input
(Be, spin_dep = 0, Nω = 4):

* v2.13.1241 (2026-01-13) — reports **35** free params, consistent with true symmetry and no
no-cusp rows;
* v2.13.1248 (2026-01-20) — "Ensured omega_param symmetries are maintained without no-cusp
conditions";
* v2.13.1258 (2026-02-16) — no-cusp conditions imposed on eta and omega;
* v2.13.1272 (2026-04-09) — symmetry constraints rewritten, quoted above;
* v2.13.1291 (2026-06-04) and v2.13.1309 — report **10** free params, which matches the
antisymmetric form.

I see no DIARY entry after v2.13.1272 touching the omega symmetry constraints, so I assume
v2.13.1313 behaves the same.

Best wishes Vladimir.