Calcite crystal RPBE force convergence

[Kit Joll]

Dear CP2K developers,

I am writing to ask for advice on a reproducible force-consistency issue that we see for bulk calcite, CaCO₃, using CP2K/Quickstep GPW.

For a periodic cell at zero external electric field, the sum of the forces over all atoms should vanish to the accuracy of the force evaluation, even when the geometry is not relaxed. For the same 30-atom calcite frame, Quantum ESPRESSO gives a net force sum close to numerical zero, whereas CP2K gives a residual of order 1e-4 Ha/Bohr. In the attached representative CP2K output, for example, the force sum is

Fx = 3.49e-5 Ha/Bohr
Fy = 1.13e-4 Ha/Bohr
Fz = -5.39e-5 Ha/Bohr

with a total net force of about 1.30e-4 Ha/Bohr.

This residual is small compared with typical individual atomic forces, but in our tests the same force-consistency issue contaminates finite-field response properties. Our intended workflow requires numerical derivatives of the forces with respect to external electric-field components, for example for atomic polar tensors and related higher-order response quantities. Other systems, including bulk water, transition-metal oxide/water interfaces, and cation solutions, behave well with the same CP2K finite-field workflow, so the problem appears to be specific to CaCO₃, or to this particular combination of pseudopotential, Gaussian basis, and GPW grid.

The attached figure shows cutoff convergence for the energy per atom and for the zero-field net force sum. The top row is CP2K, using fixed REL_CUTOFF = 120 Ry and NGRIDS = 4 while varying the MGRID CUTOFF. The bottom row is Quantum ESPRESSO for the same structure. I include the QE result mainly as a translational-invariance sanity check, not as a strict force reference, since the pseudopotentials are different. QE gives a near-zero net force sum across the tested cutoffs, while CP2K retains a much larger residual. The CP2K energy also shows noticeable oscillations with increasing MGRID cutoff, which makes me suspect a grid/multigrid or Gaussian-grid-coupling contribution, although I am not sure.

Could you advise which CP2K settings or tests would be most appropriate to diagnose this? In particular, I would be grateful for advice on whether this could be related to any of the following:

• the Ca/O/C pseudopotentials or basis sets, and their use with RPBE;

• GPW multigrid effects or Gaussian-grid coupling;

• Pulay/grid force terms;

• CUTOFF, REL_CUTOFF, NGRIDS, or less obvious QS thresholds;

• SCF/OT convergence settings.

The attached input uses TZV2P-MOLOPT-PBE-GTH basis sets, GTH-PBE pseudopotentials, RPBE, EPS_SCF = 1e-9, OUTER_SCF EPS_SCF = 1e-9, EPS_DEFAULT = 1e-18, OT/CG with 3PNT line search, CUTOFF = 2500 Ry, REL_CUTOFF = 120 Ry, and NGRIDS = 4.

I have attached a representative CP2K input/output pair for one frame. The same qualitative behaviour is observed across the other tested calcite geometries.

Thank you very much for your time. I would really appreciate any suggestions for further tests or settings to try.

Best wishes,
Kit

[Thomas Kühne]

Dear Kit,

Thank you for the detailed report and for the input/output files. I was able to
reproduce the force-sum residual for the provided calcite frame:

GPW baseline: Sum F = ( 3.48675295E-05, 1.12518883E-04, -5.39277596E-05 ) Ha/Bohr |Sum F| = 1.29554803E-04 Ha/Bohr

The behavior originate from the XC/grid treatment:

GPW + USE_FINER_GRID T, FINE_GRID_FACTOR 2.0: |Sum F| = 2.26659720E-05 Ha/Bohr GPW + finer/smoothed XC grid: |Sum F| = 1.10078684E-06 Ha/Bohr GAPW_XC: |Sum F| = 4.20880179E-05 Ha/Bohr GAPW_XC + GAPW_ACCURATE_XCINT: |Sum F| = 1.68750841E-07 Ha/Bohr GAPW + GAPW_ACCURATE_XCINT + USE_FINER_GRID T, FINE_GRID_FACTOR 2.0: |Sum F| = 3.85198029E-08 Ha/Bohr GAPW_XC + GAPW_ACCURATE_XCINT + USE_FINER_GRID T, FINE_GRID_FACTOR 2.0: |Sum F| = 1.04105369E-08 Ha/Bohr

This strongly suggests that the residual is dominated by an egg-box/grid-coupling
effect in the XC/grid integration rather than by pseudopotentials, basis sets, or
SCF noise.

For this system, the best tested setting was:

&QS EPS_DEFAULT 1.0E-18 METHOD GAPW_XC GAPW_ACCURATE_XCINT &END QS &XC DENSITY_CUTOFF 1.0E-10 GRADIENT_CUTOFF 1.0E-10 TAU_CUTOFF 1.0E-10 &XC_GRID FINE_GRID_FACTOR 2.0 USE_FINER_GRID T &END XC_GRID &XC_FUNCTIONAL NO_SHORTCUT &GGA_C_PBE &END GGA_C_PBE &GGA_X_RPBE &END GGA_X_RPBE &END XC_FUNCTIONAL &END XC

METHOD GAPW with the same GAPW_ACCURATE_XCINT and fine XC grid also performs
very well, but in this test GAPW_XC was slightly better and should be the less
intrusive change relative to the original GPW calculation.

One practical note: GAPW_ACCURATE_XCINT has been improved recently, so I would
recommend testing with a recent CP2K master or a release that includes those
changes. Also, for this calcite input current master hit a post-SCF output crash
with PRINT_LEVEL MEDIUM; using PRINT_LEVEL LOW while keeping explicit
&FORCES ON avoided that and still printed the force block.

Best wishes,
Thomas

[Kit Joll]

Dear Thomas,

Many thanks for the very quick and detailed reply! This is extremely helpful, and your diagnosis makes a lot of sense given the behaviour we were seeing.

I will try compiling the current CP2K master using the new make_cp2k.sh workflow, run the relevant regression tests, and then first reproduce your zero-field force-sum results on the same calcite frame. After that, I will check whether the same settings also resolve the sensitivity we observe in the finite-field response tensors.

Thank you also for pointing out the PRINT_LEVEL LOW workaround and the relevant GAPW_XC / GAPW_ACCURATE_XCINT settings. I will report back once I have tested these on our side.

Best wishes,
Kit

[Kit Joll]

Dear Thomas,

Thank you again for your help. I was able to reproduce your zero-field results to approximately the same accuracy with a master version cloned on Friday 22nd May. I presume the remaining small differences are due to build/library differences.

However, while the zero-field force sum is now well converged, I am not seeing the same behaviour once a finite periodic electric field is applied.

For example, using

METHOD GAPW_XC
GAPW_ACCURATE_XCINT
USE_FINER_GRID T
FINE_GRID_FACTOR 2.0

together with a periodic Berry-phase electric field of intensity 2.5e-4 a.u. along z, the force sum becomes

FORCES| Sum -6.11045615E-09 7.87254948E-09 2.04962359E-04
FORCES| Total atomic force 2.04962360E-04

So the transverse components remain well converged, but the component along the applied field direction returns to the order of 1e-4 Ha/Bohr. I see similar behaviour when applying the field along other directions. The issue also persists when increasing FINE_GRID_FACTOR from 2 to 8: the force-sum error remains orders of magnitude larger along the field axis (however, all components sum do decrease by ~1 order of magnitude relative to FINE_GRID_FACTOR 2). 

Do you know whether there are additional grid, Berry-phase, or Wannier-related settings that need to be converged for finite-field force calculations? Or is there another setting that should be used together with GAPW_XC and GAPW_ACCURATE_XCINT in the presence of a periodic electric field?

Please find attached a minimal input (start.inp) /output (output.out) pair that reproduces the issue.

Thank you again for your help and advice.

All the best,
Kit

[Thomas Kühne]

Dear Kit,

I agree with your interpretation. The important point is that the residual is now almost entirely parallel to the applied periodic field. In your z-field calculation,

FORCES| Sum -6.11045615E-09 7.87254948E-09 2.04962359E-04

the x and y components remain at the zero-field accuracy, whereas the z component is of the order of the applied field strength. This strongly suggests that 

we are no longer looking at the same XC/grid egg-box issue that was cured by GAPW_XC + GAPW_ACCURATE_XCINT. Rather, the remaining term seems 

to originate from the finite-field/Berry-phase force contribution itself.

For a neutral periodic system I would still expect the total force to vanish, also in a homogeneous Berry-phase electric field, up to numerical noise. The 

residual is also not simply a physical net-charge force. The system is neutral in terms of the GTH valence charges, and the residual is approximately 0.82 

times the field strength rather than an obvious integer-charge contribution.

I do not think there is a Wannier-localization setting that should fix this. The PERIODIC_EFIELD implementation uses the Berry-phase formulation directly; 

the printed Wannier/localization controls are not convergence parameters for this force term. Likewise, increasing the XC fine grid should only affect the 

ordinary XC/grid part. Since the transverse components are already small, and the residual tracks the field direction, I would not expect FINE_GRID_FACTOR 

alone to remove it completely.

Could you please do two quick checks, if convenient?

1. Repeat the same calculation with the field reversed, i.e. INTENSITY -2.5e-4 or POLARISATION 0 0 -1.

2. Repeat with half the field strength, e.g. 1.25e-4 a.u.

If the force-sum residual changes sign with the field and scales linearly with its magnitude, that would be very strong evidence that the issue is in the periodic 

electric-field force contribution rather than in the GAPW/XC integration.

At first sight, finite-field force code path looks like a missing or imperfect cancellation between the ionic and electronic Berry-phase force terms under a uniform 

translation. In other words, it may be a real CP2K bug or at least a limitation of the present PERIODIC_EFIELD force implementation, rather than an 

input-convergence issue.

As a practical workaround for testing internal forces, subtracting the average force from all atoms would remove the spurious center-of-mass acceleration, but 

I would not present that as a formally clean solution until we understand the origin of the term.

Best wishes,

Thomas

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[Kit Joll]

Dear Thomas,

Thank you again for the helpful suggestions. I have now repeated the finite-field calculation with both signs of the field and with the field magnitude halved.

All four calculations used the same input settings, cell, coordinates, GAPW_XC / GAPW_ACCURATE_XCINT setup, XC grid settings, basis sets and pseudopotentials. The only changes were the PERIODIC_EFIELD INTENSITY and POLARISATION. The effective field is along z, with E_z = INTENSITY * POLARISATION_z. I have attached the four corresponding input/output pairs. The attachment filenames indicate the actual field values (the PROJECT_NAME inside the input files is just a leftover label).

The resulting force sums are:

E_z / a.u.        Sum F_x          Sum F_y          Sum F_z          (Sum F_z) / E_z
-2.5e-4          -6.1085e-09      7.8702e-09     -2.04936242e-04   0.819744968
-1.25e-4         -6.1090e-09      7.8708e-09     -1.02473941e-04   0.819791528
+1.25e-4         -6.1100e-09      7.8720e-09      1.02475947e-04   0.819807576
+2.5e-4          -6.1104e-09      7.8725e-09      2.04962359e-04   0.819849436

So the absolute z-component of the force-sum residual approximately halves when the field is halved, while the ratio Sum F_z / E_z remains essentially constant at about 0.8198. The residual also changes sign when the field direction is reversed. The transverse components remain at the zero-field level, around 1e-8 Ha/Bohr.

The central finite-difference slopes are very similar for the two field magnitudes:

[ F_z(+1.25e-4) - F_z(-1.25e-4) ] / [2 * 1.25e-4] = 0.819799552

[ F_z(+2.5e-4) - F_z(-2.5e-4) ] / [2 * 2.5e-4] = 0.819797202

This seems to confirm your suggested diagnostic: the remaining force-sum residual is odd in the field, proportional to the applied field, and parallel to the field direction. It therefore looks distinct from the original XC/grid force issue.

Would you recommend any further diagnostic at this point, or would it be useful for me to prepare a more minimal test case for the periodic-electric-field force contribution?

All the best,
Kit

[Thomas Kühne]

Dear Kit,

thank you for the additional finite-field checks. Your observation was exactly the useful diagnostic: the 

residual is odd in the applied field, proportional to the field strength, and parallel to the field direction. 

This points to the periodic Berry-phase electric-field force contribution rather than to the GAPW_XC 

grid issue.

I have prepared a CP2K patch that enforces the acoustic sum rule on the isolated periodic electric-field 

force contribution for neutral periodic cells. With your +1.25e-4 a.u. calcite case, the force sum changes 

from

  (-6.11e-9, 7.87e-9, 1.02476e-4) Ha/Bohr

to

  (-6.11e-9, 7.87e-9, -3.02e-9) Ha/Bohr,

with the total force sum reduced to about 1.0e-8 Ha/Bohr.

The energy is unchanged within numerical noise. This confirms that the remaining finite-field force-sum 

residual was a center-of-mass component of the periodic electric-field force, distinct from the original 

XC/grid force issue.

A smaller reproducer would still be useful for adding a lightweight regression test, since the full calcite case 

is too expensive for the regular CP2K test suite.

Best,

Thomas

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[Kit Joll]

Dear Thomas, 

Thank you very much for fixing this issue so quickly. I'm recompiling now and then will attempt to reproduce your findings. I've also seen you've updated the APT tests, so I am presuming you don't need a minimal regtest example anymore? If you do, please let me know and I'll create something. 

Once again, thank you for your quick replies and help. 

All the best,

Kit