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I performed DFT calculations for a box of 64 water molecules under Periodic Boundary Condition using CP2K. I found that the forces on each atom do not sum up to zero, as shown in the attached figure. But the total force should be zero due to translational symmetry of the system. Any idea of what happened?

I used revPBE functional with TZV2P basis set.

enter image description here

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  • $\begingroup$ Could you include that image (or whatever data you want us to see) as a code block? Images can't be processed by screen readers and can't be copied by a user looking to analyze your data. $\endgroup$
    – Tyberius
    Jun 28, 2022 at 16:10
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    $\begingroup$ There are generally two main reasons for this (in any DFT code): firstly, your forces are only correct when the calculation has converged fully to the ground state, and so in practice there will be errors which can cause the sum to be nonzero; secondly, the sum to zero relies on a continuous translational symmetry which is not actually present in the calculation - there are finite real-space grids, local basis set centres and a whole host of objects which are not continuously translationally invariant. These latter effects are reduced with finer grids, Pulay force corrections etc. $\endgroup$ Aug 5, 2022 at 23:59
  • $\begingroup$ @PhilHasnip I guess that comment could be moved to the answer box? The OP hasn't visited for a while so perhaps we could get this out of the unanswered queue. I've added it here. $\endgroup$ Dec 31, 2022 at 3:35
  • $\begingroup$ @NikeDattani good idea $\endgroup$ Jan 1, 2023 at 0:39

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There are generally two main reasons why the total force can be nonzero (in any DFT code):

  1. Your forces are only correct when the calculation has converged fully to the ground state, and so in practice there will be errors. These errors can cause the force sum to be nonzero, and are particularly large with density-mixing (or potential-mixing) SCF methods. The errors can be reduced by tightening the convergence tolerances so that the computed solution is closer to the ground state;

  2. The physical reason forces should sum to zero is because of a continuous translational symmetry, which is not actually present in the calculation - there are finite real-space grids, local basis set centres and a whole host of objects which are not continuously translationally invariant. These errors are reduced with finer grids (eg higher plane-wave cut-off energies), Pulay force corrections etc.

For both of these reasons, the total force on the system can be a useful indicator of the quality of your calculation.

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