# Density functional theory: Total energies and forces

In DFT, forces are calculated using the Hellman-Feynman theorem, such that: $$\frac{\text{d} E_\lambda}{\text{d} \lambda} = \left \langle \psi_\lambda \left|\frac{\text{d} \hat{H_\lambda}}{\text{d}\lambda} \right |\psi_\lambda\right \rangle$$

The primary question I have is: Assuming planewave DFT using the PAW method (Vasp, GPAW), is it strictly necessary to converge total energies to obtain correct forces both in the large force (atoms closer to each other) and small force (atoms at nearly equilibrium spacing) regimes?

I've run a set of tests on systems that are relevant to me, where I've noticed the following:

For a planewave cutoffs of 600 eV - 800 eV using a particular PAW dataset (total energies converge at 1500 eV), the energies are clearly not converged, but the forces are converged to ~0.02 eV/Angs in the high force regime (~ 4 eV/Angs), and have the same direction (determined by normalizing both force vectors and projecting one onto another). In the low force regime (~0.1 eV/Angs and below), the difference in the two forces remains approximately proportional, as above.

The final geometry of my structure using these two cutoffs was also identical to a maximum of ~0.005 Angs of differences in atomic positions.

I realize that formally, the total energy would have to be converged for the Hellman-Feynman forces to be converged, but are there considerations of the implementation using planewaves, or using the PAW method that make forces converge before total energies? Such as the energy not converging, but derivative wrt parameter converging much faster?