Volume vs. energy, volume relaxations, Pulay Stress

*If you are doing energy-volume calculations or
cell shape and volume relaxations you must understand
the Pulay stress, and related problems.*

The Pulay stress arises from the fact that the plane wave basis set is not complete with respect to changes of the volume. Thus, unless absolute convergence with respect to the basis set has been achieved - the diagonal components of the stress tensor are incorrect. This error is often called ``Pulay stress''. The error is almost isotropic (i.e. the same for each diagonal component), and for a finite basis set it tends to decrease volume compared to fully converged calculations (or calculations with a constant energy cutoff).

The Pulay stress and related problems affect the behavior of
VASP and any plane wave code in several ways:
First it evidently affects the stress tensor calculated by VASP, i.e.
the diagonal components of the stress tensor are incorrect,
unless the energy cutoff is very large (ENMAX=1.3 *default is
usually a safe setting to obtain a reliable stress tensor).
In addition it should be noted that all volume/cell shape relaxation algorithms
implemented in VASP work with a constant basis set. In that
way all energy changes are strictly consistent with the calculated
stress tensor, and this in turn results in an underestimation
of the equilibrium volume unless a large plane wave cutoff is used.
Keeping the basis set constant during relaxations has also some
strange effect on the basis set. Initially all G-vectors within a sphere
are included in the basis. If the cell shape relaxation
starts the direct and reciprocal lattice vectors change.
This means that although the *number* of reciprocal G-vectors in the basis
is kept fixed, the length of the G-vectors changes, changing
indirectly the energy cutoff. Or to be more precise,
the shape of cutoff region becomes an elipsoide.
Restarting VASP after a volume relaxation
causes VASP to adopt a new ``spherical'' cutoff sphere and
thus the energy changes discontinuously (see section 6.14).

One thing which is important to understand, is that problems due
to the Pulay stress can often be neglected if only
volume conserving relaxations are performed. This is because
the Pulay stress is usually almost uniform and it therefore changes
the diagonal elements of the stress tensor only by a certain constant
amount (see below). In addition many calculations have shown
that Pulay stress related problems can
also be reduced by performing calculations at different volumes using
the same energy cutoff for each calculation (this is
what VASP does by default, see section 6.14),
and fitting the final *energies* to an equation of state.
This of course implies that
the number of basis vectors is different at each volume. But
calculations with many plane wave codes have shown that such calculations
give very reliable results for the lattice constant and
the bulk modulus and other elastic properties
even at relatively small energy cutoffs.
Constant energy cut-off calculations are less prone
to errors cause by the basis set incompleteness
than constant basis set calculations. But it should be kept in
mind that volume changes and cell shape changes must
be rather large in order to obtain reliable results from this
method, because in the limit of very small distortions the
energy changes obtained with this method are equivalent with
that obtained from the stress tensor and are therefore affected
by the Pulay stress. Only volume changes of the order of 5-10 %
guarantee that
the errors introduced by the basis set incompleteness are averaged out.

- How to calculate the Pulay stress
- Accurate bulk relaxations with internal parameters (one)
- Accurate bulk relaxations with internal parameters (two)
- FAQ: Why is my energy vs. volume plot jagged