The Pulay stress or Pulay forces (named for Peter Pulay) is an error that occurs in the stress tensor (or Jacobian matrix) obtained from self-consistent field calculations ( Hartree–Fock or

density functional theory Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...

) due to the incompleteness of the basis set.Vasp Guide
Volume vs. energy, volume relaxations, Pulay Stress
/ref> A plane-wave density functional calculation on a crystal with specified lattice vectors will typically include in the basis set all plane waves with energies below the specified energy cutoff. This corresponds to all points on the reciprocal lattice that lie within a sphere whose radius is related to the energy cutoff. Consider what happens when the lattice vectors are varied, resulting in a change in the

reciprocal lattice Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...

vectors. The points on the reciprocal lattice which represent the basis set will no longer correspond to a sphere, but an ellipsoid. This change in the basis set will result in errors in the calculated

ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...

energy change. The Pulay stress is often nearly isotropic, and tends to result in an underestimate of the equilibrium volume. Pulay stress can be reduced by increasing the energy cutoff. Another way to mitigate the effect of Pulay stress on the equilibrium cell shape is to calculate the energy at different lattice vectors with a fixed energy cutoff. Similarly, the error occurs in any calculation where the basis set explicitly depends on the position of atomic nuclei (which are to change during the geometry optimization). In this case, the

Hellmann–Feynman theorem In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem ...

– which is used to avoid derivation of many-parameter wave function (expanded in a basis set) – is only valid for the complete basis set. Otherwise, the terms in theorem's expression containing derivatives of the wavefunction persist, giving rise to additional forces – the Pulay forces: :

\begin
\frac &= \frac\langle\psi, \hat, \psi\rangle \\
&=\bigg\langle\frac\bigg, \hat\bigg, \psi\bigg\rangle + \bigg\langle\psi\bigg, \hat\bigg, \frac\bigg\rangle + \bigg\langle\psi\bigg, \frac\bigg, \psi\bigg\rangle \\
&=\underbrace_ + \bigg\langle\psi\bigg, \frac\bigg, \psi\bigg\rangle.
\end

The presence of Pulay forces makes the optimized geometry parameters converge slower with increasing basis set. The way to eliminate the erroneous forces is to use nuclear-position-independent basis functions, to explicitly calculate and then subtract them from the conventionally obtained forces, or to self-consistently optimize the center of localization of the orbitals.

References

{{reflist Density functional theory