First-principles calculation of atomic force in the LSDA+U formalism

We derive a formula for the atomic force within the LSDA+U formalism by differentiating analytically the LSDA+U total-energy functional with respect to atomic positions. The rotationally invariant form of the LSDA+U functional and the fully localized limit for the double-counting term are considered. The electronic wave functions are expanded with either plane waves or pseudoatomic orbitals (PAOs). In the PAO-basis case, the Pulay correction is also considered and included. Our formula for the atomic force is numerically tested with antiferromagnetic bulk NiO and reproduces successfully the forces obtained from numerical derivative of the total-energy values with respect to atomic displacements. As an application, we study atomic vibrations in NiO and MnO, and obtain transverse-optic phonon frequencies which are consistent with previous theoretical results. Our force formula will make it very efficient to perform large-scale calculations of atomic and phononic structures of strongly correlated materials using the LSDA+U method.