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We present a derivation of the exact expression for Pulay forces indensity-functional theory calculations augmented with extended Hubbardfunctionals, and arising from the use of orthogonalized atomic orbitals asprojectors for the Hubbard manifold. The derivative of the inverse square rootof the orbital overlap matrix is obtained as a closed-form solution of theassociated Lyapunov (Sylvester) equation. The expression for the resultingcontribution to the forces is presented in the framework of ultrasoftpseudopotentials and the projector-augmented-wave method, and using a planewave basis set. We have benchmarked the present implementation with respect tofinite differences of total energies for the case of NiO, finding excellentagreement. Owing to the accuracy of Hubbard-corrected density-functional theorycalculations - provided the Hubbard parameters are computed for the manifoldunder consideration - the present work paves the way for systematic studies ofsolid-state and molecular transition-metal and rare-earth compounds.

Pulay forces in density-functional theory with extended Hubbard functionals: From nonorthogonalized to orthogonalized manifolds