## Introduction

After almost five decades from its formulation,[1, 2] density functional theory (DFT) still represents the main computational tool to perform electronic structure calculations for systems of realistic complexity. The possibility to express all the ground state properties of a system as functionals of its electronic charge density and the existence of a variational principle for the total energy functional render DFT a practical computational tool of remarkable simplicity and efficiency. Unfortunately, the exact expression of the total energy functional is unknown and approximations are needed in order to use DFT in actual calculations. Most commonly used approximate energy functionals for DFT calculations are constructed as expansions around the homogeneous electron gas limit and fail quite dramatically in capturing the properties of systems whose ground state is characterized by a more pronounced localization of electrons. In fact, within these approximations the electron–electron interaction energy is written as the sum of the classical Coulomb coupling between electronic charge densities (Hartree term) and the so-called “exchange-correlation” (xc) term that is supposed to contain all the corrections needed to recover the many-body terms of electronic interactions, missing from the first. Due to the approximations in the latter contribution and the intrinsic difficulty in modeling its dependence on the electronic charge density, approximate functionals generally provide a quite poor representation of the many-body features of the N-electron ground state. For these reasons, correlated systems (whose physical properties are often controlled by many-body terms of the electronic interactions) still represent a formidable challenge for DFT and, despite the steady and notable progress in the definition of more accurate functionals and corrective approaches, no single scheme has been defined that is able to capture entirely the complexity of the quantum many-body problem, while maintaining a sufficiently low computational cost to perform predictive calculations on systems of realistic complexity.

Although the quantitative entity of the inaccuracy of DFT functionals depends on the details of their formulation, on the specific system being modeled, and on the physical properties under investigation, on a more general and qualitative level, the failure in describing the physics of correlated systems can be ascribed to the tendency of approximate xc functionals to over-delocalize valence electrons and to over-stabilize metalic ground states. Paradigmatic examples of problematic systems are Mott insulators[3] whose electronic localization on atomic-like states is missed by approximate DFT functionals which, instead, predict them to be metalic.

To qualitatively understand the excessive delocalization of electrons induced by approximate energy functionals, it is convenient to refer to the expression of the electron–electron interaction energy as the sum of Hartree and xc terms. The over-delocalization of electrons can be attributed to the defective (approximate) account of exchange and correlation interactions in the xc functional that fail to cancel out the electronic self-interaction contained in the classical Hartree term. In fact, the persistence of this (unphysical) self-interaction makes “fragments” of the same electron (i.e., portions of the charge density associated with it) repel each other, thus inducing an excessive delocalization of the wave functions. In light of these facts, and based on the observation that Hartree-Fock (HF) is self-interaction free many of the corrective functionals (e.g., hybrid), formulated to improve the accuracy of DFT, aim to eliminate the residual self-interaction of electrons through the explicit introduction of a (screened or approximate) Fock-exchange term. This correction often results in an insulating ground state associated with a gapped Kohn–Sham (KS) spectrum. However, two important aspects should be kept in mind. First, the KS single-particle energy spectrum is not bound to any physical quantity (so that, e.g., there is no guarantee that an insulator should have a gapped KS band structure). Second, the aforementioned difficulties arise from both exchange and correlation terms of the energy and the lack of cancellation of the electronic self-Coulomb interaction is only the single-electron manifestation of their approximate representation in current xc functionals. A better treatment of correlation effects requires a more precise description of the many-body terms of the electronic energy. Methods and corrective approaches able to handle these degrees of freedom have been formulated and developed in the last decades. DFT + Dynamical Mean Field Theory (DFT + DMFT)[4-10] and Reduced Density Matrix Functional Theory (RDMFT)[11-15] are two notable examples in this class of computational methods. Both these approaches improve quite significantly the description of correlated systems compared to most DFT functionals available. Unfortunately, while still avoiding the prohibitive cost of wave function-based tractations of the electronic problem (as, e.g., in quantum chemistry approaches), these methods are significantly more computationally intensive than DFT calculations performed with approximate energy functionals, and are both outside the realm of DFT (or even generalized KS theory), thus requiring a significant effort to be implemented in (or to be interfaced with) existing DFT codes.

In recent years, the study of complex systems and phenomena has often been based on computational methods complementing DFT with model Hamiltonians.[16] LDA+U, based on a corrective functional inspired to the Hubbard model, is one of the simplest approaches that were formulated to improve the description of the ground state of correlated systems.[17-21] Due to the simplicity of its expression, and to its low computational cost, only marginally larger than that of “standard” DFT calculations, LDA+U (if not specified otherwise, by this name we indicate a Hubbard, “+U” correction to approximate DFT functionals such as, e.g., LDA, Local Spin Density Approximation (LSDA), or GGA) has rapidly become very popular in the *ab initio* calculation community. Its use in high-throughput (HT) calculations[22-24] for materials screening and optimization is quite emblematic of both these advantages the method offers. An additional and quite distinctive advantage LDA+U offers certainly consists in the easy implementation of energy derivatives as, for example, atomic forces and stresses[25] (to be used in structural optimizations and molecular dynamics simulations[26, 27]), or second derivatives, as atomic force constants, (for the calculation of phonons[28]) or elastic constants.[29]

Although certainly important for its implementation, the simplicity of the LDA+U functional requires a clear understanding of the approximations it is based on and a precise assessment of the the conditions under which it can be expected to provide quantitatively predictive results. This analysis is the main objective of this review article together with the discussion of recent extensions to the corrective functional and of their application to selected case studies.

The reminder of this review article is organized as follows. In Theoretical Framework, Basic Formulations, and Approximations section, we will review the historical formulation of LDA+U and the most widely used implementations, discussing the theoretical background of the method in the framework of DFT. In sections Computing the Hubbard *U*, Choosing the Localized Basis Set, and The Double-Counting “Issue” and the LDA+U for Metals some open questions of the LDA+U method, namely the calculation of the Hubbard *U*, the choice of the localized basis set, and the formulation of the double counting term, will be discussed reviewing and comparing a selection of different solutions proposed in literature to date. In Extended Functionals section, we will present recent extensions to the LDA+U functional that were designed to complete the Hubbard corrective Hamiltonian with intersite and magnetic interactions. Energy Derivatives section will focus on the calculation of first and second energy derivatives (forces, stress, and dynamical matrices) of the LDA+U energy functional and will present, as an example, the calculation of the phonon spectrum of selected transition-metal oxides. Finally, in Summary and Outlook section, we will propose some conclusions and an outlook on the possible future of this method.