We present in full detail a newly developed formalism enabling density functional perturbation theory (DFPT) calculations from a $\mathrm{DFT}+U$ ground state. The implementation includes ultrasoft pseudopotentials and is valid for both insulating and metallic systems. It aims at fully exploiting the versatility of DFPT combined with the low-cost $\mathrm{DFT}+U$ functional. This allows us to avoid computationally intensive frozen-phonon calculations when $\mathrm{DFT}+U$ is used to eliminate the residual electronic self-interaction from approximate functionals and to capture the localization of valence electrons, e.g., on $d$ or $f$ states. In this way, the effects of electronic localization (possibly due to correlations) are consistently taken into account in the calculation of specific phonon modes, Born effective charges, dielectric tensors, and in quantities requiring well converged sums over many phonon frequencies, as phonon density of states and free energies. The new computational tool is applied to two representative systems, namely CoO, a prototypical transition metal monoxide and ${\mathrm{LiCoO}}_2$, a material employed for the cathode of Li-ion batteries. The results show the effectiveness of our formalism to capture in a quantitatively reliable way the vibrational properties of systems with localized valence electrons.

Electron-phonon ($e\text{−}\mathrm{ph}$) interactions are pervasive in condensed matter, governing phenomena such as transport, superconductivity, charge-density waves, polarons, and metal-insulator transitions. First-principles approaches enable accurate calculations of $e\text{−}\mathrm{ph}$ interactions in a wide range of solids. However, they remain an open challenge in correlated electron systems (CES), where density functional theory often fails to describe the ground state. Therefore reliable $e\text{−}\mathrm{ph}$ calculations remain out of reach for many transition metal oxides, high-temperature superconductors, Mott insulators, planetary materials, and multiferroics. Here we show first-principles calculations of $e\text{−}\mathrm{ph}$ interactions in CES, using the framework of Hubbard-corrected density functional theory ($\mathrm{DFT}+U$) and its linear response extension ($\mathrm{DFPT}+U$), which can describe the electronic structure and lattice dynamics of many CES. We showcase the accuracy of this approach for a prototypical Mott system, CoO, carrying out a detailed investigation of its $e\text{−}\mathrm{ph}$ interactions and electron spectral functions. While standard DFPT gives unphysically divergent and short-ranged $e\text{−}\mathrm{ph}$ interactions, $\mathrm{DFPT}+U$ is shown to remove the divergences and properly account for the long-range Fröhlich interaction, allowing us to model polaron effects in a Mott insulator. Our work establishes a broadly applicable and affordable approach for quantitative studies of $e\text{−}\mathrm{ph}$ interactions in CES, a novel theoretical tool to interpret experiments in this broad class of materials.

The design of novel cathode materials for $\mathrm{Li}$-ion batteries would greatly benefit from accurate first-principles predictions of structural, electronic, and magnetic properties as well as intercalation voltages in compounds containing transition-metal elements. For such systems, density-functional theory (DFT) with standard (semi)local exchange-correlation functionals is of limited use as it often fails due to strong self-interaction errors that are especially relevant in the partially filled $d$ shells. Here, we perform a detailed comparative study of the phospho-olivine cathode materials ${\mathrm{Li}}_x{\mathrm{MnPO}}_4$, ${\mathrm{Li}}_x{\mathrm{FePO}}_4$ and the mixed transition metal ${\mathrm{Li}}_x{\mathrm{Mn}}_{1/2}{\mathrm{Fe}}_{1/2}{\mathrm{PO}}_4$ ($x=0,1/4,1/2,3/4,1$) using four electronic-structure methods: DFT, DFT+$U$, DFT+$U$+$V$, and HSE06. We show that DFT+$U$+$V$, with onsite $U$ and intersite $V$ Hubbard parameters determined from first principles and self-consistently with respect to the structural parameters by means of density-functional perturbation theory (linear response), provides the most accurate description of the electronic structure of these challenging compounds. In particular, we demonstrate that DFT+$U$+$V$ displays very clearly “digital” changes in oxidation states of the transition-metal ions in all compounds, including the mixed-valence phases occurring at intermediate $\mathrm{Li}$ concentrations, leading to voltages in remarkable agreement with experiments. We show that the inclusion of intersite Hubbard interactions is essential for the accurate prediction of thermodynamic quantities, balancing the drive for localization induced by the onsite $U$ with intersite $V$ orbital hybridizations. At variance with other methods, DFT+$U$+$V$ describes accurately such localization-hybridization interplay, and thus opens the door for the study of more complex cathode materials as well as for a reliable exploration of the chemical space of compounds for $\mathrm{Li}$-ion batteries.

The production of hydrogen fuels, via water splitting, is of practical relevance for meeting global energy needs and mitigating the environmental consequences of fossil-fuel-based transportation. Water photoelectrolysis has been proposed as a viable approach for generating hydrogen, provided that stable and inexpensive photocatalysts with conversion efficiencies over 10% can be discovered, synthesized at scale, and successfully deployed (Pinaud et al., Energy Environ. Sci., 2013, 6, 1983). While a number of first-principles studies have focused on the data-driven discovery of photocatalysts, in the absence of systematic experimental validation, the success rate of these predictions may be limited. We address this problem by developing a screening procedure with co-validation between experiment and theory to expedite the synthesis, characterization, and testing of the computationally predicted, most desirable materials. Starting with 70 150 compounds in the Materials Project database, the proposed protocol yielded 71 candidate photocatalysts, 11 of which were synthesized as single-phase materials. Experiments confirmed hydrogen generation and favorable band alignment for 6 of the 11 compounds, with the most promising ones belonging to the families of alkali and alkaline-earth indates and orthoplumbates. This study shows the accuracy of a nonempirical, Hubbard-corrected density-functional theory method to predict band gaps and band offsets at a fraction of the computational cost of hybrid functionals, and outlines an effective strategy to identify photocatalysts for solar hydrogen generation.

Oxides containing metals or metalloids from the $p$ block of the periodic table (e.g., $\mathrm{In}$, $\mathrm{Sn}$, $\mathrm{Sb}$, $\mathrm{Pb}$, and $\mathrm{Bi}$) are of technological interest as transparent conductors and light absorbers for solar-energy conversion due to the tunability of their electronic conductivity and optical absorption. Comparatively, these oxides have found limited applications in hydrogen photoelectrolysis, primarily due to their high electronegativity, which impedes electron transfer for reducing protons into hydrogen. We have shown recently that inserting $s$-block cations into $p$-block metal oxides is effective at lowering electronegativities while affording further control of band gaps. Here, we explain the origins of this dual tunability by demonstrating the mediator role of $s$-block cations in modulating orbital hybridization while not contributing to frontier electronic states. From this result, we carry out a comprehensive computational study of 109 ternary oxides of $s$- and $p$-block metal elements as candidate photocatalysts for solar hydrogen generation. We down-select the most desirable materials using band gaps and band edges obtained from Hubbard-corrected density-functional theory, with Hubbard parameters computed entirely from first principles, evaluate the stability of these oxides in aqueous conditions, and characterize experimentally four of the remaining materials, synthesized with high phase uniformity, to validate and further develop the computational models. We thus propose nine oxide semiconductors, including $\mathrm{Cs}\mathrm{In}$${}_3$$\mathrm{O}$${}_5$, $\mathrm{Sr}$${}_2$$\mathrm{In}$${}_2$$\mathrm{O}$${}_5$, and $\mathrm{KSbO}$${}_2$, which, to the extent of our literature review, have not been previously considered as water-splitting photocatalysts.