Machine learning is increasingly used for materials discovery, yet global structure property models often lose interpretability in chemically heterogeneous spaces, where sorption, redox, and multi electron transfer depend on local microenvironments. Here we introduce hidden trend mining, integrating descriptor learning, gradient boosting regression, and subgroup discovery to identify local structure activity relationships. Applied to bimetallic thiophene 2,5 dicarboxylate metal organic frameworks, MaMb TDC, the workflow used 103 DFT calculated compositions to screen more than 800 candidates for the oxygen evolution reaction. Subgroup discovery removed high overpotential, out of trend compositions that dominated model error, reducing the left out mean absolute error from 0.25 to 0.18 V and revealing a local trend for promising catalysts. The model identified NiFe TDC, which was experimentally validated across a NiM TDC series and delivered an overpotential of 186 mV at 10 mA cm-2 with 100 h stability. Guided by the local descriptor, electronic structure analysis revealed oxidation capacity as the origin of activity variation, enabling charge delocalization in quasi one-dimensional metal oxygen chains and balanced stabilization of OH* and O* intermediates. These results show that hidden trend mining separates error prone regimes and exposes interpretable local descriptors in complex catalyst spaces. which is of great importance in the development of functional catalysts and catalytic processes.

Conventional theoretical models of heterogeneous electrocatalysis typically describe reaction mechanisms exclusively along the ground-state Born–Oppenheimer potential-energy surface (PES). However, this approximation is often insufficient for transition-metal-based catalysts with multiple coexisting spin states, where electronic excitations are energetically accessible at room temperature, and a weak spin–orbit coupling (SOC) hinders the reaction kinetics on the ground-state PES. In this work, we present a comprehensive spin-resolved reaction network for the oxygen evolution reaction (OER) on a β-NiOOH (001) surface by combining hybrid density functional theory (DFT) embedded-cluster calculations with lattice kinetic Monte Carlo (kMC) simulations. We demonstrate that strictly restricting the reaction dynamics to spin-allowed processes under adiabatic spin conservation creates a severe kinetic bottleneck due to spin-trapping in long-lived intermediates. Crucially, incorporating nonadiabatic intersystem-crossing (ISC) transitions driven by weak SOC systematically dismantles these bottlenecks. At room temperature, thermal activation drives transitions into higher-energy electronic configurations. Once populated, these excited states activate a dense network of alternative, purely spin-allowed reaction pathways featuring substantially smaller thermodynamic barriers, which successfully drops the overpotential for the multilevel system to a calculated value of 0.58 V. Our findings establish the critical role of nonadiabatic spin-flip transitions and demonstrate that accounting for multi-spin pathways is essential for the realistic quantitative evaluation and rational design of highly efficient transition-metal-based electrocatalysts.

Technetium carbides (Tc-C) are of significant interest for applications in nuclear waste management and advanced reactor technologies due to their potential to immobilize radioactive technetium-99. However, a comprehensive understanding of the Tc-C system, including phase stability, structural properties, and thermodynamic behavior, remains elusive. In this work, we present a thorough theoretical investigation of technetium carbides using a hybrid approach resting upon density functional theory calculations and machine learning methods for predicting thermodynamic properties. We explore complete compositional/configurational space for the carbon concentrations up to 20.0 at.% in two Tc lattices. By analyzing energetics of carbon interstitial defects in hexagonal (320,149 structures) and cubic (11,937 structures) Tc, we identify the most stable atomic configurations. Furthermore, we investigate thermodynamic stability as a function of temperature using phonon calculations and accounting for configurational entropy allowing to reconcile our findings with available experimental results. The developed approach reveals atomic structures of the stable Tc-C phases, provides new insights into their stability under certain conditions, and advances the fundamental understanding of the corresponding phase transitions. Moreover, this study offers valuable guidance for computational discovery of technetium-based materials, as well as materials with more complex chemical compositions using firstprinciples-based approaches in a data-efficient manner.

This manuscript introduces MatSub, an open-access software package designed to facilitate the application of Subgroup Discovery (SGD) algorithms in machine learning and data-driven scientific discovery. A key contribution of MatSub lies in the development of novel quality functions tailored to materials informatics. While existing SGD algorithms with numerical targets often emphasize statistical exceptionality, materials research typically prioritizes the identification of subgroups with extreme or optimal property values. To address this gap, MatSub incorporates quality functions that (1) guide the discovery of subgroups maximizing or minimizing a target property, (2) enforce performance-based boundary constraints to filter out undesired materials, (3) promote orthogonal subgroup discovery to reveal multiple, physically distinct mechanisms affecting material behavior, and (4) enable multitask subgroup discovery to capture subgroups that simultaneously satisfy multiple property requirements. We demonstrate the utility of these quality functions in a case study on segregation energies of single-atom alloy catalysts (SAACs), where MatSub successfully identifies diverse and interpretable subgroups linked to distinct electronic and bonding characteristics. These results highlight the software's ability to support mechanism-aware analysis and accelerate hypothesis generation in materials science and beyond. Program Title: MatSub CPC Library link to program files: https://doi.org/10.17632/9633bmtj4x.1 Developer's repository link: https://github.com/XiaojuanHu/MatSub Licensing provisions: Apache-2.0 Programming language: Java Nature of problem: In materials informatics, uncovering key property-feature relationships is critical for rational materials design. A key challenge lies in identifying subsets of materials with optimal or extreme properties-such as high catalytic activity, or strong thermoelectric performance. These high-performance materials are often scarce and governed by diverse or even competing physical mechanisms, making global models hard to interpret and frequently ineffective in capturing physically meaningful trends. Subgroup discovery (SGD), which seeks locally exceptional and interpretable patterns, is a promising alternative. However, most traditional SGD algorithms focus on statistical outliers or general exceptionality, rather than on property optimization or mechanism diversity. Moreover, existing methods typically overlook the need to discover multiple orthogonal subgroups that reflect different physical mechanisms. There is thus a critical need for tools that can identify performance-driven, interpretable, and non-overlapping subgroups in complex materials datasets to support hypothesis generation and mechanistic understanding. Solution method: We developed MatSub, a Java-based software framework designed for subgroup discovery in materials science. MatSub supports continuous target variables and provides a set of novel quality functions specifically tailored for performance-oriented applications. These include: (1) optimization-targeted functions for maximizing or minimizing material properties, (2) boundary-constrained subgroup discovery to filter out undesired property ranges, (3) orthogonal subgroup discovery to reveal multiple, distinct physical mechanisms, and (4) multitask subgroup discovery for identifying subgroups satisfying multiple property objectives simultaneously. Subgroups are defined using combinations of basic selectors derived from numerical features (discretized via k-means), categorical attributes, and inequality rules. The underlying search algorithm employs a two-step Monte Carlo sampling strategy to efficiently explore the vast search space. The output is a set of humaninterpretable subgroup rules with accompanying metrics, enabling domain experts to discover physically interpretable design patterns and multiple complementary structure-property relationships that are often hidden in global models. MatSub is extensible and especially well-suited for mechanism-aware materials analysis. Additional comments including restrictions and unusual features: MatSub is built upon the realKD software, with significant enhancements specifically developed for materials science applications. While foundational modules such as data handling and basic mining functions are inherited from realKD under its original open-source license, MatSub introduces multiple novel features: new physically motivated quality functions focused on property optimization, cutoff-constrained discovery for strict performance filtering, orthogonal subgroup discovery to expose multiple distinct mechanisms, and multitask subgroup discovery for multi-objective analysis.

We present a systematic study of how functional classification of electronic bandgaps improves subsequent machine learning modelings in a group of more than ten thousand semiconductors and insulators. In this regard, we utilize a homemade Python package, MatFeaLib, for systematic generation of 518 descriptors combining 480 composition-based statistics and 38 structural features, and then apply a hybrid parsimonious feature-selection framework to separate about 20 most important features for our final classification, clustering, and regression tasks. Three application-oriented spectral regions, including infrared, visible, and ultraviolet, are selected for our supervised classification tasks along with seven machine learning classifiers, wherein the Extreme Gradient Boost algorithm achieved the best accuracy of 81%, which increased to 94%, after incorporation of lower-fidelity generalized gradient approximation gaps. A hierarchical approach further improves the accuracy of our functional classification to 85% and 95% in the single- and multi-fidelity schemes, respectively. The SHAP analysis revealed electron number, electronegativity, and average bond length as the most influential descriptors, providing physical insights about the classification process. Class-conditioned bandgap regression exhibits a significant improvement of about 40% relative to a global regression. The relevant error diagnostics indicate that different classes may require distinct modeling approaches to capture the underlying relationships and noise characteristics. Unsupervised clustering with an iterative feature-selection technique is used to identify possible hidden patterns in the dataset, which may improve the subsequent multiclass classification and regression procedures. Our findings may open a new avenue for more accurate materials modeling and, thus, more efficient functional materials discovery.

Transition-metal compounds represent a fascinating playground for exploring the intricate relationship between structural distortions, electronic properties, and magnetic behaviour, holding significant promise for technological advancements. Among these compounds, YBaCo4O7 (Y114) is attractive due to its manifestation of a ferrimagnetic component at low temperature intertwined with distortion effect due to the charge disproportionation on Co ions, exerting profound impact on its magnetic properties. In this perspective paper, we study the structural and magnetic intricacies of the Y114 crystal using a novel first-principles methodology. Traditionally, the investigation of such materials has relied heavily on computational modelling using density-functional theory (DFT) with the on-site Coulomb interaction correction U (DFT+U) based on the Hubbard model (sometimes including Hund’s exchange coupling parameter J, DFT+U+J) to unravel their complexities. Herein, we analysed the spurious effects of magnetic-moment delocalisation and spillover to non-magnetic ions in the lattice on electronic structure and magnetic properties of Y114. To overcome this problem we have applied constrained DFT (cDFT) based on the potential self-consistency approach, and comprehensively explore the Y114 crystal’s characteristics in its ferrimagnetic order. We find that cDFT yields magnetic moments of Co ions much closer to the experimental values than Hubbard model with the parameters U and J fitted to reproduce experimental lattice constants. cDFT allows for an accurate prediction of magnetic properties using oxidation states of magnetic ions as well-defined parameters. Through this perspective, we not only enhance our understanding of the magnetic interactions in Y114 crystal, but also pave the way for future investigations into magnetic materials.

Atomic, electronic, and magnetic structure of LaSrCo1/2Fe1/2O4 mixed-metal Ruddlesden-Popper oxide is investigated theoretically using self-consistent Agapito–Curtarolo–Buongiorno–Nardelli (ACBN0) ACBN0 DFT + U approach. We show that the electronic and magnetic properties strongly depend on the distribution of transition-metal and La/Sr ions in the lattice. Fe-Fe exchange is found to be antiferromagnetic, whereas Co-Co and Fe-Co exchange is ferromagnetic. We find that Co spin states depend on the distribution in both La/Sr and transition-metal sublattices, and that all three possible spin states of Co3+ can occur depending on the distribution. The most energetically favorable configuration is found to be ferromagnetic, whereas the majority of metastable configurations are antiferromagnetic.

Elucidating the kinetics of surface reconstruction is fundamentally important yet inherently challenging due to the complex collective atomic motions occurring across high-dimensional potential-energy landscapes. Here, we combine machine-learning-based molecular dynamics simulations enhanced by well-tempered metadynamics with in situ environmental transmission electron microscopy to directly uncover the critical role of surface titanium diffusion in driving the structural evolution of the TiO_{2}(110)-(1×2) reconstruction. Under oxygen-deficient conditions, the TiO_{2}(110)-(1×2) reconstruction remains thermodynamically stable, and surface Ti migration is largely suppressed. In contrast, under oxygen-rich conditions, the surface Ti atomic rows exhibit pronounced splitting and migration, facilitated by the incorporation of additional oxygen atoms that enhance Ti mobility. Our simulations demonstrate that these oxygen-promoted processes can induce structural transformations of the TiO_{2}(110)-(1×2) reconstruction, resulting in transitions from double-row to single-row or triple-row configurations. These predictions are further validated by experimental observations. This Letter establishes a microscopic mechanism for rutile surface reconstruction kinetics and provides valuable insights for the controlled manipulation of surface structures under varying chemical environments.

Molecular adsorption plays a key role in heterogeneous catalysis, with the d-band center theory being extensively utilized for qualitative analysis of adsorption behavior of molecules on transition-metal surfaces. However, adsorption energies predicted by d-band center theory can have large errors, especially when a wide range of metals is considered. The physical mechanism responsible for the deviation has been unclear, posing significant challenges of the precise design of catalysts. Here, we have integrated density-functional theory calculations with artificial intelligence approaches to develop a unifying model that quantitatively evaluates the adsorption energy of water on various transition-metal surfaces. Data mining revealed that the large deviations between the d-band center theory predictions and first-principles calculations primarily stem from surface relaxation effects. Additionally, we discovered that the synergistic effects of electron donation and back donation, which are mediated by the d and p orbitals in metals, also play a role in these deviations, although their impact is less significant compared to that of the surface relaxation. These findings enhance our understanding of molecular adsorption behavior and are poised to influence research fields such as surface science and heterogeneous catalysis.

We propose an interpretable AI approach integrating hybrid DFT, symbolic regression, and data mining to predict chalcopyrite (ABX2) bandgaps. Key factors, including atomic size, molar volume, and electron affinity, are identified, offering insights into bandgap-composition relationship and guiding high-performance materials design.

Describing the interaction between reactive species and surfaces is crucial for designing catalyst materials. Density-functional approximation is able to quantitatively model such interaction, but its accuracy strongly depends on the choice of exchange-correlation (XC) functional approximation. In this work, we assess the performance of XC functionals for describing the interaction of C2H2 and C2H4 with (111) surfaces of Cu, Pt, Pd, and Rh by particulary focusing on RPBE and mBEEF functionals. We study the geometry and the vibrational frequencies associated with the adsorbed molecules, as well as the adsorption energies and the reaction enthalpy of semi-hydrogenation of C2H2 in the gas phase. Crucially, experimental values for vibrational frequencies of molecules adsorbed on the metal surfaces are available for more system compared to physical quantities typically used to benchmark of XC functionals, such as adsorption energies. Thus, vibrational frequencies can be utilized as reference to assess the reliability of the exchange-correlation functionals. We find that the mean percentage errors (MPEs) of RPBE and mBEEF with respect to reported experimental values of vibrational frequencies are 0.64% and -3.88 %, respectively (36 data points). For adsorption enthalpy, RPBE and mBEEF provide MPEs of 27.61 and -59.81%, respectively, with respect to reported experimental values (7 data points). Therefore, the performance of RPBE is superior to that of mBEEF for the considered systems.

The development of readily accessible and interpretable descriptors is pivotal yet challenging in the rational design of metal–organic framework (MOF) catalysts. This study presents a straightforward and physically interpretable activity descriptor for the oxygen evolution reaction (OER), derived from a dataset of bimetallic Ni-based MOFs. Through an artificial-intelligence (AI) data-mining subgroup discovery (SGD) approach, a combination of the d -band center and number of missing electrons in e g states of Ni, as well as the first ionization energy and number of electrons in e g states of the substituents, is revealed as a gene of a superior OER catalyst. The found descriptor, obtained from the AI analysis of a dataset of MOFs containing 3–5d transition metals and 13 organic linkers, has been demonstrated to facilitate in-depth understanding of structure–activity relationship at the molecular orbital level. The descriptor is validated experimentally for 11 Ni-based MOFs. Combining SGD with physical insights and experimental verification, our work offers a highly efficient approach for screening MOF-based OER catalysts, simultaneously providing comprehensive understanding of the catalytic mechanism.

Urea oxidation reaction (UOR) is a crucial process for the efficiency of urea-based energy conversion technologies, including electrocatalytic water splitting and urea fuel cells. These technologies are considered alternatives to the traditional hydrogen production methods due to lower overpotentials and higher cost-effectiveness. However, sluggish UOR kinetics currently limit the practical applications of these technologies. In this study, we investigated the influence of electrolyte composition and temperature on the performance of Ruddlesden-Popper complex oxides in UOR. Our findings demonstrate that the formation of a carbonate layer at the electrode surface is a critical factor limiting UOR kinetics. This layer is mainly composed of carbonates of alkali metal from the solution and can be eliminated at elevated temperatures due to increased solubility. Specifically, we found that LaSrNi 0 . 5 Co 0 . 5 O 4±δ catalyst exhibited superior electrocatalytic activity (6 mA cm −2 at 1.45 V vs. RHE with loading 100 µg cm −2 ) operating at 60 °C in 5 M KOH. Moreover, changing the solution to cesium hydroxide, we were able to avoid the creation of a non-conductive layer due to higher solubility of the cesium carbonate, thus stabilizing the catalytic reaction.

Despite many years studies, there is still a significant discrepancy between the first-principles calculations and experimental measurements of platinum's optical properties in the existing literature. In this work, we explain the reason for this discrepancy by combining spectroscopic ellipsometry (SE) measurements and density functional theory (DFT) calculations. Pt films were fabricated by electron beam physical vapor deposition, and their properties were measured with SE. The dielectric function and consequently plasma frequency, relaxation constant, and interband spectra were then retrieved using the Drude-Lorentz model. DFT calculations of plasma frequency and interband contribution (Lorentz part) in random-phase approximation show a good match between the experimental and theoretical parameters, providing valuable insights into the electronic and optical properties of platinum. The effect of possible defects is studied with DFT as well. The data for the dielectric function of platinum is tabulated in the spectral range of 200-3500 nm and offers important implications for its potential applications in optoelectronic devices and plasmonic systems. The long-standing discrepancy between theory and experiment is explained by insufficient and inaccurate data in the infrared region of the spectrum, which is important for reliable separation of inter and intraband transition contributions to the spectrum and determination of the plasma frequency.

Describing the interaction between reactive species and surfaces is crucial for designing catalyst materials. Density-functional approximation is able to quantitatively model such interaction, but its accuracy strongly depends on the choice of exchange-correlation (XC) functional approximation. In this work, we assess the performance of mBEEF and RPBE functionals for describing the interaction of C2H2 and C2H4 with (111) surfaces of Cu, Pt, Pd, and Rh face-center-cubic transition metals. In particular, we study the geometry and the vibrational frequencies associated with the adsorbed molecules, as well as the adsorption energies. Crucially, experimental values for vibrational frequencies of molecules adsorbed on the metal surfaces are available for more system compared to physical quantities typically used to benchmark of XC functionals, such as adsorption energies. Thus, vibrational frequencies can be utilized as reference to assess the reliability of the exchange-correlation functionals. We find that the mean percentage errors (MPEs) of RPBE and mBEEF with respect to reported experimental values of vibrational frequencies are 0.64% and -3.88 %, respectively (36 data points). For adsorption enthalpy, RPBE and mBEEF provide MPEs of 27.61 and -59.81%, respectively, with respect to reported experimental values (7 data points). Therefore, the performance of RPBE is superior to that of mBEEF for the considered systems.

Ruddlesden-Popper (RP) transition-metal oxides belong to a versatile class of functional materials whose electronic properties can be tuned by varying chemical composition in a wide range. However, the relation between chemical composition and electronic properties of multi-metal RP oxides, in particular taking into account different distributions of metal ions in the lattice, has not been investigated systematically so far. In this work, we use density-functional theory (DFT) to explore the compositional dependence of electronic, magnetic, and structural properties of La $$_{2-x}$$ Sr $$_x$$ Co $$_{1/2}$$ Fe $$_{1/2}$$ O $$_4$$ oxide series for x = 0, 1, 2. To account for localized nature of transition-metal d-orbitals, self-consistent DFT+U approach Agapito-Curtarolo-Buongiorno-Nardelli (ACBN0) is employed. We show that widely used electronic-structure-based descriptors of oxygen-defect formation energies, oxygen transport, and catalytic activity strongly depend on the distribution of transition metal ions. Nevertheless, the distribution-averaged and ground-state descriptor values for La $$_{2-x}$$ Sr $$_x$$ Co $$_{1/2}$$ Fe $$_{1/2}$$ O $$_4$$ are found to depend monotonically on the composition x. In addition, we showed that in complex systems such as multi-metal Ruddlesden-Popper phases, Hubbard U values for the same species vary significantly between crystallographically non-equivalent sites.

Symbolic-inference methods have recently found a broad application in materials science. In particular, the Sure-Independence Screening and Sparsifying Operator (SISSO) performs symbolic regression and classification by adopting compressed sensing for the selection of an optimized subset of features and mathematical operators out of a given set of candidates. However, SISSO becomes computationally unpractical when the set of candidate features and operators exceeds the size of few tens. In the present work, we combine SISSO with a genetic algorithm (GA) for the global search of the optimal subset of features and operators. We demonstrate that GA-SISSO efficiently finds more accurate predictive models than the original SISSO, due to the possibility to access a larger input feature and operator space. GA-SISSO was applied for the search of the model for the prediction of carbon-dioxide adsorption energies on semiconductor oxides. The obtained with GA-SISSO model has much higher accuracy compared to models previously discussed in the literature (based solely on the O 2p-band center). The analysis of features importance shows that, besides the O 2p-band center, the contribution of the electrostatic potential above adsorption sites and the surface formation energies are also important.

Ruddlesden-Popper (RP) transition-metal oxides belong to a versatile class of functional materials whose electronic properties can be tuned by varying chemical composition in a wide range. However, the relation between chemical composition and electronic properties of multi-metal RP oxides, in particular taking into account different distributions of metal ions in the lattice, has not been investigated systematically so far. In this work, we use density-functional theory (DFT) to explore the compositional dependence of electronic, magnetic, and structural properties of La2-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2-x}$$\end{document}Srx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_x$$\end{document}Co1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{1/2}$$\end{document}Fe1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{1/2}$$\end{document}O4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_4$$\end{document} oxide series for x = 0, 1, 2. To account for localized nature of transition-metal d-orbitals, self-consistent DFT+U approach Agapito-Curtarolo-Buongiorno-Nardelli (ACBN0) is employed. We show that widely used electronic-structure-based descriptors of oxygen-defect formation energies, oxygen transport, and catalytic activity strongly depend on the distribution of transition metal ions. Nevertheless, the distribution-averaged and ground-state descriptor values for La2-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2-x}$$\end{document}Srx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_x$$\end{document}Co1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{1/2}$$\end{document}Fe1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{1/2}$$\end{document}O4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_4$$\end{document} are found to depend monotonically on the composition x. In addition, we showed that in complex systems such as multi-metal Ruddlesden-Popper phases, Hubbard U values for the same species vary significantly between crystallographically non-equivalent sites.