Double-counting Correction Research Articles

Downfolding from ab initio to interacting model Hamiltonians: comprehensive analysis and benchmarking of the DFT+cRPA approach

Model Hamiltonians are regularly derived from first principles to describe correlated matter. However, the standard methods for this contain a number of largely unexplored approximations. For a strongly correlated impurity model system, here we carefully compare a standard downfolding technique with the best possible ground-truth estimates for charge-neutral excited-state energies and wave functions using state-of-the-art first-principles many-body wave function approaches. To this end, we use the vanadocene molecule and analyze all downfolding aspects, including the Hamiltonian form, target basis, double-counting correction, and Coulomb interaction screening models. We find that the choice of target-space basis functions emerges as a key factor for the quality of the downfolded results, while orbital-dependent double-counting corrections diminish the quality. Background screening of the Coulomb interaction matrix elements primarily affects crystal-field excitations. Our benchmark uncovers the relative importance of each downfolding step and offers insights into the potential accuracy of minimal downfolded model Hamiltonians.

Emergence of a potential charge-disproportionated insulating state in SrCrO3

We use a combination of density functional theory (DFT) and dynamical mean-field theory (DMFT) to investigate the potential emergence of a charge-disproportionated insulating phase in SrCrO3, whereby the Cr cations disproportionate according to 3Cr4+→2Cr3++Cr6+ and arrange in ordered planes perpendicular to the cubic [111] direction. We show that the charge disproportionation couples to a structural distortion where the oxygen octahedra around the nominal Cr6+ sites contract, while the octahedra surrounding the Cr3+ sites expand and distort. Our results indicate that the charge-disproportionated phase can be stabilized for realistic values of the Hubbard U and Hund's J parameters, but this requires a scaling of the DFT+DMFT double-counting correction by at least 35%. The disproportionated state can also be stabilized in DFT+U calculations for a specific magnetic configuration with antiparallel magnetic moments on adjacent Cr3+ sites and no magnetic moment on the Cr6+ sites. While this phase is higher in energy than other magnetic states that can arise in SrCrO3, the presence of an energy minimum suggests that the charge-disproportionated state represents a metastable state of SrCrO3. Published by the American Physical Society 2024

Quantum embedding methods for correlated excited states of point defects: Case studies and challenges

A quantitative description of the excited electronic states of point defects and impurities is crucial for understanding materials properties, and possible applications of defects in quantum technologies. This is a considerable challenge for computational methods, since Kohn-Sham density functional theory (DFT) is inherently a ground-state theory, while higher-level methods are often too computationally expensive for defect systems. Recently, embedding approaches have been applied that treat defect states with many-body methods, while using DFT to describe the bulk host material. We implement such an embedding method, based on Wannierization of defect orbitals and the constrained random-phase approximation approach, and perform systematic characterization of the method for three distinct systems with current technological relevance: a carbon dimer replacing a B and N pair in bulk hexagonal BN (${\mathrm{C}}_{\text{B}}{\mathrm{C}}_{\text{N}}$), the negatively charged nitrogen-vacancy center in diamond (${\mathrm{NV}}^{\ensuremath{-}}$), and an Fe impurity on the Al site in wurtzite AlN (${\text{Fe}}_{\text{Al}}$). In the context of these test-case defects, we demonstrate that crucial considerations of the methodology include convergence of the bulk screening of the active-space Coulomb interaction, the choice of exchange-correlation functional for the initial DFT calculation, and the treatment of the ``double-counting'' correction. For ${\mathrm{C}}_{\text{B}}{\mathrm{C}}_{\text{N}}$ we show that the embedding approach gives many-body states in agreement with analytical results on the Hubbard dimer model, which allows us to elucidate the effects of the DFT functional and double-counting correction. For the ${\mathrm{NV}}^{\ensuremath{-}}$ center, our method demonstrates good quantitative agreement with experiments for the zero-phonon line of the triplet-triplet transition. Finally, we illustrate challenges associated with this method for determining the energies and orderings of the complex spin multiplets in ${\text{Fe}}_{\text{Al}}$.

ΔNO and the complexities of electron correlation in simple hydrogen clusters.

The Δ natural orbital (ΔNO) two-electron density matrix (2-RDM) and energy expression are derived from a multideterminantal wave function. The approximate ΔNO 2-RDM is combined with an on-top density functional and a double-counting correction to capture electron correlation. A trust-region Newton's method optimization algorithm for the simultaneous optimization of ΔNO orbitals and occupancies is introduced and compared to the previous iterative diagonalization algorithm. The combination of ΔNO and two different on-top density functionals, Colle-Salvetti (CS) and Opposite-spin exponential cusp and Fermi-hole correction (OF), is assessed on small hydrogen clusters and compared to density functional, single-reference coupled-cluster, and multireference perturbation theory (MRMP2) methods. The ΔNO-CS and ΔNO-OF methods outperform the single-reference methods and are comparable to MRMP2. However, there is a distinct qualitative error in the ΔNO potential energy surface for H4 compared to the exact. This discrepancy is explained through analysis of the ΔNO orbitals, occupancies, and the two-electron density.

Understanding the Seebeck coefficient of LaNiO3 compound in the temperature range 300–620 K

Transition metal oxides have been attracted much attention in thermoelectric community from the last few decades. In the present work, we have synthesized LaNiO3 by a simple solution combustion process. To analyse the crystal structure and structural parameters we have used Rietveld refinement method wherein FullProf software is employed. The room temperature x-ray diffraction indicates the rhombohedral structure with space group (No. 167). The refined values of lattice parameters are a = b = c = 5.4071 Å. Temperature dependent Seebeck coefficient (S) of this compound has been investigated by using experimental and computational tools. The measurement of S is conducted in the temperature range 300–620 K. The measured values of S in the entire temperature range have negative sign that indicates n-type character of the compound. The value of S is found to be ∼−8 μV/K at 300 K and at 620 K this value is ∼−12 μV/K. The electronic structure calculation is carried out using DFT + U method due to having strong correlation in LaNiO3. The calculation predicts the metallic ground state of the compound. Temperature dependent S is calculated using BoltzTraP package and compared with experiment. The best matching between experimental and calculated values of S is observed when self-interaction correction is employed as double counting correction in spin-polarized DFT + U (=1 eV) calculation. Based on the computational results maximum power factors are also calculated for p-type and n-type doping of this compound.

Importance of charge self-consistency in first-principles description of strongly correlated systems

First-principles approaches have been successful in solving many-body Hamiltonians for real materials to an extent when correlations are weak or moderate. As the electronic correlations become stronger often embedding methods based on first-principles approaches are used to better treat the correlations by solving a suitably chosen many-body Hamiltonian with a higher level theory. The success of such embedding theories, often referred to as second-principles, is commonly measured by the quality of self-energy Σ which is either a function of energy or momentum or both. However, Σ should, in principle, also modify the electronic eigenfunctions and thus change the real space charge distribution. While such practices are not prevalent, some works that use embedding techniques do take into account these effects. In such cases, choice of partitioning, of the parameters defining the correlated Hamiltonian, of double-counting corrections, and the adequacy of low-level Hamiltonian hosting the correlated subspace hinder a systematic and unambiguous understanding of such effects. Further, for a large variety of correlated systems, strong correlations are largely confined to the charge sector. Then an adequate nonlocal low-order theory is important, and the high-order local correlations embedding contributes become redundant. Here we study the impact of charge self-consistency within two example cases, TiSe2 and CrBr3, and show how real space charge re-distribution due to correlation effects taken into account within a first-principles Green’s function-based many-body perturbative approach is key in driving qualitative changes to the final electronic structure of these materials.

Electronic structure and magnetism in UGa2 : DFT+DMFT approach

The debate whether uranium 5f electrons are closer to being localized or itinerant in the ferromagnetic compound UGa2 is not yet fully settled. The experimentally determined magnetic moments are large, approximately 3 Bohr magnetons, suggesting the localized character of the 5f electrons. In the same time, one can identify signs of itinerant as well as localized behavior in various spectroscopic observations. The band theory, employing local exchange-correlation functionals, is biased toward itinerant 5f states and severely underestimates the moments. Using material-specific dynamical mean-field theory (DMFT), we probe how a less approximate description of electron-electron correlations improves the picture. We present two variants of the theory: starting either from spin-restricted (LDA) or spin-polarized (LSDA) band structure. We show that the L(S)DA+DMFT method can accurately describe the magnetic moments in UGa2 as long as the exchange interaction between the uranium 6d and 5f electrons is preserved by a judicious choice of the spin-polarized double-counting correction. We discuss the computed electronic structure in relation to photoemission experiments and show how the correlations reduce the Sommerfeld coefficient of the electronic specific heat by shifting the 5f states slightly away from the Fermi level.

Price Variance in Hybrid-LCA Leads to Significant Uncertainty in Carbon Footprints

Hybrid Life Cycle Assessment (HLCA) methods attempt to address the limitations regarding process coverage and resolution of the more traditional Process- and Input-Output Life Cycle Assessments (PLCA, IOLCA). Due to the use of different units, HLCA methods rely on commodity price information to convert the physical units used in process inventories to the monetary units commonly used in Input-Output models. However, prices for the same commodity can vary significantly between different supply chains, or even between various levels in the same supply chain. The resulting commodity price variance in turn leads to added uncertainty in the hybrid environmental footprint. In this paper we take international trading statistics from BACI/UN-COMTRADE to estimate the variance of commodity prices, and use these in an integrated HLCA model of the process database ecoinvent with the EE-MRIO database EXIOBASE. We show that geographical aggregation of PLCA processes is a significant driver in the price variance of their reference products. We analyse the effect of price variance on process carbon footprint intensities (CFIs) and find that the CFIs of hybridised processes show a median increase of 6–17% due to hybridisation, for two different double counting scenarios, and a median uncertainty of −2 to +4% due to price variance. Furthermore, we illustrate the effect of price variance on the carbon footprint uncertainty in a HLCA study of Swiss household consumption. Although the relative footprint increase due to hybridisation is small to moderate with 8–14% for two different double counting correction strategies, the uncertainty due to price variability of this contribution to the footprint is very high, with 95% confidence intervals of (−28, +90%) and (−23, +68%) relative to the median. The magnitude and high positive skewness of the uncertainty highlights the importance of taking price variance into account when performing hybrid LCA.

Normal State of Nd1−xSrxNiO2 from Self-Consistent GW+EDMFT

The recent discovery of superconductivity in hole-doped NdNiO$_2$ thin films has captivated the condensed matter physics community. Such compounds with a formal Ni$^+$ valence have been theoretically proposed as possible analogues of the cuprates, and the exploration of their electronic structure and pairing mechanism may provide important insights into the phenomenon of unconventional superconductivity. At the modeling level, there are however fundamental issues that need to be resolved. While it is generally agreed that the low-energy properties of cuprates can to a large extent be captured by a single-band model, there has been a controversy in the recent literature about the importance of a multi-band description of the nickelates. The origin of this controversy is that studies based entirely on density functional theory (DFT) calculations miss important correlation and multi-orbital effects induced by Hund coupling, while model calculations or simulations based on the combination of DFT and (extended) dynamical mean field theory ((E)DMFT) involve ad-hoc parameters and double counting corrections that substantially affect the results. Here we use a multi-site extension of the recently developed $GW$+EDMFT method, which is free of adjustable parameters, to self-consistently compute the interaction parameters and electronic structure of hole-doped NdNiO$_2$. This full ab-initio simulation demonstrates the importance of a multi-orbital description, even for the undoped compound, and produces results for the resistivity and Hall conductance in qualitative agreement with experiment.

Electronic correlation, magnetic structure, and magnetotransport in few-layer CrI3

Using density functional theory combined with a Hubbard model (DFT+U ), the electronic band structure of CrI3 multilayers, both free-standing and enclosed between graphene contacts, is calculated. We show that the DFT+U approach, together with the 'around mean field' correction scheme, is able to describe the vertical magnetotransport in line with the experimental measurements of magnetoresistance in multi-layered CrI3 enclosed between graphene contacts. Moreover, by interpolating between different double-counting correction schemes, namely the 'around mean field' correction and the fully localized limit, we show their importance for describing both the band structure and the ground-state total energy consistently. Our description of the magnetic exchange interaction is compatible with the experimentally observed antiferromagnetic ground state in the bilayer CrI3 and the transition to a ferromagnetic arrangement in a small external magnetic field. Thus, using spin-polarized DFT+U with an 'around mean field' correction, a consistent overall picture is achieved.

Effect of charge self-consistency in DFT+DMFT calculations for complex transition metal oxides

We investigate the effect of charge self-consistency (CSC) in density functional theory plus dynamical mean-field theory (DFT+DMFT) calculations compared to simpler "one-shot" calculations for materials where interaction effects lead to a strong redistribution of electronic charges between different orbitals or between different sites. We focus on two systems close to a metal-insulator transition, for which the importance of CSC is currently not well understood. Specifically, we analyze the strain-related orbital polarization in the correlated metal CaVO$_3$ and the spontaneous electronic charge disproportionation in the rare-earth nickelate LuNiO$_3$. In both cases, we find that the CSC treatment reduces the charge redistribution compared to cheaper one-shot calculations. However, while the MIT in CaVO$_3$ is only slightly shifted due to the reduced orbital polarization, the effect of the site polarization on the MIT in LuNiO$_3$ is more subtle. Furthermore, we highlight the role of the double-counting correction in CSC calculations containing different inequivalent sites.

DFT+U in Dudarev's formulation with corrected interactions between the electrons with opposite spins: The form of Hamiltonian, calculation of forces, and bandgap adjustments.

Hubbard corrected density functional theory (DFT) methods, such as the DFT+U approach in Dudarev's approximation, are widely used for the description of energetics and electronic structure of strongly correlated materials, providing higher level of accuracy than local DFT calculations (e.g., local density approximation or generalized gradient approximation). However, the DFT+U method in Dudarev's formulation limits the introduced corrections to interactions between the electrons within the same spin channel, whereas interactions between the electrons with opposite spins are still treated using local DFT functional (e.g., Perdew-Burke-Ernzerhof). In recent years, the need for correction of these interactions between the electrons with opposite spins has been recognized and additional terms have been added to the Hubbard term to reflect it. Although such extended DFT+U functionals have been proposed, the form of respective Hamiltonian operator, defined as a total energy derivative over density with appropriate treatment of double counting corrections due to additional Hubbard terms, has not been explicitly presented. In this work, we provide an expression for such a type of Hamiltonian, which contains the respective double counting correction contributions. This formulation also allows evaluation of atomic forces, using computational settings discussed herein. In addition, we also introduce adjustments for too narrow theoretical bandgaps, using scissor operator technique. This allows for a greater level of corrections of energetics and magnetic properties of studied transition metal compounds, avoiding possible unphysical overlap between occupied and unoccupied electronic bands.

First-principles calculation of stress tensor in the LSDA+U formalism

We derive the stress-tensor formula within the $\text{LSDA}+U$ scheme by differentiating analytically the $\text{LSDA}+U$ total-energy function with respect to the strain tensor. The rotationally invariant form of the $\text{LSDA}+U$ functional is employed and the double-counting correction is considered in the fully localized limit and around mean field. The electronic wave functions are expanded with either pseudoatomic orbitals (PAOs) or plane waves. In the PAO-basis case, the orthogonality stress term is included. Our $\text{LSDA}+U$ stress-tensor formula is numerically tested with antiferromagnetic NiO and reproduces successfully the stress values obtained from numerical derivatives of the total-energy values. As an application, we study elastic constants, bulk moduli, and sound velocities of NiO and MnO, obtaining results in good agreement with experimental data.

Systematically improvable multiscale solver for correlated electron systems

The development of numerical methods capable of simulating realistic materials with strongly correlated electrons, with controllable errors, is a central challenge in quantum many-body physics. Here we describe how a hybrid between self-consistent second order perturbation theory and exact diagonalization can be used as a multi-scale solver for such systems. Using a quantum impurity model, generated from a cluster dynamical mean field approximation to the 2D Hubbard model, as a benchmark, we show that our method allows us to obtain accurate results at a fraction of the cost of typical Monte Carlo calculations. We test the behavior of our method in multiple regimes of interaction strengths and doping of the model. The algorithm avoids difficulties such as double counting corrections, frequency dependent interactions, or vertex functions. As it is solely formulated at the level of the single-particle Green's function, it provides a promising route for the simulation of realistic materials that are currently difficult to study with other methods.

Computing total energies in complex materials using charge self-consistent DFT + DMFT

We have formulated and implemented a fully charge self-consistent density functional theory plus dynamical mean-field theory methodology which enables an efficient calculation of the total energy of realistic correlated electron systems. The density functional portion of the calculation uses a planewave basis set within the projector augmented wave method enabling study of systems with large, complex unit cells. The dynamical mean-field portion of the calculation is formulated using maximally localized Wannier functions, enabling a convenient implementation which is independent of the basis set used in the density functional portion of the calculation. The importance of using a correct double-counting term is demonstrated. A generalized form of the standard double-counting correction, which we refer to as the ${U}^{\ensuremath{'}}$ form, is described in detail and used. For comparison, the density functional plus $U$ method is implemented within the same framework including the generalized double counting. The formalism is validated via a calculation of the metal-insulator and structural phase diagrams of the rare-earth nickelate perovskites as functions of applied pressure and A-site rare-earth ions. The calculated density functional plus dynamical mean-field results are found to be consistent with experiment. The density functional plus $U$ method is shown to grossly overestimate the tendency for bond disproportionation and insulating behavior.

Electronic structure, cohesive properties, and magnetism ofSrRuO3

We have performed an extensive test of the ability of density functional theory within several approximations for the exchange-correlation functional, local density approximation+Hubbard $U$ and local density approximation + dynamic mean field theory to describe magnetic and electronic properties of SrRuO$_3$. We focus on the ferromagnetic phase, illustrating differences between the orthorhombic low temperature structure vs the cubic high temperature structure. We assess how magnetism, spectral function, and cohesive properties are affected by methodology, on-site Hubbard $U$ and double counting corrections. Further, we compare the impact of the impurity solver on the quasiparticle weight $Z$, which is in turn compared to experimental results. The spectral functions resulting from the different treatments are also compared to experimental data. The impact of spin-orbit coupling is also studied, allowing us to determine the orbital moments. In the orthorhombic phase the orbital moments are found to be tilted with respect to the spin moments, emphasising the importance of taking into account the distortion of the oxygen octahedra.

Density functional plus dynamical mean-field theory of the metal-insulator transition in early transition-metal oxides

The combination of density functional theory and single-site dynamical mean-field theory, using both Hartree and full continuous-time quantum Monte Carlo impurity solvers, is used to study the metal-insulator phase diagram of perovskite transition-metal oxides of the form $AB$O$_3$ with a rare-earth ion $A$=Sr, La, Y and transition metal $B$=Ti, V, Cr. The correlated subspace is constructed from atomiclike $d$ orbitals defined using maximally localized Wannier functions derived from the full $p$-$d$ manifold; for comparison, results obtained using a projector method are also given. Paramagnetic DFT+DMFT computations using full charge self-consistency along with the standard "fully localized limit" (FLL) double counting are shown to incorrectly predict that LaTiO$_3$, YTiO$_3$, LaVO$_3$ and SrMnO$_3$ are metals. A more general examination of the dependence of physical properties on the mean $p$-$d$ energy splitting, the occupancy of the correlated $d$ states, the double-counting correction, and the lattice structure demonstrates the importance of charge-transfer physics even in the early transition-metal oxides and elucidates the factors underlying the failure of the standard approximations. If the double counting is chosen to produce a $p$-$d$ splitting consistent with experimental spectra, single-site dynamical mean-field theory provides a reasonable account of the materials properties. The relation of the results to those obtained from "$d$-only" models in which the correlation problem is based on the frontier orbital $p$-$d$ antibonding bands is determined. It is found that if an effective interaction $U$ is properly chosen the $d$-only model provides a good account of the physics of the $d^1$ and $d^2$ materials.

Gutzwiller density functional theory: a formal derivation and application to ferromagnetic nickel

We present a detailed derivation of the Gutzwiller density functional theory (DFT) that covers all conceivable cases of symmetries and Gutzwiller wave functions. The method is used in a study of ferromagnetic nickel where we calculate ground state properties (lattice constant, bulk modulus, spin magnetic moment) and the quasi-particle band structure. Our method resolves most shortcomings of an ordinary density functional calculation on nickel. However, the quality of the results strongly depends on the particular choice of the double-counting correction. This constitutes a serious problem for all methods that attempt to merge DFT with correlated-electron approaches based on Hubbard-type local interactions.

Covalency and the metal-insulator transition in titanate and vanadate perovskites

A combination of density functional and dynamical mean-field theory is applied to the perovskites SrVO$_3$, LaTiO$_3$ and LaVO$_3$. We show that DFT+DMFT in conjunction with the standard fully localized-limit (FLL) double-counting predicts that LaTiO$_3$ and LaVO$_3$ are metals even though experimentally they are correlation-driven ("Mott") insulators. In addition, the FLL double counting implies a splitting between oxygen $p$ and transition metal $d$ levels which differs from experiment. Introducing into the theory an \textit{ad hoc} double counting correction which reproduces the experimentally measured insulating gap leads also to a $p$-$d$ splitting consistent with experiment if the on-site interaction $U$ is chosen in a relatively narrow range ($\sim 6\pm 1$ eV). The results indicate that these early transition metal oxides will serve as critical test for the formulation of a general \textit{ab initio} theory of correlated electron metals.

Hellmann–Feynman Forces in the DFT+UMethod with Ultrasoft Pseudopotentials

Density functional theory (DFT) with on-site Coulomb repulsion (DFT+U) is a conventional tool to study strongly-correlated electron systems, which cannot be described properly with semilocal approximations to DFT. In many cases, the on-site Coulomb repulsion (U) and the exchange (J) energies are given as an adjustable parameter, so as to reproduce desired physical properties (such as band gap), but by determining the effective on-site Coulomb repulsion energy from first-principles, a nonempirical DFT+U calculation is also possible. Furthermore, because the total energy can be determined variationally, the Hellmann–Feynman forces are available, which allows geometry optimization and molecular dynamics simulation within DFT+U. However, the so-called double-counting correction is not defined uniquely, and as a result, formulation of the Hellmann–Feynman forces depends on the particular implementation of DFT+U energy. Here, I present an implementation of the Hellmann– Feynman forces within DFT+U scheme adopted in the STATE code, which employs the ultrasoft pseudopotentials and plane-wave basis sets. In the present implementation of DFT+U, the diagonal representation of the density matrix is employed, which differs from previous ones. Interaction energy between localized electrons is assumed to be given by the Hubbard-like expression as