Self-interaction Error Research Articles

Suppressing the gauge problem in local hybrid functionals without a calibration function: The choice of local mixing function.

Modern functionals based on the exact-exchange (EXX) energy density like local hybrid functionals (LHs) or range-separated LHs have recently received additional attention due to their advantages over established functionals when it comes to the local balance between self-interaction errors and static-correlation errors. A possible theoretical drawback of such functionals over the years has been the so-called gauge problem due to the inherent ambiguity of exchange-energy densities. Modern LHs like LH20t or more sophisticated functionals based thereon have been constructed using suitably optimized calibration functions (CFs) to minimize the mismatch of the semi-local and EXX energy densities. Here, we show that the unphysical contributions arising from the gauge problem may also be reduced significantly without a CF by tailoring the position-dependence of the EXX admixture (local mixing function, LMF) in a way to suppress spurious positive energy-density contributions locally in space. This is achieved by building the so-called x-LMFs upon the ratio between EXX and semi-local exchange-energy densities. The resulting LH24x functional provides similar accuracy, e.g., for the GMTKN55 test suite, as LH20t, but without introduction of a CF! We provide detailed comparative analyses of integrated energies and spatially resolved energy densities. The good performances of LHs for chemically relevant energy differences are to some extent due to the core nature of unphysical artifacts that cancel out efficiently.

Training Machine-Learned Density Functionals on Band Gaps.

The systematic underestimation of band gaps is one of the most fundamental challenges in semilocal density functional theory (DFT). In addition to hindering the application of DFT to predicting electronic properties, the band gap problem is intimately related to self-interaction and delocalization errors, which make the study of charge transfer mechanisms with DFT difficult. To expand the range of available tools for addressing the band gap problem, we design an approach for machine learning density functionals based on Gaussian processes to explicitly fit single-particle energy levels. We also introduce nonlocal features of the density matrix that are expressive enough to fit these single-particle levels. Combining these developments, we train a machine-learned functional for the exact exchange energy that predicts molecular energy gaps and reaction energies of a wide range of molecules in excellent agreement with reference hybrid DFT calculations. In addition, while being trained solely on molecular data, our model predicts reasonable formation energies of polarons in solids, showcasing its transferability and robustness. We discuss how this approach can be generalized to full exchange-correlation functionals, thus paving the way to the design of state-of-the-art functionals for the prediction of electronic properties of molecules and materials.

Exploiting the Correlation between the 1s, 2s, and 2p Energies for the Prediction of Core-Level Binding Energies of Si, P, S, and Cl species.

The binding energies (BEs) of the 1s, 2s, and 2p core electrons of third-period elements (Si, P, S, Cl) were calculated using Hartree-Fock (HF) and B3LYP, BH&HLYP, and LC-BOP ΔSCF, and the shifted KS ΔSCF methods. Linear relationships between two BEs were derived and compared with the Auger parameter. The derived lines are essentially parallel, with only the intercepts differing. The difference in intercepts is due to the lack of electron correlation effects in HF and the self-interaction errors (SIEs) of the functional. The slope is the slope of the linear relationship between the chemical shifts. The straight lines between the 2s and 2p BEs also coincided with the Auger parameter lines, which have a slope of 1 by definition and an intercept being the difference between the 2s and 2p BEs. The shifted KS ΔSCF scheme compensates for SIEs, yielding equations that are approximately invariant. The calculated average gaps for the 2s and 2p BEs are 51.21 eV for Si, 57.48 eV for P, 63.85 eV for S, and 70.48 eV for Cl. The straight lines representing the relationships between the BEs of the 1s and 2s and 1s and 2p electrons are also parallel to each other in ΔSCF and converge into a single line in the shifted ΔSCF scheme. However, these lines are steeper than the Auger parameter line. The derived relationships can be used to predict unknown BEs, which we have applied to many molecules. The results are highly accurate, with mean absolute errors (MAEs) of less than 0.2 eV compared to experimental values.

(Invited) Extended Hubbard Functionals for Accurate Modeling of Li-Ion Battery Cathode Materials

Designing novel cathode materials for Li-ion batteries necessitates accurate first-principles predictions of their properties. Density-functional theory (DFT) employing standard (semi-)local functionals encounters challenges due to pronounced self-interaction errors within the partially filled d shells of transition-metal (TM) elements. Here, we demonstrate the efficacy of DFT with extended Hubbard functionals in accurately predicting the "digital" change in oxidation states of TM ions for mixed-valence phases at intermediate Li concentrations in phospho-olivine and spinel cathode materials. The resulting voltages exhibit remarkable agreement with experimental data [1,2].Our approach involves the use of onsite U and intersite V Hubbard parameters computed from density-functional perturbation theory with Lowdin-orthogonalized atomic orbitals, thus circumventing empirical factors [3,4]. Notably, the incorporation of intersite Hubbard V interactions proves indispensable for the precise prediction of thermodynamic quantities, particularly when electronic localization occurs amid inter-atomic orbital hybridization.Extending our investigation to vibrational spectroscopy, we calculate Raman spectra for these materials and find very good agreement with available experimental data [5]. Leveraging the robust and accurate computational framework based on extended Hubbard functionals, we embark on a high-throughput screening of Li-containing compounds, aiming to unveil novel and promising candidates for cathodes [6]. Our findings underscore the pivotal role of advanced DFT methodologies in advancing the discovery and design of high-performance battery materials.[1] I. Timrov, F. Aquilante, M. Cococcioni, N. Marzari, PRX Energy 1, 033003 (2022).[2] I. Timrov, M. Kotiuga, N. Marzari, Phys. Chem. Chem. Phys. 25, 9061 (2023).[3] I. Timrov, N. Marzari, M. Cococcioni, Phys. Rev. B 98, 045141 (2021).[4] I. Timrov, N. Marzari, M. Cococcioni, Phys. Rev. B 103, 085127 (2018).[5] L. Bastonero, N. Marzari, I. Timrov, in preparation.[6] C. Malica, L. Bastonero, M. Bercx, N. Marzari, I. Timrov, in preparation.

Revisiting Artifacts of Kohn-Sham Density Functionals for Biosimulation.

We revisit the problem of unphysical charge density delocalization/fractionalization induced by the self-interaction error of common approximate Kohn-Sham (KS) density functional theory functionals on simulation of small to medium-sized proteins in a vacuum. Aside from producing unphysical electron densities and total energies, the vanishing of the HOMO-LUMO gap associated with the unphysical charge delocalization leads to an unphysical low-energy spectrum and catastrophic failure of most popular solvers for the KS self-consistent field (SCF) problem. We apply a robust quasi-Newton SCF solver [ Phys. Chem. Chem. Phys. 2024, 26, 6557] to obtain solutions for some of these difficult cases. The anatomy of the charge delocalization is revealed by the natural deformation orbitals obtained from the density matrix difference between the Hartree-Fock and KS solutions; the charge delocalization not only can occur between charged fragments (such as in zwitterionic polypeptides) but also involves neutral fragments. The vanishing-gap phenomenon and troublesome SCF convergence are both attributed to the unphysical KS Fock operator eigenspectra of molecular fragments (e.g., amino acids or their side chains). Analysis of amino acid pairs suggests that the unphysical charge delocalization can be partially ameliorated by the use of some range-separated hybrid functionals but not by semilocal or standard hybrid functionals. Last, we demonstrate that solutions without the unphysical charge delocalization can be located even for semilocal KS functionals highly prone to such defects, but such solutions have non-Aufbau character and are unstable with respect to mixing of the non-overlapping "frontier" orbitals. Caution should be exercised when unexpectedly small (or vanishing) HOMO-LUMO gaps and atypical SCF convergence patterns (e.g., oscillatory) are observed in KS DFT simulations in any context (bio or otherwise).

Impact of ring size on the C[sbnd]H activation reactivity of iron(IV)–oxo complexes: A computational study with TMC macrocycles

Here, a DFT investigation has been presented to demonstrate the relevance of the macrocyclic ligand ring size of the high-valent Fe(IV)O complex-catalyzed CH activation process. A nonheme iron-oxo compound bound with extensively studied macrocycle tetramethylcyclam (TMC) with varying ring size measures in terms of n = 12, 13, 14, 15, and 16 in [Fe(IV)O(n-TMC)(CH3CN)]2+ has been considered as the oxidant and dihydroanthracene as the general substrate. Counterions, as always, have been considered to avoid the self-interaction error in these DFT calculations. Computations were carried out to determine the effect of the axial ligand, acetonitrile, on the CH activation reactivity. It has been discovered that the complexes without axial ligands turned out to be more reactive compared to their axially coordinated counterparts. The most intriguing finding, however, has been that reactivity increased steadily with ring size increments, giving us the trend 12 < 13 < 14 < 15 < 16. Behind this typical pattern of reactivity, several factors played a role, including the energy of the electron acceptor orbital, which sequentially decreases, and the distortion energy to achieve the transition state, which also decreases as we move on from n = 12 to 16. The triplet-quintet energy difference of the oxidants also has a part to play, as it decreases with increased ring size, with the quintet becoming more and more dominant. The current studies have also been able to corroborate the experimental data that was published regarding Fe(IV)O(13-TMC) (without axial syn form) having a higher CH activation reactivity than Fe(IV)O(14-TMC) (with axial anti form). On the whole, this computational exploration gives us a reactivity pattern based on the ring size commutes and can lead to successful experimental results if pursued based on this reaction.

DOCSIC: A Mean-Field Method for Orbital-by-Orbital Self-Interaction Correction.

We introduce a new method to remove the one-electron self-interaction error in approximate density functional calculations on an orbital-by-orbital basis, as originally proposed by Perdew and Zunger [Phys. Rev. B 1981, 23, 5048]. This method is motivated by a recent proposal by Pederson et al. [J. Chem. Phys. 2014, 140, 121103] to remove self-interaction that employs orbitals derived from the real-space density matrix, known as FLOSIC (Fermi Löwdin orbitals self-interaction correction). However, instead of Fermi Löwdin orbitals, our scheme utilizes columns of the density matrix to determine localized orbitals, like the localization procedure proposed by Fuemmeler et al. [J. Chem. Theory Comput. 2023, 19, 8572]. The new method, dubbed DOCSIC for density matrix as orbital coefficients self-interaction correction, contrasts with traditional Perdew-Zunger or FLOSIC in that it does not incorporate additional optimization parameters, and, unlike the average density self-interaction correction of Ciofini et al. [Chem. Phys. Lett. 2003, 380, 12], it makes use of localized orbitals. Another advantage of DOCSIC is that it can be implemented as a mean-field formalism. We show details of the self-consistent generalized Kohn-Sham implementation, some illustrative results, and we finally highlight its advantages and limitations.

DFT+U study of TlXBr3 (X = Sn, Ge) perovskites as next-generation materials for optoelectronics applications

The lead-free perovskite has gained increasing importance in optoelectronic device applications. In this study, we investigate the structural, electronic, optical, mechanical, thermodynamic, and thermoelectric properties of cubic phase TlSnBr3 and TlGeBr3 using the plane wave pseudopotential method based on density functional theory (DFT). To overcome constraints of the GGA-PBE technique, we implemented the LDA + U technique to solve on-site self-interaction errors. The Hubbard parameter 'U' was identified as a suitable solution for considering on-site self-interactions and calculating better electronic energy band gaps. The obtained lattice parameters are consistent with previous published results. Our findings indicate that both materials have a direct energy band gap, with values of 1.75 eV and 1.94 eV for TlSnBr3 and TlGeBr3, respectively. We also observe the existence of covalent and ionic bonding. The optical properties of these materials demonstrate high absorption coefficients and conductivity in the visible region. The mechanical properties are derived using the Voigt-Reuss-Hill approximation and suggest strong evidence of ductile behavior. Finally, the thermoelectric properties are evaluated by analyzing thermal to electrical conductivity, Seebeck coefficient, and Figure of merit criteria. Our comparative study of TlXBr3 (X = Sn, Ge) suggests that these compounds are promising candidate materials for photovoltaic and optoelectronic devices.

Improving gas adsorption modeling for MOFs by local calibration of Hubbard U parameters.

While computational screening with density functional theory (DFT) is frequently employed for the screening of metal-organic frameworks (MOFs) for gas separation and storage, commonly applied generalized gradient approximations (GGAs) exhibit self-interaction errors, which hinder the predictions of adsorption energies. We investigate the Hubbard U parameter to augment DFT calculations for full periodic MOFs, targeting a more precise modeling of gas molecule-MOF interactions, specifically for N2, CO2, and O2. We introduce a calibration scheme for the U parameter, which is tailored for each MOF, by leveraging higher-level calculations on the secondary building unit (SBU) of the MOF. When applied to the full periodic MOF, the U parameter calibrated against hybrid HSE06 calculations of SBUs successfully reproduces hybrid-quality calculations of the adsorption energy of the periodic MOF. The mean absolute deviation of adsorption energies reduces from 0.13eV for a standard GGA treatment to 0.06eV with the calibrated U, demonstrating the utility of the calibration procedure when applied to the full MOF structure. Furthermore, attempting to use coupled cluster singles and doubles with perturbative triples calculations of isolated SBUs for this calibration procedure shows varying degrees of success in predicting the experimental heat of adsorption. It improves accuracy for N2 adsorption for cases of overbinding, whereas its impact on CO2 is minimal, and ambiguities in spin state assignment hinder consistent improvements of O2 adsorption. Our findings emphasize the limitations of cluster models and advocate the use of full periodic MOF systems with a calibrated U parameter, providing a more comprehensive understanding of gas adsorption in MOFs.

The rise and fall of stretched bond errors: Extending the analysis of Perdew-Zunger self-interaction corrections of reaction barrier heights beyond the LSDA.

Incorporating self-interaction corrections (SIC) significantly improves chemical reaction barrier height predictions made using density functional theory methods. We present a detailed orbital-by-orbital analysis of these corrections for three semi-local density functional approximations (DFAs) situated on the three lowest rungs of Jacob's ladder of approximations. The analysis is based on Fermi-Löwdin Orbital Self-Interaction Correction (FLOSIC) calculations performed at several steps along the reaction pathway from the reactants (R) to the transition state (TS) to the products (P) for four representative reactions selected from the BH76 benchmark set. For all three functionals, the major contribution to self-interaction corrections of the barrier heights can be traced to stretched bond orbitals that develop near the TS configuration. The magnitude of the ratio of the self-exchange-correlation energy to the self-Hartree energy (XC/H) for a given orbital is introduced as an indicator of one-electron self-interaction error. XC/H = 1.0 implies that an orbital's self-exchange-correlation energy exactly cancels its self-Hartree energy and that the orbital, therefore, makes no contribution to the SIC in the FLOSIC scheme. For the practical DFAs studied here, XC/H spans a range of values. The largest values are obtained for stretched or strongly lobed orbitals. We show that significant differences in XC/H for corresponding orbitals in the R, TS, and P configurations can be used to identify the major contributors to the SIC of barrier heights and reaction energies. Based on such comparisons, we suggest that barrier height predictions made using the strongly constrained and appropriately normed meta-generalized gradient approximation may have attained the best accuracy possible for a semi-local functional using the Perdew-Zunger SIC approach.

A numerical Poisson solver with improved radial solutions for a self-consistent locally scaled self-interaction correction method

The universal applicability of density functional approximations is limited by self-interaction error made by these functionals. Recently, a novel one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities was proposed. The locally scaled SIC (LSIC) method is exact for the one-electron densities, and unlike the well-known Perdew–Zunger SIC (PZSIC) method recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to PZSIC method as a special case when isoorbital indicator is set to the unity. Here, we present a numerical scheme that we have adopted to evaluate the Coulomb potential of the electron density scaled by the iso-orbital indicator required for the self-consistent LSIC calculations. After analyzing the behavior of the finite difference method (FDM) and the green function solution to the radial part of the Poisson equation, we adopt a hybrid approach that uses the FDM for the Coulomb potential due to the monopole and the GF for all higher-order terms. The performance of the resultant hybrid method is assessed using a variety of systems. The results show improved accuracy than earlier numerical schemes. We also find that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies.

Self-interaction corrected SCAN functional for molecules and solids in the numeric atom-center orbital framework.

Semilocal density-functional approximations (DFAs), including the state-of-the-art SCAN functional, are plagued by the self-interaction error (SIE). While this error is explicitly defined only for one-electron systems, it has inspired the self-interaction correction method proposed by Perdew and Zunger (PZ-SIC), which has shown promise in mitigating the many-electron SIE. However, the PZ-SIC method is known for its significant numerical instability. In this study, we introduce a novel constraint that facilitates self-consistent localization of the SIC orbitals in the spirit of Edmiston-Ruedenberg orbitals [Rev. Mod. Phys. 35, 457 (1963)]. Our practical implementation within the all-electron numeric atom-centered orbitals code FHI-aims guarantees efficient and stable convergence of the self-consistent PZ-SIC equations for both molecules and solids. We further demonstrate that our PZ-SIC approach effectively mitigates the SIE in the meta-generalized gradient approximation SCAN functional, significantly improving the accuracy for ionization potentials, charge-transfer energies, and bandgaps for a diverse selection of molecules and solids. However, our PZ-SIC method does have its limitations. It cannot improve the already accurate SCAN results for properties such as cohesive energies, lattice constants, and bulk modulus in our test sets. This highlights the need for new-generation DFAs with more comprehensive applicability.

Bond length alternation of π-conjugated polymers predicted by the Fermi-Löwdin orbital self-interaction correction method.

π-conjugated polymers have been used in a wide range of practical applications, partly due to their unique properties that originate in the delocalization of electrons through the polymer backbone. The level of delocalization can be characterized by the induced bond length alternation (BLA), with shorter BLA connected with strong delocalization and vice versa. The accurate description of this structural parameter can be considered a benchmark for testing the capability of different electronic structure methods for self-interaction error (SIE) removal and electron correlation inclusion. Density functional theory (DFT), in its local or semi-local flavors, suffers from SIE and, thus, underestimates the BLA compared to self-interaction-free methods. In this work, we utilize the Fermi-Löwdin orbital self-interaction correction (FLOSIC) method for one-electron self-interaction removal to characterize the BLA of five oligomers with increasing length extrapolated to the polymeric limit. We compare the self-interaction-free BLA to several DFT approximations, Møller-Plesset second-order perturbation theory (MP2), and the BLA obtained with the domain based local pair natural orbital CCSD(T) [DLPNO-CCSD(T)] approximation. Our findings show that FLOSIC corrects for the small BLA given by (semi-)local DFT approximations, but it tends to overcorrect with respect to CAM-B3LYP, MP2, and DLPNO-CCSD(T).

A density functional theory analysis of the C-H activation reactivity of iron(IV)-oxo complexes with an 'O' substituted tetramethylcyclam macrocycle.

In this article, we present a meticulous computational study to foresee the effect of an oxygen-rich macrocycle on the reactivity for C-H activation. For this study, a widely studied nonheme Fe(IV)O molecule with a TMC (1,4,8,11-tetramethyl 1,4,8,11-tetraazacyclotetradecane) macrocycle that is equatorially attached to four nitrogen atoms (designated as N4) and acetonitrile as an axial ligand has been taken into account. For the goal of hetero-substitution, step-by-step replacement of the N4 framework with O atoms, i.e., N4, N3O1, N2O2, N1O3, and O4 systems, has been considered, and dihydroanthracene (DHA) has been used as the substrate. In order to neutralise the system and prevent the self-interaction error in DFT, triflate counterions have also been included in the calculations. The study of the energetics of these C-H bond activation reactions and the potential energy surfaces mapped therefore reveal that the initial hydrogen abstraction, which is the rate-determining step, follows the two-state reactivity (TSR) pattern, which means that the originally excited quintet state falls lower in the transition state and the product. The reaction follows the hydrogen atom transfer (HAT) mechanism, as indicated by the spin density studies. The results revealed a fascinating reactivity order, in which the reactivity increases with the enrichment of the oxygen atom in the equatorial position, namely the order follows N4 < N3O1 < N2O2 < N1O3 < O4. The impacts of oxygen substitution on quantum mechanical tunneling and the H/D kinetic isotope effect have also been investigated. When analysing the causes of this reactivity pattern, a number of variables have been identified, including the reactant-like transition structure, spin density distribution, distortion energy, and energies of the electron acceptor orbital, i.e., the energy of the LUMO (σ*z2), which validate the obtained outcome. Our results also show very good agreement with earlier combined experimental and theoretical studies considering TMC and TMCO-type complexes. The DFT predictions reported here will undoubtedly encourage experimental research in this biomimetic field, as they provide an alternative with higher reactivity in which heteroatoms can be substituted for the traditional nitrogen atom.

Orbital dependent complications for close vs well-separated electrons in diradicals.

We investigate two limits in open-shell diradical systems: O3, in which the interesting orbitals are in close proximity to one another, and (C21H13)2, where there is a significant spatial separation between the two orbitals. In accord with earlier calculations, we find that standard density-functional approximations do not predict the open-shell character for the former case but uniformly predict the open-shell character for the latter case. We trace the qualitatively incorrect behavior in O3 predicted by these standard density functional approximations to self-interaction error and use the Fermi-Löwdin-orbital-self-interaction-corrected formalism to determine accurate triplet, closed-shell singlet, and open-shell broken-spin-symmetry electronic configurations. Analysis of the resulting many-electron overlap matrices allows us to unambiguously show that the broken-spin-symmetry configurations do not participate in the representation of the Ms = 0 triplet states and allows us to reliably extract the singlet-triplet splitting in O3 by analyzing the energy as a function of Fermi-orbital-descriptor permutations. The results of these analyses predict the percentage of open-shell character in O3, which agrees well with conventional wavefunction-based methods. While these techniques are expected to be required in cases near the Coulson-Fischer point, we find that they will be less necessary in diradical systems with well-separated electrons, such as (C21H13)2. Results based on energies from self-interaction-corrected generalized gradient, local density, and Hartree-Fock approximations and experimental results are in generally good agreement for O3. These results help form the basis for deriving extended Heisenberg-like Hamiltonians that are needed for descriptions of molecular magnets when there are competing low-energy electronic configurations.

SOS1-RSX-QIDH: A spin-opposite-scaled range-separated-exchange quadratic-integrand double-hybrid density functional

We develop and validate the SOS1-RSX-QIDH density functional, a one-parameter spin-opposite-scaled variant of the range-separated-exchange quadratic-integrand double-hybrid (RSX-QIDH) model. By entering into the family of spin-biased double hybrids, this new density functional benefits from an improved computational scaling that rivals with the one of hybrids, still conserving the accuracy of its RSX-QIDH version. As part of the latter family, this density functional is well-adapted to treat molecular systems that are particularly prone to self-interaction errors in their ground and excited states. In particular, we show that the SOS1-RSX-QIDH model is a good compromise to treat ground-state problems dealing with kinetics and has a real added value when applied to the evaluation of the excited-state properties of equilibrium and out-of-equilibrium molecular complexes. Even if spin-biased double hybrids are recognized to strongly underestimate noncovalent interactions, we notice and recommend coupling SOS1-RSX-QIDH with a nonlocal van der Waals potential, a combination that is here proved to compete with the best density-functional approximations currently in use.

Use of FLOSIC for understanding anion-solvent interactions.

An Achille's heel of lower-rung density-functional approximations is that the highest-occupied-molecular-orbital energy levels of anions, known to be stable or metastable in nature, are often found to be positive in the worst case or above the lowest-unoccupied-molecular-orbital levels on neighboring complexes that are not expected to accept charge. A trianionic example, [Cr(C2O4)3]3-, is of interest for constraining models linking Cr isotope ratios in rock samples to oxygen levels in Earth's atmosphere over geological timescales. Here we describe how crowd sourcing can be used to carry out self-consistent Fermi-Löwdin-Orbital-Self-Interaction corrected calculations (FLOSIC) on this trianion in solution. The calculations give a physically correct description of the electronic structure of the trianion and water. In contrast, uncorrected local density approximation (LDA) calculations result in approximately half of the anion charge being transferred to the water bath due to the effects of self-interaction error. Use of group-theory and the intrinsic sparsity of the theory enables calculations roughly 125 times faster than our initial implementation in the large N limit reached here. By integrating charge density densities and Coulomb potentials over regions of space and analyzing core-level shifts of the Cr and O atoms as a function of position and functional, we unambiguously show that FLOSIC, relative to LDA, reverses incorrect solute-solvent charge transfer in the trianion-water complex. In comparison to other functionals investigated herein, including Hartree-Fock and the local density approximation, the FLOSIC Cr 1s eigenvalues provide the best agreement with experimental core ionization energies.

The verification of delta SCF and Slater's transition state theory for the calculation of core ionization energy.

The core ionization energies of second- and third-period elements of the molecules C2 H5 NO2 , SiF4 , Si(CH3 )4 , PF3 , POF3 , PSF3 , CS2 , OCS, SO2 , SO2 F2 , CH3 Cl, CFCl3 , SF5 Cl, and Cl3 PS are calculated by using Hartree-Fock (HF), and Kohn-Sham (KS) with BH&HLYP, B3LYP, and LC-BOP functionals. We used ΔSCF, Slater's transition state (STS), and two previously proposed shifted STS (1) and shifted STS (2) methods, which have been developed. The errors of ΔSCF and STS come mainly from the self-interaction errors (SIE) and can be corrected with a shifting scheme. In this study, we used the shifting parameters determined for each atom. The shifted STS (1) reproduces ΔSCF almost perfectly with mean absolute deviations (MAD) of 0.02 eV. While ΔSCF and STS vary significantly depending on the functional used, the variation of shifted STS (2) is small, and all shifted STS (2) values are close to the observed ones. The deviations of the shifted STS (2) from the experiment are 0.24 eV (BH&HLYP), 0.19 eV (B3LYP), and 0.23 eV (LC-BOP). These results further support the use of shifted STS methods for predicting the core ionization energies.

Core-Level 2s and 2p Binding Energies of Third-Period Elements (P, S, and Cl) Calculated by Hartree-Fock and Kohn-Sham ΔSCF Theory.

In the present study, we investigate the use of the ΔSCF method and Slater's transition state (STS) theory to calculate the binding energies of the 2s and 2p electrons of third-period elements (P, S, and Cl). Both the Hartree-Fock (HF) and Kohn-Sham (KS) approximations are examined. The STS approximation performs well in reproducing the ΔSCF values. However, for the ΔSCF method itself, while the binding energy of the 2p electrons is accurately predicted, the results for 2s are fairly sensitive to the functional, exhibiting significant variations due to self-interaction errors (SIE). Nonetheless, the variations in chemical shifts between different species remain relatively small, and the values agree with experiments due to the cancellation of SIE. A notable observation is that the chemical shifts of the 2s and 2p electrons are similar, indicating a perturbation caused by the valence electrons. The error in the absolute binding energy of KS ΔSCF against the experiment is nearly constant for the same element in different molecules, and it depends largely on the functional owing to SIE. A shifting scheme previously developed can be employed to reproduce the experimental 2s and 2p binding energies, with dependence on the functional and atom but not on the molecule even for 2s KS ΔSCF binding energies. Upon obtaining the corrected binding energies, we find that the gap between 2s and 2p binding energy is nearly independent of chemical environment for a given element: 57.5, 63.9, and 70.9 eV for the elements P, S, and Cl, respectively.

Nonlocal Functionals Inspired by the Strongly Interacting Limit of DFT: Exact Constraints and Implementation.

Capturing strong correlation effects remains a key challenge for the development of improved exchange-correlation (XC) functionals in density functional theory. The recently proposed multiple radii functional (MRF) [J. Phys. Chem. Lett. 2017, 8, 2799; J. Chem. Theory Comput. 2019, 15, 3580] was designed to capture strong correlation effects seamlessly, as its mathematical structure draws from that of the exact XC functional in the limit of infinite correlations. The MRF functional provides a framework for building approximations along the density-fixed adiabatic connection, delivers accurate XC energy densities in the standard DFT gauge (same as that of the exact exchange energy density), and is free of one-electron self-interaction errors. To facilitate the development of XC functionals based on the MRF, we examine the behavior of the MRF functional when applied to uniform and scaled densities and consider how it can be made exact for the uniform electron gas. These theoretical insights are then used to build improved forms for the fluctuation function, an object that defines XC energy densities within the MRF framework. We also show how the MRF fluctuation function for physical correlation can be easily readjusted to accurately capture the XC functional in the limit of infinite correlations, demonstrating the versatility of MRF for building approximations for different correlation regimes. We describe the implementation of MRF using densities expanded on Gaussian basis sets, which improves the efficiency of previous grid-based MRF implementations.