Koopmans'-Type Theorem in Kohn-Sham Theory with Optimally Tuned Long-Range-Corrected (LC) Functionals

Limiting factors in the accuracy of DFT calculation for redox potentials

In the present study, we have investigated factors affecting the accuracy of computational chemistry calculation of redox potentials, namely the gas-phase ionization energy (IE) and electron affinity (EA), and the continuum solvation effect. In general, double-hybrid density functional theory methods yield IEs and EAs that are on average within ~0.1 eV of our high-level W3X-L benchmark, with the best performing method being DSD-BLYP/ma-def2-QZVPP. For lower-cost methods, the average errors are ~0.2-0.3 eV, with ωB97X-3c being the most accurate (~0.15 eV). For the solvation component, essentially all methods have an average error of ~0.3 eV, which shows the limitation of the continuum solvation model. Curiously, the directly calculated redox potentials show errors of ~0.3 eV for all methods. These errors are notably smaller than what can be expected from error propagation with the two components (IE and EA, and solvation effect). Such a discrepancy can be attributed to the cancellation of errors, with the lowest-cost GFN2-xTB method benefiting the most, and the most accurate ωB97X-3c method benefiting the least. For organometallic species, the redox potentials show large deviations exceeding ~0.5 eV even for DSD-BLYP. The large errors are attributed to those for the gas-phase IEs and EAs, which represents a major barrier to the accurate calculation of redox potentials for such systems.

Excited state properties of an A-D-A non-fullerene electron acceptor: a LC-TD-DFTB study

Understanding charge transfer processes is essential to estimate the performance of organic photovoltaic technologies. Although experimental production is on the rise, predictability strongly relies on theoretical modeling, which is limited to the size of semiconductors. As a computationally favorable approach, we benchmarked the long-range corrected (LC) time-dependent (TD) formulation of the semi-empirical density functional-based tight-binding method (DFTB) for three polycyclic aromatic hydrocarbons (PAHs) and studied the DTP-IC-4Ph molecule, a PAH-based non-fullerene electron acceptor (NFA) with an A-D-A backbone structure. After a thorough investigation into the long-range parameter (ω) tuning for naphthalene, anthracene and pyrene, the excitation energies, oscillator strengths and Natural Transition Orbitals (NTOs) were compared with the standard ωB97X-D/6-31G(d,p) level of theory and the ADC2/6-31G(d,p) multiconfigurational method. We estimated mobility-related properties of the NFA and considered 1000 thermally accessible configurations to qualitatively reproduce the experimental absorption profile and investigate the energetic disorder. Finally, we conducted a fragment-based analysis using the one-electron transition density matrix (1TDM) to determine the character of the excited states and investigate the effect of side chains on exciton formation. Our results are sensitive to the level of theory and highly dependent on the long-range parameter but suggest that the presence of alkyl chains promotes a higher average charge delocalization and allows for additional hopping mechanisms, favoring the charge transfer dynamics.

Fractional-Electron and Transition-Potential Methods for Core-to-Valence Excitation Energies Using Density Functional Theory

Methods for computing X-ray absorption spectra based on a constrained core hole (possibly containing a fractional electron) are examined. These methods are based on Slater's transition concept and its generalizations, wherein core-to-valence excitation energies are determined using Kohn-Sham orbital energies. Methods examined here avoid promoting electrons beyond the lowest unoccupied molecular orbital, facilitating robust convergence. Variants of these ideas are systematically tested, revealing a best-case accuracy of 0.3-0.4 eV (with respect to experiment) for K-edge transition energies. Absolute errors are much larger for higher-lying near-edge transitions but can be reduced below 1 eV by introducing an empirical shift based on a charge-neutral transition-potential method, in conjunction with functionals such as SCAN, SCAN0, or B3LYP. This procedure affords an entire excitation spectrum from a single fractional-electron calculation, at the cost of ground-state density functional theory and without the need for state-by-state calculations. This shifted transition-potential approach may be especially useful for simulating transient spectroscopies or in complex systems where excited-state Kohn-Sham calculations are challenging.

Comprehensive analysis (aerobic/anaerobic, molecular recognitions, band-position and degradation-mechanism) of undoped and Co-doped anatase-brookite - An experimental/theoretical evaluation of the less-studied TiO2 mixed phase

The molecular recognition (MRec) effect is required in the initial phase of organic reactions. The second stage involves molecular-orientations and molecular-orbitals energy-levels (MOrbE). The components of a reaction must be compatible in terms MRec and MOrbE. Therefore, the comprehension of photocatalytic systems applied in wastewater treatment will be improved if the MRec effect is also considered as an important factor. The purpose of this study is to provide a comprehensive understanding of the less studied anatase-brookite mixed-phase (doped and undoped). Anatase/brookite photocatalytic systems were evaluated utilizing experimental/theoretical approaches in H₂O (aerobic/anaerobic) environments with Vis-light and the organic pollutant (OrPo) methyl orange (MO). The compatibility of MRec and MOrbE of anatase-brookite mixed-phase (with the different reactive system components) confirmed this is the optimal combination for photocatalytic application. Using the sol-gel method, AM-TiO₂NP (amorphous), TiO₂NP (crystalline), and TiO₂NP-Co0.1 at% (crystalline Co-doped) anatase-brookite mixed-phase photocatalysts were obtained. The morphology and surface were characterized using XRD, BET, SEM, HR-TEM, FT-IR and XPS. Employing UV-vis DRS and PL, photo-response and electron-hole recombination were studied. LVS and Mott-Schottky plot were employed to determine photo-electrochemical activity. The results of TiO₂NP photocatalytic degradation in both aerobic and anaerobic environments are remarkable. The results of molecular dynamics (MD) simulation and Fukui Function (FF) based on density functional theory (DFT) validate the remarkable photocatalytic MO degradation.

Slater transition methods for core-level electron binding energies

Methods for computing core-level ionization energies using self-consistent field (SCF) calculations are evaluated and benchmarked. These include a "full core hole" (or "ΔSCF") approach that fully accounts for orbital relaxation upon ionization, but also methods based on Slater's transition concept in which the binding energy is estimated from an orbital energy level that is obtained from a fractional-occupancy SCF calculation. A generalization that uses two different fractional-occupancy SCF calculations is also considered. The best of the Slater-type methods afford mean errors of 0.3-0.4 eV with respect to experiment for a dataset of K-shell ionization energies, a level of accuracy that is competitive with more expensive many-body techniques. An empirical shifting procedure with one adjustable parameter reduces the average error below 0.2 eV. This shifted Slater transition method is a simple and practical way to compute core-level binding energies using only initial-state Kohn-Sham eigenvalues. It requires no more computational effort than ΔSCF and may be especially useful for simulating transient x-ray experiments where core-level spectroscopy is used to probe an excited electronic state, for which the ΔSCF approach requires a tedious state-by-state calculation of the spectrum. As an example, we use Slater-type methods to model x-ray emission spectroscopy.

The core ionization energies calculated by delta SCF and Slater's transition state theory

The core ionization energies of the second-period and third-period elements are studied by ΔSCF and Slater's transition state (STS) theory by using Hartree-Fock (HF) and Kohn-Sham (KS) approximations. Electron correlation increases the estimated core ionization energies, while the self-interaction error (SIE) decreases them, especially for the third-period elements and is a more significant factor. As a result, while HF lacks electron correlation, it is free of SIE and reasonably predicts the core ionization energies. The core ionization energies calculated by HF STS are very close to those calculated by HF ΔSCF, showing that STS reasonably describes the relaxation of the core hole. The core ionization energies calculated by KS are particularly sensitive to the SIE of the functional used, with functionals having less SIE yielding more accurate ΔSCF core ionization energies. Consequently, BH&HLYP gives better results than B3LYP and LC-BOP since BH&HLYP is the hybrid functional with high proportion of the exact HF exchange. Although the core ionization energies are underestimated by ΔSCF due to SIE, STS gives larger core ionization energies than ΔSCF due to a concave behavior of the error curves of STS, which is also related to SIE. The mean absolute deviations of STS relative to ΔSCF, and relative to the experiment, are almost constant regardless of the nuclei among the element in the second period, and likewise among those in the third period. The systematic nature suggests that shifting the STS core ionization energies may be useful. We propose the shifted STS (1) for reproducing ΔSCF values, and the shifted STS (2) to reproduce the observed ones for KS calculations. Both schemes work quite well. The calculated results of KS ΔSCF and STS vary depending on the functional. However, the variation of each species' shifted STS (2) is very small, and all shifted STS (2) values are close to the observed ones. As the shifted STS require only one SCF calculation, they are simple and practical for predicting the core ionization energies.

Can range-separated functionals be optimally tuned to predict spectra and excited state dynamics in photoactive iron complexes?

Density functional theory is an efficient computational tool to investigate photophysical and photochemical processes in transition metal complexes, giving invaluable assistance in interpreting spectroscopic and catalytic experiments. Optimally tuned range-separated functionals are particularly promising, as they were created to address some of the fundamental deficiencies present in approximate exchange-correlation functionals. In this paper, we scrutinize the selection of optimally tuned parameters and its influence on the excited state dynamics, using the example of the iron complex [Fe(cpmp)₂]²⁺ with push-pull ligands. Various tuning strategies are contemplated based on pure self-consistent DFT protocols, as well as on the comparison with experimental spectra and multireference CASPT2 results. The two most promising sets of optimal parameters are then employed to carry out nonadiabatic surface-hopping dynamics simulations. Intriguingly, we find that the two sets lead to very different relaxation pathways and timescales. While the set of optimal parameters from one of the self-consistent DFT protocols predicts the formation of long-lived metal-to-ligand charge transfer triplet states, the set in better agreement with CASPT2 calculations leads to deactivation in the manifold of metal-centered states, in better agreement with the experimental reference data. These results showcase the complexity of iron-complex excited state landscapes and the difficulty of obtaining an unambiguous parametrization of long-range corrected functionals without experimental input.

Excited-State Properties of Some Thermally Activated Delayed Fluorescence Emitters: Quest for an Accurate and Reliable Computational Method

Thermally activated delayed fluorescence (TADF) finds application in organic light-emitting diodes. The molecules exhibiting TADF are characterized by small singlet-triplet energy gaps that help reverse intersystem crossing. Recently, ionization potential (IP)-tuned range-separated (RS) density functionals have been well accepted for studying excited-state properties. In the present work, two efficient descriptor-based tuning schemes [electron localization function (ELF) and Sol] of RS density functionals have been used to accurately reproduce the excited-state properties of TADF emitters by performing a single self-consistent field calculation. The lowest singlet vertical excitation energies (E_VA(S₁)) and the vertical singlet-triplet energy gaps (ΔE_VST) are computed with ELF-, Sol-, and IP-tuned RS functionals (LC-BLYP, ωB97, ωB97X, and ωB97XD). Encouraging mean absolute deviations from the experimental values with ELF*-, Sol*-, and IP-tuned functionals are observed. Consistent performance of the non-empirical tuned functionals is noted in different solvent dielectrics. In addition to these, fractional occupation calculations have shown that our tuned functionals almost satisfy the energy linearity curve. Thus, ELF*- and Sol*-tuned functionals are promising and reliable alternatives in computing the excited-state properties. Considering the small experimental singlet-triplet gap, we recommend ELF* to calculate E_VA(S₁) and Sol* to calculate ΔE_VST.

Meta-Hybrid Density Functional Theory Prediction of the Reactivity, Stability, and IGM of Azepane, Oxepane, Thiepane, and Halogenated Cycloheptane

The application of plain cycloalkanes and heterocyclic derivatives in the synthesis of valuable natural products and pharmacologically active intermediates has increased tremendously in recent times with much attention being paid to the lower cycloalkane members. The structural and molecular properties of higher seven-membered and nonaromatic heterocyclic derivatives are less known despite their stable nature and vast application; thus, an insight into their structural and electronic properties is still needed. Appropriate quantum chemical calculations utilizing the ab initio (MP2) method, meta-hybrid (M06-2X) functional, and long-range-separated functionals (ωB97XD) have been utilized in this work to investigate the structural reactivity, stability, and behavior of substituents on cycloheptane (CHP) and its derivatives: azepane, oxepane, thiepane, fluorocycloheptane (FCHP), bromocycloheptane (BrCHP), and chlorocycloheptane (ClCHP). Molecular global reactivity descriptors such as Fukui function, frontier molecular orbitals (FMOs), and molecular electrostatic potential were computed and compared with lower members. The results of two population methods CHELPG and Atomic Dipole Corrected Hirshfeld Charges (ADCH) were equally compared to scrutinize the charge distribution in the molecules. The susceptibility of intramolecular interactions between the substituents and cycloalkane ring is revealed by natural bond orbital analysis and intramolecular weak interactions by the independent gradient model (IGM). Other properties such as atomic density of states, intrinsic bond strength index (IBSI), and dipole moments are considered. It is acclaimed that the strain effect is a major determinant effect in the energy balance of cyclic molecules; thus, the ring strain energies and validation of spectroscopic specificities with reference to the X-ray crystallographic data are also considered.

Vertical ionization potential benchmarks from Koopmans prediction of Kohn-Sham theory with long-range corrected (LC) functional

The Kohn-Sham density functional theory (KS-DFT) with the long-range corrected (LC) functional is applied to the benchmark dataset of 401 valence ionization potentials (IPs) of 63 small molecules of Chong, Gritsenko and Baerends (the CGB set). The vertical IP of the CGB set are estimated as negative orbital energies within the context of the Koopmans' prediction using the LCgau-core range-separation scheme in combination with PW86-PW91 exchange-correlation functional. The range separation parameterμof the functional is tuned to minimize the error of the negative HOMO orbital energy from experimental IP. The results are compared with literature data, includingab initioIP variant of the equation-of-motion coupled cluster theory with singles and doubles (IP-EOM-CCSD), the negative orbital energies calculated by KS-DFT with the statistical averaging of orbital potential, and those with the QTP family of functionals. The optimally tuned LC functional performs better than other functionals for the estimation of valence level IP. The mean absolute deviations (MAD) from experiment and from IP-EOM-CCSD are 0.31 eV (1.77%) and 0.25 eV (1.46%), respectively. LCgau-core performs quite well even with fixedμ(not system-dependent). Aμvalue around 0.36 bohr^-1gives MAD of 0.40 eV (2.42%) and 0.33 eV (1.96%) relative to experiment and IP-EOM-CCSD, respectively. The LCgau-core-PW86-PW91 functional is an efficient alternative to IP-EOM-CCSD and it is reasonably accurate for outer valence orbitals. We have also examined its application to core ionization energies of C(1s), N(1s), O(1s) and F(1s). The C(1s) core ionization energies are reproduced reasonably [MAD of 46 cases is 0.76 eV (0.26%)] but N(1s), O(1s) and F(1s) core ionization energies are predicted less accurately.

Taking Advantage of a Systematic Energy Non-linearity Error in Density Functional Theory for the Calculation of Electronic Energy Levels

We present an approximate approach for the calculation of ionization potential (IP) and electron affinity (EA) by exploiting the complementary energy non-linearity errors for a species M and its one-electron-ionized counterpart (M⁺). Reasonable IPs and EAs are thus obtained by averaging the orbital energies of M and M⁺, even with a low-level method such as BLYP/6-31G(d). By combining the corrected IPs and EAs, we can further obtain reasonable excitation energies. The errors in uncorrected valence IPs and uncorrected virtual-orbital energies show systematic trends. These characteristics provide a convenient and computationally efficient avenue for qualitative estimation of these properties with single corrections for multiple IPs and excitation energies.

An improved Slater's transition state approximation

We have extended Slater's transition state concept for the approximation of the difference in total energies of the initial and final states by three orbital energies of initial, final, and half-way Slater's transition states of the system. Numerical validation was performed with the ionization energies for H₂O, CO, and pyrrole by calculation using Hartree-Fock (HF) and Kohn-Sham (KS) theories with the B3LYP and LCgau-core-BOP functionals. The present extended method reproduces full ΔSCF very accurately for all occupied orbitals obtained with HF and for valence orbitals obtained with KS. KS core orbitals have some errors due to the self-interaction errors, but the present method significantly improves the core electron binding energies. In its current form, the newly derived theory may not yet be practically useful, but it is simple and conceptually useful for gaining improved understanding of SCF-type orbital theories.