Toward a correct treatment of core properties with local hybrid functionals.

In local hybrid functionals (LHs), a local mixing function (LMF) determines the position-dependent exact-exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt-LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn-Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high-density and iso-orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K-shell core excitations of second- and third-period systems including accurate state-of-the-art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange-energy densities, as well as by B95c meta-GGA correlation. The resulting LH23pt functional relates to the previous LH20t functional but specifically improves upon the core region.

Assessing locally range-separated hybrid functionals from a gradient expansion of the exchange energy density.

Locally range-separated hybrid (LRSH) functionals feature a real-space-dependent range separation function (RSF) instead of a system-independent range-separation parameter, which thus enables a more flexible admixture of exact exchange than conventional range-separated hybrid functionals. In particular, the development of suitable RSF models and exploring the capabilities of the LRSH approach, in general, are tasks that require further investigations and will be addressed in this work. We propose a non-empirical scheme based on a detailed scaling analysis with respect to a uniform coordinate scaling and on a short-range expansion of the range-separated exchange energy density to derive new RSF models from a gradient expansion of the exchange energy density. After optimizing a small set of empirical parameters introduced to enhance their flexibility, the resulting second- and fourth-order RSFs are evaluated with respect to atomic exchange energies, atomization energies, and transition barrier heights.

A Local Hybrid Functional with Wide Applicability and Good Balance between (De)Localization and Left-Right Correlation.

A new local hybrid functional, LH20t, with a position-dependent exact-exchange admixture governed by a simple local mixing function (g(r) = b·τW(r)/τ(r)), combined with gradient-corrected (PBE) exchange and meta-GGA (B95) correlation, as well as a second-order GGA-based pig2 calibration function to address the ambiguity of exchange-energy densities, has been constructed. The adjustable parameters of LH20t have been optimized in a multistep procedure based on thermochemical kinetics data and measures of spurious nondynamical correlation. LH20t has subsequently been evaluated for the full GMTKN55 main-group energetics test suite, with and without an added DFT-D4 dispersion correction. Performance of the new functional in the GMTKN55 tests is excellent, better than any global hybrid so far, approaching the best results for any rung-4 functional, without any noticeable artifacts due to the gauge ambiguity. The robust performance across the board is combined with enhanced exact-exchange admixtures of >70% near the nuclei and asymptotically (but low admixture in bonds). This helps to provide excellent performance for a wide variety of excitation classes (core, valence singlet and triplet, Rydberg, short-range intervalence charge-transfer) in TDDFT evaluations. Notably, LH20t is the first functional that provides simultaneously the correct description for the most extreme localized and delocalized cases of the MVO-10 test set of gas-phase mixed-valence systems. This outstanding performance for mixed-valence systems, which signals a very fine balance between reduced delocalization errors and a reasonable description of left-right correlation, is corroborated by tests on ground- and excited-state properties for organic and organometallic mixed-valence systems in solution.

Relativistic local hybrid functionals and their impact on 1s core orbital energies.

In this work, we report the first relativistic density functional theory calculations using relativistic local hybrid functionals. Besides outlining the construction of relativistic local hybrid exchange within a two-component-relativistic framework based on the picture-change transformation of the density matrix and a recently developed relativistic iso-orbital indicator, we investigate the influence of two-electron-relativistic effects, using relativistic functional ingredients in local hybrid functionals, and the choice of the exchange-correlation functional on atomic 1s core orbital energies of light and heavier elements. Finally, we discuss the applicability of relativistic 1s core orbital shifts for the relativistic correction of non-relativistic 1s core excitation energies.

Restoring the iso-orbital limit of the kinetic energy density in relativistic density functional theory.

In contrast to nonrelativistic density functional theory, the ratio between the von Weizsäcker and the Kohn-Sham kinetic energy density, commonly used as iso-orbital indicator t within exchange-correlation functionals beyond the generalized-gradient level, violates the exact iso-orbital limit and the appropriate parameter range, 0 ≤ t ≤ 1, in relativistic density functional theory. Based on the exact decoupling procedure within the infinite-order two-component method and the Cauchy-Schwarz inequality, we present corrections to the relativistic and the picture-change-transformed nonrelativistic kinetic energy density that restores these exact constraints. We discuss the origin of the new correction terms and illustrate the effectiveness of the current approach for several representative cases. The proposed generalized iso-orbital indicator tλ is expected to be a useful ingredient for the development of relativistic exchange-correlation functionals.

Efficient Semi-Numerical Implementation of Relativistic Exact Exchange within the Infinite-Order Two-Component Method Using a Modified Chain-of-Spheres Method.

We present an efficient implementation of relativistic exact exchange within the infinite-order two-component method (IOTC) by employing a state-of-the-art seminumerical integration technique. For accurate consideration of the picture change, inherent to two-component methods, we propose a new scheme based on a relativistic or picture-change transformation of the density matrix, which provides a simple and efficient formulation of relativistically transformed quantities such as the electron density or exact exchange and thus avoids expensive integral transformations. We show that the new scheme does not introduce additional numerical or theoretical errors beyond the approximations of the IOTC method. For the efficient implementation of exact-exchange integrals, we build upon a modified version of the chain-of-spheres exact-exchange (COSX) method. In addition to the conventional overlap and density matrix screening by S- and P-junctions, respectively, we introduce a new simple screening technique in the sense of the original COSX method by additionally considering the asymptotic decay of the integrals over the Coulomb operator within the new F-junctions. Together with the picture-change transformation of the density matrix, this modified COSX method is shown to provide superior efficiency for the calculation of relativistic exact exchange compared to a conventional analytical direct self-consistent-field implementation of exact exchange.

Lessons from the Spin-Polarization/Spin-Contamination Dilemma of Transition-Metal Hyperfine Couplings for the Construction of Exchange-Correlation Functionals.

Hyperfine couplings (HFCs) of open-shell transition-metal centers are known to often depend crucially on core-shell spin polarization (CSSP). The latter is typically underestimated by semilocal density functionals, while admixture of exact exchange (EXX) in (global) hybrid functionals enhances CSSP. Unfortunately, a metal-ligand antibonding character of one or more of the singly occupied molecular orbitals of the complex will cause substantial valence-shell spin polarization (VSSP), which for global hybrids with higher EXX admixtures may lead to substantial spin contamination, thereby deteriorating the overall electronic structure and the dipolar couplings. In view of this known dilemma, we use a subset of 3d complexes from an earlier study (M. Munzarová, M. Kaupp J. Phys. Chem. A 1999, 103, 9966-9983) to examine systematically a wide range of exchange-correlation functionals for metal HFCs, including highly parametrized (meta-)GGAs, global, and range-separated hybrid functionals not yet available in earlier studies, as well as for the first time local hybrids with real-space position-dependent EXX admixture. Both CSSP and VSSP have been carefully analyzed in terms of their orbital contributions, both for cases dominated only by CSSP and for systems influenced crucially by VSSP and spin contamination. While some more parametrized meta-GGA functionals (τ-HCTH, VSXC, partially M06-L) provide surprisingly realistic CSSP, some others (MN12-L, MN15-L) and some global hybrids (M05, M06, partly MN15) exhibit dramatic shortcomings in describing the CSSP contributions. Local hybrid functionals provide a promising way of enhancing CSSP by high EXX admixture in the core region while avoiding excessive VSSP and thus spin contamination. These analyses provide important insights that may help to construct improved functionals for HFCs and related properties (e.g., contact NMR shifts).

Development of a TDDFT-Based Protocol with Local Hybrid Functionals for the Screening of Potential Singlet Fission Chromophores.

Chromophores suitable for singlet fission need to meet specific requirements regarding the relative energies of their S0, S1, and T1 (and T2) electronic states. Accurate quantum-chemical computations of the corresponding energy differences are thus highly desirable for materials design. Methods based on density functional theory (DFT) have the advantage of being applicable to larger, often more relevant systems compared to more sophisticated post-Hartree-Fock methods. However, most exchange-correlation functionals do not provide the needed accuracy, in particular, due to an insufficient description of the T1 state. Here we use a recent singlet fission chromophore test set ( Wen , J. ; Havlas , Z. ; Michl , J. J. Am. Chem. Soc. 2015 , 137 , 165 - 172 ) to evaluate a wide range of DFT-based methods, with an emphasis on local hybrid functionals with a position-dependent exact-exchange admixture. New reference vertical CC2/CBS benchmark excitation energies for the test set have been generated, which exhibit somewhat more uniform accuracy than the previous CASPT2-based data. These CC2 reference data have been used to evaluate a wide range of functionals, comparing full linear-response TDDFT, the Tamm-Dancoff approximation (TDA), and ΔSCF calculations. Two simple two-parameter local hybrid functionals and the more empirical M06-2X global meta-GGA hybrid provide the overall best accuracy. Due to its lower empiricism and wide applicability, the Lh12ct-SsifPW92 local hybrid is suggested as the main ingredient of an efficient computational protocol for prediction of the relevant excitation energies in singlet fission chromophores. Full TDDFT for the S1, S2, and T2 excitations is combined with ΔSCF for the T1 excitations. Making use also of some error compensation with suitable DFT-optimized structures, even the most critical T1 excitations can be brought close to the target accuracy of 0.20 eV, while the other excitation energies are obtained even more accurately. This fully DFT-based protocol should become a useful tool in the field of singlet fission.

New approaches for the calibration of exchange-energy densities in local hybrid functionals.

The ambiguity of exchange-energy densities is a fundamental challenge for the development of local hybrid functionals, or of other functionals based on a local mixing of exchange-energy densities. In this work, a systematic construction of semi-local calibration functions (CFs) for adjusting the exchange-energy densities in local hybrid functionals is provided, which directly links a given CF to an underlying semi-local exchange functional, as well as to the second-order gradient expansion of the exchange hole. Using successive steps of integration by parts allows the derivation of correction terms of increasing order, resulting in more and more complicated but also more flexible CFs. We derive explicit first- and second-order CFs (pig1 and pig2) based on B88 generalized-gradient approximation (GGA) exchange, and a first-order CF (tpig1) based on τ-dependent B98 meta-GGA exchange. We combine these CFs with different long-range damping functions and evaluate them for calibration of LDA, B88 GGA, and TPSS meta-GGA exchange-energy densities. Based on a minimization of unphysical nondynamical correlation contributions in three noble-gas dimer potential-energy curves, free parameters in the CFs are optimized, and performance of various approaches in the calibration of different exchange-energy densities is compared. Most notably, the second-order pig2 CF provides the largest flexibility with respect to the diffuseness of the damping function. This suggests that higher-order CFs based on the present integration-by-parts scheme may be particularly suitable for the flexible construction of local hybrid functionals.

Validation of local hybrid functionals for TDDFT calculations of electronic excitation energies.

The first systematic evaluation of local hybrid functionals for the calculation of electronic excitation energies within linear-response time-dependent density functional theory (TDDFT) is reported. Using our recent efficient semi-numerical TDDFT implementation [T. M. Maier et al., J. Chem. Theory Comput. 11, 4226 (2015)], four simple, thermochemically optimized one-parameter local hybrid functionals based on local spin-density exchange are evaluated against a database of singlet and triplet valence excitations of organic molecules, and against a mixed database including also Rydberg, intramolecular charge-transfer (CT) and core excitations. The four local hybrids exhibit comparable performance to standard global or range-separated hybrid functionals for common singlet valence excitations, but several local hybrids outperform all other functionals tested for the triplet excitations of the first test set, as well as for relative energies of excited states. Evaluation for the combined second test set shows that local hybrids can also provide excellent Rydberg and core excitations, in the latter case rivaling specialized functionals optimized specifically for such excitations. This good performance of local hybrids for different excitation types could be traced to relatively large exact-exchange (EXX) admixtures in a spatial region intermediate between valence and asymptotics, as well as close to the nucleus, and lower EXX admixtures in the valence region. In contrast, the tested local hybrids cannot compete with the best range-separated hybrids for intra- and intermolecular CT excitation energies. Possible directions for improvement in the latter category are discussed. As the used efficient TDDFT implementation requires essentially the same computational effort for global and local hybrids, applications of local hybrid functionals to excited-state problems appear promising in a wide range of fields. Influences of current-density dependence of local kinetic-energy dependent local hybrids, differences between spin-resolved and "common" local mixing functions in local hybrids, and the effects of the Tamm-Dancoff approximation on the excitation energies are also discussed.

Efficient Semi-numerical Implementation of Global and Local Hybrid Functionals for Time-Dependent Density Functional Theory.

Local hybrid functionals with position-dependent exact-exchange admixture offer increased flexibility compared to global hybrids. For sufficiently advanced functionals of this type, this is expected to hold also for a wide range of electronic excitations within time-dependent density functional theory (TDDFT). Following a recent semi-numerical implementation of local hybrid functionals for ground-state self-consistent-field calculations (Bahmann, H.; Kaupp, M. J. Chem. Theory Comput. 2015, 11, 1540-1548), the first linear-response TDDFT implementation of local hybrids is reported, using a semi-numerical integration technique. The timings and accuracy of the semi-numerical implementation are evaluated by comparison with analytical schemes for time-dependent Hartree-Fock (TDHF) and for the TPSSh global hybrid. In combination with the RI approximation to the Coulomb part of the kernel, the semi-numerical implementation is faster than the existing analytical TDDFT/TDHF implementation of global hybrid functionals in the TURBOMOLE code, even for small systems and moderate basis sets. Moreover, timings for global and local hybrids are practically equal for the semi-numerical scheme. The way to TDDFT calculations with local hybrid functionals for large systems is thus now open, and more sophisticated parametrizations of local hybrids may be evaluated.

The vibrational spectrum of FeO2(+) isomers--theoretical benchmark and experiment.

Infrared photodissociation is used to record the vibrational spectrum of FeO2 (+)(He)2-4 which shows three bands at 1035, 980, and 506 cm(-1). Quantum chemical multi-reference configuration interaction calculations (MRCISD) of structures and harmonic frequencies show that these bands are due to two different isomers, an inserted dioxo complex with Fe in the +V oxidation state and a side-on superoxo complex with Fe in the +II oxidation state. These two are separated by a substantial barrier, 53 kJ/mol, whereas the third isomer, an end-on complex between Fe(+) and an O2 molecule, is easily converted into the side-on complex. For all three isomers, states of different spin multiplicity have been considered. Our best energies are computed at the MRCISD+Q level, including corrections for complete active space and basis set extension, core-valence correlation, relativistic effects, and zero-point vibrational energy. The average coupled pair functional (ACPF) yields very similar energies. Density functional theory (DFT) differs significantly from our best estimates for this system, with the TPSS functional yielding the best results. The other functionals tested are BP86, PBE, B3LYP, TPSSh, and B2PLYP. Complete active space second order perturbation theory (CASPT2) performs better than DFT, but less good than ACPF.