Evaluation of Local Hybrid Functionals for Electric Properties: Dipole Moments and Static and Dynamic Polarizabilities

Chemically Accurate Singlet-Triplet Gaps of Arylcarbenes from Local Hybrid Density Functionals

Singlet-triplet (ST) gaps are key descriptors of carbenes, because their properties and reactivity are strongly spin-dependent. However, the theoretical prediction of ST gaps is challenging and generally thought to require elaborate correlated wave function methods or double-hybrid density functionals. By evaluating two recent test sets of arylcarbenes (AC12 and AC18), we show that local hybrid functionals based on the "common t" local mixing function (LMF) model achieve mean absolute errors below 1 kcal/mol at a computational cost only slightly higher than that of global hybrid functionals. An analysis of correlation contributions to the ST gaps suggests that the accuracy of the common t-LMF model is mainly due to an improved description of nondynamical correlation which, unlike exchange, is not additive in each spin-channel. Although spin-nonadditivity can be achieved using the local spin polarization alone, using the "common", i.e., spin-unresolved, iso-orbital indicator t for constructing the LMF is found to be critical for consistent accuracy in ST gaps of arylcarbenes. The results support the view of LHs as vehicles to improve the description of nondynamical correlation rather than sophisticated exchange mixing approaches.

Toward the Next Generation of Density Functionals: Escaping the Zero-Sum Game by Using the Exact-Exchange Energy Density

ConspectusKohn-Sham density functional theory (KS DFT) is arguably the most widely applied electronic-structure method with tens of thousands of publications each year in a wide variety of fields. Its importance and usefulness can thus hardly be overstated. The central quantity that determines the accuracy of KS DFT calculations is the exchange-correlation functional. Its exact form is unknown, or better "unknowable", and therefore the derivation of ever more accurate yet efficiently applicable approximate functionals is the "holy grail" in the field. In this context, the simultaneous minimization of so-called delocalization errors and static correlation errors is the greatest challenge that needs to be overcome as we move toward more accurate yet computationally efficient methods. In many cases, an improvement on one of these two aspects (also often termed fractional-charge and fractional-spin errors, respectively) generates a deterioration in the other one. Here we report on recent notable progress in escaping this so-called "zero-sum-game" by constructing new functionals based on the exact-exchange energy density. In particular, local hybrid and range-separated local hybrid functionals are discussed that incorporate additional terms that deal with static correlation as well as with delocalization errors. Taking hints from other coordinate-space models of nondynamical and strong electron correlations (the B13 and KP16/B13 models), position-dependent functions that cover these aspects in real space have been devised and incorporated into the local-mixing functions determining the position-dependence of exact-exchange admixture of local hybrids as well as into the treatment of range separation in range-separated local hybrids. While initial functionals followed closely the B13 and KP16/B13 frameworks, meanwhile simpler real-space functions based on ratios of semilocal and exact-exchange energy densities have been found, providing a basis for relatively simple and numerically convenient functionals. Notably, the correction terms can either increase or decrease exact-exchange admixture locally in real space (and in interelectronic-distance space), leading even to regions with negative admixture in cases of particularly strong static correlations. Efficient implementations into a fast computer code (Turbomole) using seminumerical integration techniques make such local hybrid and range-separated local hybrid functionals promising new tools for complicated composite systems in many research areas, where simultaneously small delocalization errors and static correlation errors are crucial. First real-world application examples of the new functionals are provided, including stretched bonds, symmetry-breaking and hyperfine coupling in open-shell transition-metal complexes, as well as a reduction of static correlation errors in the computation of nuclear shieldings and magnetizabilities. The newest versions of range-separated local hybrids (e.g., ωLH23tdE) retain the excellent frontier-orbital energies and correct asymptotic exchange-correlation potential of the underlying ωLH22t functional while improving substantially on strong-correlation cases. The form of these functionals can be further linked to the performance of the recent impactful deep-neural-network "black-box" functional DM21, which itself may be viewed as a range-separated local hybrid.

Learning from the 4-(dimethylamino)benzonitrile twist: Two-parameter range-separated local hybrid functional with high accuracy for triplet and charge-transfer excitations

The recent ωLH22t range-separated local hybrid (RSLH) is shown to provide outstanding accuracy for the notorious benchmark problem of the two lowest excited-state potential energy curves for the amino group twist in 4-(dimethylamino)benzonitrile (DMABN). However, the design of ωLH22t as a general-purpose functional resulted in less convincing performance for triplet excitations, which is an important advantage of previous LHs. Furthermore, ωLH22t uses 8 empirical parameters to achieve broad accuracy. In this work, the RSLH ωLH23ct-sir is constructed with minimal empiricism by optimizing its local mixing function prefactor and range-separation parameter for only 8 excitation energies. ωLH23ct-sir maintains the excellent performance of ωLH22t for the DMABN twist and charge-transfer benchmarks but significantly improves the errors for triplet excitation energies (0.17 vs 0.24 eV). Additional test calculations for the AE6BH6 thermochemistry test set and large dipole moment and static polarizability test sets confirm that the focus on excitation energies in the optimization of ωLH23ct-sir has not caused any dramatic errors for ground-state properties. Although ωLH23ct-sir cannot replace ωLH22t as a general-purpose functional, it is preferable for problems requiring a universally good description of localized and charge-transfer excitations of both singlet and triplet multiplicity. Current limitations on the application of ωLH23ct-sir and other RSLHs to the study of singlet-triplet gaps of emitters for thermally activated delayed fluorescence are discussed. This work also includes the first systematic analysis of the influence of the local mixing function prefactor and the range-separation parameter in an RSLH on different types of excitations.

TURBOMOLE: Today and Tomorrow

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

The performance of CCSD(T) for the calculation of dipole moments in diatomics

This work analyzes the accuracy of the coupled cluster with single, double, and perturbative triple excitation [CCSD(T)] method for predicting dipole moments. In particular, we benchmark CCSD(T) predictions for the equilibrium bond length, vibrational frequency, and dipole moment versus accurate experimental data. As a result, we find that CCSD(T) leads to accurate dipole moments. However, in some cases, it disagrees with the experimental values, and the disagreement can not be satisfactorily explained via relativistic or multi-reference effects. Therefore, our results indicate that benchmark studies for energy and geometry properties do not accurately describe other electron density magnitudes.

Full Implementation, Optimization, and Evaluation of a Range-Separated Local Hybrid Functional with Wide Accuracy for Ground and Excited States

We report the first full and efficient implementation of range-separated local hybrid functionals (RSLHs) into the TURBOMOLE program package. This enables the computation of ground-state energies and nuclear gradients as well as excitation energies. Regarding the computational effort, RSLHs scale like regular local hybrid functionals (LHs) with system or basis set size and increase timings by a factor of 2-3 in total. An advanced RSLH, ωLH22t, has been optimized for atomization energies and reaction barriers. It is an extension of the recent LH20t local hybrid and is based on short-range PBE and long-range HF exchange-energy densities, a pig2 calibration function to deal with the gauge ambiguity of exchange-energy densities, and reoptimized B95c correlation. ωLH22t has been evaluated for a wide range of ground-state and excited-state quantities. It further improves upon the already successful LH20t functional for the GMTKN55 main-group energetics test suite, and it outperforms any global hybrid while performing close to the top rung-4 functional, ωB97M-V, for these evaluations when augmented by D4 dispersion corrections. ωLH22t performs excellently for transition-metal reactivity and provides good balance between delocalization errors and left-right correlation for mixed-valence systems, with a somewhat larger bias toward localized states compared to LH20t. It approaches the accuracy of the best local hybrids to date for core, valence singlet and triplet, and Rydberg excitation energies while improving strikingly on intra- and intermolecular charge-transfer excitations, comparable to the most successful range-separated hybrids available.

Local Hybrid Functional Applicable to Weakly and Strongly Correlated Systems

The recent idea (Wodyński, A.; Arbuznikov, A. V.; Kaupp M. J. Chem. Phys. 2021, 155, 144101) to augment local hybrid functionals by a strong-correlation (sc) factor obtained from the adiabatic connection in the spirit of the KP16 model has been extended and applied to generate the accurate sc-corrected local hybrid functional scLH22t. By damping small values of the ratio between nondynamical and dynamical correlation entering the correction factor, it has become possible to avoid double counting of nondynamical correlation for weakly correlated situations and thereby preserve the excellent accuracy of the underlying LH20t local hybrid for such cases almost perfectly. On the other hand, scLH22t improves substantially over LH20t in reducing fractional-spin errors (FSEs), in providing improved spin-restricted bond dissociation curves, and in treating some typical systems with multireference character. The obtained FSEs are similar to those of the KP16/B13 model and slightly larger than for B13, but performance for weakly correlated systems is better than for these two related methods, which are also difficult to use self-consistently. The recent DM21 functional based on the training of a deep neural network still performs somewhat better than scLH22t but allows no physical insights into the origins of reduced FSEs. Examination of local mixing functions (LMFs) for the corrected scLH22t and uncorrected LH20t functionals provides further insights: in weakly correlated situations, the LMF remains essentially unchanged. Strong-correlation effects manifest in a reduction of the LMF values in certain regions of space, even to the extent of producing negative LMF values. It is suggested that this is the mechanism by which also DM21, which may be viewed as a range-separated local hybrid, is able to reduce FSEs.

Benchmarking Semiempirical QM Methods for Calculating the Dipole Moment of Organic Molecules

The dipole moment is a simple descriptor of the charge distribution and polarity and is important for understanding and predicting various molecular properties. Semiempirical (SE) methods offer a cost-effective way to calculate dipole moment that can be used in high-throughput screening applications although the accuracy of the methods is still in question. Therefore, we have evaluated AM1, GFN0-xTB, GFN1-xTB, GFN2-xTB, PM3, PM6, PM7, B97-3c, HF-3c, and PBEh-3c SE methods, which cover a variety of SE approximations, to directly assess the performance of SE methods in predicting molecular dipole moments and their directions using 7211 organic molecules contained in the QM7b database. We find that B97-3c and PBEh-3c perform best against coupled-cluster reference values yielding dipole moments with a mean absolute error (MAE) of 0.10 and 0.11 D, respectively, which is similar to the MAE of density functional theory (DFT) methods (∼0.1 D) reported earlier. Analysis of the atomic composition shows that all SE methods show good performance for hydrocarbons for which the spread of error was within 1 D of the reference data, while the worst performances are for sulfur-containing compounds for which only B97-3c and PBEh-3c show acceptable performance. We also evaluate the effect of SE optimized geometry, instead of the benchmark DFT geometry, that shows a dramatic drop in the performance of AM1 and PM3 for which the range of error tripled. Based on our overall findings, we highlight that there is an optimal compromise between accuracy and computational cost using GFN2-xTB (MAE: 0.25 D) that is 3 orders of magnitude faster than B97-3c and PBEh-3c. Thus, we recommend using GFN2-xTB for cost-efficient calculation of the dipole moment of organic molecules containing C, H, O, and N atoms, whereas, for sulfur-containing organic molecules, we suggest PBEh-3c.

Assessment of hybrid functionals for singlet and triplet excitations: Why do some local hybrid functionals perform so well for triplet excitation energies?

The performance of various hybrid density functionals is assessed for 105 singlet and 105 corresponding triplet vertical excitation energies from the QUEST database. The overall lowest mean absolute error is obtained with the local hybrid (LH) functional LH12ct-SsirPW92 with individual errors of 0.11 eV (0.11 eV) for singlet (triplet) n → π* excitations and 0.29 eV (0.17 eV) for π → π* excitations. This is slightly better than with the overall best performing global hybrid M06-2X [n → π*: 0.13 eV (0.17 eV), π → π*: 0.30 eV (0.20 eV)], while most other global and range-separated hybrids and some LHs suffer from the "triplet problem" of time-dependent density functional theory. This is exemplified by correlating the errors for singlet and triplet excitations on a state-by-state basis. The excellent performance of LHs based on a common local mixing function, i.e., an LMF constructed from the spin-summed rather than the spin-resolved semilocal quantities, is systematically investigated by the introduction of a spin-channel interpolation scheme that allows us to continuously modulate the fraction of opposite-spin terms used in the LMF. The correlation of triplet and singlet errors is systematically improved for the n → π* excitations when larger fractions of the opposite-spin-channel are used in the LMF, whereas this effect is limited for the π → π* excitations. This strongly supports a previously made hypothesis that attributes the excellent performance of LHs based on a common LMF to cross-spin-channel nondynamical correlation terms.

Reliable TDDFT Protocol Based on a Local Hybrid Functional for the Prediction of Vibronic Phosphorescence Spectra Applied to Tris(2,2'-bipyridine)-Metal Complexes

An efficient computational protocol for the prediction of vibrationally resolved phosphorescence spectra is developed and validated for five tris(2,2'-bipyridine)-metal complexes ([M(bpy)₃]ⁿ⁺, where M = Zn, Ru, Rh, Os, Ir). The outstanding feature of this protocol is the use of full linear-response time-dependent density functional theory (TDDFT) for the excited-state triplet calculation, i.e., the commonly seen strategies employing the Tamm-Dancoff approximation (TDA) or unrestricted density functional theory (DFT) calculations for the T₁ state are not needed. This is achieved by the use of a local hybrid functional (LH12ct-SsirPW92) that features a real-space dependent admixture of exact exchange governed by a local mixing function. The excellent performance of this LH for triplet excitation energies known from previous studies transfers to a remarkable mean absolute error of 0.06 eV for the phosphorescence 0-0 energies investigated herein, while the popular B3PW91 functional gives an error of 0.27 eV in TDDFT and 0.09 eV in unrestricted DFT calculations, respectively. The advantages of the local hybrid are particularly apparent for excited states with a mixed-valence character. The influence of spin-orbit coupling was found to be significant for [Os(bpy)₃]²⁺ red-shifting the 0-0 energy for phosphorescence by 0.17 eV, while the effect is negligible for the other complexes (<0.03 eV). The influence of the basis-set and integration-grid sizes is evaluated, and a computationally lighter protocol is validated that leads to drastic savings in computation time with negligible loss in accuracy.

Molecular Collision Diameters and Electronic Polarizabilities: Inherent Relationship and Fast Evaluation

The collision diameter σ for a large set of molecular species is related to the static electronic polarizability α_el. A remarkable correlation between these quantities conceptually similar to the analogous one previously identified for atoms is revealed. Our recommended model is the function σ(α_el) = p₁ + p₂α_el^1/3, where p₁ = 0.768 Å and p₂ = 2.168 are the fitting parameters providing the best overall match to the reference data for collision diameter (181 data points). The obtained correlation allows one to easily find the collision diameter of molecules from the known polarizability and vice versa. These findings can be useful for many applications, where there is a need for inexpensive assessments of the collision diameters or electronic polarizabilities, for example, when developing the transport property databases for modeling of chemically reacting flows.