Importance of the correlation contribution for local hybrid functionals: range separation and self-interaction corrections

Local hybrid functionals with their position-dependent exact-exchange admixture are a conceptually simple and promising extension of the concept of a hybrid functional. Local hybrids based on a simple mixing of the local spin density approximation (LSDA) with exact exchange have been shown to be successful for thermochemistry, reaction barriers, and a range of other properties. So far, the combination of this generation of local hybrids with an LSDA correlation functional has been found to give the most favorable results for atomization energies, for a range of local mixing functions (LMFs) governing the exact-exchange admixture. Here, we show that the choice of correlation functional to be used with local hybrid exchange crucially influences the parameterization also of the exchange part as well as the overall performance. A novel ansatz for the correlation part of local hybrids is suggested based on (i) range-separation of LSDA correlation into short-range (SR) and long-range (LR) parts, and (ii) partial or full elimination of the one-electron self-correlation from the SR part. It is shown that such modified correlation functionals allow overall larger exact exchange admixture in thermochemically competitive local hybrids than before. This results in improvements for reaction barriers and for other properties crucially influenced by self-interaction errors, as demonstrated by a number of examples. Based on the range-separation approach, a fresh view on the breakdown of the correlation energy into dynamical and non-dynamical parts is suggested.     

Evaluation of range-separated hybrid density functionals for the prediction of vibrational frequencies, infrared intensities, and Raman activities

We present an assessment of different density functionals, with emphasis on range-separated hybrids, for the prediction of fundamental and harmonic vibrational frequencies, infrared intensities, and Raman activities. Additionally, we discuss the basis set convergence of vibrational properties of H2O with long-range corrected hybrids. Our results show that B3LYP is the best functional for predicting vibrational frequencies (both fundamental and harmonic); the screened-PBE hybrid (HSE) density functional works best for infrared intensities, and the long-range corrected PBE (LC-omegaPBE), M06-HF, and M06-L density functionals are almost as good as MP2 for predicting Raman activities. We show the predicted Raman spectrum of adenine as an example of a medium-size molecule where a DFT/Sadlej pVTZ calculation is affordable and compare our results against the experimental spectrum.     

Many-electron self-interaction and spin polarization errors in local hybrid density functionals

Errors for systems with noninteger occupation have been connected to common failures of density functionals. Previously, global hybrids and pure density functionals have been investigated for systems with noninteger charge and noninteger spin state. Local hybrids have not been investigated for either of those systems to the best of our knowledge. This study intends to close this gap. We investigate systems with noninteger charge to assess the many-electron self-interaction error and systems with noninteger spin state to assess the spin polarization error of recently proposed local hybrids and their range-separated variants. We find that long-range correction is very important to correct for many-electron self-interaction error in cations, whereas most full-range local hybrids seem to be sufficient for anions, where long-range-corrected density functionals tend to overcorrect. On the other hand, while all hitherto proposed long-range-corrected density functionals show large spin polarization errors, the Perdew-Staroverov-Tao-Scuseria (PSTS) functional performs best of all local hybrids in this case and shows an outstanding behavior for the dependence of the energy on the spin polarization.     

Evaluation of a combination of local hybrid functionals with DFT-D3 corrections for the calculation of thermochemical and kinetic data

Due to their position-dependent exact exchange admixture, local hybrid functionals offer a higher flexibility and thus the potential for more universal and accurate exchange correlation functionals compared to global hybrids with a constant admixture, as has been demonstrated in previous work. Yet, the local hybrid constructions used so far do not account for the inclusion of dispersion-type interactions. As a first exploratory step toward a more general approach that includes van der Waals-type interactions with local hybrids, the present work has added DFT-D3-type corrections to a number of simple local hybrid functionals. Optimization of only the s(8) and s(r,6) parameters for the S22 set provides good results for weak interaction energies but deteriorates the excellent performance of the local hybrids for G3 atomization energies and for classical reaction barriers. A combined optimization of the two DFT-D3 parameters with one of the two parameters of the spin-polarized local mixing function (LMF) of a local hybrid for a more general optimization set provides simultaneously accurate dispersion energies, improved atomization energies, and accurate reaction barriers, as well as excellent alkane protobranching ratios. For other LMFs, the improvements of such a combined optimization for the S22 energies have been less satisfactory. The most notable advantage of the dispersion-corrected local hybrids over, for example, a B3LYP-D3 approach, is in the much more accurate reaction barriers.     

Analytical nuclear excited‐state gradients for the Tamm–Dancoff approximation using uncoupled frozen‐density embedding

We report the derivation and implementation of analytical nuclear gradients for excited states using time‐dependent density functional theory using the Tamm–Dancoff approximation combined with uncoupled frozen‐density embedding using density fitting. Explicit equations are presented and discussed. The implementation is able to treat singlet as well as triplet states and functionals using the local density approximation, the generalized gradient approximation, combinations with Hartree–Fock exchange (hybrids), and range‐separated functionals such as CAM‐B3LYP. The new method is benchmarked against supermolecule calculations in two case studies: The solvatochromic shift of the (vertical) fluorescence energy of 4‐aminophthalimide on solvation, and the first local excitation of the benzonitrile dimer. Whereas for the 4‐aminophthalimide–water complex deviations of about 0.2 eV are obtained to supermolecular calculations, for the benzonitrile dimer the maximum error for adiabatic excitation energies is below 0.01 eV due to a weak coupling of the subsystems. © 2017 Wiley Periodicals, Inc.     

Local hybrid functionals: an assessment for thermochemical kinetics

Local hybrid functionals with position-dependent exact-exchange admixture are a new class of exchange-correlation functionals in density functional theory that promise to advance the available accuracy in many areas of application. Local hybrids with different local mixing functions (LMFs) governing the position dependence are validated for the heats of formation of the extended G3/99 set, and for two sets of barriers of hydrogen-transfer and heavy-atom transfer reactions (HTBH38 and NHTBH38 databases). A simple local hybrid Lh-SVWN with only Slater and exact exchange plus local correlation and a one-parameter LMF, g(r)=b(tau(W)(r)tau(r)), performs best and provides overall mean absolute errors for thermochemistry and kinetics that are a significant improvement over standard state-of-the-art global hybrid functionals. In particular, this local hybrid functional does not suffer from the systematic deterioration that standard functionals exhibit for larger molecules. In contrast, local hybrids based on generalized gradient approximation exchange tend to give rise to nonintuitive LMFs, and no improved functionals have been obtained along this route. The LMF is a real-space function and thus can be analyzed in detail. We use, in particular, graphical analyses to rationalize the performance of different local hybrids for thermochemistry and reaction barriers.     

Performance of conventional and range-separated hybrid density functionals in calculations of electronic circular dichroism spectra of transition metal complexes

A number of density functionals, including 'pure' (nonhybrid) functionals, global hybrids, and range-separated hybrids, were used to calculate the electronic circular dichroism (CD) spectra of 10 tris-bidentate transition metal complexes. The results are compared to one another and to experimental CD spectra, in an effort to illustrate the shortcomings of particular approximations in time-dependent density functional theory (TDDFT). The use of an origin invariant formalism to calculate magnetic transition dipole moments with the help of gauge-including atomic orbitals (GIAOs) is also investigated. With valence basis sets of moderate flexibility, good agreement between GIAO results and rotatory strengths calculated from the dipole-velocity representation is obtained for selected test cases. Empirically broadened vertical CD spectra calculated with the global hybrid functionals B3LYP and PBE0 are found to agree overall the best with experimental CD spectra.     

A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions

A thorough energy benchmark study of various density functionals (DFs) is carried out with the new GMTKN30 database for general main group thermochemistry, kinetics and noncovalent interactions [Goerigk and Grimme, J. Chem. Theor. Comput., 2010, 6, 107; Goerigk and Grimme, J. Chem. Theor. Comput., 2011, 7, 291]. In total, 47 DFs are investigated: two LDAs, 14 GGAs, three meta-GGAs, 23 hybrids and five double-hybrids. Besides the double-hybrids, also other modern approaches, i.e., the M05 and M06 classes of functionals and range-separated hybrids, are tested. For almost all functionals, the new DFT-D3 correction is applied in order to consistently test the performance also for important noncovalent interactions; the parameters are taken from previous works or determined for the present study. Basis set and quadrature grid issues are also considered. The general aim of the study is to work out which functionals are generally well applicable and robust to describe the energies of molecules. In summary, we recommend on the GGA level the B97-D3 and revPBE-D3 functionals. The best meta-GGA is oTPSS-D3 although meta-GGAs represent in general no clear improvement compared to numerically simpler GGAs. Notably, the widely used B3LYP functional performs worse than the average of all tested hybrids and is also very sensitive to the application of dispersion corrections. We discourage its usage as a standard method without closer inspection of the results, as it still seems to be often done nowadays. Surprisingly, long-range corrected exchange functionals do in general not perform better than the corresponding standard hybrids. However, the ωB97X-D functional seems to be a promising method. The most robust hybrid is Zhao and Truhlar's PW6B95 functional in combination with DFT-D3. If higher accuracy is required, double-hybrids should be applied. The corresponding DSD-BLYP-D3 and PWPB95-D3 variants are the most accurate and robust functionals of the entire study. Additional calculations with MP2 and and its spin-scaled variants SCS-MP2, S2-MP2 and SOS-MP2 revealed that double-hybrids in general outperform those. Only SCS-MP2 can be recommended, particularly for reaction energies. We suggest its usage when a large self-interaction error is expected that prohibits usage of double-hybrids. Perdews' metaphoric picture of Jacob's Ladder for the classification of density functionals' performance could unbiasedly be confirmed with GMTKN30. We also show that there is no statistical correlation between a functional's accuracy for atomization energies and the performance for chemically more relevant reaction energies.     

Effect of the Perdew-Zunger self-interaction correction on the thermochemical performance of approximate density functionals

The Perdew-Zunger self-interaction-corrected density functional theory (SIC-DFT) was implemented self-consistently using a quasi-Newton direct minimization method. We calculated SIC-DFT energies for a number of atoms and molecules using various approximate density functionals, including hybrids. Self-interaction errors (SIE) of these functionals were compared and analyzed in terms of contributions from valence and core orbitals. We also calculated enthalpies of formation of the standard G2-1 set of 55 molecules and found that self-interaction-correction (SIC) improves agreement with experiment only for the LSDA functional, while all other functionals show worse performance upon introducing SIC. This is the first systematic study of the effect of SIC on thermochemical properties. We found no direct connection between the magnitude of the SIE contained in a functional and its performance for thermochemistry. Approximate functionals with large self-interaction errors can accurately reproduce enthalpies of formation. Our results do not support the popular belief that a smaller SIE of hybrid functionals is the main reason for their higher accuracy.     

Modeling continuous changes in substituent electronegativity and chemical hardness using fictitious nuclear potentials

The performances of a family of recently developed generalized gradient approximation (GGA) functionals based on the Tognetti–Cortona–Adamo (TCA) family and making use of the gradient-regulated connection (GRAC) approach are here tested on an uncommon benchmark set for the prediction of transition state (TS) structures and energies of a series of four reactions involving an early transition metal (Zr, d ⁰). This benchmark test thus represents the first step in the organometallic world in which d ⁿ ions allowing complex phenomena such as spin crossover represent the higher level of complexity. The results obtained show that the performances of the GRAC-xxx functionals are comparable to those of global hybrid functionals both in the prediction of reaction barriers and of structural features of TSs. More complex functional forms (such as range-separated hybrids) in average enhance the energetic features, but not necessarily the overall accuracy on calculated structures. On the other hand, and as expected, purposely developed functionals for the prediction of chemical reactivity provide both structural and energetic features in good agreement with post-HF results. The present study, besides proving the good performances of GGA functionals of the GRAC-TCA family for the prediction of TS structural parameters and energetics of metal containing systems, also underlines the importance of the use of diversified benchmark sets to allow a fair evaluation of functionals performances.     

Assessment and validation of a screened Coulomb hybrid density functional

This paper presents a revised and improved version of the Heyd-Scuseria-Ernzerhof screened Coulomb hybrid functional. The performance of this functional is assessed on a variety of molecules for the prediction of enthalpies of formation, geometries, and vibrational frequencies, yielding results as good as or better than the successful PBE0 hybrid functional. Results for ionization potentials and electron affinities are of slightly lower quality but are still acceptable. The comprehensive test results presented here validate our assumption that the screened, short-range Hartree-Fock (HF) exchange exhibits all physically relevant properties of the full HF exchange. Thus, hybrids can be constructed which neglect the computationally demanding long-range part of HF exchange while still retaining the superior accuracy of hybrid functionals, compared to pure density functionals.     

Theoretical prediction of valence and Rydberg excited states: Minnesota exchange-correlation functionals vs symmetry adapted cluster-configuration interaction

During this contribution, we present a benchmark investigation on the applicability of several Minnesota functionals from various classes like local meta-generalized and meta-nonseparable gradient approximations, hybrids, and range-separated hybrids for describing the valence and Rydberg excitation energies of some organic compounds from different categories. Furthermore, the performances of Minnesota density functionals from density functional theory are also assessed against a wave function theory based approach in the context of excite states calculations, symmetry adapted cluster-configuration interaction (SAC-CI) method. Pragmatically, the singles and doubles linked excitation operators are considered in the SAC-CI wave functions. With more or less different accountabilities of the considered methods, it is shown that the M06-2X, M05-2X, and M11 functionals have the best performances for valence excited states. On the other hand, for Rydberg excited states although the SAC-CI method outperforms others, the statistical analyses reveal that the efficiency of some Minnesota functionals is also respectable.     

Density functional theory calculations of dynamic first hyperpolarizabilities for organic molecules in organic solvent: comparison to experiment

Against experimental values obtained from solution-phase dc electric field induced second-harmonic generation measurements at a fundamental wavelength of 1910 nm, the performance of 20 exchange-correlation functionals in density functional theory in evaluation of solvent modulated dynamic first hyperpolarizabilities of 82 organic molecules in chloroform, 1,4-dioxane, and/or dichloromethane was evaluated. The used exchange-correlation functionals consisted of generalized gradient approximation (GGA), meta-GGA, global hybrids, and range-separated hybrids. The PCM-X/6-311+G(2d,p)//PCM-B3LYP/6-31G(2df,p) level of theory was employed. The calculated results showed functionals with the exact asymptote of the exchange potential gave satisfying linear correlation with R(2) of 0.95 between experimental data and theoretical values. With a linear correction, these functionals also provided a better accuracy with mean absolute error of 5 × 10(-30) esu than other functionals. The solvent effect and solvation scheme on the calculated property were also studied.     

The role of range-separated Hartree-Fock exchange in the calculation of magnetic exchange couplings in transition metal complexes

We assess the dependence of magnetic exchange couplings on the variation of Hartree-Fock exchange (HFX) admixture in global hybrid functionals and the range-separation parameter ω in range-separated hybrid functionals in a set of 12 spin-1/2 binuclear transition metal complexes. The global hybrid PBEh (hybrid Perdew-Burke-Ernzerhof) and range-separated hybrids HSE (Heyd-Scuseria-Ernzerhof) and LC-ωPBE (long-range corrected hybrid PBE) are employed for this assessment, and exchange couplings are calculated from energy differences within the framework of the spin-projected approach. It is found that these functionals perform optimally for magnetic exchange couplings with 35% HFX admixture for PBEh, ω = 0.50 a.u.(-1) for LC-ωPBE, and ω at or near 0.0 a.u.(-1) for HSE (which corresponds to PBEh). We find that in their standard respective forms, LC-ωPBE slightly outperforms PBEh, while PBEh with 35% HFX yields exchange couplings closer to experiment than those of LC-ωPBE with ω = 0.50 a.u.(-1). Additionally, we show that the profile of exchange couplings with respect to ω in HSE is appreciably flat from 0 to 0.2 a.u.(-1). This combined with the fact that HSE is computationally more tractable than global hybrids makes HSE an attractive alternative for the evaluation of exchange couplings in extended systems. These results are rationalized with respect to how varying the parameters within these functionals affects the delocalization of the magnetic orbitals, and conclusions are made regarding the relative importance of range separation versus global mixing of HFX for the calculation of exchange couplings.     

Analytic derivatives for the XYG3 type of doubly hybrid density functionals: Theory,implementation, and assessment

We present a theoretical development of the equations required to perform an analytic geometry optimization of a molecular system using the XYG3 type of doubly hybrid (xDH) functionals. In contrast to the well‐established B2PLYP type of DH functionals, the energy expressions in the xDH functionals are constructed by using density and orbital information from another standard Kohn–Sham (KS) functional (e.g., B3LYP) for doing the self‐consistent field calculations. Thus, the xDH functionals are nonvariational in both the hybrid density functional part and the second‐order perturbation part, each of which requires formally to solve a coupled‐perturbed KS equation. An implementation is reported here which combines the two parts by defining a total Lagrangian such that only a single set of the Z‐vector equations need to be solved. The computational cost with our implementation is of the same order as those for the conventional Møller–Plesset theory to the second order (MP2) and B2PLYP. Systematic test calculations are provided for covalently bonded molecules as well as compounds involving the intramolecular nonbonded interactions for the main group elements. Satisfactory performance of the xDH functionals demonstrates that the extra computer time on top of the conventional KS procedure is well‐invested, in particular, when the standard KS functionals and MP2 as well, are problematic. © 2013 Wiley Periodicals, Inc.     

What can we learn from the adiabatic connection formalism about local hybrid functionals?

Local hybrid functionals with position-dependent exact-exchange admixture are a promising new generation of exchange-correlation functionals for a large variety of applications. So far, the local mixing functions (LMFs) determining the position dependence have been largely constructed in an ad hoc manner, albeit based on physical reasoning. Here the basic formalism of the adiabatic connection is employed to investigate the formal basis of local hybrids and to construct a priori LMFs. Both a local spin density approximation to the LMF (AC-LSDA LMF) and generalized gradient approximation approximations (AC-PW91 LMF and AC-PBE LMF) turn out to provide inferior performance when used in local hybrids to compute atomization energies and reaction barriers compared to previous semiempirical LMFs. This is rationalized by limited flexibility of these first-principles LMFs and some basic limitations of the adiabatic connection formalism in this context. Graphical analyses and formal considerations provide nevertheless important new insight into the physical background of local hybrid functionals.     

Benchmark results for empirical post-GGA functionals: difficult exchange problems and independent tests

Many of the most promising new density functionals have improved the treatment of non-local exchange effects with the help of semi-empirical information and more sophisticated recipes for combining Hartree-Fock and local exchange approximations. In order to quantify recent advancements and identify directions for improvement, we have examined a broad spectrum of test problems. We evaluate the performance of several new hybrid density functionals (ωB97, ωB97X, ωB97X-D, LRC-ωPBEh, M06, M06-2X, and M06-HF) on a variety of chemical problems, some sensitive to the treatment of exact exchange (which we have hoped to systematically improve) and some which require a balanced treatment of correlation. Since all of the functionals under consideration are parameterized with ground-state thermochemical data, the benchmark aims to determine the applicability of the new density functionals to cases that have not been considered in the optimization of the semi-empirical parameters. The first class of benchmarks includes the excitation energies of 21 molecules (83 states) primarily from a recent benchmark conducted by Tozer and co-workers, with some additional references from data made available from the groups of Thiel and Truhlar. We briefly examine the conformational preferences of a small peptide and complete our study with two recently published sets of data that have shown large, systematic errors in simple alkane thermochemistry. While our results indicate that the more general hybrids currently under development perform well for problems outside of their parameterization and improve over the standard hybrid density functionals in an essentially systematic way, there is still a significant self-interaction error in the more difficult cases. Functionals based on a range-separation of exchange and functionals depending on the kinetic-energy density both perform comparably, and there is evidence for complementary strengths.