Double-Hybrid Density Functional Theory for Core Excitations: Theory and Benchmark Calculations

The double-hybrid (DH) time-dependent density functional theory is extended to core excitations. Two different DH formalisms are presented utilizing the core–valence separation (CVS) approximation. First, a CVS-DH variant is introduced relying on the genuine perturbative second-order correction, while an iterative analogue is also presented using our second-order algebraic-diagrammatic construction [ADC(2)]-based DH ansatz. The performance of the new approaches is tested for the most popular DH functionals using the recently proposed XABOOM [J. Chem. Theory Comput.2021, 17, 1618] benchmark set. In order to make a careful comparison, the accuracy and precision of the methods are also inspected. Our results show that the genuine approaches are highly competitive with the more advanced CVS-ADC(2)-based methods if only excitation energies are required. In contrast, as expected, significant differences are observed in oscillator strengths; however, the precision is acceptable for the genuine functionals as well. Concerning the performance of the CVS-DH approaches, the PBE0-2/CVS-ADC(2) functional is superior, while its spin-opposite-scaled variant is also recommended as a cost-effective alternative. For these approaches, significant improvements are realized in the error measures compared with the popular CVS-ADC(2) method

Accurate Theoretical Thermochemistry for Fluoroethyl Radicals

An accurate coupled-cluster (CC) based model chemistry was applied to calculate reliable thermochemical quantities for hydrofluorocarbon derivatives including radicals 1-fluoroethyl (CH₃–CHF), 1,1-difluoroethyl (CH₃–CF₂), 2-fluoroethyl (CH₂F–CH₂), 1,2-difluoroethyl (CH₂F–CHF), 2,2-difluoroethyl (CHF₂–CH₂), 2,2,2-trifluoroethyl (CF₃–CH₂), 1,2,2,2-tetrafluoroethyl (CF₃–CHF), and pentafluoroethyl (CF₃–CF₂). The model chemistry used contains iterative triple and perturbative quadruple excitations in CC theory, as well as scalar relativistic and diagonal Born–Oppenheimer corrections. To obtain heat of formation values with better than chemical accuracy perturbative quadruple excitations and scalar relativistic corrections were inevitable. Their contributions to the heats of formation steadily increase with the number of fluorine atoms in the radical reaching 10 kJ/mol for CF₃–CF₂. When discrepancies were found between the experimental and our values it was always possible to resolve the issue by recalculating the experimental result with currently recommended auxiliary data. For each radical studied here this study delivers the best heat of formation as well as entropy data

High Accuracy Quantum Chemical and Thermochemical Network Data for the Heats of Formation of Fluorinated and Chlorinated Methanes and Ethanes

Reliable heats of formation are reported for numerous fluorinated and chlorinated methane and ethane derivatives by means of an accurate thermochemical protocol, which involves explicitly correlated coupled-cluster calculations augmented with anharmonic, scalar relativistic, and diagonal Born–Oppenheimer corrections. The theoretical results, along with additional experimental data, are further enhanced with the help of the thermochemical network approach. For 28 species, out of 50, this study presents the best estimates, and discrepancies with previous reports are also highlighted. Furthermore, the effects of the less accurate theoretical data on the results yielded by thermochemical networks are discussed

Accurate Diels–Alder Reaction Energies from Efficient Density Functional Calculations

We assess the performance of the semilocal PBE functional; its global hybrid variants; the highly parametrized empirical M06-2X and M08-SO; the range separated rCAM-B3LYP and MCY3; the atom-pairwise or nonlocal dispersion corrected semilocal PBE and TPSS; the dispersion corrected range-separated ωB97X-D; the dispersion corrected double hybrids such as PWPB95-D3; the direct random phase approximation, dRPA, with Hartree–Fock, Perdew–Burke–Ernzerhof, and Perdew–Burke–Ernzerhof hybrid reference orbitals and the RPAX2 method based on a Perdew–Burke–Ernzerhof exchange reference orbitals for the Diels–Alder, DARC; and self-interaction error sensitive, SIE11, reaction energy test sets with large, augmented correlation consistent valence basis sets. The dRPA energies for the DARC test set are extrapolated to the complete basis set limit. CCSD­(T)/CBS energies were used as a reference. The standard global hybrid functionals show general improvements over the typical endothermic energy error of semilocal functionals, but despite the increased accuracy the precision of the methods increases only slightly, and thus all reaction energies are simply shifted into the exothermic direction. Dispersion corrections give mixed results for the DARC test set. Vydrov–Van Voorhis 10 correction to the reaction energies gives superior quality results compared to the too-small D3 correction. Functionals parametrized for energies of noncovalent interactions like M08-SO give reasonable results without any dispersion correction. The dRPA method that seamlessly and theoretically correctly includes noncovalent interaction energies gives excellent results with properly chosen reference orbitals. As the results for the SIE11 test set and H₂⁺ dissociation show that the dRPA methods suffer from delocalization error, good reaction energies for the DARC test set from a given method do not prove that the method is free from delocalization error. The RPAX2 method shows good performance for the DARC, the SIE11 test sets, and for the H₂⁺ and H₂ potential energy curves showing no one-electron self-interaction error and reduced static correlation errors at the same time. We also suggest simplified DARC6 and SIE9 test sets for future benchmarking

Optimization of the Linear-Scaling Local Natural Orbital CCSD(T) Method: Improved Algorithm and Benchmark Applications

An optimized implementation of the local natural orbital (LNO) coupled-cluster (CC) with single-, double-, and perturbative triple excitations [LNO–CCSD­(T)] method is presented. The integral-direct, in-core, highly efficient domain construction technique of our local second-order Møller–Plesset (LMP2) scheme is extended to the CC level. The resulting scheme, which is also suitable for general-order LNO–CC calculations, inherits the beneficial properties of the LMP2 approach, such as the asymptotically linear-scaling operation count, the asymptotically constant data storage requirement, and the completely independent domain calculations. In addition to integrating our recent redundancy-free LMP2 and Laplace-transformed (T) algorithms with the LNO–CCSD­(T) code, the memory demand, the domain and LNO construction, the auxiliary basis compression, and the previously rate-determining two-external integral transformation have been significantly improved. The accuracy of all of the approximations is carefully examined on medium-sized to large systems to determine reasonably tight default truncation thresholds. Our benchmark calculations, performed on molecules of up to 63 atoms, show that the optimized method with the default settings provides average correlation and reaction energy errors of less than 0.07% and 0.34 kcal/mol, respectively, compared to the canonical CCSD­(T) reference. The efficiency of the present LNO–CCSD­(T) implementation is demonstrated on realistic, three-dimensional examples. Using the new code, an LNO–CCSD­(T) correlation energy calculation with a triple-ζ basis set is feasible on a single processor for a protein molecule including 2380 atoms and more than 44000 atomic orbitals

An Integral-Direct Linear-Scaling Second-Order Møller–Plesset Approach

An integral-direct, iteration-free, linear-scaling, local second-order Møller–Plesset (MP2) approach is presented, which is also useful for spin-scaled MP2 calculations as well as for the efficient evaluation of the perturbative terms of double-hybrid density functionals. The method is based on a fragmentation approximation: the correlation contributions of the individual electron pairs are evaluated in domains constructed for the corresponding localized orbitals, and the correlation energies of distant electron pairs are computed with multipole expansions. The required electron repulsion integrals are calculated directly invoking the density fitting approximation; the storage of integrals and intermediates is avoided. The approach also utilizes natural auxiliary functions to reduce the size of the auxiliary basis of the domains and thereby the operation count and memory requirement. Our test calculations show that the approach recovers 99.9% of the canonical MP2 correlation energy and reproduces reaction energies with an average (maximum) error below 1 kJ/mol (4 kJ/mol). Our benchmark calculations demonstrate that the new method enables MP2 calculations for molecules with more than 2300 atoms and 26000 basis functions on a single processor

Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional

We analyzed various possibilities to improve upon the SCAN meta-generalized gradient approximation density functional obeying all known properties of the exact functional that can be satisfied at this level of approximation. We examined the necessity of locally satisfying a strongly tightened lower bound for the exchange energy density in single-orbital regions, the nature of the error cancellation between the exchange and correlation parts in two-electron regions, and the effect of the fourth-order term in the gradient expansion of the correlation energy density. We have concluded that the functional can be modified to separately reproduce the exchange and correlation energies of the helium atom by locally releasing the strongly tightened lower bound for the exchange energy density in single-orbital regions, but this leads to an unbalanced improvement in the single-orbital electron densities. Therefore, we decided to keep the <i>F</i>_X ≤ 1.174 exact condition for any single-orbital density, where <i>F</i>_X is the exchange enhancement factor. However, we observed a general improvement in the single-orbital electron densities by revising the correlation functional form to follow the second-order gradient expansion in a wider range. Our new revSCAN functional provides more-accurate atomization energies for the systems with multireference character, compared to the SCAN functional. The nonlocal VV10 dispersion-corrected revSCAN functional yields more-accurate noncovalent interaction energies than the VV10-corrected SCAN functional. Furthermore, its global hybrid version with 25% of exact exchange, called revSCAN0, generally performs better than the similar SCAN0 for reaction barrier heights. Here, we also analyzed the possibility of the construction of a local hybrid from the SCAN exchange and a specific locally bounded nonconventional exact exchange energy density. We predict compatibility problems since this nonconventional exact exchange energy density does not really obey the strongly tightened lower bound for the exchange energy density in single-orbital regions

Dissociation of the Fluorine Molecule

The primary purpose of the present study is to resolve the discrepancy that exists between the two most recently published dissociation energies for the fluorine molecule [<i>D</i>₀(F₂)] and, consequently, for the associated heats of formation of the fluorine atom [Δ_f<i>H</i>₀^°(F)]. We hope to provide a reliable, well-established theoretical estimate for these thermochemical quantities. To this end, a high-accuracy coupled-cluster-based composite ab initio model chemistry has been utilized. The protocol involves contributions of up to pentuple excitations in coupled-cluster theory augmented with basis set extrapolation techniques and additional corrections beyond the nonrelativistic and Born–Oppenheimer approximations. The augmented core–valence correlation consistent basis set families, aug-cc-pCV<i>X</i>Z, have been successively used, in some cases, up to octuple-ζ quality. Our best theoretical results for <i>D</i>₀(F₂) and Δ_f<i>H</i>₀^°(F) obtained in this study are 154.95 ± 0.48 and 77.48 ± 0.24 kJ/mol, respectively. Because conflicting theoretical results are also reported about the existence of a barrier along the dissociation curve of F₂, extensive multireference configuration interaction and coupled-cluster calculations have been performed using reference orbitals taken from all-electron complete active space self-consistent field computations. Extrapolations from the results obtained with the aug-cc-pCV<i>X</i>Z (<i>X</i> = T, Q, 5) basis sets clearly indicate that the barrier indeed exists. It is located at 3.80 ± 0.20 Å along the dissociation curve with a height of 42 ± 10 μE_h (∼0.11 ± 0.03 kJ/mol). Because of the neglect of this effect during the evaluation of the raw experimental data used to obtain <i>D</i>₀(F₂) = 154.52 ± 0.12 kJ/mol and Δ_f<i>H</i>₀^°(F) = 77.26 ± 0.06 kJ/mol [Stevens; et al. J. Phys. Chem. A 2010, 114, 13134], an additional error should be attached to these latter values. Obviously, the barrier does not affect either the experimental results, <i>D</i>₀(F₂) = 154.92 ± 0.10 kJ/mol and Δ_f<i>H</i>₀^°(F) = 77.46 ± 0.05 kJ/mol [Yang; et al. J. Chem. Phys. 2005, 122, 134308; 2007, 127, 209901], which are based on the ion-pair dissociation of the molecule, or the data calculated theoretically. It is also noteworthy that our best estimates are in excellent agreement with those obtained from the ion-pair dissociation experiment

Solvation and Protonation of Coumarin 102 in Aqueous Media: A Fluorescence Spectroscopic and Theoretical Study

The ground- and excited-state protonation of Coumarin 102 (C102), a fluorescent probe applied frequently in heterogeneous systems with an aqueous phase, has been studied in aqueous solutions by spectroscopic experiments and theoretical calculations. For the dissociation constant of the protonated form in the ground state, p<i>K</i>_a = 1.61 was obtained from the absorption spectra; for the excited-state dissociation constant, p<i>K</i>_a^* = 2.19 was obtained from the fluorescence spectra. These values were closely reproduced by theoretical calculations via a thermodynamic cycle (the value of p<i>K</i>_a^* also by calculations via the Förster cycle) using an implicit–explicit solvation model (polarized continuum model + addition of a solvent molecule). The theoretical calculations indicated that (i) in the ground state, C102 occurs primarily as a hydrogen-bonded water complex, with the oxo group as the binding site, (ii) this hydrogen bond becomes stronger upon excitation, and (iii) in the ground state, the amino nitrogen atom is the protonation site, and in the excited state, the carboxy oxygen atom is the protonation site. A comprehensive analysis of fluorescence decay data yielded the values <i>k</i>_pr = 3.27 × 10¹⁰ M^–1 s^–1 for the rate constant of the excited-state protonation and <i>k</i>_dpr = 2.78 × 10⁸ s^–1 for the rate constant of the reverse process (<i>k</i>_pr and <i>k</i>_dpr were treated as independent parameters). This, considering the relatively long fluorescence lifetimes of neutral C102 (6.02 ns) and its protonated form (3.06 ns) in aqueous media, means that a quasi-equilibrium state of excited-state proton transfer is reached in strongly acidic solutions

Construction and Application of a New Dual-Hybrid Random Phase Approximation

The direct random phase approximation (dRPA) combined with Kohn–Sham reference orbitals is among the most promising tools in computational chemistry and applicable in many areas of chemistry and physics. The reason for this is that it scales as <i>N</i>⁴ with the system size, which is a considerable advantage over the accurate ab initio wave function methods like standard coupled-cluster. dRPA also yields a considerably more accurate description of thermodynamic and electronic properties than standard density-functional theory methods. It is also able to describe strong static electron correlation effects even in large systems with a small or vanishing band gap missed by common single-reference methods. However, dRPA has several flaws due to its self-correlation error. In order to obtain accurate and precise reaction energies, barriers and noncovalent intra- and intermolecular interactions, we construct a new dual-hybrid dRPA (hybridization of exact and semilocal exchange in both the energy and the orbitals) and test the performance of this new functional on isogyric, isodesmic, hypohomodesmotic, homodesmotic, and hyperhomodesmotic reaction classes. We also use a test set of 14 Diels–Alder reactions, six atomization energies (AE6), 38 hydrocarbon atomization energies, and 100 reaction barrier heights (DBH24, HT-BH38, and NHT-BH38). For noncovalent complexes, we use the NCCE31 and S22 test sets. To test the intramolecular interactions, we use a set of alkane, cysteine, phenylalanine-glycine-glycine tripeptide, and monosaccharide conformers. We also discuss the delocalization and static correlation errors. We show that a universally accurate description of chemical properties can be provided by a large, 75% exact exchange mixing both in the calculation of the reference orbitals and the final energy