The equilibrium association free enthalpies ΔGa for typical supramolecular complexes in solution are calculated by ab initio quantum chemical methods. Ten neutral and three positively charged ...complexes with experimental ΔGa values in the range 0 to −21 kcal mol−1 (on average −6 kcal mol−1) are investigated. The theoretical approach employs a (nondynamic) single‐structure model, but computes the various energy terms accurately without any special empirical adjustments. Dispersion corrected density functional theory (DFT‐D3) with extended basis sets (triple‐ζ and quadruple‐ζ quality) is used to determine structures and gas‐phase interaction energies (ΔE), the COSMO‐RS continuum solvation model (based on DFT data) provides solvation free enthalpies and the remaining ro‐vibrational enthalpic/entropic contributions are obtained from harmonic frequency calculations. Low‐lying vibrational modes are treated by a free‐rotor approximation. The accurate account of London dispersion interactions is mandatory with contributions in the range −5 to −60 kcal mol−1 (up to 200 % of ΔE). Inclusion of three‐body dispersion effects improves the results considerably. A semilocal (TPSS) and a hybrid density functional (PW6B95) have been tested. Although the ΔGa values result as a sum of individually large terms with opposite sign (ΔE vs. solvation and entropy change), the approach provides unprecedented accuracy for ΔGa values with errors of only 2 kcal mol−1 on average. Relative affinities for different guests inside the same host are always obtained correctly. The procedure is suggested as a predictive tool in supramolecular chemistry and can be applied routinely to semirigid systems with 300–400 atoms. The various contributions to binding and enthalpy–entropy compensations are discussed.
Unprecedented accuracy for computed association free enthalpies of supramolecular host–guest complexes (some examples are shown here) in solution has been achieved by a combination of high‐level quantum chemical procedures. The approach is quite general, includes all basic physical effects quantitatively and requires no special empirical adjustments.
A theoretical disaster: In contrast to general opinion, modern density functional methods are not able to even qualitatively describe the overall energetics of simple alkylation reactions (see ...scheme). The main reason is that the currently used hybrid functionals almost completely neglect the generally relevant electron correlation effects for medium‐range electron–electron distances.
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.
We propose a fully-automated composite scheme for the accurate and numerically stable calculation of molecular entropies by efficiently combining density-functional theory (DFT), semi-empirical ...methods (SQM), and force-field (FF) approximations. The scheme is systematically expandable and can be integrated seamlessly with continuum-solvation models. Anharmonic effects are included through the modified rigid-rotor-harmonic-oscillator (msRRHO) approximation and the Gibbs-Shannon formula for extensive conformer ensembles (CEs), which are generated by a metadynamics search algorithm and are extrapolated to completeness. For the first time, variations of the ro-vibrational entropy over the CE are consistently accounted-for through a Boltzmann-population average. Extensive tests of the protocol with the two standard DFT approaches B97-3c and B3LYP-D3 reveal an unprecedented accuracy with mean deviations <1 cal mol
−1
K
−1
(about <1-2%) for the total gas phase molecular entropy of medium-sized molecules. Even for the hardship case of extremely flexible linear alkanes (C
14
H
30
-C
16
H
34
), errors are only about 3 cal mol
−1
K
−1
. Comprehensive tests indicate a relatively strong variation of the conformational entropy on the underlying level of theory for typical drug molecules, inferring the complex potential energy surfaces as the main source of error. Furthermore, we show some application examples for the calculation of free energy differences in typical chemical reactions.
A novel scheme for the automated calculation of the conformational entropy together with a modified thermostatistical treatment provides entropies with unprecedented accuracy even for large, complicated molecules.
The inclusion of dynamical and static electron correlation (SEC) is mandatory for accurate quantum chemistry (QC). SEC is particularly difficult to calculate and hence a qualitative understanding is ...important to judge the applicability of approximate QC methods. Existing scalar SEC diagnostics, however, lack the important information where the SEC effects occur in a molecule. We introduce an analysis tool based on a fractional occupation number weighted electron density (ρFOD) that is plotted in 3D for a pre‐defined contour surface value. The scalar field is obtained by finite‐temperature DFT calculations with pre‐defined electronic temperature (e.g. TPSS at 5000 K). FOD plots only show the contribution of the “hot” (strongly correlated) electrons. We discuss illustrative plots for a broad range of chemical systems from small molecules to large conjugated molecules with polyradicaloid character. Spatial integration yields a single number which can be used to globally quantify SEC.
Hot FOD: The inclusion of static electron correlation (SEC) is mandatory for accurate quantum chemistry yet is particularly difficult to calculate. An analysis tool is developed based on a fractional occupation number weighted electron density (ρFOD) that is plotted as an isosurface and shows the “hot” (strongly correlated) electrons. Spatial integration of ρFOD yields a single number which can be used to globally quantify SEC.
A new test set (S12L) containing 12 supramolecular noncovalently bound complexes is presented and used to evaluate seven different methods to account for dispersion in DFT (DFT-D3, DFT-D2, DFT-NL, ...XDM, dDsC, TS-vdW, M06-L) at different basis set levels against experimental, back-corrected reference energies. This allows conclusions about the performance of each method in an explorative research setting on “real-life” problems. Most DFT methods show satisfactory performance but, due to the largeness of the complexes, almost always require an explicit correction for the nonadditive Axilrod–Teller–Muto three-body dispersion interaction to get accurate results. The necessity of using a method capable of accounting for dispersion is clearly demonstrated in that the two-body dispersion contributions are on the order of 20–150% of the total interaction energy. MP2 and some variants thereof are shown to be insufficient for this while a few tested D3-corrected semiempirical MO methods perform reasonably well. Overall, we suggest the use of this benchmark set as a “sanity check” against overfitting to too small molecular cases.
The S12L test set for supramolecular Gibbs free energies of association ΔG a ( Grimme S. Chem. Eur. J. 2012, 18, 9955−9964 ) is extended to 30 complexes (S30L), featuring more diverse interaction ...motifs, anions, and higher charges (−1 up to +4) as well as larger systems with up to 200 atoms. Various typical noncovalent interactions like hydrogen and halogen bonding, π–π stacking, nonpolar dispersion, and CH−π and cation–dipolar interactions are represented by “real” complexes. The experimental Gibbs free energies of association (ΔG a exp ) cover a wide range from −0.7 to −24.7 kcal mol–1. In order to obtain a theoretical best estimate for ΔG a , we test various dispersion corrected density functionals in combination with quadruple-ζ basis sets for calculating the association energies in the gas phase. Further, modern semiempirical methods are employed to obtain the thermostatistical corrections from energy to Gibbs free energy, and the COSMO-RS model with several parametrizations as well as the SMD model are used to include solvation contributions. We investigate the effect of including counterions for the charged systems (S30L-CI), which is found to overall improve the results. Our best method combination consists of PW6B95-D3 (for neutral and charged systems) or ωB97X-D3 (for systems with counterions) energies, HF-3c thermostatistical corrections, and Gibbs free energies of solvation obtained with the COSMO-RS 2012 parameters for nonpolar solvents and 2013-fine for water. This combination gives a mean absolute deviation for ΔG a of only 2.4 kcal mol–1 (S30L) and 2.1 kcal mol–1 (S30L-CI), with a mean deviation of almost zero compared to experiment. Regarding the relative Gibbs free energies of association for the 13 pairs of complexes which share the same host, the correct trend in binding affinities could be reproduced except for two cases. The MAD compared to experiment amounts to 1.2 kcal mol–1, and the MD is almost zero. The best-estimate theoretical corrections are used to back-correct the experimental ΔG a values in order to get an empirical estimate for the “experimental”, zero-point vibrational energy exclusive, gas phase binding energies. These are then utilized to benchmark the performance of various “low-cost” quantum chemical methods for noncovalent interactions in large systems. The performance of other common DFT methods as well as the use of semiempirical methods for structure optimizations is discussed.
Double‐hybrid density functionals (DHDFs) are reviewed in this study. In DHDFs parts of conventional density functional theory (DFT) exchange and correlation are replaced by contributions from ...nonlocal Fock‐exchange and second‐order perturbative correlation. The latter portion is based on the well‐known MP2 wave‐function approach in which, however, Kohn–Sham orbitals are used to calculate its contribution. First, related methods preceding this idea are reviewed, followed by a thorough discussion of the first modern double‐hybrid B2‐PLYP. Parallels and differences between B2‐PLYP and its various successors are then outlined. This discussion is rounded off with representative thermochemical examples demonstrating that DHDFs belong to the most robust and accurate DFT approaches currently available. This analysis also presents hitherto unpublished results for recently developed DHDFs. Finally, how double‐hybrids can be combined with linear‐response time‐dependent DFT is also outlined and the value of this approach for electronically excited states is shown. WIREs Comput Mol Sci 2014, 4:576–600. doi: 10.1002/wcms.1193
This article is categorized under:
Electronic Structure Theory > Density Functional Theory
Mean-field electronic structure methods like Hartree–Fock, semilocal density functional approximations, or semiempirical molecular orbital (MO) theories do not account for long-range electron ...correlation (London dispersion interaction). Inclusion of these effects is mandatory for realistic calculations on large or condensed chemical systems and for various intramolecular phenomena (thermochemistry). This Review describes the recent developments (including some historical aspects) of dispersion corrections with an emphasis on methods that can be employed routinely with reasonable accuracy in large-scale applications. The most prominent correction schemes are classified into three groups: (i) nonlocal, density-based functionals, (ii) semiclassical C 6-based, and (iii) one-electron effective potentials. The properties as well as pros and cons of these methods are critically discussed, and typical examples and benchmarks on molecular complexes and crystals are provided. Although there are some areas for further improvement (robustness, many-body and short-range effects), the situation regarding the overall accuracy is clear. Various approaches yield long-range dispersion energies with a typical relative error of 5%. For many chemical problems, this accuracy is higher compared to that of the underlying mean-field method (i.e., a typical semilocal (hybrid) functional like B3LYP).
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