On the accuracy of density-functional theory exchange-correlation functionals for H bonds in small water clusters: benchmarks approaching the complete basis set limit

The ability of several density-functional theory (DFT) exchange-correlation functionals to describe hydrogen bonds in small water clusters (dimer to pentamer) in their global minimum energy structures is evaluated with reference to second order Moller-Plesset perturbation theory (MP2). Errors from basis set incompleteness have been minimized in both the MP2 reference data and the DFT calculations, thus enabling a consistent systematic evaluation of the true performance of the tested functionals. Among all the functionals considered, the hybrid X3LYP and PBE0 functionals offer the best performance and among the nonhybrid generalized gradient approximation functionals, mPWLYP and PBE1W perform best. The popular BLYP and B3LYP functionals consistently underbind and PBE and PW91 display rather variable performance with cluster size.     

Optical rotation calculated with time-dependent density functional theory: the OR45 benchmark

Time-dependent density functional theory (TDDFT) computations are performed for 42 organic molecules and three transition metal complexes, with experimental molar optical rotations ranging from 2 to 2 × 10(4) deg cm(2) dmol(-1). The performances of the global hybrid functionals B3LYP, PBE0, and BHLYP, and of the range-separated functionals CAM-B3LYP and LC-PBE0 (the latter being fully long-range corrected), are investigated. The performance of different basis sets is studied. When compared to liquid-phase experimental data, the range-separated functionals do, on average, not perform better than B3LYP and PBE0. Median relative deviations between calculations and experiment range from 25 to 29%. A basis set recently proposed for optical rotation calculations (LPol-ds) on average does not give improved results compared to aug-cc-pVDZ in TDDFT calculations with B3LYP. Individual cases are discussed in some detail, among them norbornenone for which the LC-PBE0 functional produced an optical rotation that is close to available data from coupled-cluster calculations, but significantly smaller in magnitude than the liquid-phase experimental value. Range-separated functionals and BHLYP perform well for helicenes and helicene derivatives. Metal complexes pose a challenge to first-principles calculations of optical rotation.     

Influence of the exchange‐correlation functional in all‐electron calculations of the vibrational frequencies of corundum (α‐Al2O3)

The equilibrium structural parameters, high‐ and low‐frequency dielectric tensors, Born effective charges, and Γ‐point vibrational frequencies of bulk Al₂O₃ corundum are calculated by using the periodic, ab initio program CRYSTAL, which adopts an all‐electron Gaussian‐type basis set. The effect of basis set and the performance of three different functionals, i.e., LDA, PW91, and B3LYP, are discussed. The mean absolute deviation from the measured frequencies is as small as 7 cm^?1 for both the LDA and B3LYP functionals, indicating that these functionals perform extremely well in this case. The mean absolute deviation increases to 18 cm^?1 when the PW91 functional is used. All three functionals reproduce the equilibrium geometry of corundum to a high level of accuracy, with LDA and B3LYP outperforming PW91 slightly. The comparison of the current all‐electron calculations with previous plane‐wave, pseudo‐potential calculations shows an overall similar performance. The results of isotopic substitution for both Al and O are also presented. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

A DFT conformational analysis and VCD study on methyl tetrahydrofuran-2-carboxylate

DFT calculations were performed on (S)-methyl tetrahydrofuran-2-carboxylate to facilitate the interpretation of IR and VCD spectra. The potential energy surface could not be described unambiguously using the 6-31G* basis set in combination with different density functionals including B1LYP, B3LYP, B3P86, B3PW91, B98, BHandH, BHandHLYP, MPW1PW91 and PBE1PBE. In contrast, a uniform conformational picture could be found using the cc-pVTZ basis set. Using this large basis set and the collection of nine functionals from above, the dipole and rotational strengths were calculated, and compared to experimental values which were extracted from the experimental IR and VCD spectra for (+)-(S)-methyl tetrahydrofuran-2-carboxylate. A detailed analysis on the agreement between experiment and simulated spectra was performed by assigning the experimental bands based on the harmonic fundamentals obtained for all functionals except BHandH, which performs badly over the whole line. Assessing the dipole strengths, all tested functionals perform equally well. For the rotational strengths, differences can be observed: B3LYP, B1LYP and B98 give the highest correlation with experiment, while PBE1PBE gives the lowest correlation. Comparable conclusions are obtained using a neighborhood similarity measure.     

DFT calculations on the spin-crossover complex Fe(salen)(NO): a quest for the best functional

DFT calculations on the spin-crossover complex Fe(salen)(NO) provide a striking illustration of the comparative performance of different exchange-correlation functionals vis-à-vis the issue of transition metal spin state energetics. Thus, although the "classic" pure functionals PW91 and BLYP favor the S = 1/2 state by about 10 kcal/mol, relative to the S = 3/2 state, the hybrid functional B3LYP favors the latter state by nearly the same margin. In contrast, the newer pure functionals OLYP and OPBE, based on the OPTX exchange functional, as well as the B3LYP* hybrid functional (which has 15% Hartree-Fock exchange, compared with 20% for B3LYP) predict nearly isoenergetic S = 1/2 and 3/2 states, as required for a spin-crossover complex. Intriguingly, the OLYP and B3LYP* spin density profiles for the S = 1/2 state of Fe(salen)(NO) are substantially dissimilar.     

Assessing alkyl-, silyl-, and halo-substituent effects on the electron affinities of silyl radicals

Neutral anion energy differences for a large class of alpha-substituted silyl radicals have been computed to determine the effect of alkyl, silyl, and halo substituents on their electron affinities. In particular, we report theoretical predictions of the adiabatic electron affinities (AEAs), vertical electron affinities (VEAs), and vertical detachment energies (VDEs) for a series of methyl-, silyl-, and halo-substituted silyl radical compounds. This work utilizes the carefully calibrated DZP++ basis set, in conjunction with the pure BLYP and OLYP functionals, as well as with the hybrid B3LYP, BHLYP, PBE1PBE, MPW1K, and O3LYP functionals. Bromine has the largest effect in stabilizing the anions, and the BLYP/DZP++ AEA for SiBr(3) is 3.29 eV. The other predicted electron affinities are for SiH(3) (1.37 eV), SiH(2)CH(3) (1.09 eV), SiH(2)F (1.54 eV), SiH(2)Cl (1.94 eV), SiH(2)Br (2.05 eV), SiH(2)(SiH(3)) (1.77 eV), SiH(CH(3))(2) (0.92 eV), SiHF(2) (1.86 eV), SiHCl(2) (2.53 eV), SiHBr(2) (2.67 eV), Si(CH(3))(3) (0.86 eV), SiF(3) (2.66 eV), SiCl(3) (3.21 eV), Si(SiH(3))(3) (2.25 eV), and SiFClBr (3.13 eV). For the five silyl radicals where experimental data are available, the BLYP functional gives the most accurate determination of AEAs; the average absolute error is 0.04(1) eV, whereas the corresponding errors for the O3LYP, MPW1K, PBE1PBE, B3LYP, OLYP, and BHLYP functionals are 0.05(8), 0.06(0), 0.06(3), 0.08(5), 0.11(5), and 0.15(3) eV, respectively.     

Energy landscapes of nucleophilic substitution reactions: a comparison of density functional theory and coupled cluster methods

We have carried out a detailed evaluation of the performance of all classes of density functional theory (DFT) for describing the potential energy surface (PES) of a wide range of nucleophilic substitution (SN2) reactions involving, amongst others, nucleophilic attack at carbon, nitrogen, silicon, and sulfur. In particular, we investigate the ability of the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA as well as hybrid DFT to reproduce high-level coupled cluster (CCSD(T)) benchmarks that are close to the basis set limit. The most accurate GGA, meta-GGA, and hybrid functionals yield mean absolute deviations of about 2 kcal/mol relative to the coupled cluster data, for reactant complexation, central barriers, overall barriers as well as reaction energies. For the three nonlocal DFT classes, the best functionals are found to be OPBE (GGA), OLAP3 (meta-GGA), and mPBE0KCIS (hybrid DFT). The popular B3LYP functional is not bad but performs significantly worse than the best GGA functionals. Furthermore, we have compared the geometries from several density functionals with the reference CCSD(T) data. The same GGA functionals that perform best for the energies (OPBE, OLYP), also perform best for the geometries with average absolute deviations in bond lengths of 0.06 A and 0.6 degrees, even better than the best meta-GGA and hybrid functionals. In view of the reduced computational effort of GGAs with respect to meta-GGAs and hybrid functionals, let alone coupled cluster, we recommend the use of accurate GGAs such as OPBE or OLYP for the study of SN2 reactions.     

Systematic studies on the computation of nuclear magnetic resonance shielding constants and chemical shifts: the density functional models

We present a systematic density functional investigation on the prediction of the 13C, 15N, 17O, and 19F NMR properties of 23 molecules with 21 density functionals. Extensive comparisons are made for both 13C magnetic shieldings and chemical shifts with respect to the gas phase experimental data and the best CCSD(T) results. We find that the OPBE and OPW91 exchange-correlation functionals perform significantly better than some popular functionals such as B3LYP and PBE1PBE, even surpassing, in many cases, the standard wavefunction-based method MP2. Further analysis has been performed to explore the individual role played by various exchange and correlation functionals. We find that the B88 and PBE exchange functionals have a too strong tendency of deshielding, leading to too deshielded magnetic shielding constants; whereas the OPTX exchange functional performs remarkably well. We claim that the main source of error arises from the exchange functional, but correlation functional also makes important contribution. We find that the correlation functionals may be grouped into two classes. class A, such as LYP and B98, leads to deshielded NMR values, deteriorating the overall performance; whereas class B, such as PW91 and PBE, generally increases the absolute shieldings, which complements the exchange functionals, leading to improved results in the calculation of NMR data.     

Assessment of new meta and hybrid meta density functionals for predicting the geometry and binding energy of a challenging system: the dimer of H2S and benzene

Noncovalent interactions of a hydrogen bond donor with an aromatic pi system present a challenge for density functional theory, and most density functionals do not perform well for this kind of interaction. Here we test seven recent density functionals from our research group, along with the popular B3LYP functional, for the dimer of H 2S with benzene. The functionals considered include the four new meta and hybrid meta density functionals of the M06 suite, three slightly older hybrid meta functionals, and the B3LYP hybrid functional, and they were tested for their abilities to predict the dissociation energies of three conformations of the H 2S-benzene dimer and to reproduce the key geometric parameters of the equilibrium conformation of this dimer. All of the functionals tested except B3LYP correctly predict which of the three conformations of the dimer is the most stable. The functionals that are best able to reproduce the geometry of the equilibrium conformation of the dimer with a polarized triple-zeta basis set are M06-L, PWB6K, and MPWB1K, each having a mean unsigned relative error across the two experimentally verifiable geometric parameters of only 8%. The success of M06-L is very encouraging because it is a local functional, which reduces the cost for large simulations. The M05-2X functional yields the most accurate binding energy of a conformation of the dimer for which a binding energy calculated at the CCSD(T) level of theory is available; M05-2X gives a binding energy for the system with a difference of merely 0.02 kcal/mol from that obtained by the CCSD(T) calculation. The M06 functional performs well in both categories by yielding a good representation of the geometry of the equilibrium structure and by giving a binding energy that is only 0.19 kcal/mol different from that calculated by CCSD(T). We conclude that the new generation of density functionals should be useful for a variety of problems in biochemistry and materials where aromatic functional groups can serve as hydrogen bond acceptors.     

Comparative assessment of density functional methods for 3d transition-metal chemistry

In the present study, we comparatively assessed the newly developed M05 functional against a data set of reaction energies for transition-metal chemistry. The functionals to which we compare are BLYP, B3LYP, B97-2, MPWLYP1M, TPSS, and TPSSh. We draw the following conclusions: (1) TPSS gives the best performance for calculating the binding energies of three transition-metal dimers (Sc(2), Ni(2), and V(2)) that have severe multireference character, (2) B97-2 gives the best performance for calculating the binding energies of the nine metal-ligand diatomics (three monohydrides, three monoxide, and three monofluorides), and (3) M05 gives the overall best performance for all 18 data in the assessment, and it has a mean unsigned error 55% lower than the popular B3LYP functional. Since the M05 functional also gives good performance for main-group thermochemistry, for noncovalent chemistry, and for calculating barrier heights, M05 can be applied to a wide range of problems where nonhybrid functionals or functionals designed for kinetics fail.     

Assessment of a composite method based on selected density functional theory methods and complete basis set extrapolation formulas

Extrapolation formulas based on power and exponential formulas, as well as alternatives from a Taylor series, were tested and used with density functional theory (DFT) for the calculation of enthalpies of formation. The following four functionals were analyzed: B3LYP, BMK, M06-2X, and B2PLYP. Preliminary tests pointed to B2PLYP and B3LYP as the best and worst functionals, respectively. Taylor series expressions were as accurate as the power formulas and presented better performance than the exponential equation. The power formula (Equation (2)) and one of the simplest Taylor expressions (Equation (13)) were selected for the calculations with B3LYP and B2PLYP, with further empirical adjustments based on the higher level correction (HLC) and scaling of the experimental atomization energies used to calculate enthalpies of formation. HLC improved the B3LYP mean absolute error (MAE) from approximately 4.3 to 3.5 kcal mol⁻¹ using both extrapolation alternatives. For B2PLY, the MAEs were improved from 2.7 to 2.6 kcal mol⁻¹. Regarding the G3/05 test set, a significant improvement in the MAEs around 2.5 and 1.5 kcal mol⁻¹ were achieved using B3LYP and B2PLYP, respectively. The accuracy obtained from these empirical corrections was equivalent to other composite methods. The MAEs from B3LYP and B2PLYP may be suggested as ranges for the possible accuracy to be achieved by some DFT methods. The empirical corrections suggested in this work are improvements that may be considered to provide acceptable accuracy for enthalpies of formation and possibly other properties.     

DFT study of the structural characteristics of the yttrium(3+) aqua ion

Density functional theory (DFT) using SVWN5, B3LYP, B3P86, O3LYP, B3PW91, B1LYP, B971, MPW1PW91, PBE1PBE, BHandH, and BHandHLYP density functionals was employed to study the structural characteristics of the Y(H₂O) ₈ ³⁺ yttrium aqua ion. The nonlocal hybrid GGA functionals show worse predictive ability in structural calculations of the Y(H₂O) ₈ ³⁺ aqua ion compared to the relatively simple combined functional BHandH and to the simplest SVWN5 functional in LSDA theory.     

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.     

Performance of density functional theory in computing nonresonant vibrational (hyper)polarizabilities

A set of exchange‐correlation functionals, including BLYP, PBE0, B3LYP, BHandHLYP, CAM‐B3LYP, LC‐BLYP, and HSE, has been used to determine static and dynamic nonresonant (nuclear relaxation) vibrational (hyper)polarizabilities for a series of all‐trans polymethineimine (PMI) oligomers containing up to eight monomer units. These functionals are assessed against reference values obtained using the Møller–Plesset second‐order perturbation theory (MP2) and CCSD methods. For the smallest oligomer, CCSD(T) calculations confirm the choice of MP2 and CCSD as appropriate for assessing the density functionals. By and large, CAM‐B3LYP is the most successful, because it is best for the nuclear relaxation contribution to the static linear polarizability, intensity‐dependent refractive index second hyperpolarizability, static second hyperpolarizability, and is close to the best for the electro‐optical Pockels effect first hyperpolarizability. However, none of the functionals perform satisfactorily for all the vibrational (hyper)polarizabilities studied. In fact, in the case of electric field‐induced second harmonic generation all of them, as well as the Hartree–Fock approximation, yield the wrong sign. We have also found that the Pople 6–31+G(d) basis set is unreliable for computing nuclear relaxation (hyper)polarizabilities of PMI oligomers due to the spurious prediction of a nonplanar equilibrium geometry. © 2013 Wiley Periodicals, Inc.     

Numerical examination of performance of some exchange-correlation functionals for molecules containing heavy elements

The performance of 17 exchange-correlation functionals for molecules containing heavy elements are numerically examined through four-component relativistic density DFT calculations. The examined functionals show the similar accuracy as they do for the molecules containing light elements only except for bond lengths. LDA and OP86 produce good results for bond lengths and frequencies but bad bond energies. Different functionals do not show much different performance for bond energies except LDA. BP86 and GP86 produce results with average accuracy while LYP does not perform well. Although encouraging results are obtained with functional B97GGA-1, other heavily parameterized and meta-GGA functionals do not produce impressive results.     

Calculation of the Vibrationally Resolved,Circularly Polarized Luminescence of d‐Camphorquinone and (S,S)‐trans‐β‐Hydrindanone

Circularly polarized luminescence (CPL), the differential emission of left‐ and right‐handed circularly polarized light from a molecule, is modeled by using time‐dependent density functional theory. Calculations of the CPL spectra for the first electronic excited states of d‐camphorquinone and (S,S)‐trans‐β‐hydrindanone under the Franck–Condon approximation and using various functionals are presented, as well as calculations of absorption, emission, and circular dichroism spectra. The functionals B3LYP, BHLYP, and CAM‐B3LYP are employed, along with the TZVP and aug‐cc‐pVDZ Gaussian‐type basis sets. For the lowest‐energy transitions, all functionals and basis sets perform comparably, with the long‐range‐corrected CAM‐B3LYP better reproducing the excitation energy of camphorquinone but leading to a blue shift with respect to experiment for hydrindanone. The vibrationally resolved spectra of camphorquinone are very well reproduced in terms of peak location, widths, shapes, and intensities. The spectra of hydrindanone are well reproduced in terms of overall envelope shape and width, as well as the lack of prominent vibrational structure in the emission and CPL spectra. Overall the simulated spectra compare well with experiment, and reproduce the band shapes, emission red shifts, and presence or absence of visible vibrational fine structure.     

Stacking of Metal Chelates with Benzene: Can Dispersion‐Corrected DFT Be Used to Calculate Organic–Inorganic Stacking?

CCSD(T)/CBS energies for stacking of nickel and copper chelates are calculated and used as benchmark data for evaluating the performance of dispersion‐corrected density functionals for calculating the interaction energies. The best functionals for modeling the stacking of benzene with the nickel chelate are M06HF‐D3 with the def2‐TZVP basis set, and B3LYP‐D3 with either def2‐TZVP or aug‐cc‐pVDZ basis set, whereas for copper chelate the PBE0‐D3 with def2‐TZVP basis set yielded the best results. M06L‐D3 with aug‐cc‐pVDZ gives satisfying results for both chelates. Most of the tested dispersion‐corrected density functionals do not reproduce the benchmark data for stacking of benzene with both nickel (no unpaired electrons) and copper chelate (one unpaired electron), whereas a number of these functionals perform well for interactions of organic molecules.