Further evidences of the quality of double-hybrid energy functionals for π-conjugated systems

Despite numerous interesting efforts along decades to improve the accuracy of density functionals with broad applicability, such as B3LYP, there are still large sets of molecular systems where improvements are badly needed. We select π-conjugated systems as an example of the latter due to the subtle interplay between some physical effects affecting possibly most of the calculations: self-interaction or delocalization error, medium-range correlation signatures, dispersive-like weak interactions, and static correlation effects. We further assess a recently proposed modification of the B2-PLYP double-hybrid functional, called B2π-PLYP, that is expected to yield substantial progress for this kind of systems. Generally speaking, when compared with other more popular and older density functionals, double hybrids behave particularly accurate for π-conjugated systems without suffering the large errors that are common in former yet conventional methods.     

Generalized nonlocal kinetic energy density functionals based on the von Weizsäcker functional

We generalize the ideas behind the procedure for the construction of kinetic energy density functionals with a nonlocal term based on the structure of the von Weizs?cker functional, and present several types of nonlocal terms. In all cases, the functionals are constructed such that they reproduce the linear response function of the homogeneous electron gas. These functionals are designed by rewriting the von Weizs?cker functional with the help of a parameter β that determines the power of the electron density in the expression, a strategy we have previously used in the generalization of Thomas-Fermi nonlocal functionals. Benchmark calculations in localized systems have been performed with these functionals to test both their relative errors and the quality of their local behavior. We have obtained competitive results when compared to semilocal and previous nonlocal functionals, the generalized nonlocal von Weizs?cker functionals giving very good results for the total kinetic energies and improving the local behavior of the kinetic energy density. In addition, all the functionals discussed in this paper, when using an adequate reference density, can be evaluated as a single integral in momentum space, resulting in a quasilinear scaling for the computational cost.     

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.     

An examination of density functionals on aldol, Mannich and ��-aminoxylation reaction enthalpy calculations

The reaction enthalpies of aldol, Mannich and ??-aminoxylation reactions were calculated by density functional theory (DFT) using long-range-corrected (LC), hybrid B3LYP and other up-to-date functionals to show why conventional DFT including B3LYP has given poor enthalpies for these reactions. As a result, we found that long-range exchange interactions significantly affect the reaction enthalpies. We therefore proposed that the poor enthalpies of B3LYP are due to its insufficient long-range exchange effect. On the other hand, LC functionals accurately reproduce reaction enthalpies for these reactions. However, we noticed that even LC functionals present poor reaction enthalpies for specific reactions, in which many branches are produced or very small molecules such as methane molecule participate.     

How well can new-generation density functionals describe the energetics of bond-dissociation reactions producing radicals?

The performance of various density functionals has been tested for three sets of reaction energies involving radicals. It is shown that two recently designed functionals, M05-2X and M06-2X, provide the best performance. These functionals provide useful and affordable methods for future mechanistic studies involving organic radicals.     

How reliable are DFT transition structures? Comparison of GGA, hybrid-meta-GGA and meta-GGA functionals

There have been many comparisons of computational methods applied to ground states, but studies of organic reactions usually require calculations on transition states, and these provide a different test of the methods. We present calculations of the geometries of nineteen covalent-bond forming transition states using HF and twelve different functionals, including GGA, hybrid-GGA and hybrid meta-GGA approaches. For the calculation of the TS geometries, the results suggest that B3LYP is only slightly less accurate than newer, computationally more expensive methods, and is less sensitive to choice of integration grid. We conclude that the use of B3LYP and related functionals is still appropriate for many studies of organic reaction mechanisms.     

Assessment of density functionals and force field methods on anion–π interaction in heterocyclic calix complexes

The performance of ten density functionals and four force field methods in describing non-covalent interactions have been assessed by studying the interaction energies and structures of the four anion–π complexes involving tetraoxacalix[2]arene[2]triazine and various anions. Their structures are optimized at MP2/6-311++G(d,p) level, and interaction energies are obtained at DF-MP2-F12/aug-cc-pVDZ level. The result shows that the functional M06-2X predicts the most reliable interaction energy, followed by wB97XD and BHandH. B97D slightly overestimates the interaction energy. Other functionals and force field methods seriously overestimate the interaction energy. For the structures, three functionals M06-2X, wB97XD and BH and H predict the most reliable results, followed by B97D. The force field methods predict the largest deviations. The present work suggests that the functional M06-2X is a reliable method to describe energies and structural properties of the large molecules involving the anion–π interactions.     

Rényi entropy of the infinite well potential in momentum space and Dirichlet-like trigonometric functionals

The momentum entropic moments and Rényi entropies of a one-dimensional particle in an infinite well potential are found by means of explicit calculations of some Dirichlet-like trigonometric integrals. The associated spreading lengths and quantum uncertainty-like sums are also provided.     

Can B3LYP be improved by optimization of the proportions of exchange and correlation functionals?

B3LYP is the most famous hybrid density functional theory model, which includes Hartree–Fock exchange, local exchange, gradient exchange correction, local correlation, and gradient correlation correction. Historically, the relative weight of each component in B3LYP, which is controlled by three empirical parameters (a₀, a_x, a_c), has not been optimized. In this work, we perform global optimization against accurate experimental reference, optimal empirical parameters, and the better version of B3LYP are obtained and denoted as OpB3LYP. The performance of OpB3LYP is widely tested over many species and chemical properties, the results show that the computational accuracy is significantly improved as compared to original B3LYP and the serious size dependence of B3LYP is remarkably overcome by the employment of OpB3LYP. The comparative assessment of OpB3LYP and other prevalent functionals indicates that OpB3LYP is a promising functional for large molecules. © 2015 Wiley Periodicals, Inc.     

Long-range corrected hybrid functionals for π-conjugated systems: dependence of the range-separation parameter on conjugation length

Long-range corrected (range-separated hybrid) functionals represent a relatively new class of functionals for generalized Kohn-Sham theory that have proven to be very successful, for instance, when it comes to predicting ionization potentials and energy gaps for a wide range of molecules and solids. The results obtained from long-range corrected density functional theory approaches can be improved dramatically, if the range-separation parameter (ω) is optimized for each system separately. In this work, we have optimized ω for a series of π-conjugated molecular systems of increasing length by forcing the resulting functionals to obey the ionization potential-theorem, i.e., that their highest occupied eigenvalue be equal to the ΔSCF ionization potential. The optimized ω values are observed to vary substantially from their default values for the functionals. For highly conjugated chains such as oligoacenes and polyenes, we find that the characteristic length scale of the range-separation, i.e., 1/ω, grows almost linearly with the number of repeat units, for saturated alkane chains, however, 1/ω quickly saturates after 5-6 repeat units. For oligothiophenes, we find that 1/ω grows linearly for the shorter oligomers but then saturates at around 10 repeat units. Our results point to a close relation between the optimal range-separation parameter and the degree of conjugation in the system.     

Use of γ-irradiation to produce films from whey,casein and soya proteins: structure and functionals characteristics

γ-irradiation and thermal treatments have been used to produce sterilized cross-linked films. Formulations containing variable concentrations of calcium caseinate and whey proteins (whey protein isolate (WPI) and commercial whey protein concentrate) or mixture of soya protein isolate (SPI) with WPI was investigated on the physico-chemical properties of these films. Results showed that the mechanical properties of cross-linked films improved significantly the puncture strength for all types of films. Size-exclusion chromatography showed for no cross-linked proteins, a molecular mass of around 40 kDa. The soluble fractions of the cross-linked proteins molecular distributions were between 600 and 3800 kDa. γ-irradiation seems to modify to a certain extent the conformation of proteins which will adopt structures more ordered and more stable, as suggested by X-ray diffraction analysis. Microstructure observations showed that the mechanical characteristics of these films are closely related to their microscopic structure. Water vapor permeability of films based on SPI was also significantly decreased when irradiated. Microbial resistance was also evaluated for cross-linked films. Results showed that the level of biodegradation of cross-linked films was 36% after 60 d of fermentation in the presence of Pseudomonas aeruginosa.     

How reliable is DFT in predicting relative energies of polycyclic aromatic hydrocarbon isomers? comparison of functionals from different rungs of jacob's ladder

Density functional theory (DFT) is the only quantum‐chemical avenue for calculating thermochemical/kinetic properties of large polycyclic aromatic hydrocarbons (PAHs) such as graphene nanoflakes. Using CCSD(T)/CBS PAH isomerization energies, we find that all generalized gradient approximation (GGA) and meta GGA DFT functionals have severe difficulties in describing isomerization energies in PAHs. The poor performance of these functionals is demonstrated by the following root‐mean‐square deviations (RMSDs) obtained for a database of C₁₄H₁₀ and C₁₈H₁₂ isomerization energies. The RMSDs for the GGAs range between 6.0 (BP86‐D3) and 23.0 (SOGGA11) and for the meta GGAs they range between 3.5 (MN12‐L) and 11.9 (τ‐HCTH) kJ mol⁻¹. These functionals (including the dispersion‐corrected methods) systematically and significantly underestimate the isomerization energies. A consequence of this behavior is that they all predict that chrysene (rather than triphenylene) is the most stable C₁₈H₁₂ isomer. A general improvement in performance is observed along the rungs of Jacob's Ladder; however, only a handful of functionals from rung four give good performance for PAH isomerization energies. These include functionals with high percentages (40–50%) of exact Hartree–Fock exchange such as the hybrid GGA SOGGA11‐X (RMSD = 1.7 kJ mol⁻¹) and the hybrid‐meta GGA BMK (RMSD = 1.3 kJ mol⁻¹). Alternatively, the inclusion of lower percentages (20–30%) of exact exchange in conjunction with an empirical dispersion correction results in good performance. For example, the hybrid GGA PBE0‐D3 attains an RMSD of 1.5 kJ mol⁻¹, and the hybrid‐meta GGAs PW6B95‐D3 and B1B95‐D3 result in RMSDs below 1 kJ mol⁻¹. © 2016 Wiley Periodicals, Inc.     

Overestimation of the stability of the π-delocalized versus the σ-localized configuration in radicals by current density functionals: the case of vinylacyl radicals

 Calculations with the density functional theory (DFT) method using the most popular functional, Becke's three parameter hybrid with the Lee, Yang and Parr correlation functional, predict the π-delocalized configuration of the vinylacyl radical, CH₂=CH—C(•)=O, to be more stable than the σ-localized configuration in contrast with ab initio unrestricted quadratic configuration interaction with single and double excitations calculations as previously found for the isoelectronic vinyl radical, Y-C(•)=CH₂, bearing π-type α substituents. Experimental evidence on the electronic configuration adopted by vinyl radicals is contrasting. In the present case comparison with experiment indicates firmly that the currently available density functionals overestimate the stability of π-delocalized versus σ-localized configurations in radicals since they favor the π configuration for the γ-methylvinylacyl radical, CH₃—CH=CH—C(•)=O, in contrast with unequivocal electron spin resonance data. This failure is mainly due to an incorrect estimate of dynamic correlation energy with DFT functionals. Received: 13 January 2000 / Accepted: 16 March 2000 / Published online: 21 June 2000     

Assessing the performance of density functional theory for the electronic structure of metal-salens: the M06 suite of functionals and the d⁴-metals

We have systematically investigated the electronic structure of the d? metal-salen complexes including the Cr(II)-, Mn(III)-, Fe(IV)-, Mo(II)-, Tc(III)-, and Ru(IV)-salen complexes. Density functional theory (DFT) has been employed, using the BP86 and B3LYP functionals, and the entire M05 and M06 suites of meta-generalized gradient functionals. These results are compared to robust complete active-space self-consistent field (CASSCF) optimized geometries and complete active-space third-order perturbation theory (CASPT3) energies for the lowest singlet, triplet, and quintet states. Although the M06 and M06-L DFT functionals have been generally recommended for computations on complexes that contain main group and transition metals, none of the M0-functionals appear statistically better than the B3LYP functional for the computation of spin-state energy gaps. DFT- and CASSCF-optimized geometries normally agree to within 0.3 ? least root mean squared deviation (LRMSD) for the singlet and triplet structures and less than 0.1 ? LRMSD for the quintet structures. It can be concluded that no DFT functional tested here can be considered reliable over all metal-salen complexes and it is highly recommended that the accuracy of any given DFT functional should be assessed on a case-by-case basis.     

Why are time-dependent density functional theory excitations in solids equal to band structure energy gaps for semilocal functionals, and how does nonlocal Hartree-Fock-type exchange introduce excitonic effects?

We examine the time-dependent density functional theory (TD-DFT) equations for calculating excitation energies in solids with Gaussian orbitals and analytically show that for semilocal functionals, their lowest eigenvalue collapses to the minimum band orbital energy difference. With the introduction of nonlocal Hartree-Fock-type exchange (as in hybrid functionals), this result is no longer valid, and the lowest TD-DFT eigenvalue reflects the appearance of excitonic effects. Previously reported "charge-transfer" problems with semilocal TD-DFT excitations in molecules can be deduced from our analysis by taking the limit to infinite lattice constant.     

Which DFT functional performs well in the calculation of methylcobalamin? Comparison of the B3LYP and BP86 functionals and evaluation of the impact of empirical dispersion correction

Controversy remains regarding the suitable density functionals for the calculation of vitamin B(12) systems that contain cobalt. To identify the optimum functionals, geometry optimization calculations were performed on a full-size model of methylcobalamin (MeCbl) using the B3LYP, B3LYP-D, BP86, and BP86-D methods in conjunction with the 6-31G* basis set. Single-point energy evaluations were also performed with the 6-311+G(2d,p) basis set. Consistent with previous studies, the BP86-optimized geometry showed fairly good agreement with the experimental geometry. Various factors that may influence the homolytic bond dissociation energy (BDE) of the Co-C bond of MeCbl were systematically evaluated with these methods. Our analysis demonstrated that dispersion was the largest correction term that influenced the magnitude of BDE. Previous studies have shown that B3LYP significantly underestimates BDE, whereas BP86 gives BDE values that are fairly close to the experimental values (36-37 kcal/mol). The same trend in the relative magnitudes of the BDEs was observed in the present calculations. However, BP86 underestimated the BDE for a full model of MeCbl. When the amount of Hartree-Fock exchange in the B3LYP functional was reduced to 15% and the dispersion correction was made (i.e., B3LYP*-D), the calculated BDE was in good accord with experimental values. B3P86-D also performed well. A detailed analysis was undertaken to determine which atoms in cobalamin have large dispersion interactions with a methyl fragment of MeCbl.     

Is the uniform electron gas limit important for small Ag clusters? Assessment of different density functionals for Ag(n) (n < or = 4)

Twenty-three density functional theory (DFT) methods, including the second- and the third-generation functionals, are tested in conjunction with two basis sets (LANL2DZ and SDD) for studying the properties of neutral and ionic silver clusters. We find that DFT methods incorporating the uniform electron gas limit in the correlation functional, namely, those with Perdew's correlation functionals (PW91, PBE, P86, and TPSS), Becke's B95, and the Van Voorhis-Scuseria functional VSXC, generally perform better than the other group of functionals, e.g., those incorporating the LYP correlation functional and variations of the B97 functional. Strikingly, these two groups of functionals can produce qualitatively different results for the Ag3 and Ag4 clusters. The energetic properties and vibrational frequencies of Ag(n) are also evaluated by the different functionals. The present study shows that the choice of DFT methods for heavy metals may be critical. It is found that the exact-exchange-incorporated PBE functional (PBE1PBE) is among the best for predicting the range of properties.