Long‐range corrected functionals satisfy Koopmans' theorem: Calculation of correlation and relaxation energies

In this article, we show that the long‐range‐corrected (LC) density functionals LC‐BOP and LCgau‐BOP reproduce frontier orbital energies and highest‐occupied molecular orbital (HOMO)—lowest‐unoccupied molecular orbital (LUMO) gaps better than other density functionals. The negative of HOMO and LUMO energies are compared with the vertical ionization potentials (IPs) and electron affinities, respectively, using CCSD(T)/6‐311++G(3df,3pd) for 113 molecules, and we found LC functionals to satisfy Koopmans' theorem. We also report that the frontier orbital energies and the HOMO‐LUMO gaps of LC‐BOP and LCgau‐BOP are better than those of recently proposed ωM05‐D (Lin et al., J. Chem. Phys. 2012, 136 , 154109). We express the exact IP in terms of orbital relaxation, and correlation energies and hence calculate the relaxation and correlation energies for the same set of molecules. It is found that the LC functionals, in general, includes more relaxation effect than Hartree–Fock and more correlation effect than the other density functionals without LC scheme. Finally, we scan μ parameter in LC scheme from 0.1 to 0.6 bohr^?1 for the above test set molecules with LC‐BOP functional and found our parameter value, 0.47 bohr^?1, is usefully applicable to our tested systems. © 2013 Wiley Periodicals, Inc.     

Theoretical and Structural Analysis of Long CC Bonds in the Adducts of Polycyanoethylene and Anthracene Derivatives and Their Connection to the Reversibility of Diels–Alder Reactions

X‐ray structure determinations on four Diels–Alder adducts derived from the reactions of cyano‐ and ester‐substituted alkenes with anthracene and 9,10‐dimethylanthracene have shown the bonds formed in the adduction to be particularly long. Their lengths range from 1.58 to 1.62 Å, some of the longest known for Diels–Alder adducts. Formation of the four adducts is detectably reversible at ambient temperature and is associated with free energies of reaction ranging from ?2.5 to ?40.6 kJ mol^?1. The solution equilibria have been experimentally characterised by NMR spectroscopy. Density‐functional‐theory calculations at the MPW1K/6‐31+G(d,p) level with PCM solvation agree with experiment with average errors of 6 kJ mol^?1 in free energies of reaction and structural agreement in adduct bond lengths of 0.013 Å. To understand more fully the cause of the reversibility and its relationship to the long adduct bond lengths, natural‐bond‐orbital (NBO) analysis was applied to quantify donor–acceptor interactions within the molecules. Both electron donation into the σ*‐anti‐bonding orbital of the adduct bond and electron withdrawal from the σ‐bonding orbital are found to be responsible for this bond elongation.     

Activation of the Unreactive Bond in C70 Fullerene toward Diels‐Alder Reaction by Encapsulation of a Lithium Atom

Diels‐Alder cycloaddition reaction is useful for generation of covalent derivatives of fullerenes. Diels‐Alder reactions of C₇₀ and dienes usually take place at the carbon‐carbon bond that has a short bond length in C₇₀, while the bonds with long lengths are generally unreactive. In this paper, we investigated the reactivities of Li⁺@C₇₀ and Li@C₇₀ toward Diels‐Alder reactions with cyclohexadiene by means of density functional theory calculations. We found that the thermodynamic and kinetic reactivities of the fullerene cage are changed significantly after the encapsulation of the lithium ion or atom. The encapsulated lithium ion causes a remarkable decrease of the activation barrier for the cycloaddition reaction, which can be ascribed to the enhanced orbital interaction between cyclohexadiene and the fullerene cage. The unreactive bond with a long length in C₇₀ is activated efficiently after the encapsulation of the lithium atom. According to the activation‐strain model analysis, the improved reactivity of the long bond is associated with the small deformation energy and large interaction energy of the reactants. Unlike conventional Diels‐Alder reactions that proceed through concerted mechanism, the reaction of Li@C₇₀ and cyclohexadiene undergoes an unusual stepwise mechanism because of the open‐shell electronic structure of Li@C₇₀.     

How Alkali Cations Catalyze Aromatic Diels‐Alder Reactions

We have quantum chemically studied alkali cation‐catalyzed aromatic Diels‐Alder reactions between benzene and acetylene forming barrelene using relativistic, dispersion‐corrected density functional theory. The alkali cation‐catalyzed aromatic Diels‐Alder reactions are accelerated by up to 5 orders of magnitude relative to the uncatalyzed reaction and the reaction barrier increases along the series Li⁺ < Na⁺ < K⁺ < Rb⁺ < Cs⁺ < none. Our detailed activation strain and molecular‐orbital bonding analyses reveal that the alkali cations lower the aromatic Diels‐Alder reaction barrier by reducing the Pauli repulsion between the closed‐shell filled orbitals of the dienophile and the aromatic diene. We argue that such Pauli mechanism behind Lewis‐acid catalysis is a more general phenomenon. Also, our results may be of direct importance for a more complete understanding of the network of competing mechanisms towards the formation of polycyclic aromatic hydrocarbons (PAHs) in an astrochemical context.     

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.     

Bioinspired Intramolecular Diels–Alder Reaction: A Rapid Access to the Highly‐Strained Cyclopropane‐Fused Polycyclic Skeleton

A bioinsipred gold‐catalyzed tandem Diels–Alder/Diels–Alder reaction of an enynal and a 1,3‐diene, forming the highly‐strained benzotricyclo[3.2.1.0^2,7]octane skeleton, was reported. In contrast, a Diels–Alder/Friedel–Crafts tandem reaction occurred instead when silver salts were used as the catalyst. Although both reactions experienced the similar Diels–Alder reaction of a pyrylium intermediate with a 1,3‐diene, they have different reaction mechanisms. The former proceeded with a stepwise Diels–Alder reaction, while the latter one with a concerted one.     

Synthesis of a Double Helicene by a Palladium‐Catalyzed Cross‐Coupling Reaction: Structure and Physical Properties

For this study, twisted π‐extended helicene 1 and double helicene 2 with a helicene framework were synthesized through palladium‐catalyzed C?H arylation or Suzuki–Miyaura coupling reaction. X‐ray crystallography revealed grossly twisted structures that were soluble in various conventional organic solvents. Optical properties based on UV/Vis and fluorescence spectra were measured. Electrochemical properties were also studied by measurements of cyclic voltammetry in 1 and 2 , which revealed their HOMO and the LUMO energies. Theoretical calculation supports their HOMO and LUMO energies and molecular orbitals. Furthermore, a racemization process of 2 predicted that the activation free energy at 300 K would be 31.8 kcal mol^?1 by DFT calculation, which indicated the static helicity at 300 K.     

Relationship between orbital energy gaps and excitation energies for long‐chain systems

The difference between the excitation energies and corresponding orbital energy gaps, the exciton binding energy, is investigated based on time‐dependent (TD) density functional theory (DFT) for long‐chain systems: all‐trans polyacetylenes and linear oligoacenes. The optimized geometries of these systems indicate that bond length alternations significantly depend on long‐range exchange interactions. In TDDFT formalism, the exciton binding energy comes from the two‐electron interactions between occupied and unoccupied orbitals through the Coulomb‐exchange‐correlation integral kernels. TDDFT calculations show that the exciton binding energy is significant when long‐range exchange interactions are involved. Spin‐flip (SF) TDDFT calculations are then carried out to clarify double‐excitation effects in these excitation energies. The calculated SF‐TDDFT results indicate that double‐excitation effects significantly contribute to the excitations of long‐chain systems. The discrepancies between the vertical ionization potential minus electron affinity (IP–EA) values and the HOMO–LUMO excitation energies are also evaluated for the infinitely long polyacetylene and oligoacene using the least‐square fits to estimate the exciton binding energy of infinitely long systems. It is found that long‐range exchange interactions are required to give the exciton binding energy of the infinitely long systems. Consequently, it is concluded that long‐range exchange interactions neglected in many DFT calculations play a crucial role in the exciton binding energies of long‐chain systems, while double‐excitation correlation effects are also significant to hold the energy balance of the excitations. © 2016 Wiley Periodicals, Inc.     

Bifunctional Hydrogen Bond Donor-Catalyzed Diels–Alder Reactions: Origin of Stereoselectivity and Rate Enhancement

The selectivity and rate enhancement of bifunctional hydrogen bond donor-catalyzed Diels–Alder reactions between cyclopentadiene and acrolein were quantum chemically studied using density functional theory in combination with coupled-cluster theory. (Thio)ureas render the studied Diels–Alder cycloaddition reactions exo selective and induce a significant acceleration of this process by lowering the reaction barrier by up to 7 kcal mol⁻¹. Our activation strain and Kohn–Sham molecular orbital analyses uncover that these organocatalysts enhance the Diels–Alder reactivity by reducing the Pauli repulsion between the closed-shell filled π-orbitals of the diene and dienophile, by polarizing the π-orbitals away from the reactive center and not by making the orbital interactions between the reactants stronger. In addition, we establish that the unprecedented exo selectivity of the hydrogen bond donor-catalyzed Diels–Alder reactions is directly related to the larger degree of asynchronicity along this reaction pathway, which is manifested in a relief of destabilizing activation strain and Pauli repulsion.     

The effect of conjugated spacer on novel carbazole derivatives for dye‐sensitized solar cells: Density functional theory/time‐dependent density functional theory study

The ground‐state structure and frontier molecular orbital of D‐π‐A organic dyes, CFT1A, CFT2A, and CFT1PA were theoretically investigated using density functional theory (DFT) on B3LYP functional with 6‐31G(d,p) basis set. The vertical excitation energies and absorption spectra were obtained using time‐dependent DFT (TD‐DFT). The adsorptions of these dyes on TiO₂ anatase (101) were carried out by using a 38[TiO₂] cluster model using Perdew–Burke–Ernzerhof functional with the double numerical basis set with polarization (DNP). The results showed that the introduction of thiophene–thiophene unit (T–T) as conjugated spacer in CFT2A could affect the performance of intramolecular charge transfer significantly due to the inter‐ring torsion of T–T being decreased compared with phenylene–phenylene (P–P) spacer of CFP2A in the researhcers' previous report. It was also found that increasing the number of π‐conjugated unit gradually enhanced charge separation between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of these dyes, leading to a high‐efficiency photocurrent generation. The HOMO–LUMO energy gaps were calculated to be 2.51, 2.37, and 2.50 eV for CFT1A, CFT2A, and CFT1PA respectively. Moreover, the calculated adsorption energies of these dyes on TiO₂ cluster were ～14 kcal/mol, implying that these dyes strongly bind to TiO₂ surface. Furthermore, the electronic HOMO and LUMO shapes of all dye–TiO₂ complexes exhibited injection mechanism of electron via intermolecular charge‐transfer transition. © 2012 Wiley Periodicals, Inc.     

One‐Pot Synthesis of 2‐Arylthio‐2‐cyclohexenone Derivatives by the Diels–Alder Reaction of 4‐Arylthio‐3‐hydroxy‐2‐pyrones

The base‐catalyzed Diels–Alder reactions of 4‐arylthio‐3‐hydroxy‐2‐pyrones are reported. Treatment of 4‐arylthio‐3‐hydroxy‐2‐pyrones and dienophiles with triethylamine gave 2‐arylthio‐2‐cyclohexenone derivatives by the Diels–Alder reaction involving a decarboxylation in excellent to reasonable yields.     

Dual Activation of Aromatic Diels–Alder Reactions

The unusually fast Diels–Alder reactions of [5]cyclophanes were analyzed by DFT at the BLYP-D3(BJ)/TZ2P level of theory. The computations were guided by an integrated activation-strain and Kohn–Sham molecular orbital analysis. It is revealed why both [5]metacyclophane and [5]paracyclophane exhibit a significant rate enhancement compared to their planar benzene analogue. The activation strain analyses revealed that the enhanced reactivity originates from 1) predistortion of the aromatic core resulting in a reduced activation strain of the aromatic diene, and/or 2) enhanced interaction with the dienophile through a distortion-controlled lowering of the HOMO–LUMO gap within the diene. Both of these physical mechanisms and thus the rate of Diels–Alder cycloaddition can be tuned through different modes of geometrical distortion (meta versus para bridging) and by heteroatom substitution in the aromatic ring. Judicious choice of the bridge and heteroatom in the aromatic core enables effective tuning of the aromatic Diels–Alder reactivity to achieve activation barriers as low as 2 kcal mol⁻¹, which is an impressive 35 kcal mol⁻¹ lower than that of benzene.     

Origin of the “endo rule” in Diels–Alder reactions

Detailed density functional theory calculations definitively rationalize the preference for the endo cycloadduct (also known as endo rule) in text‐book thermal Diels–Alder reactions involving maleic anhydride and cyclopentadiene or butadiene. This selectivity is mainly caused by an unfavorable steric arrangement in the transition‐state region of the exo pathway which translates into a more destabilizing activation strain. In contrast with the widely accepted, orbital‐interaction‐based explanation for the endo rule, it is found that neither the orbital interactions nor the total interaction between the deformed reactants contributes to the endo selectivity. © 2013 Wiley Periodicals, Inc.     

Comparison of DFT methods for molecular orbital eigenvalue calculations

We report how closely the Kohn-Sham highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenvalues of 11 density functional theory (DFT) functionals, respectively, correspond to the negative ionization potentials (-IPs) and electron affinities (EAs) of a test set of molecules. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin density approximation (LSDA), five generalized gradient approximation (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH&HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with average absolute errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, respectively. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (approximately 0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calculated HOMO eigenvalue and the experimental -IP and calculated HOMO-LUMO gap and experimental lowest excitation energy enables us to derive a simple correction formula.     

Manganese(I)‐Catalyzed C−H Activation/Diels–Alder/retro‐Diels–Alder Domino Alkyne Annulation featuring Transformable Pyridines

Complexity‐increasing Domino reactions comprising C?H allenylation, a Diels–Alder reaction, and a retro‐Diels–Alder reaction were realized by a versatile catalyst derived from earth‐abundant, non‐toxic manganese. The C?H activation/Diels–Alder/retro‐Diels–Alder alkyne annulation sequence provided step‐economical access to valuable indolone alkaloid derivatives through a facile organometallic C?H activation manifold with transformable pyridines.     

Oxidative Copolymerisation of para‐Functionalized Phenols Catalyzed by Horseradish peroxidase and Thermocrosslinking via Diels‐Alder and (1+3) Cycloaddition

The horseradish peroxidase‐ (HRP‐) catalyzed polyrecombination of N‐(4‐hydroxyphenyl)‐2‐furamide ( 2 ) and 4‐hydroxyphenylmaleimide ( 1 ) is described. The resulting copolymer was used to build crosslinked materials via Diels‐Alder and cycloaddition reactions. We followed the enzymatic copolymerization process of an equimolar mixture of 1 and 2 using high pressure liquid chromatography (HPLC), size‐exclusion chromatography (SEC) and by matrix assisted laser desorption/ionization‐time of flight mass spectroscopy (MALDI‐TOF MS) analysis and found that the polymerization of 2 , which has a significant higher highest occupied molecular orbital (HOMO) energy, proceeds much faster. The HOMO energies of 1 and 2 were calculated using a differential Fourier transform (DFT) method. Furthermore, we tested copolymers consisting of the monomer units 4‐hydoxybenzaldehyde‐methylnitrone ( 3 ) and N‐methacryloyl‐11‐aminoundecanoyl‐4‐hydroxyanilide ( 4 ) on their ability to form networks via 1,3‐dipolaric cycloaddition reactions upon heating. The crosslinking of all copolymers was proven by Fourier transform infrared (FT‐IR) spectroscopy.     

3D Photofixation Lithography in Diels–Alder Networks

3D structures are written and developed in a crosslinked polymer initially formed by a Diels–Alder reaction. Unlike conventional liquid resists, small features cannot sediment, as the reversible crosslinks function as a support, and the modulus of the material is in the MPa range at room temperature. The support structure, however, can be easily removed by heating the material, and depolymerizing the polymer into a mixture of low‐viscosity monomers. Complex shapes are written into the polymer network using two‐photon techniques to spatially control the photoinitiation and subsequent thiol–ene reaction to selectively convert the Diels–Alder adducts into irreversible crosslinks.     

Benzocyclobutadienes: An Unusual Mode of Access Reveals Unusual Modes of Reactivity

The reaction of an aryne with an alkyne to generate a benzocyclobutadiene (BCB) intermediate is rare. We report here examples of this reaction, revealed by Diels–Alder trapping of the BCB by either pendant or external electron‐deficient alkynes. Mechanistic delineation of the reaction course is supported by DFT calculations. A three‐component process joining the benzyne first with an electron‐rich and then with an electron‐poor alkyne was uncovered. Reactions in which the BCB functions in a rarely observed role as a 4π diene component in Diels–Alder reactions are reported. The results also shed new light on aspects of the hexadehydro‐Diels–Alder reaction used to generate the benzynes.     

Natural bond orbital‐based energy density analysis for correlated methods: Second‐order Møller–Plesset perturbation and coupled‐cluster singles and doubles

Natural bond orbital‐based energy density analysis (NBO‐EDA), which split energies into atomic and bonding contributions, is proposed for correlated methods such as coupled‐cluster singles and doubles (CCSD) and second‐order Møller–Plesset (MP2) perturbation. Applying NBO‐EDA for CCSD and MP2 to ethylene and the Diels–Alder reaction, we are successful in obtaining useful knowledge regarding electron correlation of π‐ and σ‐type orbitals, and clarifying the difference of the reaction barriers and heat of reaction calculated by CCSD and MP2. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008