•The vibronic coupling constant and density are extended to a mixed state.•The relation between chemical potential and vibronic coupling is obtained.•Vibronic coupling enhances inter-molecular charge transfer.Vibronic coupling constant (VCC) and density (VCD) defined for a pure state, which have been successfully applied for reactions of fullerenes and nanographenes as reactivity indices, are extended for a mixed state. The extended VCC and VCD, thermodynamical vibronic coupling constant (ThVCC) and density (ThVCD), are formulated in the finite-temperature grand-canonical ensemble. ThVCD can be applied for charge transfer of a fractional number of electron. Based on the total differential of chemical potential, the relationship between chemical potential, absolute hardness, and vibronic coupling in a bimolecular reaction is discussed.

The concepts of symmetry-controlled thermally activated delayed fluorescence (SC-TADF) and inverted singlet–triplet (iST) structure are proposed. Molecules that can exhibit SC-TADF or have an iST structure can be employed as light-emitting molecules in organic light-emitting diodes. The molecular symmetry plays crucial roles in these concepts since they are based on the selection rules for the electric dipole transition, intersystem crossing, and nonradiative vibronic (electron-vibration) transitions. In addition to the symmetry conditions for the SC-TADF and iST molecules, the molecules should have small diagonal and off-diagonal vibronic coupling constants for suppressing vibrational relaxations and nonradiative vibronic transitions, respectively, and a large transition dipole moment for the fluorescence process. Analyses using the vibronic coupling and transition dipole moment densities are employed to reduce the vibronic coupling constants and to increase the transition dipole moment. The preferable point groups in the development of SC-TADF and iST molecules are discussed on the basis of the ratios of forbidden pairs of irreducible representations. It is found that the existence of the inversion symmetry is preferable for designing SC-TADF and iST molecules. On the basis of these guiding principles, we designed some anthracene and pyrene derivatives as candidate iST molecules. Their electronic structures, spin–orbit couplings, transition dipole moments, and vibronic couplings are discussed.

The cycloaddition reactivities of perylene (C20H12), bisanthene (C28H14), and tribenzoperylene (C30H16) are theoretically investigated with vibronic coupling density analysis. Their HOMOs are strongly delocalized over the molecules, and therefore the reactive sites are smeared. Vibronic coupling density analysis clearly shows that the bay regions of armchair edges are reactive. This is consistent with experimental findings that maleic anhydride attacks the bay regions.

The regioselectivities of La2@C80 in thermal nucleophilic and electrophilic attacks were theoretically investigated using vibronic coupling density (VCD) analysis. Nucleophilic and electrophilic cycloadditions to La2@C80 were experimentally reported to yield [6,6] and [6,5] adducts, respectively, as major products. VCD analysis provided a clear explanation for these experimental results. For nucleophilic reactions, it was found that the reactive [6,6] bonds did not have a large lowest unoccupied molecular orbital (LUMO) density and Fukui function but a large potential derivative with respect to a reaction mode. The VCD illustrates the origin of the interaction between the electronic and vibrational states. On the other hand, conventional reactivity indices such as frontier orbital density take only the electronic state into account. The result suggested that the stabilization due to vibronic couplings plays an important role in the regioselectivity of nucleophilic cycloadditions. The VCD with respect to the effective mode could provide a picture of the functional groups, which are the double bonds of ethylene moieties. VCD analysis with respect to hypothetical localized modes enabled the quantitative prediction of regioselectivities.

The anionic state of the icosahedral W@Au12 cluster offers a rare example of a Jahn–Teller (JT) instability in an icosahedral fourfold degenerate Γ8 spinor level. The JT energy splittings of the ground Γ8 and excited sixfold degenerate Γ9 splittings in the vicinity of the degeneracy point are calculated with relativistic density functional theory. The results are very well explained by a first-order coupling model, based on the orbital instability of the spherical d-shell of the cluster. In addition the pentagonal JT minimum has been determined. It presents a remarkable example of an auro-sandwich type compound.

A theoretical design principle for enhancement of the quantum yield of light-emitting molecules is desired. For the establishment of the principle, we focused on the S1 states of blue-emitting anthracene derivatives: 2-methyl-9,10-di(2′-naphthyl)anthracene (MADN), 4,9,10-bis(3′,5′-diphenylphenyl)anthracene (MAM), 9-(3′,5′-diphenylphenyl)-10-(3′′,5′′-diphenylbiphenyl-4′′-yl) anthracene (MAT), and 9,10-bis(3′′′,5′′′-diphenylbiphenyl-4′-yl) anthracene (TAT) [Kim et al., J. Mater. Chem., 2008, 18, 3376]. The vibronic coupling constants and transition dipole moments were calculated and analyzed by using the concepts of vibronic coupling density (VCD) and transition dipole moment density (TDMD), respectively. It is found that the driving force of the internal conversions and vibrational relaxations originate mainly from the anthracenylene group. On the other hand, fluorescence enhancement results from the large torsional distortion of the side groups in the S1 state. The torsional distortion is caused by the diagonal vibronic coupling for the lowest-frequency mode in the Franck–Condon (FC) S1 state, which originates from a small portion of the electron density difference on the side groups. These findings lead to the following design principles for anthracene derivatives with a high quantum yield: (1) reduction in the electron density difference and overlap density between the S0 and S1 states in the anthracenylene group to suppress vibrational relaxation and radiationless transitions, respectively; (2) increase in the overlap density in the side group to enhance the fluorescence.

•Anthracene derivatives with small vibronic coupling constants and large transition dipole moment are theoretically designed.•The designed molecule was synthesized. The molecule exhibited a high quantum yield of 96%.•The observed absorption and photoluminescence spectra are consistent with calculations.•The molecules are the first examples of designed fluorescent molecules from the view of vibronic couplings.•The present approach can contribute to the development of light-emitting materials.5,11-Bis(phenylethynyl)benzo[1,2-f:4,5-f′]diisoindole-1,3,7,9(2H,8H)-tetraone 1H was designed as an application of the theoretical design principle for fluorescent molecules which is derived from the vibronic coupling density analysis. For solubility reasons, tertiary-butylated 1H, 2,8-di-tert-butyl-5,11-bis(phenylethynyl)benzo[1,2-f:4,5-f′]diisoindole-1,3,7,9(2H,8H)-tetraone 1 was synthesized and its fluorescence properties were measured. It is found that the photoluminescence quantum yield of 1 was 96%. We discuss the rationale for designing 1H as a highly efficient fluorescent molecule, and compare the theoretical calculations for 1 with the observed absorption and photoluminescence spectra.

•The deformation of the cylindrical structure in [6]cycloparaphenylene is studied.•This deformation is ascribed to orbital vibronic couplings of the deformation modes.•We calculated orbital vibronic coupling constants and vibronic coupling densities.•The pair of frontier orbitals and sigma-type orbitals contributes to the deformation.The instability of the cylindrical D6hD6h structure in [6]cycloparaphenylene, the non-zero dihedral angles between adjacent paraphenylene units, is investigated from the viewpoint of the pseudo Jahn–Teller effect, wherein the driving forces are orbital vibronic couplings of the deformation modes between occupied and unoccupied orbitals. We show that the overlap densities between the frontier orbitals including the highest occupied molecular orbital (HOMO) and a certain unoccupied/occupied molecular orbital mainly contribute to the instability. The results of vibronic coupling density analysis justify the strong coupling of the unoccupied orbital with the HOMO.

The vibronic coupling constants and reorganization energies of oligofluorenes OF(n) (n = 1–6) are calculated for their cationic states (hole transport). Those of oligothiophenes OT(2n) (n = 1–6) are also calculated for comparison. The vibronic coupling constants of OF(n) are smaller than those of OT(2n), and decrease with increasing n. For the elucidation of the small vibronic couplings of the oligofluorenes, the calculated vibronic coupling constants are analyzed on the basis of the concept of vibronic coupling density. The vibronic coupling density of OF(n) becomes small in the middle of the chain with increasing n because of the reduction in the electron-density difference between the neutral and cationic states. It is found that orbital relaxation plays a crucial role in the distribution of the electron-density difference. From the fragment molecular orbital analyses, the large orbital relaxation in OF(n) is found to originate from the small transfer integral between the fragment molecular orbitals. These findings led to a design principle for a carrier-transporting oligomer/polymer with small vibronic couplings, or small reorganization energy, as follows: the orbital interaction between the monomers should be small from the view of vibronic couplings.

The reaction mechanism for mechanochemical synthesis of dibenzophenazine was theoretically investigated in terms of the vibronic coupling density, which describes the interactions between electrons and nuclear motions. The concept theoretically indicates experimentally observed reactive sites that cannot be explained by the conventional frontier orbital theory. The results of vibronic coupling density analysis suggested the difference between reaction mechanisms under thermal and mechanochemical conditions.

•Phosphorescence of a photocatalyst, highly dispersed vanadium oxide on silica, is simulated.•The irregular gap between peaks is explained including three vibrational modes.•In the three modes, Jahn–Teller mode is included.•The presence of the Jahn–Teller effect which induces the photocatalysis is confirmed.Vibronic progressions in the phosphorescence spectrum of highly dispersed vanadium oxide on silica support, V2O5/SiO2, measured by Hazenkamp and Blasse [18] is simulated. The irregular peak spacing is revealed to be evidence of the existence of a few active vibrational modes coupled with the electronic excited T1T1 state, which is assigned to the active state as a photocatalyst. Among the active modes, the Jahn–Teller mode is responsible for the photocatalytic reactivity.

The chemical reactivity in nucleophilic cycloaddition to C70 is investigated on the basis of vibronic (electron-vibration) coupling density and vibronic coupling constants. Because the e1″ LUMOs of C70 are doubly degenerate and delocalized throughout the molecule, it is difficult to predict the regioselectivity by frontier orbital theory. It is found that vibronic coupling density analysis for the effective mode as a reaction mode illustrates the idea of a functional group embedded in the reactive sites. Furthermore, the vibronic coupling constants for localized stretching vibrational modes enable us to estimate the quantitative reactivity. These calculated results agree well with the experimental findings. The principle of chemical reactivity proposed by Parr and Yang is modified as follows: the preferred direction is the one for which the initial vibronic coupling density for a reaction mode of the isolated reactant is a minimum.

The vibronic coupling constants and reorganization energies of oligofluorenes OF(n) (n = 1–6) are calculated for their cationic states (hole transport). Those of oligothiophenes OT(2n) (n = 1–6) are also calculated for comparison. The vibronic coupling constants of OF(n) are smaller than those of OT(2n), and decrease with increasing n. For the elucidation of the small vibronic couplings of the oligofluorenes, the calculated vibronic coupling constants are analyzed on the basis of the concept of vibronic coupling density. The vibronic coupling density of OF(n) becomes small in the middle of the chain with increasing n because of the reduction in the electron-density difference between the neutral and cationic states. It is found that orbital relaxation plays a crucial role in the distribution of the electron-density difference. From the fragment molecular orbital analyses, the large orbital relaxation in OF(n) is found to originate from the small transfer integral between the fragment molecular orbitals. These findings led to a design principle for a carrier-transporting oligomer/polymer with small vibronic couplings, or small reorganization energy, as follows: the orbital interaction between the monomers should be small from the view of vibronic couplings.

The concepts of symmetry-controlled thermally activated delayed fluorescence (SC-TADF) and inverted singlet–triplet (iST) structure are proposed. Molecules that can exhibit SC-TADF or have an iST structure can be employed as light-emitting molecules in organic light-emitting diodes. The molecular symmetry plays crucial roles in these concepts since they are based on the selection rules for the electric dipole transition, intersystem crossing, and nonradiative vibronic (electron-vibration) transitions. In addition to the symmetry conditions for the SC-TADF and iST molecules, the molecules should have small diagonal and off-diagonal vibronic coupling constants for suppressing vibrational relaxations and nonradiative vibronic transitions, respectively, and a large transition dipole moment for the fluorescence process. Analyses using the vibronic coupling and transition dipole moment densities are employed to reduce the vibronic coupling constants and to increase the transition dipole moment. The preferable point groups in the development of SC-TADF and iST molecules are discussed on the basis of the ratios of forbidden pairs of irreducible representations. It is found that the existence of the inversion symmetry is preferable for designing SC-TADF and iST molecules. On the basis of these guiding principles, we designed some anthracene and pyrene derivatives as candidate iST molecules. Their electronic structures, spin–orbit couplings, transition dipole moments, and vibronic couplings are discussed.

A theoretical design principle for enhancement of the quantum yield of light-emitting molecules is desired. For the establishment of the principle, we focused on the S1 states of blue-emitting anthracene derivatives: 2-methyl-9,10-di(2′-naphthyl)anthracene (MADN), 4,9,10-bis(3′,5′-diphenylphenyl)anthracene (MAM), 9-(3′,5′-diphenylphenyl)-10-(3′′,5′′-diphenylbiphenyl-4′′-yl) anthracene (MAT), and 9,10-bis(3′′′,5′′′-diphenylbiphenyl-4′-yl) anthracene (TAT) [Kim et al., J. Mater. Chem., 2008, 18, 3376]. The vibronic coupling constants and transition dipole moments were calculated and analyzed by using the concepts of vibronic coupling density (VCD) and transition dipole moment density (TDMD), respectively. It is found that the driving force of the internal conversions and vibrational relaxations originate mainly from the anthracenylene group. On the other hand, fluorescence enhancement results from the large torsional distortion of the side groups in the S1 state. The torsional distortion is caused by the diagonal vibronic coupling for the lowest-frequency mode in the Franck–Condon (FC) S1 state, which originates from a small portion of the electron density difference on the side groups. These findings lead to the following design principles for anthracene derivatives with a high quantum yield: (1) reduction in the electron density difference and overlap density between the S0 and S1 states in the anthracenylene group to suppress vibrational relaxation and radiationless transitions, respectively; (2) increase in the overlap density in the side group to enhance the fluorescence.