Internal Conversion Rate Constants Research Articles

Fast calculation of internal conversion rate constant using the time-dependent formalism

A new method for calculating the internal conversion rate constant, which takes into account all possible vibrational transitions between two adiabatic electronic states, is proposed. The simple analytical form of the correlation function obtained in the harmonic and anharmonic approximations and taking into account the Franck-Condon term makes it possible to perform fast calculations. In addition, the method takes into account temperature effects within harmonic kind, which extends its area of application to explain experimental processes. The approach has been applied to the study of molecules important for organic electronics, such as tetraoxa[8]circulene, free-base porphyrin and poly[n]fluorenes. The method provides valuable information on IC processes and facilitates the design of efficient molecular systems.

Internal conversion induced by external electric and magnetic fields.

We have developed a new methodology for calculating contributions to the rate constants (kIC) of internal conversion that are induced by external electric (kIC-E) or magnetic (kIC-M) fields. The influence of the external electric and magnetic fields on the kIC was estimated for seven representative molecules. We show that the kIC-E contribution calculated at a field strength of 1011 V m-1 is generally as large as the kIC rate constant in the absence of the external field. For indocyanine green, azaoxa[8]circulene, and pyromitene 567, the kIC-E contribution is as large as kIC already at a field strength of 109 V m-1. Such electric-field strengths occur for example in plasmonic studies and in strong laser-field experiments. The induced effect on the kIC rate constant should be accounted for in calculations of photophysical properties of molecules involved in such experiments. The induced effect of an external magnetic field on kIC can be neglected in experiments on Earth because the magnetic contribution becomes significant only at very strong magnetic fields of 104-105 T that cannot be achieved on Earth. However, the magnetic effect on the rate constant of internal conversion can be important in astrophysical studies, where extremely strong magnetic fields occur near neutron stars and white dwarfs.

Intramolecular rate-constant calculations based on the correlation function using temperature dependent quantum Green's functions.

A theoretical method for calculating rate constants for internal conversion (IC), intersystem crossing (ISC) and radiative (R) electronic transitions is presented. The employed method uses temperature-dependent quantum Green's functions, which give the opportunity to consider almost any nth-order polynomial perturbation operator and the influence of external electromagnetic fields on the rate constants. The rate constants of the IC, ISC and R processes are calculated for two important indocyanine molecules namely indocyanine green (ICG) and heptamethine cyanine (IR808) at the Franck-Condon level using the temperature-dependent quantum Green's function approach. Calculations at the time-dependent density functional theory level with the MN15 functional show that ICG and IR808 have only one triplet state below the S1 state. The main deactivation channel of the S1 state is the IC process with a large (kIC(S1 → S0)) rate constant of ∼109-1011 s-1. The estimated quantum yield of fluorescence (φfl) is ∼0.001-0.24 for the two studied molecules, which agrees rather well with experimental values. Thus, the present approach enables calculations of the three kinds of rate constants and the quantum yield of fluorescence using the same computational methodology.

Role of Vibronic Couplings and Energy Gap in the Internal Conversion Process of a Molecule

Abstract An analytical expression for the rate constant of internal conversion (IC) in a molecule was derived based on the crude adiabatic representation. All vibrational modes were considered to be on an equal footing in the rate constant expression. Based on this expression, we investigated the role of vibronic couplings and electronic energy gap in IC processes, using 9-fluorenone as an illustrative example. Vibrational modes with strong off-diagonal vibronic coupling constants (VCCs) give rise to non-radiative transitions. In contrast, vibrational modes with strong diagonal VCCs constitute the final vibronic states that accept the excess electronic energy between the initial and final electronic states. Therefore, vibrational modes are classified into promoting and accepting modes based on their roles. We identified important promoting modes responsible for one-phonon emission/absorption and accepting modes that contribute greatly to the final state. A Franck-Condon envelope, which describes the density of final vibronic states, explains the dependence of the rate constant on the electronic energy gap. VCC can be visualised as a spatial distribution of its density form, i.e., vibronic coupling density (VCD). The VCD concept is expected to facilitate the design of functional molecules with IC processes understood in terms of electronic states and vibrational modes.

Calculation of Excited State Internal Conversion Rate Constant Using the One-Effective Mode Marcus-Jortner-Levich Theory.

In this article, the one-effective mode Marcus-Jortner-Levich (MJL) theory and the classical Marcus theory for electron transfer were applied to estimate the internal conversion rate constant, kIC, of organic molecules and a Ru-based complex, all belonging to the Marcus inverted region. For this, the reorganization energy was calculated using the minimum energy conical intersection point to account for more vibrational levels, correcting the density of states. The results showed good agreement with experimental and theoretically determined kIC, with a small overestimation by the Marcus theory. Also, molecules less dependent on the solvent effects, like benzophenone, presented better results than molecules with an expressive dependence, like 1-aminonaphthalene. Moreover, the results suggest that each molecule possesses unique normal modes leading to the excited state deactivation that does not necessarily match the X-H bond stretching, as previously suggested.

Internal conversion rate constant calculations considering Duschinsky, anharmonic and Herzberg-Teller effects.

A novel method for calculating rate constants for internal conversion (kIC) that simultaneously accounts for Duschinsky, anharmonic and Herzberg-Teller effects has been developed and implemented. This method has been applied to robust planar molecules like tetraoxa[8]circulene (4B), free-base porphyrin (H2P) and pyrometene (PM567) with small Duschinsky rotation (i.e. with almost identical normal coordinates in the ground and excited states) and to poly[n]fluorenes (P[n]F) (n = 2-14) with a substantial Duschinsky rotation. The obtained results show that the Duschinsky effect is large in the harmonic approximation, whereas it is in general much smaller in the anharmonic approximation. The Duschinsky effect is found to be large for high frequency vibrational modes with energies of ∼3300 cm-1 such as the X-H (X = C, N and O) stretching modes that mix in the S1 → S0 electronic transition. However, even in this case, the increase in kIC due to the Duschinsky effect does not exceed one order of magnitude. The calculations show that anharmonic contributions to kIC are larger than Herzberg-Teller contributions which in turn are larger than contributions from the Duschinsky effect ANH > HT > Du. We also show that an approximation, where only X-H bonds are considered in the kIC calculation, is accurate even for P[n]F (n = 2-14).

Interexcited State Photophysics I: Benchmarking Density Functionals for Computing Nonadiabatic Couplings and Internal Conversion Rate Constants.

We present the first benchmarking study of nonadiabatic matrix coupling elements (NACMEs) calculated using different density functionals. Using the S1 → S0 transition in perylene solvated in toluene as a case study, we calculate the photophysical properties and corresponding rate constants for a variety of density functionals from each rung of Jacob's ladder. The singlet photoluminescence quantum yield (sPLQY) is taken as a measure of accuracy, measured experimentally here as 0.955. Important quantum chemical parameters such as geometries, absorption, emission, and adiabatic energies, NACMEs, Hessians, and transition dipole moments were calculated for each density functional basis set combination (data set) using density functional theory based multireference configuration interaction (DFT/MRCI) and compared to experiment where possible. We were able to derive simple relations between the TDDFT and DFT/MRCI photophysical properties; with semiempirical damping factors of ∼0.843 ± 0.017 and ∼0.954 ± 0.064 for TDDFT transition dipole moments and energies to DFT/MRCI level approximations, respectively. NACMEs were dominated by out-of-plane derivative components belonging to the center-most ring atoms with weaker contributions from perturbations along the transverse and longitudinal axes. Calculated theoretical spectra compared well to both experiment and literature, with fluorescence lifetimes between 7.1 and 12.5 ns, agreeing within a factor of 2 with experiment. Internal conversion (IC) rates were then calculated and were found to vary wildly between 106-1016 s-1 compared with an experimental rate of the order 107 s-1. Following further testing by mixing data sets, we found a strong dependence on the method used to obtain the Hessian. The 5 characterized data sets ranked in order of most promising are PBE0/def2-TZVP, ωB97XD/def2-TZVP, HCTH407/TZVP, PBE/TZVP, and PBE/def2-TZVP.

Theory of Exciton Dynamics in Thermally Activated Delayed Fluorescence

AbstractThermally activated delayed fluorescence (TADF) is an upconversion photophysical process based on the population of a bright excited singlet state from an excited triplet via reverse intersystem crossing (rISC), which might be employed to overcome the statistical limitation of electron‐hole recombination in organic light‐emitting diodes. In this work, we describe the intricacies of TADF dynamics by means of a quantum master equation and through perturbative approaches to rate constants of internal conversion and ISC. Simulation of the time evolution from the lowest excited triplet in systems with small energy gaps allows to disentangle the role of different electronic states mediating the population of the singlet manifold and the importance of various rISC mechanisms. Energy gaps between charge transfer (CT) and local excitons (LE), and interstate (hyperfine, spin‐orbit and vibronic) couplings are key factors tuning the feasibility of rISC. Since in molecules LE/CT spin‐orbit coupling is, in general, larger than hyperfine interactions, the rISC rate is largely increased in those situations where either the direct or mediated triplet‐singlet interconversion between states of different character is energetically available. Interestingly, temperature dependence of rISC rate is stronger for the spin‐vibronic coupling mechanism than via hyperfine interaction.

Controlling the fluorescence quantum yields of benzothiazole-difluoroborates by optimal substitution.

Precise tuning of the fluorescence quantum yield, vital for countless applications of fluorophores, remains exceptionally challenging due to numerous factors affecting energy dissipation phenomena often leading to its counterintuitive behavior. In contrast to the absorption and emission wavelength which can be precisely shifted to the desired range by simple structural changes, no general strategy exists for controllable modification of the fluorescence quantum yield. The rigidification of the molecular skeleton is known to usually enhance the emission and can be practically realized via the limiting molecular vibrations by aggregation. However, the subtle balance between the abundant possible radiative and non-radiative decay pathways makes the final picture exceptionally sophisticated. In the present study, a series of nine fluorophores obtained by peripheral substitution with two relatively mild electron donating and electron withdrawing groups are reported. The obtained fluorescence quantum yields range from dark to ultra-bright and the extreme values are obtained for the isomeric molecules. These severe changes in emission efficiency have been shown to arise from the complex relationship between the Franck-Condon excited state and conical intersection position. The experimental findings are rationalized by the advanced quantum chemical calculations delivering good correlation between the measured emission parameters and theoretical radiative and internal conversion rate constants. Therefore, the described substituent exchange provides a method to rigorously adjust the properties of molecular probes structurally similar to thioflavin T.

Internal conversion of singlet and triplet states employing numerical DFT/MRCI derivative couplings: Implementation, tests, and application to xanthone.

We present an efficient implementation of nonadiabatic coupling matrix elements (NACMEs) for density functional theory/multireference configuration interaction (DFT/MRCI) wave functions of singlet and triplet multiplicity and an extension of the Vibes program that allows us to determine rate constants for internal conversion (IC) in addition to intersystem crossing (ISC) nonradiative transitions. Following the suggestion of Plasser et al. [J. Chem. Theory Comput. 12, 1207 (2016)], the derivative couplings are computed as finite differences of wave function overlaps. Several measures have been taken to speed up the calculation of the NACMEs. Schur's determinant complement is employed to build up the determinant of the full matrix of spin-blocked orbital overlaps from precomputed spin factors with fixed orbital occupation. Test calculations on formaldehyde, pyrazine, and xanthone show that the mutual excitation level of the configurations at the reference and displaced geometries can be restricted to 1. In combination with a cutoff parameter of tnorm = 10-8 for the DFT/MRCI wave function expansion, this approximation leads to substantial savings of cpu time without essential loss of precision. With regard to applications, the photoexcitation decay kinetics of xanthone in apolar media and in aqueous solution is in the focus of the present work. The results of our computational study substantiate the conjecture that S1 T2 reverse ISC outcompetes the T2 ↝ T1 IC in aqueous solution, thus explaining the occurrence of delayed fluorescence in addition to prompt fluorescence.

Efficiency enhancement in orange red thermally activated delayed fluorescence OLEDs by using a rigid di-indolocarbazole donor moiety

The development of orange-red thermally activated delayed fluorescence (TADF) materials with good optoelectronic and electroluminescence properties are full of challenges due to limited molecular designs. Herein, we investigated a series of materials using dimethylacridine (DMAC) and diindolocarbazole (DI) donor with dibenzo [a, c] phenazine (DBP) acceptor, and compared the characteristics of all synthesized materials as emitters. Due to lower singlet energy level, the rate constant of internal conversion (kIC) contribute to nonradiative decay cannot be ignored at the orange-red region. Therefore, employing rigid donor and acceptor moieties play a major role in suppressing kIC. We observed that employing a rigid donor of DI at β position enhanced the PLQY of 96.1% when compared to acridine based emitters. 5-(dibenzo [a, c] phenazin-11-yl)-10, 15-diphenyl-10, 15-dihydro-5H-diindolo [3, 2-a:3′, 2′-c] carbazole (β-DI-DBP) based OLED device showed better current efficiency (CEmax) of 47.1 cd/A and maximum external quantum efficiency (EQE) of 23.8%. This work demonstrates a better approach towards high PLQY orange red TADF material development.

Fast estimation of the internal conversion rate constant in photophysical applications.

An efficient method for estimating non-adiabatic coupling matrix elements (NACME) and rate constants for internal conversion (kIC) is presented. The method, based on Plotnikov's theory, requires only calculations of the electronic wave functions and the corresponding electronic excitation energies. Computationally expensive calculations of the derivatives of the electronic wave function with respect to the nuclear coordinates are avoided. When the main accepting modes of the electronic excitation energy are X-H vibrations, the present method can be used for estimating the efficiency of the energy transfer between donor and acceptor molecules. It can also be used in studies of the influence of hydrogen bonding or solvent effect on fluorescence quenching, in studies of vibronic effects of TADF (thermally activated delayed fluorescence) emitters, and for calculating kIC. Here, kIC and NACME are calculated for free-base porhyrin, magnesium porphyrin, azulene, naphthalene, pyrene and fluorenone interacting with a solvent molecule. Reverse kIC and NACME are further calculated for the T1→ T2 transition of dibenzothiophene-S,S-dioxide (PTZ-DBTO2), which is used in TADF applications. Finally, we estimate the efficiency of the energy transfer between two large porphyrinoid dimers.

Electrochemical Properties and Excited-State Dynamics of Azaperylene Derivatives.

A series of azaperylene derivatives such as monoazaperylene (MAPery), 1,6-diazaperylene (1,6-DiAPery), 1,7-diazaperylene (1,7-DiAPery), 1,12-diazaperylene (1,12-DiAPery), triazaperylene (TriAPery), and tetraazaperylene (TetAPery) was synthesized by changing the position and number of nitrogen atoms at the bay region of a perylene skeleton in 1, 6, 7, and 12 positions. The density functional theory (DFT) calculations and electrochemical measurements suggested that the energies of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) states significantly become stabilized with increasing the number of nitrogen atoms, whereas the estimated HOMO-LUMO gaps approximately remain constant. This result is in good agreement with the absorption and fluorescence spectral measurements. Additionally, these steady-state spectroscopic measurements demonstrate the broadened spectra as compared to pristine perylene (Pery). In photophysical measurements, the fluorescence quantum yields (ΦFL) significantly decreased as the number of nitrogen atoms increased, whereas much enhanced quantum yields and rate constants of internal conversion (ΦIC and kIC) were observed. Especially, the increased kIC values of TriAPery (kIC: ∼108 s-1) and TetAPery (kIC: ∼109 s-1) are much larger than those of diazaperylene and monoazaperylene derivatives (kIC: ∼107 s-1). These photophysical trends were successfully explained by time-dependent DFT (TD-DFT) calculations. Finally, the characteristic protonated and deprotonated processes of nitrogen atoms in azaperylenes under acidic conditions were monitored utilizing absorption and fluorescence measurements. The binding constants demonstrate that the nitrogen atoms at 1 and 12 positions of a perylene skeleton are essential for the increased values.

Excited-state hydrogen bonding: Detecting ammonia using an HHTP-DPB covalent organic framework

The hydrogen bonding between ammonia (NH3) and a two-dimensional material 2,3,6,7,10,11-hexahydroxytriphenylene (HHTP)-4,4′-diphenyl-butadiynebis (boronic acid) (DPB) covalent organic framework (HHTP-DPB COF) in its electronically excited state was investigated using density functional theory and time-dependent density functional theory. The formation of the hydrogen bonding changed the luminescence mechanism of the material, which was confirmed by analysis of the frontier molecular orbitals and the corresponding electronic configurations. The comparison of the properties of hydrogen bonds in the ground state (S0) and excited state (S1) revealed that in the S1 state the hydrogen bonds were strengthened, which was adverse to the radiation transition process, and resulting in the weakening of the fluorescence intensity. After the introduction of NH3, the fluorescence rate constant of the COF was decreased and the internal conversion rate constant was increased. Therefore, these findings indicated that HHTP-DPB COF has great application prospects for detecting the indoor air pollutant NH3.

First-principles calculations of anharmonic and deuteration effects on the photophysical properties of polyacenes and porphyrinoids.

A new method for calculating internal conversion rate constants (k[combining low line]IC), including anharmonic effects and using the Lagrangian multiplier technique, is proposed. The deuteration effect on k[combining low line]IC is investigated for naphthalene, anthracene, free-base porphyrin (H2P) and tetraphenylporphyrin (H2TPP). The results show that anharmonic effects are important when calculating k[combining low line]IC for transitions between electronic states that are energetically separated (ΔE) by more than 20 000-25 000 cm-1. Anharmonic effects are also important when ΔE < 20 000-25 000 cm-1 and when the accepting modes are X-H stretching vibrations with a frequency larger than 2000 cm-1. The calculations show that there is mixing between the S1 and S2 states of naphthalene induced by non-adiabatic interactions. The non-adiabatic interaction matrix element between the S1 and S2 states is 250 cm-1 and 50 cm-1 for the normal and fully deuterated naphthalene structure and this difference significantly affects the estimated fluorescence quantum yield. Besides aromatic hydrocarbons H2P and H2TPP, the k[combining low line]IC rate constant is also calculated for pyrometene (PM567) and tetraoxa[8]circulene (4B) with a detailed analysis of the effect of the vibrational anharmonicity.

Luminescence and Geometric and Electronic Structures of Porphyrazines with Annelated 1,4-Diazepine Rings

Absorption, fluorescence, and phosphorescence spectra of metal complexes of 4-tert-butylphenyl-substituted tetra(1,4-diazepino)porphyrazine MPADz4Ph8′ (M = Mg, Zn; PA = porphyrazine; Dz = diazepine; Ph = phenyl; Ph′ = 4-tert-butylphenyl) were recorded in DMF at 293 and 77 K. The Q(0–0) absorption bands of MgPADz4Ph8′ and ZnPADz4Ph8′ did not show signs of splitting, i.e., corresponded to the monomeric form. Singlet–triplet gaps of 4700 cm–1 for the Mg complex and 4800 cm–1 for the Zn complex were determined from fluorescence and phosphorescence spectra. Fluorescence quantum yields and lifetimes were measured. Rate constants of S1,2 → S0 radiative transitions and total rate constants of nonradiative deactivation of the S1,2 states were determined. Internal conversion rate constants were estimated. Conformer geometries of MgPADz4Ph8 (in the 6H tautomeric form) were calculated by density functional theory in the PBE/TZVP version. The conformer of symmetry S4v (D2d ) had the lowest energy. Nonplanar diazepine rings induced small but perceptible out-of-plane distortions in the central porphyrazine ring of MgPADz4Ph8. This feature correlated with the observed Stokes shifts of 400 and 350 cm–1 (at 293 K) for MgPADz4Ph8′ and ZnPADz4Ph8′ whereas the metal phthalocyanines had shifts of ~50 cm–1. Excited electronic states of MgPADz4Ph8 were calculated using INDO/Sm. The Q-state energy of 15,200 cm–1 agreed well with the observed value of 14,800 cm–1. Strong featureless absorption in the range 330–450 nm (Soret band analog) with maxima at ~25,000 and ~29,000 cm–1 was due mainly to two strong electronic transitions with calculated energies of 24,100 and 31,500 cm–1; a shoulder on the long-wavelength side of the Soret band at ~23,000 cm–1, to a transition of calculated energy 23,800 cm–1.

Photophysical Constants of the Tetraoxa[8]Circulene Molecule

The rate constants of photophysical processes (radiation rate constant, internal conversion rate constant, and quantum fluorescence yield) have been calculated for the tetraoxa[8]circulene molecule by the INDO/S, TDDFT, and CC2 methods. It is confirmed that the doubly degenerate triplet level Т2 and the level Т1 are located below the first singlet excited level S1. According to the selection rules for the angular momentum, the spin-orbit interaction between the levels S1 and T1 (S1 and T2) is equal to zero. Therefore, the internal conversion is the only nonradiative channel in this molecule. The results of calculations also demonstrate that values of the internal conversion rate constant and the quantum fluorescence yield obtained by the INDO/S method are incorrect.

Calculating rate constants for intersystem crossing and internal conversion in the Franck-Condon and Herzberg-Teller approximations.

Effective and fast algorithms for calculating rate constants for internal conversion (IC) and intersystem crossing (ISC) in the Franck-Condon and Herzberg-Teller approximations have been developed and implemented. The methods have been employed for calculating IC and ISC rate constants for the pyrromethene-567 dye (PM567), hetero[8]circulene (4B) and free-base porphyrin (H2P). The fluorescence quantum yields obtained by comparing calculated rate constants for the radiative and non-radiative processes are in good agreement with experimental data.

Controlling the excited-state dynamics of low band gap, near-infrared absorbers via proquinoidal unit electronic structural modulation.

While the influence of proquinoidal character upon the linear absorption spectrum of low optical bandgap π-conjugated polymers and molecules is well understood, its impact upon excited-state relaxation pathways and dynamics remains obscure. We report the syntheses, electronic structural properties, and excited-state dynamics of a series of model highly conjugated near-infrared (NIR)-absorbing chromophores based on a (porphinato)metal(ii)-proquinoidal spacer-(porphinato)metal(ii) (PM-Sp-PM) structural motif. A combination of excited-state dynamical studies and time-dependent density functional theory calculations: (i) points to the cardinal role that excited-state configuration interaction (CI) plays in determining the magnitudes of S1 → S0 radiative (kr), S1 → T1 intersystem crossing (kISC), and S1 → S0 internal conversion (kIC) rate constants in these PM-Sp-PM chromophores, and (ii) suggests that a primary determinant of CI magnitude derives from the energetic alignment of the PM and Sp fragment LUMOs (ΔEL). These insights not only enable steering of excited-state relaxation dynamics of high oscillator strength NIR absorbers to realize either substantial fluorescence or long-lived triplets (τT1 > μs) generated at unit quantum yield (ΦISC = 100%), but also crafting of those having counter-intuitive properties: for example, while (porphinato)platinum compounds are well known to generate non-emissive triplet states (ΦISC = 100%) upon optical excitation at ambient temperature, diminishing the extent of excited-state CI in these systems realizes long-wavelength absorbing heavy-metal fluorophores. This work highlights approaches to: (i) modulate low-lying singlet excited-state lifetime over the picosecond-to-nanosecond time domain, (ii) achieve NIR fluorescence with quantum yields up to 25%, (iii) tune the magnitude of S1-T1 ISC rate constant from 109 to 1012 s-1 and (iv) realize T1-state lifetimes that range from ∼0.1 to several μs, for these model PM-Sp-PM chromophores, and renders new insights to evolve bespoke photophysical properties for low optical bandgap π-conjugated polymers and molecules based on proquinoidal conjugation motifs.

Application of vibrational correlation formalism to internal conversion rate: case study of Cu(n) (n = 3, 6, and 9) and H2/Cu3.

This work reports non-radiative internal conversion (IC) rate constants obtained for Cun with n = 3, 6, and 9 and H2 on Cu3. The Time-Dependent Density Functional Theory (TDDFT) method was employed with three different functionals in order to investigate the electronic structures and the absorption spectra. The performance of the generalized gradient approximation of Perdew, Burke and Ernzerhof (PBE) and the hybrid B3LYP and PBE0 exchange correlation functionals in combination with the SVP and the def2-TZVP basis sets was examined. TDDFT results were used as input data to compute internal conversion rate constants. For this purpose, we have developed a program package. A description of the theoretical background used in our numerical implementation and the program input file is presented. In view of future applications of this program package in photoinduced catalysis, we present the analysis of the IC rate processes for the photodissociation of H2 on Cu3. These results showed the applicability of the method and the computational program to identify the vibrational modes in transition metal clusters giving rise to the largest IC rate constant due to their interactions with the excited electronic states occurring in the hot-electron induced dissociation phenomena.