First-Principles Method for Calculating the Rate Constants of Internal-Conversion and Intersystem-Crossing Transitions

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First-principles method for calculating the rate constants of internal-conversion and Cite this: Phys. Chem. Chem. Phys., 2018, 20,6121 intersystem-crossing transitions†

R. R. Valiev, *ab V. N. Cherepanov,a G. V. Baryshnikov ac and D. Sundholm b

A method for calculating the rate constants for internal-conversion (kIC) and intersystem-crossing (kISC) processes within the adiabatic and Franck–Condon (FC) approximations is proposed. The applicability of the

method is demonstrated by calculation of kIC and kISC for a set of organic and organometallic compounds with experimentally known spectroscopic properties. The studied molecules were pyrromethene-567 dye, psoralene, hetero[8]circulenes, free-base porphyrin, naphthalene, and larger polyacenes. We also studied

fac-Alq3 and fac-Ir(ppy)3, which are important molecules in organic light emitting diodes (OLEDs). The excitation energies were calculated at the multi-configuration quasi-degenerate second-order perturbation theory (XMC-QDPT2) level, which is found to yield excitation energies in good agreement with Creative Commons Attribution-NonCommercial 3.0 Unported Licence. experimental data. Spin–orbit coupling matrix elements, non-adiabatic coupling matrix elements, Huang–Rhys factors, and vibrational energies were calculated at the time-dependent density functional Received 31st December 2017, theory (TDDFT) and complete active space self-consistent field (CASSCF) levels. The computed fluorescence Accepted 31st January 2018 quantum yields for the pyrromethene-567 dye, psoralene, hetero[8]circulenes, fac-Alq3 and fac-Ir(ppy)3 DOI: 10.1039/c7cp08703a agree well with experimental data, whereas for the free-base porphyrin, naphthalene, and the polyacenes, the obtained quantum yields significantly differ from the experimental values, because the FC and adiabatic rsc.li/pccp approximations are not accurate for these molecules. This article is licensed under a 1. Introduction in deactivating excited electronic states at high molecular concentrations,10 whereas the intramolecular processes depend Molecular photophysics is a rapidly developing research area of almost completely on the intrinsic properties of the molecule.11 Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. molecular and chemical physics.1–3 Interaction of the electro- Photophysical properties like the quantum yields of fluorescence

magnetic radiation with molecules, molecular luminescence (jfl) and phosphorescence (jphos) are determined by the ratio properties, and transformations of excited electronic energy between the rate constants of radiative (kr) and nonradiative (knr) into vibrational energy are important photophysical processes. intramolecular processes.9,10 Detailed knowledge about these photophysical properties is When a photon is absorbed, molecular systems transfer needed when designing optical molecular devices such as organic from the ground state to an excited electronic state, and they light-emitting diodes,4,5 laser applications,6 and other devices that can be de-excited via diﬀerent channels. Radiative deactivation convert light energy or other forms of energy.7 The mechanism of channels comprise emission of a photon with an energy that is light-induced processes such as the light absorption of photo- smaller than or equal to the excitation energy. Nonradiative synthesis can also be understood by studying the photophysical channels do not involve any photon emission but the excess properties of the chlorophylls and other involved molecules.8 energy transfers to vibrational energy leading to an increase Photophysical processes in molecules can be divided into of the temperature. Internal conversion (IC) and the intersystem intramolecular and intermolecular ones.9 The intermolecular crossing (ISC) are main intramolecular nonradiative processes.9–11 processes play a key role in luminescence quenching and The electronic energy in both processes is converted into molecular vibrations. In the case of IC, the spin multiplicity a Tomsk State University, 36 Lenin Avenue, Tomsk, Russia. of the molecule is conserved, whereas ISC involves the initial E-mail: [email protected] and final states of different spin multiplicities.1 The IC process b Department of Chemistry, University of Helsinki, A. I. Virtanens plats 1, occurs due to nonadiabatic coupling interactions. ISC is caused P.O. Box 55, FI-00014 Helsinki, Finland c Division of Theoretical Chemistry and Biology, School of Biotechnology, by the spin–orbit coupling interaction. When the concentration KTH Royal Institute of Technology, SE-10691 Stockholm, Sweden of molecules is low, it can be assumed that the total rate † Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp08703a constant for the nonradiative process is the sum of the rate

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constants of the two channels knr = kIC + kISC, where kIC and kISC research areas of photonics. The reference values for the rate are the rate constants for IC and ISC, respectively.9,12 The total constants are also available for these molecules,34–40 since the

rate constant for the nonradiative process knr can be estimated photophysical properties of these compounds have previously 29,31,33 experimentally from the quantum yields (jfl, jphos) and from been studied by other groups. 9,10 the rate constant of the radiative processes (kr). However, the 13,14 experimental determination of kIC and kISC is very difficult. 2. Theory The rate constants for the radiative transitions can be calculated quantum chemically15–17 employing nonadiabatic mole- 2.1. Non-radiative electronic transitions 18–22 cular dynamics simulations at ab initio levels of theory or The general expression for the rate constant of non-radiative by using specific approximations considering empirical electronic transitions derived by Plotnikov, Robinson, Jortner 1,15,23 corrections. Molecular dynamics simulations are compu- reads11,24 tationally expensive, which limits their application to smaller X 2 1 molecular systems. The semiempirical approach by Plotnikov, 2 2 Gfn knr ¼ Vi0;fn Gfn Dif þ ; (1) Artyukhov and Maier, which is based on the incomplete n 4 neglected differential overlap (INDO) method with a spectro- where i is the initial electronic state, f is the final electronic scopic parameterization, can be used for calculating the photo- state, n is a vibrational level of f, G is the relaxation width of physical properties of large organic molecules.24–31 Maier and fn the vibronic level |fni, D =|E E | is the energy diﬀerence Artyukhov employed computational methods based on the if i0 fn between the initial and final vibronic states, and V is theory developed by Robinson, Jortner11 and Plotnikov24 to i0,fn the matrix element of the perturbation operator. Only the estimate k and k at the INDO level of theory.25–28 They IC ISC lowest vibronic state is considered for the initial state. The calculated the matrix elements of the nonadiabatic coupling perturbation operator is the spin–orbit coupling interaction and spin–orbit coupling operators for the singlet and triplet for ISC transitions and the non-adiabatic coupling interaction

Creative Commons Attribution-NonCommercial 3.0 Unported Licence. electronic states at the INDO level of theory, whereas the for the IC process. eqn (1) written in atomic units holds at vibrational integrals were estimated from the experimental ambient temperatures (T r 300 K) when k { G . The spectroscopic data. The INDO approach renders routine calcu- nr fn conditions are generally fulfilled in experimental studies of lations of k and k for large organic molecules consisting of IC ISC luminescence properties. G of about 1014 s1 is generally up to 200 atoms feasible. fn much larger than the k of about 107–1012 s1.24 The D However, the semiempirical INDO approach of Artyukhov nr if value is not larger than 100 cm1 for polyatomic molecules.24 and Maier can be applied only to molecules consisting of light The expression can be simplified to eqn (2) when assuming atoms such as H, C, N, O, F, S, and Cl, whereas it cannot that Gfn depends only weakly on the vibrational level n and that be employed in studies of photophysical properties of organo- 24

This article is licensed under a D { G : metallic compounds. The semiempirical method has also been if fn X found to lead to large errors in the rate constants kIC and kISC for 4 2 knr ¼ Vi0;fn : (2) porphyrins and [8]circulenes, because the INDO calculations Gf n are not able to provide accurate electronic excitation energies Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. for these molecules. A better accuracy has been obtained 2.2. IC rate constant for porphyrins and [8]circulenes by combining the INDO method with excitation energies calculated at the time- In the framework of the adiabatic and Franck–Condon (FC) 24,41 dependent density functional theory (TDDFT) and correlated approximation, Vi0,fn can be written as 29–33 ab initio levels of theory. However, for some [8]circulenes Vi0;fn ¼ hiijjTR f hi0jn X the matrix elements of the nonadiabatic coupling (NACME) and 1 @ @ Mn 0 n i f ; (3) the spin–orbit coupling (SOCME) operators calculated at the n @Rn @Rn INDO level were found to be inaccurate. In this work, the electronic excitation energies, the coupling matrix elements, where TR is the kinetic energy operator of the electrons, n is a and the vibrational integrals are calculated at the density func- @ nuclear index, Mn is the mass of nucleus n, i f is the tional theory (DFT) and correlated ab initio levels of theory, @Rn opening new possibilities of employing the theory of Plotnikov, nonadiabatic coupling matrix element between the electronic Robinson, and Jortner without relying on any semiempirical states |ii and |fi,|0i is the lowest vibration state of |ii and |ni is parameters. a vibration state of the final electronic state |fi, Rn is the nuclear In this work we employ the theory of Plotnikov, Robinson, @ 24 coordinates of nucleus n, and 0 n is the nonadiabatic Jortner using NACME and SOCME calculated at the DFT and @Rn ab initio levels to obtain accurate values for the k and k @ IC ISC coupling matrix element between |0i and |ni. The i f rate constants for nonradiative intramolecular transitions. The @Rn methods have been used in studies of pyrromethene-567 dye, @ and 0 n integrals are calculated for the equilibrium psoralene, acenes, fac-Alq3, fac-Ir(ppy)3, hetero[8]circulenes, @Rn

and free-base porphyrin, which are important in the diﬀerent geometry (R = R0) of the |fi state. Since the first term in eqn (3)

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is usually much smaller than the second term, eqn (3) can be The IC rate constant can be calculated using written as24,42 4 Eif ¼n1o1þn2oX2þ...þn3N6o3N6 X 2 @ @ kIC ¼ Vi0;ffn1n2::n3N6g ; (11) 1 Gf Vi0;fn ¼ Mn 0 n i f ; (4) n1;n2...;n3N6 n @Rn @Rn where the expression for V is given in eqn (10). E which can be expressed in Cartesian coordinates as i0;ffgn1n2...n3N6 if is the energy gap between the electronic states |ii and |fi. X X 1 @ @ Vi0;fn ¼ Mn 0 n i f : (5) @R @R 2.3. ISC rate constant n q¼x;y;z qv qv @ The spin–orbit coupling is the perturbation operator of the ISC 0 n can be written using normal coordinates when @Rqv process. The spin–orbit coupling matrix elements depend only the rotational and translational motions have been identified on the space and spin coordinates of the electrons but not on and removed 1,24 *+ the nuclear ones. The expression for the ISC rate constant @ 3XN6 @ can then be written as 0j jn ¼ 0102 ...03N6 Bnqj n1n2 ...n3N6 ; X @Rqn @Qj 4 j¼1 k ¼ hiijH jf 2 hi0jn 2 ISC G SO (6) f n 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 In eqn (6), |0ki and |nki are the harmonic oscillator wave Eif ¼n1o1þn2oX2þ...þn3N6o3N6 3YN6 nk 4 2@ expðykÞyk A functions of the initial and final states, nk is the nth excitation ¼ hiijHSOjf : Gf nk! n1;n2::;n3N6 k¼1 of the kth oscillator, Qj is the normal coordinate of the jth

oscillator, Bnqj are matrix elements that describe the connection (12) between the displacements in Cartesian coordinates of nth atom (DR ) in respect to the equilibrium of and normal The excitations of the normal vibrations with the high Creative Commons Attribution-NonCommercial 3.0 Unported Licence. qn frequency at o = B1000–1800 cm1 and significant Huang–Rhys coordinate (Qj). By inserting eqn (6) into eqn (5) one obtains X X factor (y 4 0.1) are the largest contributions to the summation 1 @ Vi0;f n n ...n ¼ Mn i f (Eif = n1o1 + n2o2++ n3N6o3N6) in the expressions for the f 1 2 3N6g @R n q¼x;y;z qn FC factor in eqn (11) and (12).6,12,24 2 3 3XN6 3YN6 2.4. Parameters for the calculation of the ISC and IC rate 6 @ 7 4 Bnqj 0j nj hi0kjnk 5 @Q constants j¼1 j kaj k¼1 To calculate the ISC and IC rate constants, values for This article is licensed under a (7) @ i f , hi|HSO|fi, yj and oj, Eif and Gf are needed. In this @ @R 0j nj can be expressed in harmonic approximation as qv @Q j @ work, i f is calculated for molecules consisting of more @Rqv Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. @ 2 1 2 nj 1 42 0j nj ¼ oj nj yj y exp yj ; (8) than 50 atoms at the TDDFT level using Turbomole and for @Q 2n ! j j j smaller molecules they are calculated at the CASSCF level of 43 where yj is the Huang–Rhys factor and oj is the vibration theory in GAMESS-US. The Huang–Rhys factors and the

frequency of the jth mode. The Franck–Condon factor h0k|nki vibration frequencies (oj) are calculated at the DFT and corre- 41 44 can then be written as lated ab initio levels of theory. The hi|HSO|fi matrix elements 45,46 nk have been computed at the CASSCF and TDDFT levels. The 2 expðÞyk yk hi0kjnk ¼ : (9) line width (Gf) has been estimated using the Lax and Pekar nk! models47–49 in the simulation of molecular vibronic spectra The final expression is obtained by inserting eqn (8) and (9) using the displaced oscillator model. into eqn (7): X X 1 @ 3. Computational methods and studies Vi0;ffn1n2::n3N6g ¼ Mn i f n q¼x;y;z @Rqv 3.1 Studied molecules 20 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The computational methods described above have been employed 3YN6 nk 6B expðykÞy C on a number of molecules with interesting photophysical properties. 4@ k A n ! kaj k The studied molecules shown in Fig. 1 comprise the commercial k¼1 laser dye 1,3,5,7,8-pentamethyl-2,6-diethylpyrromethene-difluoro- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# 34 35 3XN6 borate (PM567), 7H-furo[3,2-g]chromen-7-one (psoralene), 1 2 n 1 B o n y y j exp y : fac-tris(8-hydroxyquinolinato)aluminium ( fac-Alq )38 and fac- vqj 2n ! j j j j j 3 j¼1 j 39 tris(2-phenylpyridine)iridium ( fac-Ir(ppy)3), tetraoxa[8]circulene (10) (4B)36 and its NH substituted and benzoannelated derivatives

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Fig. 1 The molecular structures of the studied molecules.

29 (AOC) and (1B3N), polyacenes (naphthalene, anthracene, tetra- We chose the oB97xD functional for Alq3 and Ir(ppy)3, because 37 42 cene, pentacene, and hexacene), and free-base porphyrin (H2P). difficulties arise when using the B3LYP functional to properly The molecules were chosen for the following reasons. PM567 describe excited states of metal–ligand compounds with sig-

has a very large fluorescence quantum yield of jfl = 0.9–1.0 and is nificant charge-transfer characteristics, whereas the oB97xD therefore often used in laser devices.38 Psoralene has a very small functional is expected to be able to describe such states better, B 37 39 Creative Commons Attribution-NonCommercial 3.0 Unported Licence. fluorescence quantum yield of jfl 0.1 and a large kISC and is since it has the correct long-range shape of the potential. 37 therefore used in photodynamic therapy applications. Alq3 and The molecular structure optimizations were carried out using 38,39 56 Ir(ppy)3 are used in OLED devices. The electroluminescence Gaussian-09 software. Only the lowest isomers of fac-Alq3 and

of Alq3 and Ir(ppy)3 is due to fluorescence and phosphorescence, fac-Ir(ppy)3 were considered. respectively.38,39 The photophysical properties of [8]circulenes Calculations of singlet and triplet excitation energies were have previously been studied by us at the INDO level of theory carried out using the extended multi-configuration quasi- using the method of Plotnikov, Robinson, Jortner.29 degenerate perturbation theory at the second order (XMC- The chosen compounds have very diﬀerent fluorescence quan- QDPT2),57 becausethisleveloftheoryhasbeenfoundtoyield

tum yields jfl. The fluorescence spectrum of the polyacenes has a the correct relative order of the lowest singlet and triplet electronic This article is licensed under a complicated vibronic fine structure requiring calculations of transi- states for organic and organometallic compounds.58 The XMC- tion moments between many excited states with diﬀerent spins.50 QDPT2 calculations were performed using the optimized mole-

The energy gap between the first excited singlet state (S1)andthe cular structures of the S1 state. In the XMC-QDPT2 calculations,

ground state (S0) systematically reduces from the ultraviolet to 30 states were included in the eﬀective Hamiltonian. The number Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. infrared spectral region when increasing the number of benzoic of active electrons (e), number of active orbitals (o), and the rings in the polyacene.50 The calculations on the polyacenes yield a number of states (s) of the state-average (SA) complete active

relation between kIC and the optical gap. The present approach is self-consistent space field (CASSCF) calculations are PM567

also applied to free-base porphyrin (H2P), because it is well known (8 e, 6 o, 5 s), psoralene (10 e, 9 o, 5 s), Alq3 (12 e, 9 o, 5 s), 1 that the calculation of kIC and kISC for H2P is challenging. Ir(ppy)3 (12 e, 9 o, 5 s), tetraoxa[8]circulene (4B) (10 e, 10 o, 10 s), o-dinaphthalene-containing azatrioxa[8]circulene (AOC) (10 e, 3.2 Computational methods 10 o, 10 s), trinaphthalene-containing tetraoxa[8]circulene (1B3N),

The molecular structures of the ground electronic states (S0)of (10 e, 11 o, 10 s), free-base porphyrin (H2P) (11 e, 14 o, 10 s),

all the molecules except Alq3 and Ir(ppy)3 were optimized at the naphthalene (12 e, 9 o, 10 s), anthracene (10 e, 10 o, 10 s), density functional theory (DFT) level51 using the B3LYP52 tetracene (10 e 11 o, 10 s), pentacene (8 e, 8 o, 10 s), and exchange–correlation functional and def2-TZVP53 basis sets. hexacene (8 e, 8 o, 10 s). The XMC-QDPT2 calculations were 59 For Alq3 and Ir(ppy)3, the oB97xD functional was employed carried out using Firefly. in the DFT optimization of the molecular structures.54 The The discrepancy between the TDDFT and XMC-QDPT2 excita-

def2-TZVP basis sets were used for Alq3. For Ir(ppy)3,weusedthe tion energies for the S1 - S0 electronic transitions does not LANL2DZ basis set and effective core potentials for Ir and the exceed 1000–1500 cm1 for any of the studied compounds. Thus, 6-31G(d,p) basis sets for the rest of the atoms.55 The equilibrium a combination of the XMC-QDPT2 and TDDFT levels of theory

geometry of the first excited electronic state (S1) was obtained can be used in the calculation of photophysical properties. at the TDDFT/B3LYP/def2-TZVP level of theory for PM567, psoralene, polyacenes, and free-base porphyrin. The molecular 3.3 Huang–Rhys factors and vibrational relaxation widths

structures of the first excited state (S1) of Alq3 and Ir(ppy)3 were The yj, oj, and Gf values were obtained using the Lax and optimized at the TDDFT level using the oB97xD functional.54 Pekar model,47–49 which is described in detail in ref. 48 and 49.

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The potential energy surfaces of S0 and S1 are assumed to be 3.4 Calculation of matrix elements harmonic with the same vibrational energies (o )forthetwostates. j @ The energy functions for the two states can then be written as The nonadiabatic coupling matrix elements S0 S1 were @Rqv X X i i 0 i 0 1 2 0 2 calculated with Turbomole at the TDDFT level of theory using E ðQÞ¼E ðQ Þþ Vj Qj Qj þ oj ðQj Qj Þ ; 42,60 j 2 j the perturbation theory. The spin–orbit coupling matrix elements hS1|HSO|Tii between the S1 state and the energetically X 1 X f f 0 f 0 2 0 2 lower ith triplet state T were calculated at the CASSCF level using E ðQÞ¼E ðQ Þþ Vj Qj Qj þ oj ðQj Qj Þ i 2 61 j j Gamess-US. The excitation energies calculated at the XMC- (13) QDPT2 level were used as the zeroth-order values. In order to calculate kISC, the one-electron spin–orbit coupling operator of where {Q} is the set of normal coordinates of the harmonic the Pauli-Breit Hamiltonian was used.62 The contributions from i;f oscillators, V ¼ @Ei;f @Q is the gradient of the potential j Q¼Q0 the two-electron part of the spin–orbit coupling operator are energy surface along the jth mode at a chosen point Q0. From beyond the Franck–Condon approximation.1 However, previous eqn (13) it is easily seen that calculations have shown that the use of the one-electron spin–

i;f orbit operator within the FC approximation leads to accurate kISC @E ðQÞ i;f 2 0 29,58,63 ¼ Vj þ oj Qj Qj values for the organic and organometallic compounds. @Qj

f i 3.5 Radiative rate constants and fluorescence quantum yields @E ðQÞ @E ðQÞ f i ¼ Vj Vj ¼ const (14) @Qj @Qj When contributions from higher excited states can be neglected, the fluorescence quantum yield from the S state can be i;f 1 9,12 i;f 0 Vj obtained as Qj Qj ¼ 2 : oj kr Creative Commons Attribution-NonCommercial 3.0 Unported Licence. P jfl ¼ ; (16) f i 2 Thus, DQ = (V V )/o and finally the Huang–Rhys kr þ kIC þ kISTi j j j j i factors (yj) are obtained as 2 where kISTi is a ISC rate constant between S1 and energetically Vf Vi lower triplet states T , k and k are the radiative and IC rate 1 2 j j i r IC yj ¼ ojDQj ¼ 3 : (15) 2 2oj constants of the electronic transition from S1 to S0, respectively. 64 The kr can be estimated using the Strickler–Berg equation The line width Gf is taken from the generation function for 1 2 the simulation of the profile of the vibration spectra within kr ¼ f E ðÞS1 ! S0 ; (17) 1:5

This article is licensed under a 47–49 the Lax-Pekar model. The calculated value for Gf is about 14 1 - 1.6 10 s for the S1 state of the studied molecules. where f is the oscillator strength and E(S1 S0) is the de-excitation

The optimized molecular structures of the S0 and S1 states as energy from S1 to S0.

well as the gradient and the Hessian calculated for the S0 state The table illustrating the employed level of theory is given for Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. i,f were used in the calculation of oj and Vj . In the calculation clear understanding of the subsequent discussions (Table 1).

of the IC rate constants, the Hessian was calculated for the S0 state, whereas the gradient for the S1 state was calculated using 4. Results and discussion the molecular structure of the S0 state. We assume that the 4.1. Pm567 vibrational energies (oj) and the Huang–Rhys factors yi are the

same for the IC and ISC processes. Our previous calculations The optimized molecular structure of the S1 state is shown in showed that this assumption yields good estimates for the ISC Fig. 2a. The energy levels of the lowest singlet and triplet states 29,58 rate constants for several molecules. and the kIC and kISC rate constants of the transitions are shown

Table 1 The employed levels of theory for the compounds under investigation

PM567, psoralene, [8]circuelenes Properties and H2P, polyacenes Alq3 Ir(ppy)3

Geometries of S0 and S1 states DFT/B3LYP/def2-TZVP DFT/oB97xD/def2- DFT/oB97xD/LANL2DZ and 6- TZVP 31G(d,p) Energies of S1 - S0 electronic transition and triplet XMC-QDPT2/def2-TZVP XMC-QDPT2/ XMC-QDPT2/oB97xD/ states, oscillator strength ( f ) oB97xD/def2-TZVP LANL2DZ and 6-31G(d,p) @ TD-DFT/B3LYP/def2-TZVP TDDFT/B3LYP/def2- TDDFT/B3LYP/def2-TZVP S0 S1 @Rqv TZVP

hS1|HSO|Tii MCSCF/def2-TZVP MCSCF/oB97xD/ MCSCF/oB97xD/LANL2DZ and def2-TZVP 6-31G(d,p) y, o and G (TD)-DFT/B3LYP/def2-TZVP (TD)-DFT/oB97xD/ (TD)-DFT/oB97xD/LANL2DZ def2-TZVP and 6-31G(d,p)

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Fig. 2 (a) The optimized molecular structure of PM567. (b) The energy level diagram for PM567 including the computed values for the radiative and nonradiative rate constants.

Table 2 The calculated excitation energies (in eV) of the studied mole- agrees with the experimental one of 0.9.38 Thus, the present cules are compared to available experimental data. Calculated oscillator approach is able to predict the large (jfl) observed for PM567. strengths are given in parenthesis The largest values for the Huang–Rhys factors (yj) and corres- Compound State XMC-QDPT2 Exp. ponding wavenumbers (oj) are given in Table 3. The vibrational 1 a mode at 1255 cm has a large y value of 0.1. Since only one PM567 S1 2.13(0.2) 2.22 T1 1.75 — vibrational mode has a large y value in that energy range the T2 2.08 — nonradiative rate constants k and k are small as compared Creative Commons Attribution-NonCommercial 3.0 Unported Licence. IC ISC

b with the rate constant of the radiative transition kr. Psoralene S1 3.09(0.2) 3.02 T 2.75 2.73b 1 4.2. Psoralene T2 3.02 —

c The optimized molecular structure of the S1 state of psoralene 4B S1 2.64 2.44 T1 1.89 is shown in Fig. 3a. The energies of the lowest singlet and T2 (T3) 2.58 triplet states and the kIC and kISC rate constants are shown in

c Fig. 3b. The excitation energies listed in Table 2 agree well with AOC S1 3.00(0.3) 3.02 T1 1.89 the experimental values. The dominating deactivated channel This article is licensed under a T2 2.9 of the S1 state is via the kISC1 channel, because of the strong

c spin–orbit coupling interaction between S1 and T1. See Table S1 1B3N S1 2.68(0.5) 3.08 T1 2.1 in the ESI.† The calculated and experimental quantum yields of 35 T2 2.24 0.08 are in perfect agreement. The calculated excitation energies Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. T3 2.34 were found to agree very well with the experimental ones, even T4 2.64 though the solvent effect can be as large as 1% for psoralene.35 d H2PS1 1.92 2.01 The calculated yj and oj values for psoralene are given in Table 3, d T1 1.62 1.58 where one sees that many vibrational modes with energies T2 1.8 1 larger than 1000 cm have large yj values leading to a high e Alq3 S1 2.33(0.1) 2.38–2.48 density of FC factors and fast nonradiative transitions. The T1 1.86 — calculated rate constants explain the low value and the high f Ir(ppy)3 S1 2.74 2.69 triplet quantum yield of psoralene. f T1 2.37 2.44 4.3. [8]circulenes 4B, AOC and 1B3N a Ref. 38. b Ref. 37. c Ref. 29. d Ref. 40. e Ref. 34. f Ref. 35. The optimized molecular structures of the S1 state of tetraoxa[8]circulene (4B) and its NH substituted and benzoannelated in Fig. 2b. Table 2 contains the calculated excitation energy of derivatives (AOC) and (1B3N) are shown in Fig. 3a, c, and e. The

the S1 state and the excitation energies calculated for the triplet energy levels of their lowest singlet and triplet states and the kIC

states that are energetically below S1. The calculated excitation and kISC rate constants are shown in Fig. 3b, d and f, respectively.

energy for S1 agrees well with the experimental value with a The lowest excitation energies are given in Table 2. The intensity deviation of only 0.09 eV. The rate constants in Fig. 2b show for the radiative S0–S1 transition for tetraoxa[8]circulene (4B)

that kr is a main deactivation channel of the S1 state. The largest vanishes in the FC approximation. The transition becomes 65 rate constant among the nonradiative transitions is kISC2 due allowed in the Herzberg–Teller approximation. An oscillator to the small energy gap between S1 and T2. The calculated strength for 4B of 0.03 is obtainedÐ from the experimental fluorescence quantum yield (j) is 0.79, which qualitatively extinction coefficient f ¼ 4:32 109 eðvÞdv.66 The spin–orbit

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1 Table 3 Calculated wavenumbers oj (in cm ) for a few relevant vibrational modes. The largest Huang–Rhys factors (yj)fortheS1 - S0 and S1 - Tj nonradiative transitions are given in parenthesis

Compound oj (yj) PM567 210 (0.05); 571 (0.12); 1255 (0.1) Psoralene 234 (0.29); 396 (0.65); 742 (0.42); 1103 (0.15); 1164 (0.25); 1190 (0.54); 1615 (0.14); 1689 (0.24); 1877 (0.08) 4B 662 (0.17); 1051 (0.07); 1276 (0.35); 1453 (0.19); 1536 (0.16); 1703 (0.84) AOC 1308 (0.05); 1473 (0.14); 1675 (0.15); 1678 (0.05) 1B3N 232 (0.05); 1474 (0.15); 1518 (0.05); 1687 (0.05) H2P 157 (0.05); 310 (0.14); 1393 (0.03) Alq3 141 (0.36); 151 (0.25); 164 (0.08); 185 (0.08); 213 (0.06); 357 (0.54); 422 (0.09); 458 (0.10); 506 (0.12); 554 (0.34); 585 (0.11); 587 (0.13); 661 (0.33); 670 (0.21); 1434 (0.1) 1634 (0.07); 1637 (0.29); 1659 (0.07) Ir(ppy)3 121 (0.28); 146 (0.46); 187 (0.08); 258 (0.31); 269 (0.06); 1040 (0.01); 1512 (0.14); 1615 (0.11) Creative Commons Attribution-NonCommercial 3.0 Unported Licence.

Fig. 3 (a) The optimized molecular structure of psoralene. (b) The energy level diagram for psoralene including the computed values for the radiative and nonradiative rate constants.

4.4. Free-base porphyrin coupling matrix elements between S1 and Tj vanish due to

symmetry reasons since the molecule belongs to the D4h point Calculation of the nonradiative rate constants kIC and kISC for H2P 1 group. Thus, IC is the main deactivation channel of the S1 state. is a challenge. Several previous works have shown that the FC

This article is licensed under a 1,15 The calculated value for jfl is 0.13, which agrees well with the approximation is not enough to obtain accurate rate constants. experimental value of 0.09.36 AOC has the largest quantum TheintensityoftheQbandintheelectronicabsorptionspectrum

yield (jfl) of about 1.0, because the rate constant kISC is close to for H2P can be accurately simulated only when considering 67 zero and kIC is significantly smaller than kr. The experimental vibrational eﬀects at the Herzberg–Teller approximation. In this Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. 29 jfl value for AOC is 0.91. 1B3N has a smaller calculated jfl of work,weadoptedtheexperimental value 0.03 for the oscillator 0.56 which is in reasonable agreement with the experimental strength ( f )inthecalculationofkr using eqn (17). The molecular 29 value of 0.36. The reason for the smaller jfl value is the structure of free-base porphyrin is shown in Fig. 5a and the presence of several low-lying triplet states. The main deactiva- calculated energy levels and rate constants are presented in

tion channel of the S1 state is then the ISC process. The present Fig. 5b. The calculated rate constants for the nonradiative transi- calculations explain why the studied [8]circulenes have tions kIC and kISC are underestimated, because the present

different quantum yields. The largest yj values and the corres- calculations yield a quantum yield of 0.96, which is an order of 40 ponding vibrational energies (oj) are listed in Table 3. The main magnitude larger than the experimental value of 0.08. Apre- 7 1 15 vibrational mode for the nonradiative transition is at about viously calculated value for kISC1 is B10 s . The largest 1 1400 cm for AOC and 1B3N. We have previously calculated Huang–Rhys factors (yj) and the corresponding vibrational fre-

kIC and kISC at the INDO level of theory using one vibrational quencies are listed in Table 3. The calculated yj values are very mode with an energy of 1400 cm1.29 The obtained results were small leading to an underestimation of the rate constants of the

in good agreement with the present ones except for kIC of tetra- nonradiative transitions. The vibrational modes with energies in 1 1 oxa[8]circulenes (4B), because the NACMEs calculated at the the range 1000 cm o oj o 3000 cm contribute significantly

INDO level are inaccurate. The reason for the deviation can to the FC factor. The yj values of the vibrational modes in this

be traced back to the large yj factor for the vibrational mode at energy range should be about 0.1–0.4. However, the largest yj 1703 cm1, which consists of vibrational motion inside the value is 0.03 for the vibrational mode at 1393 cm1,which macrocycle.65 For the other [8]circulenes (AOC and 1B3N), this is about one order of magnitude too small. In our previous 1 mode has a smaller yj value. The present calculations show that INDO study considering one vibrational mode at 1400 cm ,an the INDO approach does not yield accurate nonradiative rate empirical yj value of 0.3 was used leading to the rather accurate 33 constants for tetraoxa[8]circulene (4B) (Fig. 4). values for kIC and kISC. Thus, one has to go beyond the FC

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Fig. 4 The optimized molecular structures of (a) 4B, (c) AOC and (e) 1B3N. The energy levels and rate constants are given for the same molecules in (b), (d), and (f), respectively.

approximation in order to obtain accurate values for the non- T1 states are compared with the experimental data in Table 2. radiative rate constants of free-base porphyrin. The calculated excitation energies agree well with the experi-

mental data. Only the lowest triplet state lies below the S1 state. 4.5. Alq3 and Ir(ppy)3 For Alq3, the radiative and the two nonradiative deactivation

The optimized molecular structures of the S1 state of fac-tris(8- channels of the S1 state have almost the same rate constants.

hydroxyquinolinato)aluminium ( fac-Alq3) and fac-tris(2-phenyl- The calculated quantum yield (jfl)of0.25agreeswiththe

pyridine)iridium ( fac-Ir(ppy)3) are shown in Fig. 6a and c. experimental values of 0.04–0.3 that strongly depend on environ- 38 The energy levels of the lowest singlet and triplet states and mental effects. For Ir(ppy)3, the ISC process dominates the

the kIC and kISC rate constants of the transitions are shown in deactivation of the S1 state due to the strong spin–orbit

Fig. 6b and d. The calculated excitation energies of the S1 and coupling effects. See Table S1 in the ESI.† The rate constant

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Fig. 5 (a) The optimized molecular structure of free-base porphyrin (H2P). (b) The energy level diagram for H2P including the computed values for the radiative and nonradiative rate constants. Creative Commons Attribution-NonCommercial 3.0 Unported Licence. This article is licensed under a Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM.

Fig. 6 (a) The optimized molecular structure of fac-Alq3. (b) The energy-level diagram for fac-Alq3 including the computed values for the radiative and

nonradiative rate constants. (c) The optimized molecular structure of fac-Ir(ppy)3. (d) The energy-level diagram for fac-Ir(ppy)3 including the computed values for the radiative and nonradiative rate constants.

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Fig. 7 The calculated energy levels for (a) naphthalene, (b) anthracene, (c) tetracene, (d) pentacene, (e) hexacene and the computed values of the radiative and nonradiative rate constants.

kISC1 is much larger than kr leading to a very small quantum Table 4 The calculated excitation energies (in eV) of the polyacenes are yield, which agrees with the previously calculated value of compared with the experimental data 12 1 68 about 10 s for kISC1 and the experimental results. The Compound State XMC-QDPT2 Exp.a largest yj values and the corresponding vibrational energies (oj)

are reported in Table 3. Alq3 has many more vibrational Naphthalene S1 3.66(0.003) 3.86 T1 2.67 2.63 modes with large yj values than Ir(ppy)3, which explains why Anthracene S1 2.97(0.2) 3.3 Alq3 has a larger kIC rate constant than Ir(ppy)3. T1 1.57 1.85 Tetracene S1 2.7(0.1) 2.6 4.6. Polyacenes T1 1.16 1.26 Pentacene S1 1.9(0.1) 2.1 The energy levels of the lowest singlet and triplet states and the kIC T1 0.7 0.95 and kISC rate constants of the transitions of the polyacenes are Hexacene S1 1.6(0.1) 1.8 T1 0.5 0.81 shown in Fig. 7. The calculated excitation energies of the S1 and T1 states are compared with the experimental data in Table 4. a The experimental data are taken from ref. 37.

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1 Table 5 Calculated wavenumbers (oj in cm ) of the significant vibrational modes for the nonradiative transitions and the corresponding Huang–Rhys

factors (yj)fortheS1 to S0 nonradiative transition of the polyacenes

Compound oj (yj) Naphthalene 519 (0.06); 775 (0.23); 1055 (0.1); 1413 (0.36) Anthracene 398 (0.37); 640 (0.02); 765 (0.02); 1039 (0.01); 1194 (0.08); 1301 (0.16); 1443 (0.44); 1606 (0.40) Tetracene 318 (0.26); 633 (0.03); 765 (0.04); 1032 (0.01); 1190 (0.04); 1243 (0.17); 1425 (0.15); 1441 (0.19); 1573 (0.02); 1589 (0.25) Pentacene 263 (0.18); 615 (0.02); 764 (0.05); 1190 (0.03); 1228 (0.17); 1426 (0.24); 1449 (0.08); 1570 (0.15) Hexacene 225 (0.15); 579 (0.01); 765 (0.1); 1030 (0.01); 1183 (0.05); 1212 (0.11); 1216 (0.01); 1217 (0.06); 1418 (0.25); 1560 (0.06)

The calculated quantum yields (jfl) for naphthalene, anthracene, For polyacenes and H2P, it is necessary to go beyond the FC and tetracene, pentacene, hexacene, and heptacene are 0.94, 0.99, adiabatic approximations. The FC approximation breaks down, 0.99, 0.03, and 0.001 respectively, which can be compared with when the Huang–Rhys factor (y) is about 0.01–0.04 for modes the experimental values of 0.23, 0.24, 0.21, 0.08, and 0.01.41 with energies of about 1000–1800 cm1 leading to strongly The calculated quantum yields diﬀer from the experimental underestimated FC factors. Small values for the FC factor lead

ones, whereas a similar trend is obtained for the polyacenes. to too small calculated values for kIC and kISC. The value of y can

The calculated jfl values are not very accurate, because the be used as a criterion for the applicability of our approach. For 1 S1 state is not a pure adiabatic state but consists of a mixture of H2P, the y value of the vibrational mode at 1393 cm has a small 50 higher energy states. The estimated quantum yields (jfl)agree Huang–Rhys factor suggesting that the present approach cannot

qualitatively with the experimental ones only when kIC is larger be applied to free-base porphyrin.

than kISC. For the polyacenes, the calculated kISC contribution to The computations show that in the IC process the excitation

thenonradiativerateconstantknr is underestimated. The calcu- energy transfers into the vibrations of the X–H (X = C, N and O) 24–28 lated kIC and kISC rate constants are expected to be rather accurate, bonds supporting the theory by Plotnikov, Maier and Artyukhov. whereas they cannot be used for estimating the quantum yield However, when estimating the FC factors of the ISC process, all the Creative Commons Attribution-NonCommercial 3.0 Unported Licence.

when kIC o kISC.Thevaluesofyj and oj are of the vibrational vibrational modes have to be taken into account, because many modes that are most important for the nonradiative transitions are vibrational modes with energies in the range of 1000–1800 cm1 listed in Table 5. The vibrational mode at about 1400 cm1 has a y contribute significantly to the FC factor.

value of about 0.3 for all the polyacenes except for tetracene. This The accuracy of the calculated kIC and kISC rate constants

mode yields the main contribution to the kISC rate constant. In the depends strongly on the accuracy of the calculated excitation

calculation of kISC for the polyacenes, it is necessary to consider energies of the excited singlet and triplet states. Since the 25–28 onlythismode,asdonebyMaierandArtyukhov. obtained kIC and kISC rate constants depend strongly on the accuracy of the calculated values for the singlet and triplet

This article is licensed under a 4.7. The analysis of nonadiabatic integrals excitation energies, we calculated the excitation energies at the Calculations of the nonadiabatic integrals (8) show that the largest XMC-QDPT2 level of theory, which is computationally expen- integral value of 0.007 a.u. is obtained for the vibrational modes sive but a feasible level for molecules of this size.69,70 with energies of about 3000 cm1,becausethey value is very The present approach makes it possible to estimate rate Open Access Article. Published on 16 February 2018. Downloaded 10/2/2021 10:25:10 AM. small for them. For the other vibrational modes, the integrals (10) constants for nonradiative transitions (kIC and kISC) for organic are a factor of 2–22 times smaller. The high-frequency modes with and organometallic compounds within the FC and adiabatic B3000 cm1 usually correspond to X–H stretching vibrations, approximation by using calculated values for the matrix ele- whereX=C,N,Ooranotherlightatom.Thepresentstudyshows ments of the spin–orbit coupling interaction and nonadiabatic

that in the IC channel, the excitation energy of the S1 state coupling interaction. transforms into stretching vibrations of X–H modes, which supports the theory by Plotnikov, Maier and Artyukhov.24–28 Conflicts of interest

5. Summary and conclusions There are no conflicts to declare.

The proposed computational approach has been used for calculating the rate constants for internal conversion (kIC) and inter- Acknowledgements system crossing (kISC) of a number of organic and organometallic compounds. The calculations show that the radiative process The research was carried out using a grant from the Russian dominates for PM567 and AOC. The non-radiative ISC process Science Foundation (project No. 17-73-20012). This research has

dominates for psoralene and Ir(ppy)3, whereas the non-radiative been supported by the Academy of Finland through two projects

IC process dominates for tetraoxa[8]circulene. For Alq3, the rate (275845 and 1309394). DS thanks the Magnus Ehrnrooth Foun- constants of the radiative and the non-radiative (IC and ISC) dation for financial support. CSC – the Finnish IT Center for processes are of the same size. When the Franck–Condon (FC) Science – as well as the Finnish Grid and Cloud Infrastructure and adiabatic approximations are valid, the proposed approach (persistent identifier urn:nbn:fi:research-infras-2016072533) are

can be used for estimating the fluorescence quantum yield (jfl). acknowledged for computational resources.

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