Ab initio study on the excited states of pyrene and its derivatives using multi-reference perturbation theory methods

Low-lying singlet excited states of pyrene derivatives originated from the 1La and 1Lb states of pyrene have decisive influences on their absorption and fluorescence emission behaviors. Calculation of these excited states with quantitative accuracy is required for the theoretical design of pyrene derivatives tailored to target applications; this has been a long-standing challenge for ab initio quantum chemical calculations. In this study, we explore an adequate computational scheme through calculations of pyrene and its phenyl-substituted derivatives using multi-reference perturbation theory (MRPT) methods. All valence π orbitals on the pyrene moiety were assigned to the active orbitals. Computational load was reduced by restricting the electron excitations within the active orbitals in the preparation of reference configuration space. A generalized multi-configuration quasi-degenerate perturbation theory (GMCQDPT) was adopted to treat the reference space other than the complete active space. The calculated 1La and 1Lb excitation energies of pyrene are in good agreement with the experimental values. Calculations of 1,3,6,8-tetraphenyl pyrene suggest that the energetic ordering of 1La and 1Lb is inverted through tetraphenyl substitution and its lowest singlet excited state is the 1La parentage of pyrene, which is consistent with the experimentally deduced scheme. These results are not readily obtained by MRPT calculations with a limited number of active orbitals and single-reference theory calculations. Diphenyl pyrenes (DPPy) were also calculated at the same level of theory to investigate the dependence on the substitution positions of phenyl groups.


Introduction
Pyrene is one of the most well-studied polycyclic aromatic hydrocarbons (PAHs) because of its characteristic photophysical properties, such as prominent absorption bands and a uorescence emission. 17][28][29] An aromatic excimer is a dimeric complex of the same aromatic molecules that is formed in the excited state.The uorescence emission of an excimer was rstly observed for a pyrene solution by Förster and Kasper in 1954. 27The excimer uorescence emission band is signicantly red-shied, broad, and structureless, so that it is clearly distinguishable from that of the monomer.][61][62][63][64] Pyrene has two important excited states, 1 L a and 1 L b in Platt's notation. 65These excited states are closely relevant to the absorption and uorescence emission behavior of pyrene.Excitation from the ground state to the 1 L a state gives a prominent absorption band around 340 nm in the UV/Vis spectrum because of its large oscillator strength.In contrast, the oscillator strength of the 1 L b state is negligibly small; the 1 L b absorption band is barely visible in the absorption spectrum. 66The 1 L b state is the lowest singlet excited state and the 1 L a state is the second lowest; therefore, the uorescence emission of pyrene is of 1 L b parentage according to Kasha's rule. 67In substituted pyrene derivatives, the excitation energies and absorbance of these excited states are perturbed by the substituents.In some cases, even an energetic ordering of the 1 L a and 1 L b states is inverted as a result of the energy-level shis of these excited states.1,3,6,8-Tetraphyenyl pyrene (TPPy) is recognized as a typical example. 68,69TPPy exhibits a relatively short uorescence lifetime (s z 3 ns) 68,69 and a high uorescence quantum yield (q F ¼ 0.9) 70 compared to pyrene (s > 300 ns and q F ¼ 0.32). 68,69,71These conspicuous changes in the photophysical properties can be attributed to an inversion of the energetic ordering of the 1 L a and 1 L b states through tetraphenyl substitution.9][70] Although this hypothetical mechanism based on the experimental ndings was supported by theoretical studies that employed semi-empirical methods 68,69,72 and has been widely accepted, the 1 L a -1 L b inversion in TPPy still remains to be corroborated by ab initio calculations.Given such a background, a dependable computational scheme that enables accurate calculations of these excited states is required for the theoretical design of pyrene derivatives customized to target applications; therefore, the prediction of their energetic ordering is of particular importance.
Calculation of the 1 L a and 1 L b excited states of pyrene with quantitative accuracy has been a long-term challenge for ab initio quantum chemistry. 73Time-dependent density functional theory (TDDFT) [74][75][76][77][78] has been recognized as an efficient approach to the excited states of large molecules 79,80 and is thus utilized in investigations of pyrene and its derivatives.Behind numerous successful results, comprehensive assessments indicate that TDDFT calculations yield inconsistencies in accuracy between the 1 L a and 1 L b excitation energies; in the worse cases, even the energetic ordering is incorrectly predicted. 73,81,824][85][86][87][88][89][90] The second-order approximate coupled cluster singles and doubles (CC2) method is also widely used in the calculation of large molecules as an ab initio wave function approach. 91The energetic ordering of 1 L a and 1 L b of pyrene given by CC2 calculation is generally consistent with the experimental results.However, the incorporation of multi-conguration characters of excited states using singlereference methods such as CC2 is possibly insufficient, and could thus be responsible for the calculation errors.In this regard, multi-reference methods are expected to make up for this shortcoming. 92Bito et al. conducted multi-reference conguration interaction (MRCI) calculations as pioneering work, 93 followed by theoretical studies that utilized multireference perturbation theory (MRPT) methods, 73,81,94 such as multi-conguration quasi-degenerate perturbation theory (MCQDPT) 95 and complete active space second-order perturbation theory (CASPT2). 96These methods adopt a complete active space self-consistent eld (CASSCF) wavefunction 97 as their reference.The CAS involves all electron congurations generated by distributing active electrons among active orbitals.Consequently, the dimension rapidly increases with the number of active orbitals and electrons, which makes the routine computation impossible.Therefore, in these studies, a limited number of p orbitals and p electrons were selected from a total of 16 valence p orbitals and 16 valence p electrons to construct the reference CAS.Although the energetic ordering of 1 L a and 1 L b is correctly predicted in most of these calculations, absolute errors in the excitation energies of 0.2-0.7 eV remain, and the calculation accuracy is inconsistent between the 1 L a and 1 L b states.The discrepancies at the CASSCF level are much larger because dynamical electron correlation is not sufficiently incorporated.
Recent advances in theory and computational techniques have yielded signicant progress.Freidzon et al. demonstrated that the 1 L a and 1 L b excitation energies of pyrene can be accurately predicted 98 using extended MCQDPT (XMCQDPT) method. 99Nenov et al. performed second-order perturbation theory restricted active space (RASPT2) calculations 100 with the reference conguration space constructed using 16 p orbitals and 16 p electrons. 101The calculated excitation energies were in good agreement with the experimental values, even though the computational load was reduced by restricting the electron excitations within the active orbitals.Most recently, Noble et al. conducted multi-state CASPT2 calculations with the Cholesky decomposition technique 102,103 in their study of electronic relaxations from the S 3 state of pyrene. 104The reference conguration space was prepared using 16 p orbitals and 16 p electrons, and the results were quite accurate.Lischka et al. carried out the calculations of paradigmatic aromatic molecules including pyrene using both multi-reference and singlereference methods. 105In their results, the 1 L a and 1 L b excitation energies of pyrene calculated using the DFT/MRCI method 106 were in good agreement with the experimental values.The same authors successfully applied their calculation scheme to the study of large-sized aromatic dimers. 107These studies suggest that the excited states of pyrene derivatives could also be calculated with quantitative accuracy by utilizing these advanced methods.Successful results with the full p valence reference space also imply that the incorporation of multi-conguration character is of key importance for accurate calculation.
In this study, we explore an adequate computational scheme that is suitable for the excited states of pyrene derivatives through calculations of pyrene and its phenyl substitutedderivatives shown in Fig. 1.Generalized multi-conguration quasi-degenerate perturbation theory (GMCQDPT) 108 was adopted in addition to the original MCQDPT.In contrast to the conventional multi-reference theories where the reference space is limited to the CAS, GMCQDPT allows more general types of reference.Accordingly, it has the potential to reduce the computational load without a signicant decrease of accuracy and could enable the handling of pyrene derivatives with large system sizes.

Reference wave functions for GMCQDPT
In the preparation of the reference wave function for GMCQDPT, the molecular orbitals are divided into three groups: inactive, active, and external orbitals.Whereas the inactive orbitals are always doubly occupied, external orbitals are always vacant.The electrons involved in the active orbitals are regarded as active electrons and the electron congurations are generated by distributing active electrons among active orbitals.The procedure so far is the same as that adopted in the construction of the CAS.][111] In this study, the MRX type of reference was employed because of its simplicity.In the MRX framework, parent congurations and the electron excitation level, n, are dened.The excited congurations having n or less excited electrons from the parent congurations are involved into the reference space, in addition to the parent congurations themselves.If n is equal to the number of active electrons, then the reference space is identical with the CAS.The dimension of the reference space can be reduced by limiting the number of parent congurations and specifying n less than the number of active electrons.
In the GMCQDPT calculations of pyrene in this study, all 16 valence p orbitals were selected as active orbitals.Similarly, in the calculations of phenyl-substituted derivatives, the 16 p orbitals on their pyrene moieties were treated as active orbitals.The Hartree-Fock type ground state conguration with eight doubly-occupied p orbitals and eight unoccupied p* orbitals was dened as a parent conguration, and the n value was set at 2, 3 and 4. Henceforth, these reference spaces are denoted as MRX(n).GMCQDPT calculations were performed in the following two steps as well as the conventional CASSCF-MCQDPT procedure: multi-conguration self-consistent eld (MCSCF) calculation with MRX(n) conguration space was rstly performed; the second-order perturbation calculation was subsequently conducted employing the obtained MCSCF function as its reference.

Calculations
The molecular geometrical structures were optimized using DFT with the B3LYP functional. 112,113The ground and excited states were subsequently calculated at the optimized geometries.The absence of an imaginary number frequency was conrmed for all optimized structures by vibrational analyses.In addition to the GMCQDPT calculations, conventional CASSCF-MCQDPT calculations were also performed for comparison; CAS(4pe, 4po), CAS(8pe, 8po), and CAS(12pe, 12po) were used as reference spaces.CAS{(2m)pe, (2m)po} (m ¼ 2, 4, and 6) was constructed using the m highest occupied and the m lowest unoccupied p orbitals on the pyrene moiety.The ground state, and the 1 L a and 1 L b excited states were averaged with even weights in the MCSCF calculations, and these three states were simultaneously perturbed in the MRPT calculations.Equations of motion coupled cluster singles and doubles (EOM-CCSD), 114,115 and TDDFT calculations with B3LYP, CAM-B3LYP, 116 and uB97XD 117 functionals were also performed.The DPPy derivatives were calculated using GMCQDPT with MRX(4) and TDDFT.The cc-pVDZ basis set was used throughout the calculations. 118The molecular symmetries assumed were D 2h , D 2 , C 2 , and C 2h for pyrene, 119 TPPy, 120 1,6-DPPy, and 2,7-DPPy, 121 respectively.MCSCF and MRPT calculations were performed using the GAMESS program. 122,123Other calculations were conducted using Gaussian09. 124

Calculation results
125][126][127][128] According to the previous studies, the 1 L a state of pyrene is characterized by two singly excited congurations: one is the highest occupied molecular orbital (HOMO) / the lowest unoccupied molecular orbital (LUMO) single excitation, and the other is the HOMOÀ1 / LUMO+1 single excitation.In contrast, HOMOÀ1 / LUMO and HOMO / LUMO+1 single excitations are dominant in the 1 L b state. 119,129he excited states were then identied on the basis of these congurations and related molecular orbitals.Fig. 2 shows the natural orbitals from HOMOÀ1 to LUMO+1 obtained using MCSCF with the MRX(4) conguration space.All natural orbitals from the MCSCF wave functions are presented in Fig. S1-S3.† Electronic state total energies and MCSCF excitation energies are listed in Table S1.† 3.1.1.Pyrene.The GMCQDPT-calculated excitation energies of pyrene exhibit systematic improvements with increasing n.The calculated 1 L b excitation energy with MRX(2) is underestimated by approximately 0.2 eV; the deviation is suppressed with an increase of the n value; the 1 L a and 1 L b excitation energies calculated with MRX(3) and MRX(4) are both in good agreement with the experimental values.In contrast, the MCQDPT results largely uctuate depending on the reference space.Both the 1 L a and 1 L b excitation energies with CAS(8pe, 8po) are relatively close to the corresponding experimental values, whereas both values are signicantly underestimated with CAS(4pe, 4po).The 1 L a excitation energy calculated with CAS(12pe, 12po) is close to the experimental values; however, the 1 L b excitation energy is overestimated by as much as ca.0.6 eV, so that the predicted energetic ordering of these two states is contrary to the experimental results.The 1 L b excitation energy is more sensitive to the reference space than the 1 L a excitation energy, which suggests its considerable multi-conguration character.The EOM-CCSD calculation overestimates the excitation energies by approximately 0.5 eV for both 1 L a and 1 L b .The calculation results with the TDDFT method are similar to those reported in pioneering works. 73,81,82,105,130,131In the B3LYP results, the 1 L a excitation energy is underestimated, whereas the 1 L b excitation energy is overestimated.The predicted ordering of these states in energy is consequently inconsistent with the experimental ordering.Although the calculated ordering is marginally correct in the CAM-B3LYP and uB97XD results, the excitation energies of the 1 L a and 1 L b states are quite close to each other due to serious overestimations of the 1 L b energy level.Thus, reliable calculations of the 1 L a and 1 L b excitation energies could not be accomplished using the exchange-correlation functionals examined here.The calculated oscillator strengths of the 1 L a state are much larger than those of the 1 L b state in all cases, which is in agreement with the experimental observations. 66.1.2.TPPy.The excitation energies calculated for TPPy are generally decreased from those for pyrene.The GMCQDPT calculations suggest that the 1 L a excitation energy decreases to a greater extent than the 1 L b excitation energy from that of Paper RSC Advances pyrene through tetraphenyl substitution.As a result, the 1 L a energy level approaches 1 L b in the results with MRX(2) and MRX(3).However, the 1 L a state is still higher-lying than the 1 L b state in these results.In contrast, the energetic ordering of 1 L a and 1 L b is inverted in the calculation results with MRX(4), and the 1 L a state is determined to be the lowest singlet excited state. 68,69In addition, the 1 L a excitation energies calculated with GMCQDPT are close to the experimental value.The MCQDPT results largely vary depending on the reference space, similar to the calculations for pyrene.The results with CAS(4pe, 4po) and CAS(8pe, 8po) are similar to the GMCQDPT calculations with MRX(2) and MRX(3): the 1 L a state approaches 1 L b state in terms of energy; however, it is still higher-lying than the 1 L b state and the 1 L a -1 L b inversion is not predicted.The calculations with CAS(12pe, 12po) give confusing results: the incorrectlypredicted energetic ordering of 1 L a and 1 L b for pyrene is inverted, which results in another inconsistency with the experimentally-deduced scheme.The EOM-CCSD calculation fails to predict the 1 L a -1 L b inversion through tetraphenyl substitution and overestimates the 1 L a excitation energy.Given the overestimations of the excitation energies in pyrene, the 1 L b excitation energy of TPPy may also be overestimated.The TDDFT calculations predict that the 1 L a state is lower-lying than the 1 L b state in TPPy, which is similar to the GMCQDPT calculations with MRX(4).The 1 L a excitation energy with B3LYP is in good agreement with the experimental value, whereas CAM-B3LYP and uB97XD give overestimated values.The 1 L b excitation energies are higher than that obtained using GMCQDPT with MRX(4) by 0.25 eV for B3LYP and by ca.0.5 eV for the other two functionals.
Overall, provided that the GMCQDPT method along with MRX(4) is adopted, the excitation energies of pyrene and TPPy can be accurately calculated, and the 1 L a -1 L b inversion through tetraphenyl substitution is predicted.The calculated oscillator strength of the 1 L a state for TPPy is consistently larger than that of pyrene, whereas that of the 1 L b state is still vanishingly small.
3.1.3.1,6-DPPy.The calculations suggest that the excitation energies of the DPPy derivatives are generally lower than those of pyrene.Let us rst examine the calculation results of 1,6-DPPy.In the calculation results of GMCQDPT with MRX(4), the 1 L a energy level shis largely downward from that of pyrene and approaches the 1 L b energy level.Although the energetic ordering of the 1 L a and 1 L b states is unchanged, their energy levels are quite close to each other.The calculated 1 L a oscillator strength of 1,6-DPPy is larger than that of pyrene, which results in the ordering of pyrene < 1,6-DPPy < TPPy.
The TDDFT calculations also predict a downward shi of the 1 L a energy level without signicant change of the 1 L b energy level.However, in contrary to the GMCQDPT results, the TDDFT calculations suggest that the 1 L a state is the lowest singlet excited state.As mentioned above, the 1 L b excitation energy of pyrene is signicantly overestimated by the TDDFT calculations.The 1 L a energy level calculated using TD-B3LYP is estimated to be lower than the 1 L b energy level even for nonsubstituted pyrene.The gap between 1 L a and 1 L b is extremely underestimated using TD-CAM-B3LYP and TD-uB97XD; because of this imbalanced alignment of the 1 L a and 1 L b energy levels for non-substituted pyrene, the energetic ordering of the 1 L a and 1 L b states is easily inverted through the downward shi of the 1 L a energy level.Meanwhile, the TDDFT calculations suggest an increase of the 1 L a oscillator strength through the diphenyl substitution as well as the GMCQDPT calculations.The oscillator strengths of the 1 L b state are much smaller than those of the 1 L a state, as with the cases of pyrene and TPPy.Yet, 1,6-DPPy exhibits the largest 1 L b oscillator strength among the molecules calculated here.
3.1.4.2,7-DPPy.In the results of GMCQDPT with MRX(4), the excitation energies of 2,7-DPPy only slightly decrease from those of pyrene for both 1 L a and 1 L b .Consequently, the energy gap between the 1 L a and 1 L b states (0.34 eV) is only slightly decreased from the value in pyrene (0.44 eV).Contrary to the other two derivatives, 2,7-DPPy exhibits a decreased 1 L a oscillator strength compared to pyrene.Thus, the inuence of phenyl substituents is dependent not only on the excited state but also on the substitution position.
The results of TDDFT calculations show a different behavior from the GMCQDPT results; the 1 L b energy level is more largely shied downward without signicant change of the 1 L a energy level.The behavior is also in contrast to the case of 1,6-DPPy which exhibits a largely downward shi of the 1 L a energy level without a major change in the 1 L b energy level.Meanwhile, a decrease of the 1 L a oscillator strength through diphenyl substitution is predicted using TDDFT, as with GMCQPDT.

Discussion
3.2.1.Dependence on the reference space.The excitation energies calculated using GMCQDPT with MRX(4) are plotted in Fig. 3 as a visualization of the energy level shis of the 1 L a and 1 L b states through phenyl substitutions.The phenyl groups at 2,7-positions have less impact on the excitation energies.In contrast, the 1 L a energy level shis largely downward in 1,6-DPPy without a signicant change of the 1 L b energy level.The further downward shi of the 1 L a energy level causes the energetic ordering of the 1 L a and 1 L b states in TPPy to be inverted from that in pyrene.Therefore, the ab initio calculation results corroborate the 1 L a -1 L b inversion that was hypothesized based on the experimentally-observed photophysical properties and supported by semi-empirical calculations. 68,69,72The results indicate that the uorescence emission of TPPy has an 1 L a parentage, which is consistent with the high q F and short s of TPPy with respect to pyrene.Let us rst discuss the dependence of the calculation results on the reference space.
The accuracy of the MRPT calculation reveals a strong dependence on the reference conguration space.Therefore, the requirements for the reference space to achieve reliable calculations can be determined from these results.The excitation energies with MCQDPT are highly dependent on the reference CAS, whereas those with GMCQDPT vary only in a narrow range.Thus, the calculation accuracy is sensitive to the number of active orbitals.There is no guarantee that MRPT with less active orbitals will be successful, even though its reference space is the CAS.In contrast, the incorporation of multiplyexcited congurations into the reference space leads to a systematic improvement within the framework of a full p valence reference space.RASPT2 calculations by Nenov et al. implied that the reference space with incorporation of up to quadruple excitations is necessary to compute the excitation energies of pyrene with high accuracy. 101It should be noted that involving quadruply-excited congurations is also critically important for prediction of the 1 L a -1 L b inversion.
The improvement of the calculation accuracy through the expansion of the reference space implies a signicant multi-conguration character of these excited states.Therefore, the main congurations of the excited states in the MCSCF wave functions and their weights were analyzed (Table 2).The total weights of the main congurations for 1 L a and 1 L b , denoted by s a and s b , are also presented in Table 2.A smaller s value indicates a greater contribution of the electron congurations other than the main congurations.Therefore, the s value is correlated with the degree of the multi-conguration character of the excited state.The weights of the main congurations are relatively reduced as the contributions of more electron congurations are incorporated; therefore, the s values are decreased with expansion of the reference space.Let us review the results obtained for pyrene with MRX(n); the s b values are generally smaller than the s a values and more largely varied depending on the reference space.Such behavior can be attributed to the signicant multi-conguration character of 1 L b compared to 1 L a , which is consistent with a high dependence of the calculated 1 L b excitation energy on the reference space (Table 1).The calculations with MRX(4) give the smallest s values among the examined reference spaces for both L a and L b ; this is expected to be more suitable to deal with the complexity of these excited states.Although the s values with CAS have similar trends to that with MRX(n), their uctuations with the reference space are more pronounced and the values are generally larger than those with MRX(4).These results suggest that multi-conguration character is only insufficiently incorporated with these CAS references, even at the maximum CAS(12pe, 12po).The s values and their trends for TPPy, 1,6-DPPy and 2,7-DPPy are quite similar to those of pyrene, which indicates a comparable degree of the multi-conguration character of their excited states.3.2.2.Effects of phenyl groups on the excitation energies.We move on to the discussion on the effects of phenyl substitutions.The s values analyzed above indicate that the 1 L a and 1 L b excited states have considerable multi-conguration characters.Nevertheless, the main congurations of these states still have large weights (Table 2), suggesting that the effects of phenyl groups might be essentially understood based on the molecular orbitals which are relevant to the main congurations: HOMOÀ1, HOMO, LUMO and LUMO+1.In this regard, the correlation between these molecular orbitals and the calculation results is explored.
The GMCSCF wave function with MRX(n) was constructed in terms of a linear combination of electron congurations and the GMCSCF calculations were carried out with a stateaveraging scheme.Consequently, the GMCSCF orbital energies do not have clear physical meanings.The orbital energies obtained using the DFT calculations are dependent on the exchange-correlation functionals.Therefore, we analyzed the Hartree-Fock orbital energies collected in Table 3 where the energy gap between the p and q orbitals is denoted by D p/q ; the energy shis from the corresponding value of pyrene are shown in parentheses.The molecular orbitals obtained from the Hartree-Fock calculations are shown in Fig. S4.† In TPPy and 1,6-DPPy, the HOMO and LUMO energy levels are largely shied compared to the HOMOÀ1 and LUMO+1 levels; the HOMO energy level is shied upward, whereas the LUMO energy level is shied downward.In 2,7-DPPy, in contrast, the HOMOÀ1 and LUMO+1 energy levels of 2,7-DPPy are largely shied without signicant changes of HOMO and LUMO energies.These variations of the orbital energies are consistent with the orbital distributions shown in Fig. 2 and S4.† The HOMO and LUMO of pyrene have large coefficients at positions 1, 3, 6, and 8; therefore, these orbitals are sensitive to phenyl substitution at these positions.In contrast, the HOMOÀ1 and LUMO+1 have large coefficients at positions 2 and 7; the phenyl substitution at positions 2 and 7 has a large impact on these orbitals.The electronic interactions between pyrene moieties and phenyl substituents can be also visually conrmed in the molecular orbitals of pyrene derivatives; the HOMO and LUMO of TPPy

RSC Advances Paper
and 1,6-DPPy are partially extended to the phenyl substituents, whereas HOMOÀ1 and LUMO+1 are extended in 2,7-DPPy.The orbital extensions are more noticeable in the Hartree-Fock orbitals (Fig. S4 †).As a result of the energetic changes of the orbitals, the D HOMO/LUMO values are in the order of pyrene z 2,7-DPPy > 1,6-DPPy > TPPy, which is in good agreement with that of the 1 L a excitation energies obtained using GMCQDPT with MRX(4) (Table 1).Although the D HOMO/LUMO value of 2,7-DPPy is almost the same as that with pyrene, D HOMOÀ1/LUMO+1 of 2,7-DPPy is much smaller than that of pyrene.The HOMOÀ1 / LUMO+1 single excitation is another main conguration of the 1 L a states; therefore, a reduced D HOMOÀ1/LUMO+1 could be responsible for the slightly decreased 1 L a excitation energy.A similar correlation between the D values and excitation energies can be also found for the TDDFT results (Table 1).Thus, as for 1 L a , the calculated excitation energies are well correlated with the D values related to the main congurations.
The calculated D HOMO/LUMO+1 and D HOMOÀ1/LUMO values of phenyl substituted derivatives are lower than those of pyrene, which is consistent with their lower 1 L b excitation energies obtained using GMCQDPT with MRX(4).The uctuations of the D HOMO/LUMO+1 and D HOMOÀ1/LUMO values are rather milder than those of the D HOMO/LUMO and D HOMOÀ1/LUMO+1 values.This behavior is also consistent with the insensitivity of the 1 L b excitation energies to phenyl substitution compared to the 1 L a excitation energies, which results in a reduction of the 1 L a -1 L b gap in 1,6-DPPy and the 1 L a -1 L b inversion in TPPy (Fig. 3).However, the D HOMO/LUMO+1 and D HOMOÀ1/LUMO values are in the order of pyrene > 1,6-DPPy > TPPy > 2,7-DPPy, which is not in agreement with the ordering of the 1 L b excitation energies with GMCQDPT: pyrene > 1,6-DPPy z 2,7-DPPy > TPPy (Table 1).The 1 L b state has signicant multi-conguration character, as suggested from the less s values (Table 2), which could be responsible for the weak correlation between the D values and the 1 L b excitation energies, and also for smaller uctuations of the 1 L b excitation energies.In contrast to the GMCQDPT results, the 1 L b excitation energies with TDDFT exhibit a stronger correlation with the D HOMO/LUMO+1 and D HOMOÀ1/LUMO values; these values are in the same order: pyrene > 1,6-DPPy > TPPy > 2,7-DPPy.Since TDDFT is a one-particle theory, the excitation energies obtained using TDDFT are susceptible to the energies of one-electron orbitals.Instead, the 1 L b excitation energy tends to be overestimated because of insufficient incorporation of multi-conguration character.
3.2.3.Effects of phenyl groups on the oscillator strengths.The excitation energies are reasonably correlated with the D values, suggesting that the 1 L a and 1 L b states are primarily characterized by their main congurations.Therefore, the effects of phenyl substitution on the oscillator strength could be also understood based on the orbitals related to the main congurations.As for allowed transitions, spatial expansion of the relevant molecular orbitals generally leads to an increased oscillator strength through enlargement of the transition dipole moment.The HOMO and LUMO are spatially extended to four phenyl groups in TPPy and two phenyl groups in 1,6-DPPy (Fig. 2).Therefore, the calculated 1 L a oscillator strengths are in the order of TPPy > 1,6-DPPy > pyrene.In contrast, the calculated 1 L a oscillator strength of 2,7-DPPy is decreased from that of pyrene.The mechanism can also be understood based on the main congurations. 129,132A linear combination of HOMO / LUMO and HOMOÀ1 / LUMO+1 singly excited congurations results in 1 L a as a lower-energy component and 1 B a as a higher-energy component. 129The transition moments derived from these main congurations are counteracted with each other in 1 L a , whereas they are reinforced in 1 B a .The weights of these congurations are different; therefore, a transition moment still remains aer their mutual cancellation, and the 1 L a state can thus give a visible absorption band.The difference in weights between these main congurations of 2,7-DPPy is reduced from that of pyrene (Table 2).The weight of the HOMOÀ1 / LUMO+1 excited conguration of 2,7-DPPy is specically increased from that of pyrene, while that of the HOMO / LUMO excited conguration is decreased.The origin of such behavior is a reduced D HOMOÀ1/LUMO+1 without a major change in D HOMO/LUMO , so that the weight of the HOMOÀ1 / LUMO+1 excitation is relatively increased.In addition, the transition moment originated from the HOMOÀ1 / LUMO+1 single excitation is enlarged because these orbitals are partially extended to phenyl groups at positions 2 and 7. Therefore, the transition moments derived from these congurations are largely canceled by each other, which results in a decreased oscillator strength.Qiao et al. measured the UV/Vis absorption spectra of pyrene and its 2,7-substituted derivatives in dichloromethane. 121The results showed that 2,7-DPPy exhibited an absorption band with l max of 342 nm and this band is expected to have parentage from 1 L a of pyrene.The molar extinction coefficient is slightly smaller than that of the pyrene 1 L a absorption band.The present calculation results are consistent with the experimental observations.The linear combination of HOMO / LUMO+1 and HOMOÀ1 / LUMO singly excited congurations results in 1 L b as a lower-energy component and 1 B b as a higher-energy component. 129Therefore, the transition moments derived from these main congurations are counteracted each other in 1 L b , whereas they are reinforced in 1 B b .These congurations have almost the same weights in 1 L b (Table 2); therefore, the transition dipole moments are almost completely canceled, which results in its negligibly small oscillator strength.The cancellation is slightly suppressed in 1,6-DPPy because of its lower symmetry.Consequently, the 1 L b oscillator strength of 1,6-DPPy is high when compared to pyrene and other derivatives with higher symmetries.
The results of the TDDFT calculations are similar to those obtained from the GMCQDPT calculations; the calculated 1 L a oscillator strengths are in the order of TPPy > 1,6-DPPy > pyrene; the calculated 1 L b oscillator strengths are nearly zero except for 1,6-DPPy.Overall, the effects of substituents on the 1 L a excitation energy, the 1 L a and 1 L b oscillator strengths can be reasonably evaluated using TDDFT; however, the 1 L b excitation energy cannot be accurately predicted; in the worse cases, even the energetic ordering is incorrectly predicted.Thus, a multireference treatment is essential for an accurate calculation of the 1 L b excitation energy, the ordering of the 1 L a and 1 L b states, and the energy gap between these states.

Conclusion
The absorption and uorescence emission behavior of pyrene derivatives is characterized by the low-lying excited states derived from 1 L a and 1 L b of pyrene.In this study, pyrene, TPPy, 1,6-DPPy, and 2,7-DPPy as typical examples were calculated using MRPT methods to explore a dependable computational scheme for these excited states.When all valence p and p* orbitals on the pyrene moiety were incorporated into the reference space, the calculated 1 L a and 1 L b excitation energies of pyrene were in good agreement with the experimental values.The uorescence emission of TPPy was predicted to have pyrene 1 L a parentage, which theoretically corroborates the experimental observations.A reference space involving singly, doubly, triply, and quadruply excited congurations was required to predict the 1 L a -1 L b inversion through tetraphenyl substitution.
MRPT calculations with smaller reference spaces and singlereference theory calculations exhibit inconsistencies with these results, which suggests that adequate treatment of the multi-conguration character is essential.The effect of phenyl substitution was dependent not only on the excited state but also on the substitution position.The detailed mechanism was claried by examination of the MCSCF wave functions.These calculations were successfully conducted at a tractable computational cost by the adoption of GMCQDPT to enable the handling of general types of reference conguration space.

Conflicts of interest
There are no conicts of interest to declare.

Table 2
Main configurations of the excited states and their weights.HOMO and LUMO are denoted as H and L, respectively.The values are rounded off to four decimal places