Theoretical Molecular Design of Heteroacenes for ... - ACS Publications

Article pubs.acs.org/JPCC

Theoretical Molecular Design of Heteroacenes for Singlet Fission: Tuning the Diradical Character by Modifying π‑Conjugation Length and Aromaticity Soichi Ito and Masayoshi Nakano* Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan S Supporting Information *

ABSTRACT: A theoretical molecular design for efficient singlet fission (SF) is performed for several heteroacene models involving nitrogen (N) atoms based on the diradical character criterion of the energy level matching conditions. This criterion is found to be closely related to the relative contributions of diradical and zwitterionic resonance structures of the heteroacenes, i.e., the aromaticity of the central ring(s). From the analysis of the diradical characters of these heteroacene models, the increase in the aromaticity of the central ring(s) is found to prefer the diradical form to the zwitterionic form. From the comparison of the excitation energies evaluated by multireference secondorder perturbation theory calculations, two promising candidates, chosen based on the diradical character criterion, are found to satisfy the energy level matching conditions and to possess high triplet energies of ∼1.1 eV, which are suitable for an application in organic photovoltaic cells. The proposed two candidates are shown to have mutually different types of the first excited singlet states, which are distinguished by the primary excitation configurations. These results suggest that the proposed two candidates exhibit different singlet fission dynamics due to the different amplitude of the electronic coupling. that for finding molecules which exhibit singlet fission the following energy level matching conditions must be satisfied25

1. INTRODUCTION Recently, singlet fission has attracted many researchers since Nozik and co-workers suggested that singlet fission improves the power conversion efficiency of organic photovoltaic (OPV) cells.1,2 With a detailed balance model, Hanna and Nozik quantitatively showed that singlet fission photovoltaic cells overcome the Shockley−Queisser limit (33.7%) in single junction photovoltaic cells when a properly chosen cell is combined with a singlet fission cell and that the limit reaches up to 45−47%, depending on the cell combination.3 Singlet fission occurs in a system of two coupled molecules or chromophores at least, where a singlet exciton generated by light absorption splits into two triplet excitons sharing its energy via a tripletcoupled singlet state. Using triplet excitons rather than singlet excitons, singlet fission photovoltaic cells could be more efficient in OPV application because they have longer lifetimes than singlet excitons due to slight spin−orbit coupling in typical organic molecules. Thus, longer diffusion lengths of triplet excitons than those of singlet excitons are expected. This feature of triplet excitons indicates that they have many opportunities to reach the donor/acceptor interface, where they are separated into free charge carriers. Although lots of experimental and theoretical studies have been performed for exploring the mechanism of singlet fission in several systems composed of pentacene,4−10 tetracene,9−13 rubrene,14−16 etc., the number of molecules which exhibit singlet fission is still limited.17−24 Smith and Michl presented © XXXX American Chemical Society

2E(T1) − E(S1) ∼ 0 or < 0

(1)

and 2E(T1) − E(T2) < 0

(2)

where E(S1) and E(Tn) indicate excitation energies of the first singlet and the nth triplet states, respectively. Equations 1 and 2 are necessary for making singlet fission isoergic or exoergic, and triplet−triplet annihilation endoergic, respectively. Although these conditions are not completely sufficient for achieving efficient singlet fission because singlet fission is a bimolecular process, these conditions are considered to be the most important among contributing factors, e.g., electronic coupling between S1S0 and 1(T1T1), which depends on the relative molecular configuration in dimer25 and nonadiabatic coupling mediated by intermolecular motion between the opticallyallowed singlet and a dark (optically-forbidden) triplet-coupled singlet states.10 On the basis of these conditions, they predicted that “alternant hydrocarbons” or “diradicaloids” are promising candidates for singlet fission.25 In quantum chemistry, the relationship among these excitation energies is found to be Received: October 15, 2014 Revised: November 26, 2014

A

DOI: 10.1021/jp5103737 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C described by the diradical character,26,27 which is a quantitative index of open-shell singlet nature.28 The multiple diradical character yi (0 ≤ yi ≤ 1, i = 0, 1, 2, ...) is defined as the occupation number of the lowest unoccupied natural orbital (LUNO)+i.29 Indeed, in our previous papers,30−33 we have clarified that an essential factor for designing singlet fission molecules is the diradical character: the energy level matching conditions are satisfied when a molecule has weak/intermediate diradical character (indicated by y0) as well as a quite weak tetraradical character (indicated by y1). This simple guideline based on the diradical character has been substantiated by our previous studies on several antiaromatic condensed-ring hydrocarbons with 4nπ electrons,31 a series of oligorylenes,32 and a series of typical alternant/nonalternant hydrocarbons, i.e., phenacenes, acenes, and isobenzofulvenes.33 It has been found from these results that a weak/intermediate diradical character y0 ranges from ∼0.2 to ∼0.4, and a quite weak tetraradical character satisfies y1/y0 < 0.2, the ranges of which are suited for efficient singlet fission.30−33 In this study, we perform the diradical character based theoretical molecular design for singlet fission in heteroacene systems involving several nitrogen (N) atoms, which have two types of resonance structures, i.e., open-shell (diradical) and closed-shell (zwitterionic) forms (see Figure 1). We here pay

theoretical chemists due to their possibility to realize high-spin organic compounds.36−38 Among them, m-xylylene is known to have the triplet ground state and has been well investigated experimentally and theoretically.39−41 In addition, non-Kekulé heteroacenes involving m-xylylene-like structure have also been intensively studied from the same perspective.42−46 Figure 1 shows heteroacene model molecules Nm-i examined in this study, where m (= 0, 1, and 2) represents the number of N atoms involved per ring in the square bracket and i (= 3 and 4) denotes the number of six-membered rings. They have similar structure to m-xylylene except that the two radical sites indicated by dots (•) are substituted by N atoms. Because mxylylene is known to have the triplet ground state,39−41 heteroacene molecules had also been expected to be promising candidates for molecular magnets.42−46 In spite of the structural similarity, N0-3 and N1-3, which were synthesized by Hutchison et al.42 and by Langer et al.,43 respectively, were found to have the singlet ground states experimentally and theoretically.42−45 In contrast, Rajca et al.46 reported the synthesis and measurement of high-spin molecules involving similar structure to that in Figure 1, though their molecule has only two N atoms on the radical sites. Amiri et al.45 have theoretically investigated the effect of the number of N atoms substituted in the heteroacene framework on the electronic structures and have clarified the impact of changing the number and position of N atoms on the singlet−triplet energy gap. Although these molecules could have two resonance structures, the previous papers 42,43 predicted that the contribution of zwitterionic closed-shell form is dominant in the ground state on the basis of the atomic charge distributions and carbon (C)−carbon (C) bond lengths obtained from the X-ray diffraction results. Hutchison et al. explained the stabilization of the zwitterionic form by partitioning N0-3 into the two independent charged cyanines and by comparing their structures with their parent ones.42 Nevertheless, from a previous study,47 we expect that some of these systems still have the possibility of exhibiting weak/intermediate diradical characters in the singlet ground state based on their resonance structures and that T1 is a relatively high-lying state, which is substantiated for N0-3 by the experimental ESR results.42 The quantitative evaluation and tuning of the diradical character in the intermediate region is found to be important for searching and designing candidate molecules for efficient singlet fission because the energy level matching conditions (eqs 1 and 2) have strong dependences on the diradical characters.30−33 As mentioned above, molecules with weak/intermediate diradical character are expected to be suited for efficient singlet fission from the viewpoint of the energy level matching conditions. Thus, we predict that the molecules shown in Figure 1 are good candidates for singlet fission with high E(T1), or even if not, they present a good starting point for further chemical modifications to realize efficient singlet fission through tuning the diradical character. We here examine the effect of two types of chemical modifications on the diradical characters: the first one is the π-conjugation extension (increasing i from 3 to 4) because the π-conjugation extension increases the diradical character,31−34,48−50 while the second one is the substitution of CH by a N atom (m = 0, 1, and 2). The electronic structural changes of the central aromatic ring(s) in these molecules are speculated to affect the diradical character through the change in aromaticity in the central benzene ring(s): the increase in aromaticity in the central ring(s) leads to the increase in the relative contribution of diradical form in the resonance

Figure 1. Resonance structures of heteroacene models Nm-i.

attention to the role of the aromaticity related to the diradical character through the resonance structures. The chemical structure of the model molecules predicts the increase in the diradical contribution with increasing the aromaticity of the central ring(s),34 which corresponds to nonzero values of yi. From the viewpoint of the energy level matching conditions, therefore, we present tuning schemes for the yi of these heteroacene models by several chemical modifications, which cause a change in the aromaticity of the central rings based on the resonance structures. We also clarify that the exchange interaction between molecular orbitals,27,33,35 which relates to the singlet and triplet energy splittings, is closely connected with the classification of the electronic nature of the first singlet excited state, which is useful for the classification of the singlet fission molecules. This classification is related to the relative molecular configuration in the dimer maximizing the electronic coupling for efficient singlet fission.25 The present results demonstrate high potential of heteroacenes as a new class of efficient singlet fission molecules and contribute to building concrete design guidelines for efficient singlet fission with high triplet energy, which is shown to be achieved by tuning yi through appropriate chemical modifications, i.e., tuning of the molecular π-conjugation size and the number of N atoms involved.

2. MODEL MOLECULES AND COMPUTATIONAL DETAILS 2.1. Heteroacene Model Molecules. Non-Kekulé hydrocarbons, which are a class of hydrocabons with two unpaired electrons at least, have been fascinating to experimental and B


Article

The Journal of Physical Chemistry C

lowest two A1 states and the lowest two B2 states, is employed in both singlet and triplet calculations in order to achieve the well-balanced description of these excitation energies. To confirm the adequacy of the choice of the active space in the RASPT2, we compare E(T1) values calculated using the RASPT2 method with those using the SF-PBE50 method, which is known to well reproduce experimental E(T1) values53 and has no arbitrary parameters such as the choice of active space in the RASPT2 method. Because we mainly focus on the relationship between the diradical character and excitation energy (the energy level matching conditions), only vertical excitation energies without zero-point vibrational energy corrections are examined. Multiple diradical character y i is calculated by the approximately spin-projected (AP) scheme27,28

structures, i.e., the increase in the diradical character (see Figure 1). 2.2. Computational Details. The spin-flip time-dependent density functional theory (SF-TDDFT) method 51 was employed with the collinear and Tamm−Dancoff approximation using the BHandHLYP functional (50% HF plus 50% Becke exchange52 with Lee−Yang−Parr correlation53) for the ground-state geometry optimization of each molecule in order to explicitly include static electron-correlation effects because the model molecules are predicted to have non-negligible diradical character, i.e., near-degenerate HOMO and LUMO. This method is hereinafter referred to as the SF-BHHLYP method. Triplet excitation energy E(T1) of each model system with the equilibrium geometry was calculated using the noncollinear spin-flip TDDFT,54 where the PBE50 exchangecorrelation functional (50% HF plus 50% Becke exchange with Perdew−Burke−Ernzerhof correlation55) is employed since this functional is found to well reproduce singlet−triplet energy gaps of a number of benchmark molecular sets.54 This method is referred to as the SF-PBE50 method in this study. These SFPBE50 results for E(T1) are employed for the comparison with the results by multireference calculations, i.e., restricted active space second-order perturbation theory (RASPT2)56 calculations, in order to determine the reasonable active space size for a balanced description of plural excited states because the SF-PBE50 method is shown to give reliable E(T1) values for many diradical molecules.54 Although the SF-TDDFT method is known to formally describe singlet excited states primarily including single and double excitation configurations from the HOMO to LUMO and to give a good theoretical estimation for the singlet−triplet energy gap, it is unable to describe some important singlet excited states of the present model molecules as shown below. Thus, we employ the noncollinear SF-PBE50 method only for choosing the appropriate active space for the model molecules. All the SF-TDDFT calculations were performed with reference state of Ms = 1 triplet computed from the restricted Kohn−Sham equation. We evaluated the electronic excitation energies, E(S1), E(T1), and E(T2), of each molecule using the RASPT2 method. This method allows us to choose a more flexible active space in multiconfiguration selfconsistent-field (MCSCF) calculation rather than the complete active space SCF (CASSCF), which performs a full configuration interaction in the active space together with the orbital relaxation. The active space in the RASSCF and RASPT2 methods is divided into three subspaces, RAS1, RAS2, and RAS3: RAS2 is identical to CAS, while RAS1 and RAS3 are orbital subspaces, which are located outside RAS2 and are composed of doubly occupied and virtual orbitals, respectively. The latter two subspaces are used for generating additional configurations with the restrictions concerning the maximum number of excitations from RAS1 into RAS2 and/or into RAS3 and of excitations from RAS1 and/or RAS2 into RAS3. The active space in the RASSCF and orbitals in the wholeRASPT2 is described as (ne, no)/(ne2, no2)//(n1, n3), where the first parentheses indicate ne electrons in no orbitals in the whole active space including RAS1, RAS2, and RAS3, the second ones ne2 electrons in no2 orbitals in RAS2, and the third ones n1 (n3) electron excitation at most from (into) RAS1 (RAS3) are allowed. Here, we employ the active spaces (16, 14)/(6, 6)// (1, 1) for three-ring molecules (i = 3) and (20, 18)/(6, 6)//(1, 1) for four-ring molecules (i = 4), respectively, both of which are referred to as simply the RASPT2 method hereafter. The state average procedure for the lowest four states, e.g., the

yi = 1 −

2Ti 1+

Ti2

, where Ti =

nHONO − i − nLUNO + i 2

(3)

Ti represents the overlap integral between the corresponding orbitals, and nHONO−i and nLUNO+i indicate the occupation numbers of the highest occupied natural orbital (HONO)−i (i = 0, 1, ...) and of the LUNO+i, respectively. In this study, the occupation numbers in eq 3 are calculated at the spinunrestricted Hartree−Fock (UHF) level of theory, which are found to be in good agreement with those at the highly electron-correlated level of theory (see the Supporting Information in ref 57). In order to quantify the relationship between the diradical character and the relative contribution of the resonance structure, we employ the nucleus-independent chemical shift (NICS(1)zz)58 value as an index for aromaticity, which is calculated by the spin-unrestricted DFT method with the long-range-corrected BLYP functional (LC-(U)BLYP with range-separating parameter μ was set to 0.33 bohr−1 59−61) using the gauge-independent atomic orbitals (GIAO) approach.62 It has been found that there is a correlation between NICS(1)zz and diradical character for open-shell singlet condensed-ring molecules34 (see the Supporting Information for details). The exchange integrals, which are used in Section 3.4, are evaluated in the same manner as that in ref 34 except for using the restricted Hartree−Fock MOs in this study. The geometry optimization is performed with the 6-311G* basis set, while all the single-point calculations are done with the 6-31G* basis set. The above calculations were performed by several quantum chemistry packages, i.e., GAMESS201263 for the SFBHHLYP, Q-Chem 4.164 for the SF-PBE50, Molpro2010.165 for the RASPT2, and Gaussian0966 for UHF and NICS(1)zz calculations.

3. RESULTS AND DISCUSSION 3.1. Relationship between the Diradical Character and the Geometry of Heteroacenes (Nm-i). The SFBHHLYP calculation shows that all of the model molecules (Nm-i) have the open-shell singlet ground state (Table S1 in the Supporting Information), which is indeed supported partly by the experimental facts that N0-3 and N1-3 have singlet ground states. This implies that the relative contribution of zwitterionic and diradical resonance forms in the ground state of these molecules is predicted to be characterized by the diradical character (Figure 1). We here discuss the relationship between the diradical character and the geometry, i.e., the C−C bond lengths R1 and R2 (see Figure 2), which are related to the relative contribution C


Article


diradical character, we examine the aromaticity using the NICS(1)zz value in the next section. 3.2. Relationship between the Diradical Character and the Aromaticity. The aromaticity of these molecules is evaluated by using the NICS(1)zz value in relation to the decrease of the diradical character y0 with increasing the number of N atoms (m) substituted especially into the threering model molecules, where we cannot find any clear correlations between the diradical character and the C−C bond length R1 with increasing m. The NICS(1)zz value is defined by the magnetic shielding tensor σzz 1 Å above the ring plane of a molecule, where the z-axis is perpendicular to the ring plane. It is known that strongly aromatic molecules show a negative NICS(1)zz value of large magnitudes, e.g., −29.1 ppm for benzene at the LC-BLYP/6-31G*//B3LYP/6-311G* level of theory, while strongly antiaromatic molecules show positive NICS(1)zz values with large amplitudes. Note here that we focus on the local aromaticity of the model molecules in relation to the resonance structures (Figure 1) though all of the model molecules are formally 4nπ antiaromatic systems. All NICS(1)zz values in Table 1 are calculated at 1 Å above the center of the central ring in the three-ring systems (i = 3) and of the second ring from the terminal ring in the four-ring systems (i = 4). Although we performed spin-unrestricted calculations for evaluating NICS(1)zz, broken-symmetry solutions for N1-3 and N2-3 were found to reduce to the spin-restricted solutions. Both the NICS(1)zz values obtained from the spin-restricted and broken-symmetry wavefunctions are found to give the same tendency: with increasing m, the NICS(1)zz values increase for both i = 3 and 4 molecules. On the other hand, in the molecules with the same m, the NICS(1)zz values are found to decrease with increasing i from 3 to 4. As seen from Table 1, the diradical character y0 increases when the NICS(1)zz value decreases. This relationship clearly shows that the aromaticity of the central ring(s) is closely correlated to the diradical character through the resonance structures shown in Figure 1. 3.3. Energy Level Matching Conditions. We now examine the energy level matching conditions, eqs 1 and 2, for the model molecules as well as their relationships with the diradical characters. As seen from Table 1, weak/intermediate diradical molecules N0-3 ((y0, y1) = (0.296, 0.002)) and N2-4 ((y0, y1) = (0.228, 0.005)) are expected to be good candidates for singlet fission.30−33 On the other hand, although N0-4 and N1-4 are also expected to satisfy these conditions, their relatively large diradical characters (y0 = 0.620 and 0.464, respectively) are predicted to cause small E(T1) as compared to N0-3 and N2-4. In order to confirm our prediction based on the diradical character, we evaluate the vertical excitation energies, E(S1), E(T1), and E(T2), of these molecules. We have to employ the calculation method so as to treat both static and dynamical electron correlations in a balanced way since the model molecules have non-negligible diradical characters, the feature of which implies the importance of static electron correlation, and their typical singlet π−π* excited states could be significantly affected by dynamical electron correlation, i.e., σ−π polarization effect67−69 and contraction of p-orbitals.69 Here, we discuss an appropriate choice of the active space for a balanced description of excited states using the RASPT2 method. Comparison of E(T1) values calculated using the SFPBE50 and the RASPT2 methods with two types of active spaces is shown in Table S3 in the Supporting Information. From Table S3 (Supporting Information), we found

Figure 2. C−C bond length indices (R1, R2) in Nm-3 (a) and Nm-4 (b), where C atoms and bond structure are omitted for simplicity.

of zwitterionic and diradical resonance forms: the longer the bond length R1 and/or R2, the more the zwitterionic, and vice versa (see Figure 1). Table 1 shows the diradical characters y0 Table 1. Diradical Character yi [-], the C−C Bond Lengths [Å], and the NICS(1)zz [ppm] Value molecule

y0

y1

R1

R2

N0-3 N1-3 N2-3 N0-4 N1-4 N2-4

0.296 0.163 0.112 0.620 0.464 0.228

0.002 0.002 0.002 0.021 0.014 0.005

1.447 1.454 1.453 1.445 1.452 1.447

− − − 1.453 1.468 1.484

NICSzz(1) 1.2 1.6 9.9 −8.3 −3.4 −0.2

(−3.0)a (−)a (−)a (−15.8)a (−9.2)a (−0.8)a

a

Values in parentheses are obtained from the broken-symmetry calculations (LC-UBLYP/6-31G*).

and y1 and the C−C bond lengths R1 and R2 for these molecules at their ground-state geometries. The smallest diradical character is observed in N2-3 (y0 = 0.112) and the largest one in N0-4 (y0 = 0.620). The y0 and y1 values are shown to increase (decrease) with increasing i with keeping m constant (increasing m with keeping i constant). In all the model molecules, y1 is found to be negligible ( y0(N1-4, R2 = 1.468 Å) > y0(N2-4, R2 = 1.484 Å) (see Figure 1). In order to verify this relationship between the aromaticity and the D


Article

The Journal of Physical Chemistry C qualitatively the same result for change in m or i between SFPBE50 and RASPT2 calculations: E(T1) increases (decreases) with increasing m with keeping i constant (increasing i with keeping m constant). The relative magnitude of E(T1) between N0-3 and N2-4 is inversed between the SF-PBE50 and RASPT2 results. This indicates that it is difficult to reproduce the relative excitation energies for molecules with different i and m within the accuracy of ∼0.1−0.2 eV in the present RASPT2 or SF-PBE50 calculation. Nevertheless, the comparison of RASPT2 calculation results between the active space (16, 14)/(0, 0)//(4, 4), which is possible only for i = 3 molecules, and (16, 14)/(6, 6)//(1, 1) confirms the validity of our active space choice of (16, 14)/(6, 6)//(1, 1). Thus, we predict that the RASPT2 calculations with active space (16, 14)/(6, 6)//(1, 1) for i = 3 and (20, 18)/(6, 6)//(1, 1) for i = 4, at least, qualitatively reproduce the effect of chemical modification of the model molecules except for the above case (the comparison between N0-3 and N2-4). As seen from Tables 1 and 2 and Figure 3, E(T1) rapidly decreases with increasing y0. In other words, E(T1) is well

described as a monotonical decreasing function of y0 within a series of molecules with the same m or i. The relationship between y0 and E(T1) was well investigated in the analytical model system26,27 and real molecular systems,31−33 which are found to be in good agreement with the present results. Tables 1 and 2 and Figure 3 also show the decrease of E(S1) with increasing y0, the feature of which is in agreement with that in the previous results.26,27,30−33 Note that the gradient of 2E(T1) as a function of y0 is slightly steeper than that of E(S1) in our model molecules, the tendency of which is the same as that in our previous studies.30−33 As a result, by tuning diradical character y0 in the weak/intermediate region through the chemical modifications shown in this study, it should be possible to present potential candidate molecules for satisfying 2E(T1) − E(S1) ∼ 0 or 0.2 as well as negligible y1 are found to satisfy eq 2. As a result, N0-3 and N2-4 are found to satisfy both the energy level matching conditions (eqs 1 and 2). Starting from N0-3, two chemical modifications which have opposite effects on the diradical character, i.e., increasing the number of N atoms (m) and extending the π-conjugation (increasing i), turn out to keep the diradical character of N2-4 in the weak/ intermediate region. 3.4. Classification of Singlet Fission Molecules Based on Characteristics of the First Excited Singlet State. We next discuss the classification of the singlet fission based on the wavefunction nature of the first excited singlet state S1. Smith and Michl classified chromophores into three classes I, II, and III based on the wavefunction nature of the S1 state in a singlet fission study:25 Class I indicates the molecules whose S1 state is

Table 2. Vertical Excitation Energies [eV] Calculated Using the RASPT2 Method molecule

E(S1)

E(T1)

E(T2)

N0-3 N1-3 N2-3 N0-4 N1-4 N2-4

2.41 2.57 2.71 1.75 2.13 2.60

1.10 1.36 1.56 0.48 0.80 1.08

2.51 2.66 2.79 1.74 2.21 2.75

Figure 3. Singlet [E(S1)] and triplet [E(T1)] excitation energies of heteroacene models. Red and blue circles represent three- (i = 3) and four- (i = 4) ring molecules, respectively. The diradical character y0 values are shown in parentheses. In the colored region, the energy level matching condition (eq 1) is satisfied. Darker (lighter) region shows the smaller (larger) energy-loss |2E(T1) − E(S1)| region. Two promising candidates, N0-3 and N2-4, which have 0.2 < y0 < 0.4, for efficient singlet fission are surrounded by a dotted circle.

E


Article


Figure 4. Schematic picture of the energy splitting between singlet and triplet states due to the exchange integral 2Kij. The left-hand side shows single electron excitation H (HOMO) → L (LUMO), while the right-hand side H → L + 1. The brown dotted arrows represent single electron excitations in the Hückel picture without exchange interaction, while the purple two-way arrows represent the exchange energy splitting 2Kij. G, S, and T indicate the singlet ground, singlet excited, and triplet excited states, respectively. Two typical cases are shown: (a) the molecule has a small H−L gap and moderate amplitudes of 2KHL and 2KH,L+1, and (b) the molecule has a much smaller L − L + 1 gap than that of (a) and much larger 2KHL than 2KH,L+1.

(= [(H(α) → L(α)) ± (H(β) → L(β))]/√2) single excitation configuration state function (CSF), where the plus/minus sign represents triplet/singlet configuration. The energy splitting between the triplet and singlet CSFs is 2KHL, and the situation is the same for any pairs of orbitals. Thus, whether the lowest singlet excited state S1 is dominated by H → L excitation or others depends on both the orbital energy and the amplitude of the exchange interaction. Two typical situations of the energy diagram are shown in Figure 4: case (a) has a small H−L gap and moderate amplitudes of 2KHL and 2KH,L+1, where S1 is dominated by H → L excited CSF, while case (b) has a much smaller L − L + 1 gap and much larger 2KHL than 2KH,L+1, where S1 is dominated by H → L + 1 rather than H → L. We next discuss which type of models shown above well describe the electronic structure of heteroacene molecules. First, we examine the wavefunction nature of the first excited singlet state S1. It is found that all Nm-3 molecules have the largest weight of H → L + 1 (54−63%) single excitation configuration in the S1 wavefunction and that the second largest weight of H → L (9−18%) excitation configuration is not negligible. Since this indicates significant interaction between these two configurations, Nm-3 is shown to approximately belong to Class II. The situation is more complicated in Nm-4 molecules: S1 state wavefunctions are primarily described by the superposition of H → L (27%) and H − 1 → L (38%) in N0-4,

primarily described by single excitation configuration H → L, Class II whose S1 state is by single excitation configuration H − 1 → L and/or H → L + 1, and Class III whose S1 state is by double excitation H,H → L,L. This classification is related to the optimal conformation of molecules for maximizing the electronic coupling primarily contributing to singlet fission,25 leading to the different singlet fission dynamics between different classes of molecules with similar chemical structure. As shown below, the different nature of the S1 wavefunction in the model molecules is closely related to different values of orbital energy difference, Δεij = εi − εj, and of twice the exchange integral, 2Kij, where i and j indicate H − 1, H, L, and L + 1 and the pair indices are represented by (i, j) = (H, L), (H, L + 1), and (H − 1, L). In a typical approximation of the π−π* excited state, its wavefunction is described by single excitation configuration H → L since ΔεHL is the smallest energy difference in the whole orbital energy differences. This simple assumption is known to be sometimes invalid, e.g., in the case of a large difference in 2Kij. In nonrelativistic quantum chemistry, an eigenfunction of the electronic Hamiltonian must also be an eigenfunction of the spin-angular momentum operator, S2. Since the H(σ) → L(σ) singly excited determinant (where σ in parentheses represents α/β spin) itself is not an eigenfunction of S2, we have to consider spin-adapted configurations rather than determinants, such as the H → L F


Article


differences, Δεij, are found to increase with increasing m (see Figure S3 in the Supporting Information). Note that ΔεH,L+1 is smaller than ΔεH−1,L in i = 3, while the situation is inversed in i = 4, where ΔεH,L+1 is larger than ΔεH−1,L. This is due to the lack of orbital energy stabilization in L + 1 caused by the πconjugation extension, though L is stabilized, while H − 1 and H are destabilized, since L + 1 has almost no nearest-neighbor orbital interaction, which leads to smaller orbital energy change than other MOs with increasing the π-conjugation. Finally, we discuss the relationship between wavefunction nature of S1, 2Kij, and Δεij. The S1 state of the model molecules with i = 3 is mainly described by H → L + 1 CSF because of the much smaller 2KH,L+1 than 2KHL (see Table 3). In addition, 2KH,L+1 is shown to be smaller than 2KH−1,L, and ΔεH,L+1 is shown to be smaller than ΔεH−1,L (see Figure S3, Supporting Information). This feature further confirms that the model molecules with i = 3 belong to Class II dominated by H → L + 1. In i = 4 model molecules, the large amplitude of 2KHL promotes significant mixing of H → L + 1 and H − 1 → L CSFs in the low-lying singlet excited states as in i = 3 model molecules (see Table 3). In the S1 state of N0-4, due to the smaller ΔεH−1,L than ΔεH,L+1 (see Figure S3, Supporting Information), H − 1 → L configuration tends to interact with H → L more than H → L + 1, which leads to a small contribution of H → L + 1. In N1-4 and N2-4, with increasing m, the decrease in the difference between ΔεH‑1,L and ΔεH,L+1 (see Figure S3, Supporting Information) promotes significant mixing not only with H − 1 → L but also with H → L + 1. On the other hand, the larger contribution of H → L in N2-4 than in N1-4 (see Table S5, Supporting Information) is predicted to come from the difference ΔεH−1,L − ΔεH,L between these two molecules (see Figure S3, Supporting Information).

H → L + 1 (20%) and H − 1 → L (44%) in N1-4, and H → L (46%) and H → L + 1 (21%) in N2-4. It is therefore found that N1-4 belongs to Class II, while N0-4 and N2-4 might belong to Class II and I, respectively, despite there remains uncertainty of classification. Second, we examine twice the exchange integral 2Kij (Table 3), which is closely related to the classification25 of the singlet Table 3. Exchange Integral 2Kij [eV] between MOs i and j, i.e., (i, j) = (H, L), (H, L + 1), and (H − 1, L) molecule

(H, L)

(H, L + 1)

(H − 1, L)

N0-3 N1-3 N2-3 N0-4 N1-4 N2-4

2.98 2.76 2.59 2.90 2.58 2.27

0.36 0.41 0.43 0.28 0.31 0.34

1.12 1.28 1.42 0.93 0.99 1.08

fission as explained below. As seen from Table 3, 2KHL lies in the range 2.2−3.0 eV, while 2KH,L+1 is much smaller (0.2−0.5 eV) than 2KHL. The significantly large 2KHL as compared to 2KH,L+1 of all the model molecules originates from the much larger overlap between the HOMO and LUMO than that between the HOMO and LUMO+1 because the HOMO has significant amplitudes on the nodal sites of LUMO+1 (see Figure S1 in the Supporting Information). This feature of MO distributions may be understood by the Coulson−Rushbrooke paring theorem70 in alternant hydrocarbons, though the model molecules have 4nπ electrons. The large difference between 2KHL and 2KH,L+1 is shown to play an important role for characterizing the wavefunction of the S1 state as discussed below. The 2KHL is found to be larger in the case of smaller i with keeping m constant and of smaller m with keeping i constant as shown in Table 3. The former feature is understood as the result of decreasing the π-electron delocalization by reducing the π-conjugation size, while the latter one is understood as follows. The effect of the N atom substitution clearly appears in the HOMO, not in the LUMO (see Figure S1 in the Supporting Information). The HOMO distribution of nonsubstituted molecules has larger amplitudes on sites X1/X2 than those on radical (N•) sites. N atom(s) on sites X1/X2 are predicted to interact with other N atoms more strongly in the substituted model molecules than in nonsubstituted molecules (X1/X2 = CH) because of the similar orbital energy level of the central region in the substituted models to that of the side rings involving N atoms. It is predicted that this significant interaction between N atoms in the side rings and those on X1/X2 sites decreases the HOMO distribution on X1/X2 sites and increases that on the neighboring two radical N• sites. Third, we examine orbital energies εi and their differences Δεij of the model molecules. With increasing m for each i = 3 or 4 molecules, orbital energies εi are found to decrease except for εL (see Figure S2 in the Supporting Information). This stabilization of the orbital energies with increasing m, especially in H and H − 1, is predicted to mainly come from the same origin of decreasing exchange integral, i.e., π-orbital delocalization induced by N atom substitution, as discussed in the above paragraph. With increasing i from 3 to 4 with keeping m constant, π-conjugation extension is turned out to destabilize the orbital energies εH−1 and εH and to stabilize εL, while not to affect εL+1. From these observations, the orbital energy

4. CONCLUSION In this paper, we have investigated a theoretical molecular design for efficient singlet fission using heteroacene models Nm-i by tuning the diradical characters so as to satisfy the energy level matching conditions (eqs 1 and 2). The tuning of the diradical character into the weak/intermediate region is found to be achieved by chemical modifications, i.e., substitution of N atoms and/or π-conjugation extension. The analysis of C−C bond length and NICS(1)zz value showed that the aromaticity of the central ring(s) of heteroacenes is correlated to the change in the diradical character through the relative contribution of the zwitterionic and diradical resonance structures. Two promising candidates, N0-3 and N2-4, are expected to exhibit singlet fission exothermically (2E(T1) − E(S1) < 0) with E(T1) = 1.10 and 1.08 eV, respectively, which are known to be an optimal energy gap for OPV application of singlet fission. It is also found that these two candidates, N0-3 and N2-4, belong to Class II and I, respectively, while there remains ambiguity for the classification of the latter molecule due to the plural configuration contributions in the S1 excited state. From the detailed analysis, the exchange integrals between frontier orbitals and orbital energy differences are clarified to play important roles for the classification of singlet fission molecules. This result is useful for designing a different class of singlet fission molecules with similar structure, e.g., similar HOMO and LUMO distributions, as shown in the proposed two candidates, which are predicted to cause the difference in the amplitude of the electronic coupling primarily contributing to singlet fission and thus to exhibit different singlet fission dynamics. G


Article


■

(9) Burdett, J. J.; Berdeen, C. J. The Dynamics of Singlet Fission in Crystalline Tetracene and Covalent Analogues. Acc. Chem. Res. 2013, 46, 1312−1320. (10) Zimmerman, P. M.; Bell, F.; Casanova, D.; Head-Gordon, M. Mechanism for Singlet Fission in Pentacene and Tetracene: From Singlet Exciton to Two Triplets. J. Am. Chem. Soc. 2011, 133, 19944− 19952. (11) Chan, W.-L.; Tritsch, J. R.; Zhu, X. Y. Harvesting Singlet Fission for Solar Energy Conversions: One- versus Two-Electron Transfer from the Quantum Mechanical Superposition. J. Am. Chem. Soc. 2012, 134, 18295−18302. (12) Chan, W.-L.; Berkelbach, T. C.; Provorse, M. R.; Monahan, N. R.; Tritsch, H.; Hybertsen, M.; Reichman, D.; Gao, J.; Zhu, X. Y. The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory. Acc. Chem. Res. 2013, 46, 1321−1329. (13) Tayabjee, M. J. Y.; Clady, R. G. C. R.; Schmidt, T. W. The Exciton Dynamics in Tetracene Thin Films. Phys. Chem. Chem. Phys. 2013, 15, 14797−14805. (14) Ma, L.; Zhang, K.; Kloc, C.; Sun, H.; Soci, C.; Michel-Beyerle, M. E.; Gurzadyan, G. G. Fluorescence from Rubrene Singlet Crystals: Interplay of Singlet Fission and Energy Trapping. Phys. Rev. B 2013, 87, 201203/1−201203/5. (15) Ma, L.; Galstyan, G.; Zhang, K.; Kloc, C.; Sun, H.; Soci, C.; Michel-Beyerle, M. E.; Gurzadyan, G. G. Two-Photon-Induced Singlet Fission in Rubrene Singlet Crystal. J. Chem. Phys. 2013, 138, 184508/ 1−184508/6. (16) Piland, G. B.; Burdett, J. J.; Kurunthu, D.; Bardeen, C. J. Magnetic Field Effects on Singlet Fission and Fluorescence Decay Dynamics in Amorphous Rubrene. J. Phys. Chem. C 2013, 117, 1224− 1236. (17) Paci, I.; Johnson, J. C.; Chen, X.; Rana, G.; Popović, D.; Cavid, D. E.; Nozik, A. J.; Ratner, M. A.; Michl, J. Singlet Fission for DyeSensitized Solar Cells: Can a Suitable Sensitizer Be Found? J. Am. Chem. Soc. 2006, 128, 16546−16553. (18) Johnson, J. C.; Nozik, A. J.; Michl, J. High Triplet Yield from Singlet Fission in a Thin Film of 1,3-Diphenylisobenzofuran. J. Am. Chem. Soc. 2010, 132, 16302−16303. (19) Schwerin, A. F.; Johnson, J. C.; Smith, M. B.; Smith, P. S.; Popović, D.; Č erný, J.; Havlas, Z.; Paci, I.; Akdag, A.; MacLeod, M. K.; et al. Toward Designed Singlet Fission: Electronic States and Photophysics of 1,3-Diphenylisobenzofuran. J. Phys. Chem. A 2010, 114, 1457−1473. (20) Musser, A. J.; Al-Hashimi, M.; Maiuri, M.; Brida, D.; Heeney, M.; Cerullo, G.; Friend, R. H.; Clark, J. Activated Singlet Exciton Fission in a Semiconducting Polymer. J. Am. Chem. Soc. 2013, 135, 12747−12754. (21) Eaton, S. W.; Shoer, L. E.; Karlen, S. D.; Dyar, S. M.; Margulies, E. A.; Veldkamp, B. S.; Ramanan, C.; Hartzler, D. A.; Savikhin, S.; Marks, T. J.; et al. Singlet Exciton Fission in Polycrystalline Thin Films of a Slip-Stacked Perylenediimide. J. Am. Chem. Soc. 2013, 135, 14701−14712. (22) Lefler, K. M.; Brown, K. E.; Salamant, W. A.; Dyar, S. M.; Knowles, K. E.; Wasielewski, M. R. Triplet State Formation in Photoexcited Slip-Stacked Pelylene-3,4−9,10-bis(dicarboximide) Dimers on a Xanthene Scoffold. J. Phys. Chem. A 2013, 117, 10333−10345. (23) Roberts, S. T.; McAnally, R. E.; Mastron, J. N.; Webber, D. H.; Whited, M. T.; Brutchey, R. L.; Thompson, M. E.; Bradforth, S. E. Efficient Singlet Fission Discovered in a Disordered Acene Film. J. Am. Chem. Soc. 2012, 134, 6388−6400. (24) Dillon, R. J.; Piland, G. B.; Bardeen, C. J. Different Rates of Singlet Fission in Monoclinic versus Orthorhombic Crystal Froms of Diphenylhexatriene. J. Am. Chem. Soc. 2013, 135, 17278−17281. (25) Smith, M. B.; Michl, J. Singlet Fission. Chem. Rev. 2010, 110, 6891−6936. (26) Nakano, M.; Kishi, R.; Ohta, S.; Takahashi, H.; Kubo, T.; Kamada, K.; Ohta, K.; Botek, E.; Champagne, B. Relationship between Third-Order Nonlinear Optical Properties and Magnetic Interactions

ASSOCIATED CONTENT

S Supporting Information *

Total energies at the equilibrium geometries of singlet and triplet states at the SF-BHHLYP/6-311G* level of theory. Comparison of excitation energies E(S1) and E(T1) calculated by the RASPT2 and SF-PBE50 methods. Excitation energy E(T2) calculated by the RASPT2 method. CI coefficients of S1, S2, T1, and T2 states calculated by the RASSCF method. Molecular orbitals at the RHF/6-31G* level of theory. Orbital energies and orbital energy gaps at the RHF/6-31G* level of theory. Relationships between aromaticity, NICS(1)zz, and diradical character. Molecular geometries optimized at the SFBHHLYP/6-311G* level of theory are given by text files “NmiX.xyz”, where X = s(singlet) or t(triplet). This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*Fax: +81-6-6850-6268. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported by JSPS Research Fellowship for Young Scientists (No. 02604505), a Grant-in-Aid for Scientific Research (No. 25248007) from JSPS, a Grant-in-Aid for Scientific Research on Innovative Areas (Nos. A24109002a and A26107004a), MEXT, the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan. Theoretical calculations are partly performed using Research Center for Computational Science, Okazaki, Japan.

■

REFERENCES

(1) Nozik, A. J.; Ellingson, R. J.; Micic, O. I.; Blackburn, J. L.; Yu, P.; Murphy, J. E.; Beard, M. C.; Rumbles, G. Proceedings of the 27th DOE Solar Photochemistry Research Conference; Airline Conference Center, Warrenton, VA, June 6−9, 2004; pp 63−66. (2) Michl, J.; Chen, X.; Rana, G.; Popović, D. B.; Downing, J.; Nozik, A. J.; Johnson, J. C.; Ratner, M. A.; Paci, I. Book of Abstracts, DOE Solar Program Review Meetings, Denver, CO, October 24−28, 2004; U.S. Department of Energy: Washington, DC, 2004; p 5. (3) Hanna, M. C.; Nozik, A. J. Solar Conversion Efficiency of Photovoltaic and Photoelectrolysis Cells with Charge Multiplication Absorbers. J. Appl. Phys. 2006, 100, 074510/1−074510/8. (4) Wilson, M. W. B.; Rao, A.; Ehrler, B.; Friend, R. H. Singlet Exciton Fission in Polycrystalline Pentacene: From Photophysics toward Devices. Acc. Chem. Res. 2013, 46, 1330−1338. (5) Rao, A.; Wilson, M. W. B.; Albert-Seifried, S.; Pietro, R. D.; Friend, R. H. Photophysics of Pentacene Thin Films: The Role of Exciton Fission and Heating Effects. Phys. Rev. B 2011, 84, 195411/1− 195411/8. (6) Wilson, M. W. B.; Rao, A.; Clark, J.; Kumar, S. S.; Brida, D.; Cerullo, G.; Friend, R. H. Ultrafast Dynamics of Exciton Fission in Polycrystalline Pentacene. J. Am. Chem. Soc. 2011, 133, 11830−11833. (7) Thorsmølle, V. K.; Averitt, R. D.; Demsar, J.; Smith, D. L.; Tretiak, S.; Martin, R. L.; Chi, X.; Crone, B. K.; Ramirez, A. P.; Taylar, A. J. Morphology Effectively Controls Singlet-Triplet Excitation Relaxastion and Charge Transport in Organic Semiconductors. Phys. Rev. Lett. 2009, 102, 017401/1−017401/4. (8) Marciniak, H.; Pugliesi, I.; Nickel, B.; Lochbrunner, S. Ultrafast Singlet and Triplet Dynamics in Microcrystalline Pentacene Films. Phys. Rev. B 2009, 79, 235318/1−235318/8. H


Article

The Journal of Physical Chemistry C in Open-shell Systems: A New Paradigm for Nonlinear Optics. Phys. Rev. Lett. 2007, 99, 033001/1−033001/4. (27) Nakano, M. Excitation Energies and Properties of Open-Shell Singlet Molecules: Applications to a New Class of Molecules for Nonlinear Optics and Singlet Fission; Springer: Berlin, 2014. (28) Yamaguchi, K. The Electronic Structures of Biradicals in the Unrestricted Hartree-Fock Approximation. Chem. Phys. Lett. 1975, 33, 330−335. (29) Yamaguchi, K. In Self-Consistent Field: Theory and Applications; Carbo, R., Klobukowski, M., Eds.; Elsevier: Amsterdam, The Netherlands, 1990; pp 727−828. (30) Minami, T.; Nakano, M. Diradical Character View of Singlet Fission. J. Phys. Chem. Lett. 2012, 3, 145−150. (31) Ito, S.; Minami, T.; Nakano, M. Diradical Character Based Design for Singlet Fission of Condensed-Ring Systems with 4nπ Electrons. J. Phys. Chem. C 2012, 116, 19729−19736. (32) Minami, T.; Ito, S.; Nakano, M. Theoretical Study of Singlet Fission in Oligorylenes. J. Phys. Chem. Lett. 2012, 3, 2719−2723. (33) Minami, T.; Ito, S.; Nakano, M. Fundamental of DiradicalCharacter-Based Molecular Design for Singlet Fission. J. Phys. Chem. Lett. 2013, 4, 2133−2137. (34) Motomura, S.; Nakano, M.; Fukui, H.; Yoneda, K.; Kubo, T.; Carlon, R.; Champagne, B. Size Dependences of the Diradical Character and the Second Hyperpolarizabilities in DicyclopentaFused Acenes: Relationships with Their Aromaticity/Antiaromaticity. Phys. Chem. Chem. Phys. 2011, 13, 20575−20583. (35) Inoue, Y.; Yamada, T.; Champagne, B.; Nakano, M. Equatorial Ligand Effects on the Diradical Character Dependence of the Second Hyperpolarizabilities of Open-Shell Singlet Transition-Metal Dinuclear Complexes. Chem. Phys. Lett. 2013, 570, 75−79. (36) Doughrty, D. A. Spin Control in Organic Molecules. Acc. Chem. Res. 1991, 24, 88−94. (37) Rajca, A. Organic Diradicals and Polyradicals: From Spin Coupling to Magnetism? Chem. Rev. 1994, 94, 871−893. (38) Abe, M. Diradicals. Chem. Rev. 2013, 113, 7011−7088. (39) Wright, B. B.; Platz, M. S. Electron Spin Resonance Spectroscopy of the Triplet State of m-Xylylene. J. Am. Chem. Soc. 1983, 105, 628−630. (40) Wenthold, P. G.; Kim, J. B.; Lineberger, W. C. Photoelectron Spectroscopy of m-Xylylene Anion. J. Am. Chem. Soc. 1997, 119, 1354−1359. (41) Wang, T.; Krylov, A. I. The Effect of Substituents on Electronic States’ Ordering in Meta-Xylylene Diradicals: Qualitative Insights from Quantitative Studies. J. Chem. Phys. 2005, 123, 104304/1−104304/6. (42) Hutchison, K.; Srdanov, G.; Hicks, R.; Yu, H.; Wudl, F. Tetraphenylhexaazaanthracene: A Case for Dominance of Cyanine Ion Stabilization Overwhelming 16π Antiaromaticity. J. Chem. Soc. 1998, 120, 2989−2990. (43) Langer, P.; Bodtke, A.; Saleh, N. N. R.; Görls, H.; Schreiner, P. R. 3,5,7,9-Tetraphenylhexaazaacridines: A Highly Stable, Weakly Antiaromatic Species with 16 π Electrons. Angew. Chem., Int. Ed. 2005, 44, 5255−5259. (44) Langer, P.; Amiri, S.; Bodtke, A.; Saleh, N. N. R.; Weisz, K.; Görls, H.; Schreiner, P. R. 3,5,7,9-Substituted Hexaazaacridines: Toward Structures with Nearly Degenerate Singlet-Triplet Energy Separations. J. Org. Chem. 2008, 73, 5048−5063. (45) Amiri, S.; Schreiner, P. R. Non-Kekulé N-substituted mPhenylenes: N-Centered Diradicals versus Zwitterions. J. Phys. Chem. A 2009, 113, 11750−11757. (46) Rajca, A.; Shitaishi, K.; Pink, M.; Rajca, S. Triplet (S = 1) Ground State Aminyl Diradical. J. Chem. Am. Soc. 2007, 129, 7232− 7233. (47) Nakano, M.; Kishi, R.; Takebe, A.; Nate, M.; Takahashi, H.; Kubo, T.; Kamada, K.; Ohta, K.; Champagne, B.; Botek, E. Second Hyperpolarizabilities (γ) of 1,3-Dipole Systems: Diradical Character Dependence of γ, CP963, Part A. In Computational Methods in Science and Engineering, Theory and Computation: Old Problems and New Challenges; Maroulis, G., Simos, T., Eds.; AIP: Melville, NY, 2007; pp 102−105.

(48) Nakano, M.; Nagai, H.; Fukui, H.; Yoneda, K.; Kishi, R.; Takahashi, H.; Shimizu, A.; Kubo, T.; Kamada, K.; Ohta, K.; et al. Theoretical Study of Third-Order Nonlinear Optical Properties in Square Nanographenes with Open-Shell Singlet Ground States. Chem. Phys. Lett. 2008, 467, 120−125. (49) Konishi, A.; Hirao, Y.; Matsumoto, K.; Kurata, H.; Kishi, R.; Shigeta, Y.; Nakano, M.; Tokunaga, K.; Kamada, K.; Kubo, T. Synthesis and Characterization of Quateranthene: Elucidating the Characteristics of the Edge State of Graphene Nanoribbons at the Molecular Level. J. Am. Chem. Soc. 2012, 135, 1430−1437. (50) Sun, Z.; Lee, S.; Park, K. H.; Zhu, X.; Zhang, W.; Zheng, B.; Hu, P.; Zeng, Z.; Das, S.; Li, Y.; et al. Dibenzoheptazethrene Isomers with Different Biradical Characters: An Exercise of Clar’s Aromatic Sextet Rule in Singlet Biradicaloids. J. Am. Chem. Soc. 2013, 135, 18229− 18236. (51) Shao, Y.; Head-Gordon, M.; Krylov, A. I. The Spin-Flip Approach with Time-Dependent Density Functional Theory: Theory and Applications to Diradicals. J. Chem. Phys. 2003, 118, 4807−4818. (52) Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. Phys. Rev. A 1988, 38, 3098−3100. (53) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formulas into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785−789. (54) Bernard, Y. A.; Shao, Y.; Krylov, A. I. General Formulation of Spin-Flip Time-Dependent Density Functional Theory Using NonCollinear Kernels: Theory, Implementation, and Benchmarks. J. Chem. Phys. 2012, 136, 204103/1−204103/17. (55) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (56) Celani, P.; Werner, H.-J. Multireference Perturbation Theory for Large Restricted and Selected Active Space Reference Wavefunctions. J. Chem. Phys. 2000, 112, 5546−5557. (57) Fukui, H.; Nakano, M.; Shigeta, Y.; Champagne, B. Origin of the Enhancement of the Second Hyperpolarizabilities in Open-Shell Singlet Transition-Metal Systems with Metal-Metal Multiple Bonds. J. Phys. Chem. Lett. 2011, 2, 2063−2066. (58) Fallar-Bagher-Shaidael, H.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; Schleyer, P. v. R. Which NICS Aromaticity Index for Planer π Rings Is Best? Org. Lett. 2006, 8, 863−866. (59) Tawada, Y.; Tsuneda, T.; Yanagisawa, S. A Long-RangeCorrected Time-Dependent Density Functional Theory. J. Chem. Phys. 2004, 120, 8425−8433. (60) Kishi, R.; Bonness, S.; Yoneda, K.; Takahashi, H.; Nakano, M.; Botek, E.; Champagne, B.; Kubo, T.; Kamada, K.; Ohta, K.; et al. Long-Range Corrected Density Functional Theory Study on Static Second Hyperpolarizabilities of Singlet Diradical Systems. J. Chem. Phys. 2010, 32, 094107/1−094107/11. (61) Yoneda, K.; Nakano, M.; Inoue, Y.; Inui, T.; Fukuda, K.; Shigeta, Y. Impact on Antidot Structure on the Multiradical Characters, Aromaticities, and Third-Order Nonlinear Optical Properties of Hexagonal Graphene Nanoflakes. J. Phys. Chem. C 2012, 116, 17787−17795. (62) Schreckenbach, G.; Ziegler, T. Calculation of NMR Shielding Tensors Using Gauge-Including Atomic Orbitals and Modern Density Functional Theory. J. Phys. Chem. 1995, 99, 606−611. (63) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J.; et al. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347−1363. (64) Shao, Y.; Fusti-Molnar, L.; Jung, Y.; Kussmann, J.; Ochsenfeld, C.; Brown, S. T.; Gilbert, A. T. B.; Slipchenko, L. V.; Levchenko, S. V.; O’Neill, D. P. et al., Q-CHEM, Version 4.0; Q-Chem. Inc.: Pittsburgh, PA, 2008. (65) Werner, H.-J.; Knowles, P. J.; Knizia G.; Manby, F. R.; Schütz, M.; Celani, P.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G. et al. MOLPRO, version 2010.1, a package of ab initio programs. (66) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, I


Article

The Journal of Physical Chemistry C B.; Petersson, G. A. et al. Gaussian 09, Revision A.1; Gaussian, Inc.: Wallingford, CT, 2009. (67) Hashimoto, T.; Nakano, H.; Hirao, K. Theoretical Study of the Valence π→π* Excited States of Polyacenes: Benzene and Naphthalene. J. Chem. Phys. 1996, 104, 6244−6258. (68) Kawashima, Y.; Hashimoto, T.; Nakano, H.; Hirao, K. Theoretical Study of the Valence π→π* Excited States of Polyacenes: Anthracene and Naphthacene. Theor. Chem. Acc. 1999, 102, 49−64. (69) Angeli, C. On the Nature of the π→π* Ionic Excited States: The V State of Ethene as a Prototype. J. Comput. Chem. 2009, 30, 1319− 1333. (70) Coulson, C. A.; O’Leary, B.; Mallion, R. B. Hückel Theory for Orgnic Chemists; Academic Press: London, U.K., 1978.

J


Theoretical Molecular Design of Heteroacenes for ... - ACS Publications

Recommend Documents