Supporting information on. Singlet Diradical Character from Experiment

Transcription

1 Supporting information on Singlet Diradical Character from Experiment Kenji Kamada,,* Koji Ohta, Akihiro Shimizu, Takashi Kubo,,* Ryohei Kishi, Hideaki Takahashi, Edith Botek, Benoît Champagne,,* and Masayoshi Nakano,* Photonics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Ikeda, Osaka , Japan Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka, Osaka , Japan Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka , Japan Laboratoire de Chimie Théorique (LCT), Facultés Universitaires Notre-Dame de la Paix (FUNDP), Rue de Bruxelles, 61, B-5000 Namur (Belgium) * Corresponding authors. for M.N.: mnaka@cheng.es.osaka-u.ac.jp 1

2 Contents 1. Derivation of Eq. (1) and electronic states (Figure 1S) as well as its illustration in the case of a twosite model like the stretched H 2 molecule. 2. Calculation formula of diradical character from the occupation numbers of spin-unrestricted Hartree- Fock natural orbitals (UNOs) using the 6-31G** basis set 3. Molecular structure (Figure 2S) and cartesian coordinates (Table 1S) of 1 optimized at the UB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** 4. Molecular structure (Figure 3S) and cartesian coordinates (Table 2S) of 2 optimized at the UB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** 5. Molecular structure (Figure 4S) and cartesian coordinates (Table 3S) of anthracene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** 6. Molecular structure (Figure 5S) and cartesian coordinates (Table 4S) of naphthalene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** 7. Molecular structure (Figure 6S) and cartesian coordinates (Table 5S) of chrysene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** 8. Molecular structure (Figure 7S) and cartesian coordinates (Table 6S) of fluorene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** 9. Molecular structure (Figure 8S) and cartesian coordinates (Table 7S) of TIPS-pentacene using the geometry, isopropyl groups of which are replaced by H atoms, derived from the crystal structure (Ref. 31) and the occupation numbers of frontier UNOs/6-31G**. 10. All the calculations in 3-9 are carried out by Gaussian 03. The reference is given. 2

3 1. Derivation of Eq. (1). We briefly provide the derivation of Eq. (1) according to Refs. [21] and [24], which employs a twosite model A B with two electrons in two orbitals as the simplest example of symmetric diradical molecules. The magnetic orbitals can be either symmetry-adapted, g and u, or localized natural orbitals (LNO), a and b, which are related through a(x) 1 2 [ g(x)+ u(x) ], and b(x) 1 [ g(x) u(x) ]. (s1.1) 2 For M S = 0 (singlet and triplet), using LNOs there are two neutral, ab and b a, and two ionic, aa and bb, determinants, where the upper-bar (non-bar) indicates the β (α) spin. The configuration interaction (CI) matrix in the LNO representation { ab, ba, aa, bb } takes the form 0 K ab t ab t ab K ab 0 t ab t ab, (s1.2) t ab t ab U K ab t ab t ab K ab U where the energy of the neutral valence bond (VB) determinants is taken as the energy origin. The U( U aa U ab ) indicates the difference between on-site and inter-site Coulomb integrals, K ab ( ab 1 r 12 ba 0) is a direct exchange integral, and t ab ( a F b ) is a transfer integral, where F is the Fock operator. The diagonalization of this matrix gives the four electronic states: (I) an essentially neutral lowest-energy singlet state of g symmetry S 1g = κ( ab + ba )+ η( aa + bb ) (κ > η > 0), (s1.3) with energy 1 E 1g = K ab + U U t ab 2, (s1.4) (II) an ionic singlet state with u symmetry S 1u = ( aa bb ) 2 (s1.5) with energy 1 E 1u =U K ab, (s1.6) (III) an essentially ionic higher-energy singlet state of g symmetry S 2g = η( ab + ba )+κ( aa + bb ) (κ > η > 0) (s1.7) with energy 1 E 2g = K ab + U + U t ab 2, (s1.8) 3

4 and (IV) a neutral triplet state with u symmetry with energy T 1u ( = ( ab ba ) 2) (s1.9) 3 E 1u = K ab. (s1.10) Here, and κ = U U t ab, (s1.11) η = 2 t ab U + U 2 2 ( +16t ab ) U t ab. (s1.12) On the other hand, the singlet ground state is also described using the symmetry-adapted MOs, g and u, [see Eq. (s1.1)]: S 1g = ξ gg ζ uu, (s1.13) where we get the following relation from Eqs. (s1.3), (s1.11), (s1.12) and (s1.13): ξ = κ +η = U U t ab + 4 t ab U + U 2 2 ( +16t ab ) U t ab (s1.14) and ζ = κ η = U U t ab 4 t ab U + U 2 2 ( +16t ab ) U t ab. (s1.15) Since the diradical character (y) is defined as the twice the weight of double excitation configuration in the ground state (s1.13), y 2ζ 2 =1 4 t ab U t ab. (s1.16) From Eqs. (s1.4), (s1.6), (s1.8) and (s1.10), we obtain the relations between parameters and observables: t ab = 1 4 ( 1 E 2g 1 E 1g ) 2 ( 1 E 1u 3 E 1u ) 2, (s1.17) 4

5 and U= 1 E 1u 3 E 1u. (s1.18) Substituting (s1.17) and (s1.18) into (s1.16), we obtain 1 E 1u 3 E 1u y =1 1 1 E 2g 1 E 1g 2 =1 1 E 2 S 1u,S 1g E T1u,S 1g, (s1.19) E S2g,S 1g where E S1u,S 1g, E S2g,S 1g and E T1u,S 1g represent the excitation energies of higher singlet state with g symmetry (two-photon allowed excited state), of lower singlet state with u symmetry (one-photon allowed excited state), and of triplet state with u symmetry, respectively (see Figure S1). S 2g ( 1 E 2g ) S 1u ( 1 E 1u ) E S 2g, S 1g E S1u, S 1g T 1u ( 3 E 1u ) E T1u, S 1g S 1g ( 1 E 1g ) Figure 1S. Four electronic states for a symmetric two-site model with two electrons in two orbitals. We now consider the right-hand side of Eq. (s1.19) for the H 2 molecule (this is also valid for any two-site model). From Fig. 1S and Eq. (s1.18), in the case of small U (corresponding to the H 2 model near its equilibrium geometry), 1 E 1u approaches 3 E 1u, i.e., the parentheses inside the square root become small (<< 1), which leads to y 0. On the contrary, in case of large U (U >> t ab ) (corresponding to a stretched H 2 model), from Eqs. (s1.4), (s1.6), (s1.8) and (s1.10), the relations become: 1 E 2g U + K ab, 1 E 1u U K ab, 1 E 1g K ab and 3 E 1u = K ab, so that we get the relation: 1 E 2g 1 E 1g 1 E 1u 3 E 1u, leading to y 1. More details on the H 2 dissociation model are available in the following reference: Nakano, M.; Kishi, R.; Ohta, S.; Takebe, A.; Takahashi, H.; Furukawa, S.; Kubo, T.; Morita, Y.; Nakasuji, K.; Yamaguchi, K. et al. Origin of the Enhancement of the Second Hyperpolarizability of Singlet Diradical Systems with Intermediate Diradical Character, J. Chem. Phys. 2006, 125,

6 2. Calculation formula of diradical character from the occupation numbers of spin-unrestricted Hartree-Fock natural orbitals (UNOs) using the 6-31G** basis set The diradical character y i related to HOMO-i and LUMO+i is defined by the weight of the doublyexcited configuration in the multi-configurational (MC)-SCF theory and is formally expressed in the spinprojected UHF (PUHF) theory as [23] y i =1 2T i 1+T i 2, (s2.1) where T i is the orbital overlap between the corresponding orbital pairs [23] ( χ HOMO i and η HOMO i ) and can also be represented using the occupation numbers ( n i ) of UHF natural orbitals (UNOs): T i = n HOMO i n LUMO+i 2, (s2.2) The diradical character y i obtained from the UNO occupation numbers takes a value between 0 and 1, which correspond to the closed-shell and pure diradical systems, respectively. The present calculation scheme using the UNOs is the simplest but it can well reproduce the diradical character calculated by other methods such as the ab initio configuration interaction (CI) method [S1]. It is noted that the present formula employs the UHF NOs and not the UDFT NOs, which would lead to incorrect lower diradical character in the present formula. Reference (S1) D. Herebian, K. E. Wieghardt, F. Neese, J. Am. Chem. Soc. 2003, 125,

7 3. Molecular structure and cartesian coordinates of 1 Top view Figure 2S. Molecular structure of 1 7

8 Table 1S: Cartesian coordinates [Å] of 1 optimized at the UB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** Atom x y z C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C H H C C H H H H C C H H H H H H C C C C C

9 C C C C C C C H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H Occupation numbers of UNOs/6-31G** n HOMO = n LUMO =

10 4. Molecular structure and cartesian coordinates of 2 Top view Side view Figure 3S. Molecular structure of 2 10

11 Table 2S: Cartesian coordinates [Å] of 2 optimized at the UB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** Atom x y z C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C H H H H H H H H H H H H C C C C C

12 C C C C C C C C C C C C C C C C C C C C C C C C C C C H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H

13 H H H H H * Maximum displacement in the optimization procedure is au, which is slightly over the standard threshold in Gaussian03 ( au), although other convergence criteria in optimization procedure are all satisfied. Using several nearly optimized geometries, such slight changes in geometries are found to cause negligible changes (less than 0.01) in the diradical character. Occupation numbers of UNOs/6-31G** n HOMO = n LUMO =

14 5. Molecular structure and cartesian coordinates of anthracene Figure 4S. Molecular structure of anthracene 14

15 Table 3S: Cartesian coordinates [Å] of anthracene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** Atom x y z C C C C C C C C C C C C C C H H H H H H H H H H Occupation numbers of UNOs/6-31G** n HOMO = n LUMO =

16 6. Molecular structure and cartesian coordinates of naphthalene Figure 5S. Molecular structure of naphthalene 16

17 Table 4S: Cartesian coordinates [Å] of naphthalene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** Atom x y z C C C C C C C C C C H H H H H H H H Occupation numbers of UNOs/6-31G** n HOMO = n LUMO =

18 7. Molecular structure and cartesian coordinates of chrysene Figure 6S. Molecular structure of chrysene 18

19 Table 5S: Cartesian coordinates [Å] of chrysene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** Atom x y z C C C C C C C C C C C C C C C C C C H H H H H H H H H H H H Occupation numbers of UNOs/6-31G** n HOMO = n LUMO =

20 8. Molecular structure and cartesian coordinates of fluorene Figure 7S. Molecular structure of fluorene 20

21 Table 6S: Cartesian coordinates [Å] of fluorene optimized at the RB3LYP/6-31G** level of approximation and the occupation numbers of frontier UNOs/6-31G** Atom x y z C C C C C C C C C C C C C H H H H H H H H H H Occupation numbers of UNOs/6-31G** n HOMO = n LUMO =

22 9. Molecular structure and cartesian coordinates of TIPS-pentacene Figure 8S. Molecular structure of TIPS-pentacene 22

23 Table 7S: Cartesian coordinates [Å] of TIPS-pentacene using the geometry, isopropyl groups of which are replaced by H atoms, derived from the crystal structure (Ref. 31) and the occupation numbers of frontier UNOs/6-31G** Atom x y z Si C C C C C C C C C C C C C Si C C C C C C C C C C C C C H H H H H H H H H H H H H H H H H H Occupation numbers of UNOs/6-31G** 23

24 n HOMO = n LUMO =

25 10. Full reference of Gaussian 03 program package. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian 03, Revision C.02, Gaussian, Inc., Wallingford CT,