Supporting Information

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1 Supporting Information Computational Evidence of Inversion of 1 L a and 1 L b -Derived Excited States in Naphthalene Excimer Formation from ab Initio Multireference Theory with Large Active Space: DMRG-CASPT2 Study Soichi Shirai, * Yuki Kurashige, and Takeshi Yanai * DOI: /acs.jctc.6b00210 * To whom correspondence should be addressed. shirai@mosk.tytlabs.co.jp; yanait@ims.ac.jp. Toyota Central R&D Laboratories, Inc., Nagakute, Aichi , Japan. Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, Aichi , Japan. Japan Science and Technology Agency, PRESTO, Honcho, Kawaguchi, Saitama , Japan. S1

2 (1.421) (1.377) (118.8 ) (1.434) (1.417) (120.1 ) (118.7 ) (1.087) (1.088) Figure S1. Comparison between the calculated (B3LYP/6-31G(d)) and experimental geometrical parameters of the naphthalene monomer. The experimental values are given from Baba, M.; Kowaka, Y.; Nagashima, U.; Ishimoto, T.; Goto, H.; Nakayama, N. J. Chem. Phys. 2011, 135, (Ref. 30 in the main text). The calculated values are in parentheses. Bond lengths are given in Å. S2

3 Figure S2. Localized orbitals obtained from the DMRG-CASSCF wavefunction of the 1 1 B 3g excited state at r(r R) = 3.1 Å. S3

4 (a) HOMO 1 (b) HOMO (c) LUMO (d) LUMO+1 (e) LUMO+1 (f) LUMO+1 LUMO HOMO LUMO HOMO + HOMO-1 HOMO-1 1 L a 1 L b Figure S3. Molecular orbitals of the naphthalene obtained by using B3LYP/6-31G(d): (a) HOMO 1, (b) HOMO, (c) LUMO and (d) LUMO+1, (e) main configuration of the monomer 1 L a : HOMO LUMO single electron excitation (f) the monomer 1 L b characterized by a superposition of HOMO LUMO+1 and HOMO 1 LUMO singly excited configurations S4

5 HOMO 9 HOMO 8 HOMO 7 HOMO 6 HOMO 5 (1a g ) (1b 1u ) (1b 3u ) (1b 2g ) (1b 2u ) HOMO 4 HOMO 3 HOMO 1 HOMO 2 HOMO (1b 3g ) (2a g ) (2b 1u ) (1b 1g ) (1a u ) LUMO LUMO+2 LUMO+1 LUMO+3 LUMO+4 (2b 3u ) (2b 2g ) (2b 2u ) (2b 3g ) (3a g ) LUMO+5 LUMO+6 LUMO+7 LUMO+8 LUMO+9 (3b 1u ) (2b 1g ) (2a u ) (3b 2u ) (3b 3g ) Figure S4. Natural orbitals of the naphthalene dimer obtained from the DMRG-CASSCF wavefunction of the ( 1 A g ) at r(r R) = 3.6 Å. S5

6 HOMO 9 HOMO 8 HOMO 7 HOMO 5 HOMO 6 (1a g ) (1b 1u ) (1b 3u ) (1b 2g ) (1b 2u ) HOMO 3 HOMO 4 HOMO 1 HOMO 2 HOMO (1b 3g ) (2a g ) (2b 1u ) (1b 1g ) (1a u ) LUMO LUMO+2 LUMO+1 LUMO+3 LUMO+4 (2b 3u ) (2b 2g ) (2b 2u ) (2b 3g ) (3a g ) LUMO+5 LUMO+6 LUMO+7 LUMO+8 LUMO+9 (3b 1u ) (2b 1g ) (2a u ) (3b 2u ) (3b 3g ) Figure S5. Natural orbitals of the naphthalene dimer obtained from the DMRG-CASSCF wavefunction of the 1 L a ( 1 B 3g ) excited state at r(r R) = 3.1 Å. S6

7 HOMO 9 HOMO 8 HOMO 7 HOMO 5 HOMO 6 (1a g ) (1b 1u ) (1b 3u ) (1b 2g ) (1b 2u ) HOMO 4 HOMO 3 HOMO 1 HOMO 2 HOMO (1b 3g ) (2a g ) (2b 1u ) (1b 1g ) (1a u ) LUMO LUMO+2 LUMO+1 LUMO+3 LUMO+4 (2b 3u ) (2b 2g ) (2b 2u ) (2b 3g ) (3a g ) LUMO+5 LUMO+6 LUMO+7 LUMO+8 LUMO+9 (3b 1u ) (2b 1g ) (2a u ) (3b 2u ) (3b 3g ) Figure S6. Natural orbitals of the naphthalene dimer obtained from the DMRG-CASSCF wavefunction of the 1 L b ( 1 B 2g ) excited state at r(r R) = 3.2 Å. S7

8 Table S1. Occupation numbers of the natural orbitals (NOONs) in the CASSCF wave function of the ( 1 A g ) at r(r R)=3.6 Å. orbital (8πe, 8πo) (10πe, 10πo) (12πe, 12πo) (20πe, 20πo) 3b 3g empty empty empty b 2u empty empty empty a u empty empty empty b 1g empty empty empty b 1u empty empty a g empty b 3g b 2u b 2g b 3u a u b 1g b 1u a g b 3g closed b 2u closed closed b 2g closed closed closed b 3u closed closed closed b 1u closed closed closed a g closed closed closed S8

9 Table S2. Occupation numbers of the natural orbitals (NOONs) in the CASSCF wave function of 1 L a ( 1 B 3g ) at r(r R)=3.1 Å. orbital (8πe, 8πo) (10πe, 10πo) (12πe, 12πo) (20πe, 20πo) 3b 3g empty empty empty b 2u empty empty empty a u empty empty empty b 1g empty empty empty b 1u empty empty a g empty b 3g b 2u b 2g b 3u a u b 1g b 1u a g b 3g closed b 2u closed closed b 2g closed closed closed b 3u closed closed closed b 1u closed closed closed a g closed closed closed S9

10 Table S3. Occupation numbers of the natural orbitals (NOONs) in the CASSCF wave function of 1 L b ( 1 B 2g ) at r(r R)=3.2 Å. orbital (8πe, 8πo) (10πe, 10πo) (12πe, 12πo) (20πe, 20πo) 3b 3g empty empty empty b 2u empty empty empty a u empty empty empty b 1g empty empty empty b 1u empty empty a g empty b 3g b 2u b 2g b 3u a u b 1g b 1u a g b 3g closed b 2u closed closed b 2g closed closed closed b 3u closed closed closed b 1u closed closed closed a g closed closed closed S10

11 Table S4. Electronic state total energies and basis set super position errors (BSSE) for naphthalene dimer in the with r(r R) = 3.1 Å obtained by utilizing MP2 method. basis set sum of monomer energy in monomer basis a (hartree) sum of monomer energy in dimer basis b (hartree) BSSE (hartree) (ev) cc-pvdz cc-pvtz a Twice of monomer energy calculated by using basis functions for the monomer. b Twice of monomer energy calculated by using basis functions for the dimer. S11

12 Table S5. Electronic state total energies of the naphthalene dimer in the with and without incorporating the solvent effect of n-heptane (ε = ) obtained by using MP2. The solvent effect was considered by utilizing IEFPCM. basis set r(r R) = 3.1 Å r(r R) = 10.0 Å correction to binding energy, D e total energy (hartree) solvation free energy (hartree) total energy (hartree) solvation free energy (hartree) (hartree) (ev) with solvent without solvent with solvent without solvent cc-pvdz cc-pvtz S12

13 Energy / ev Energy / ev (a) (b) r(r R) / Å r(r R) / Å Figure S7. DMRG-CASSCF potential energy curves of the ( 1 A g, ), 1 L a ( 1 B 3g, ) and 1 L b ( 1 B 2g, ) excited states of the naphthalene dimer obtained by using (a) cc-pvdz and (b) cc-pvtz basis sets, respectively. The energies relative to the energy at r(r R) = 10.0 Å are plotted. S13

14 Table S6. Occupation numbers based on the mean-field (MF) approximated description of the 1 A g (), and 1 L a ( 1 B 3g ), and 1 L b ( 1 B 2g ) wavefunctions. orbital 1 A g a L a b L b 3b 3g b 2u a u b 1g b 1u a g b 3g b 2u (LUMO+1) b 2g b 3u (LUMO) a u (HOMO) b 1g b 1u (HOMO 1) a g b 3g b 2u b 2g b 3u b 1u a g a HOMO LUMO single electron excitation. b The superposition of HOMO LUMO+1 and HOMO 1 LUMO singly excited configurations. S14

15 Natural orbital occupation number (NOON) analysis In order to analyze the nature of electron correlation captured by the CASSCF wavefunction for the 1 L a ( 1 B 3g ) and 1 L b ( 1 B 2g ) states, we examine the differences in electron occupancies for orbitals between the CASSCF and mean-field (MF) wavefunctions. The deviation from the MF description is associated with the effect of electron correlation. To measure the relative degree of the correlation effect of each excited state compared to that of the ( 1 A g ), we evaluated the following indexes defined as Δ I ( 1 L a ) = {n I ( 1 L a ) n I ( 1 A g ) } { n I MF ( 1 L a ) n I MF ( 1 A g )}, (S1a) Δ I ( 1 L b ) = {n I ( 1 L b ) n I ( 1 A g ) } { n I MF ( 1 L b ) n I MF ( 1 A g )}, (S1b) where n I (X) and n I MF (X) refer to the occupancies of the orbital I of the state X for the CASSCF and MF wavefunctions, respectively. Eqs (S1a) and (S1b) gauge the degree of the role of orbital I in describing the electron correlation that affects the change in occupancy caused by the excitation. The values of n I MF (X) are shown in Table S6. The values of Δ I ( 1 L a ) (Eq. (S1a)) and Δ I ( 1 L b ) (Eq. (S1b)) obtained using the occupation numbers provided in Tables S1 S3 and S6 are shown in Table S7 for the CASSCF wavefunctions with CAS(8πe, 8πo), CAS(10πe, 10πo), CAS(12πe, 12πo), and CAS(20πe, 20πo). S15

16 Table S7. Analysis of natural orbital occupation numbers (NOONs) to examine the nature of electron correlation captured by the CASSCF wavefunctions with various CAS references for the 1 L a and 1 L b states. The values of eqs. (S1a) and (S1b) in page s13 obtained for each orbital using the occupancies in Tables S1 S3 and S6 are shown. They indicate the degree of the role of each orbital in describing the electron correlation of the excited state. a orbital I Δ I ( 1 L a ) [Eq. (S1a)] Δ I ( 1 L b ) [Eq. (S1b)] (8πe, 8πo) (10πe, 10πo) (12πe, 12πo) (20πe, 20πo) (8πe, 8πo) (10πe, 10πo) (12πe, 12πo) (20πe, 20πo) 3b 3g b 2u a u b 1g b 1u a g b 3g b 2u b 2g b 3u a u b 1g b 1u a g b 3g b 2u b 2g b 3u b 1u a g sum a Plotted in Figure 4 in the manuscript. S16

17 Energy / ev Energy / ev (a) (b) r(r R) / Å r(r R) / Å Figure S8. EOM-CCSD potential energy curves of the ( 1 A g, ), 1 L a ( 1 B 3g, ) and 1 L b ( 1 B 2g, ) excited states of the naphthalene dimer obtained by using (a) cc-pvdz and (b) aug-cc-pvdz basis sets. The plotted energies are given relative to the energy at r(r R) = 10.0 Å. S17

18 Energy / ev Energy / ev Energy / ev Energy / ev (a) (b) r(r R) / Å r(r R) / Å (c) (d) r(r R) / Å r(r R) / Å Figure S9. CIS(D) potential energy curves of the ( 1 A g, ), 1 L a ( 1 B 3g, ) and 1 L b ( 1 B 2g, ) excited states of the naphthalene dimer obtained by using (a) cc-pvdz, (b) aug-cc-pvdz, (c) cc-pvtz and (d) aug-cc-pvtz basis sets. The plotted energies are given relative to the energy at r(r R) = 10.0 Å. S18

19 Energy / ev Energy / ev Energy / ev (a) (b) (c) r(r R) / Å r(r R) / Å r(r R) / Å Figure S10. TDDFT potential energy curves of the ( 1 A g, ), 1 L a ( 1 B 3g, ) and 1 L b ( 1 B 2g, ) excited states of the naphthalene dimer obtained by using (a) TD-B3LYP, (b) TD-CAM-B3LYP, and (c) TD-ωB97XD. The plotted energies are given relative to the energy at r(r R) = 10.0 Å. The basis set used was cc-pvtz. S19

20 Table S8: Electronic state total energies in hartree calculated by DMRG-CASSCF. r(r R) (Å) cc-pvdz cc-pvtz ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) S20

21 Table S9: Electronic state total energies in hartree calculated by DMRG-CASPT2. r(r R) (Å) cc-pvdz cc-pvtz ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) S21

22 Table S10: Electronic state total energies in hartree calculated by CASSCF/cc-pVDZ with (8πe, 8πo), (10πe, 10πo) and (12πe, 12πo). r(r R) (Å) (8πe, 8πo) (10πe, 10πo) (12πe, 12πo) ( 1 A g) L a L b L a L b L a L b ( 1 B 3g) ( 1 B 2g) ( 1 A g) ( 1 B 3g) ( 1 B 2g) ( 1 A g) ( 1 B 3g) ( 1 B 2g) S22

23 Table S11: Electronic state total energies in hartree calculated by CASPT2/cc-pVDZ with (8πe, 8πo), (10πe, 10πo) and (12πe, 12πo). r(r R) (Å) (8πe, 8πo) (10πe, 10πo) (12πe, 12πo) ( 1 A g) 1 L a ( 1 B 3g) 1 L b ( 1 B 2g) ( 1 A g) 1 L a ( 1 B 3g) 1 L b ( 1 B 2g) ( 1 A g) 1 L a ( 1 B 3g) 1 L b ( 1 B 2g) S23

24 Table S12: Electronic state total energies in hartree calculated by EOM-CCSD. r(r R) (Å) cc-pvdz aug-cc-pvdz ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) S24

25 Table S13: Electronic state total energies in hartree calculated by CIS(D) with cc-pvdz and aug-cc-pvdz basis set. r(r R) (Å) cc-pvdz aug-cc-pvdz ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) S25

26 Table S14: Electronic state total energies in hartree calculated by CIS(D) with cc-pvtz and aug-cc-pvtz basis set. r(r R) (Å) cc-pvtz aug-cc-pvtz ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) ( 1 A g ) L a ( 1 B 3g ) L b ( 1 B 2g ) S26

27 Table S15: Electronic state total energies of a naphthalene molecule (values are in hartree). method basis set excited state 1 L a 1 L b RHF cc-pvtz CASSCF with CAS(10πe,10πo) cc-pvtz CASPT2 with CAS(10πe,10πo) cc-pvtz EOM-CCSD cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz CIS(D) cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz S27