Landscapes of Four-Enantiomer Conical Intersections for

Article pubs.acs.org/JPCA

Landscapes of Four-Enantiomer Conical Intersections for Photoisomerization of Stilbene: CASSCF Calculation Yibo Lei,†,‡ Le Yu,† Bo Zhou,†,§ Chaoyuan Zhu,†,* Zhenyi Wen,§ and Sheng Hsien Lin† †

Department of Applied Chemistry, Institute of Molecular Science and Center for Interdisciplinary Molecular Science, National Chiao-Tung University, Hsinchu 300, Taiwan ‡ Key Laboratory of Synthetic and Natural Functional Molecule Chemistry of Ministry of Education, The College of Chemistry & Materials Science, Shaanxi key Laboratory of Physico-Inorganic Chemistry, Northwest University, Xi’an 710069, P. R. China § Institute of Modern Physics, Northwest University, Xi’an, 710069, P. R. China S Supporting Information *

ABSTRACT: The photoisomerization of cis- and transstilbene through conical intersections (CI) is mainly governed by four dihedral angles around central CC double bonds. The two of them are CCCC and HCCH dihedral angles that are found to form a mirror rotation coordinate, and the mirror plane appears at the two dihedral angles equal to zeros with which the middle state is defined through partial optimization. There exist the first-type of hulatwist-CI enantiomers, the second-type of hula-twist-CI enantiomers, the first-type of one-bond-flip-CI enantiomers, and the second type of one-bond-flip-CI enantiomers as well as cis-enantiomers and trans-enantiomers with respect to this mirror plane. The complete active space self-consistent field method is employed to calculate minimum potential energy profile along the mirror rotation coordinate for each enantiomers, and it is found that the left-hand manifold and the right-hand manifold of potential energy surfaces can be energetically transferred via photoisomerization. Furthermore, two-dimensional potential energy surfaces in terms of the branching plane g−h coordinates are constructed at vicinity of each conical intersection, and the landscapes of conical intersections show distinct feature, and in excited-state four potential wells separated in different section of g−h plane related to different conical intersections which indicate different photoisomerization pathways. conical intersections which are named as OBF-CI7,8 and HTCI,9 respectively. The stilbene has cis- and trans-isomers that can be photoinduced and transferred from one to the other in accompanying with electronically nonadiabatic transition among various electronic states. Traditionally, the isomerization of cis- and trans-conformations has been investigated by computing the ground-state S0 and the lowest excited-state S1. The dominant excitation from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO) corresponds to the S1 transition, which was confirmed to be a bright state.8 In the recent years, the photoisomerization involving the conical intersections plus fluorescence spectra has been extensively investigated by both experimentalists and theoreticians.7−28 The OBF mechanism describes isomerization mainly depending on phenyl rotation in

1. INTRODUCTION Conical intersections that play a significant role in many photochemical and photophysical processes are essential for the relaxation of a molecule from electronic excited states to ground state.1 There are many examples of photochemical isomerisations about CC double bonds that have been shown to involve conical intersections and stilbene has been studied as a prototype for these processes. The isomerization of cis- and trans-stilbene has been extensively studied for more than 40 years and the quantum yields of the photoisomerization were measured experimentally.2,3 Three mechanisms were characterized for the isomerization so far; they are the conventional one-bond flip (OBF),4 hula-twist (HT),5 and the aborted HT mechanisms.6 The OBF mechanism involves a 180° rotation around the central ethylenic CC bond. The HT mechanism is considered as the concerted torsion around the adjacent vinyl−phenyl bond and a remarkable bend of in plane CCC angle in accompanying with the central ethylenic CC bond rotation. The aborted HT mechanism is explained as the rotation of one of two phenyl rings in stilbene is aborted and turned back. The cis-to-trans and trans-to-cis isomerizations are taken place by going through pathway of © 2014 American Chemical Society

Special Issue: International Conference on Theoretical and High Performance Computational Chemistry Symposium Received: February 26, 2014 Revised: June 30, 2014 Published: June 30, 2014 9021

dx.doi.org/10.1021/jp5020109 | J. Phys. Chem. A 2014, 118, 9021−9031

The Journal of Physical Chemistry A

Article

Figure 1. Standard numbering and definitions of internal coordinates for stilbene: (a) middle-S0, (b) HT1-CI-L, (c) HT2-CI-L, (d) OBF1-CI-L, and (e) OBF2-CI-L. D1 is defined as dihedral angle C11−C8C1C2, D2 as H10C8C1H9, D3 as C8C1C2C3, and D4 as C12 C11C8C1. Note that every arrowhead directs to the atom that moves when the internal coordinate varies.

OBF mechanism.17 Fuß and co-workers6,18,19 suggested that the HT and the aborted HT mechanisms were deduced from the systematic features of the cis- and trans-photoisomerizations of nonpolar conjugated molecules and these were investigated by the experimental measurements as well.21−23 Starting from the Franck−Condon (FC) regions, PESs along the reaction coordinate (curved one-dimension) were also investigated.17,25,28 It was analyzed that one-dimensional PESs with respect to the main twist of phenyl groups around the central CC bond reflects the main dynamical effect for isomerization at OBF-CI as mentioned above. The two torsion

which two phenyl rings twist around the central ethylene CC bond, but some other torsions might be also included.7−14 The conical intersection OBF-CI associated with the OBF mechanism was calculated to be the lowest in energy, so that the photoisomerization reactions were predicted to be in favor of this mechanism.8 The further dynamical simulations were done by calculating the evolution of HOMO and LUMO orbitals along with the nuclear motions including all nuclear coordinates.15,16 The vibrational normal-mode analysis with specified PES involving the steep route indicated that the vibrational mode with frequency 240 cm−1 also strengthens the 9022



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Table 1. Optimized Geometrical Parameters and Potential Energies of Stationary Points Separately on the S0 and S1 Potential Energy Surfacesa indexb C1C2 C1C8 C1H9 C8H10 C8C11 C8C1C2 H9C1C8 H10C8C1 C11C8C1 H9C1C8C2 D10 D20 D30 D40 DD10 DD20 ΔE (eV)c

cis-S0

TS-S0

trans-S0

cis-S1

twist-S1

TS-S1

trans-S1

1.477767 1.480 64 1.334 76 1.335 51 1.077 06 1.076 77 1.077 06 1.076 76 1.477 83 1.480 67 130.45 129.93 115.95 116.21 115.95 116.20 130.44 129.93 178.98 178.96 −5.79 −5.63 −3.74 −3.65 39.40 41.29 −144.06 −141.71 −4.77 −4.64 −1.02 −0.99 0.220 0.209

1.439 63 1.449 20 1.457 20 1.455 41 1.077 74 1.077 68 1.077 64 1.077 56 1.440 66 1.450 65 124.95 124.91 118.15 118.38 118.13 118.36 125.05 125.02 −177.91 −178.06 −91.10 −91.21 −86.63 −87.21 2.34 2.43 −177.33 −177.61 −88.87 −89.21 −2.24 −2.00 2.367 2.344

1.468 35 1.472 13 1.336 82 1.336 91 1.075 35 1.075 19 1.075 35 1.075 18 1.468 35 1.472 15 126.59 126.39 118.91 119.07 118.91 119.07 126.60 126.39 −178.19 −178.08 −179.47 −179.50 −175.87 −175.70 16.34 17.77 −164.09 −162.44 −177.67 −177.60 −1.80 −1.90 0.000 0.000

1.405 67 1.404 46 1.396 83 1.402 20 1.075 44 1.076 35 1.075 53 1.076 35 1.405 37 1.404 44 123.08 124.28 118.79 118.27 118.77 118.27 123.10 124.28 −179.22 −177.84 −19.18 −27.31 −17.63 −23.00 −14.64 −13.97 160.95 163.92 −18.41 −25.16 −0.78 −2.15 4.875 4.995

1.446 28 1.432 14 1.429 52 1.429 95 1.085 65 1.082 99 1.098 42 1.098 34 1.415 67 1.417 14 119.51 120.24 110.72 112.14 121.71 121.57 126.04 126.18 −138.34 −143.94 −110.64 −107.10 −68.48 −70.68 21.71 19.63 −179.24 −179.32 −89.56 −88.89 −21.08 −18.21 4.435 4.598

1.396 35 1.405 22 1.429 01 1.419 40 1.074 25 1.076 90 1.083 83 1.086 13 1.398 98 1.404 99 121.49 122.90 119.09 118.64 118.23 118.84 126.57 126.39 −179.76 −178.95 −136.14 −132.35 −134.98 −130.06 0.36 1.95 −175.02 −174.81 −135.56 −131.21 −0.58 −1.14 5.188 5.281

1.404 04 1.404 91 1.403 13 1.404 98 1.072 47 1.073 42 1.072 47 1.073 43 1.404 04 1.404 89 125.83 125.28 118.34 118.45 118.34 118.45 125.83 125.28 −179.32 −178.03 −176.87 −168.47 −175.50 −164.53 1.96 5.57 −178.17 −174.98 −176.19 −166.50 −0.69 −1.97 5.167 5.183

The first and second rows are data calculated from CASSCF(6,6) and CASSCF(2,2). Bond lengths are in angstroms (Å); bond and dihedral angles are in degrees. bDD10 = (D10 + D20)/2 and DD20 = (D10 − D20)/2. cΔE is all relative to trans-S0. a

plane appears at D1 = D2 = 0°. There exist four enantiomer conical intersections with respect to this mirror plane. Actually, there exist a cis-enantiomer and trans-enantiomer with respect to this mirror plane as well. The rest of this paper is organized as follows: Section 2 first introduces the ab initio quantum chemistry method employed in the present calculation and provides a definition of the mirror rotation coordinate that turns out to be a very important new concept for the photoisomerization of stilbene. Section 3 presents how these four enantiomer conical intersections are connected by the mirror rotation coordinate, and how the cisenantiomer and trans-enantiomer are connected by the mirror rotation coordinate. To figure out landscapes of the four enantiomer conical intersections, we plot the potential energy surfaces of ground-state S0 and the first-excited-state S1 in terms of the branching plane h−g coordinates29−31 in the vicinity of each conical intersection configuration. Concluding remarks are mentioned in section 4.

angles in terms of phenyl rotation and the pyramidalization coordinate were utilized for constructing two-dimensional PESs around the OBF-CI.8 These studies told that at the OBF-CI the two-dimensional PESs can describe trans−cis nonadiabatic dynamics for stilbene. It is suggested that the branching ratio is mainly determined by potential energy surfaces near the OBFCI. However, the HT-CI is also energetically accessible. Furthermore, are there only these two conical intersections involved in photoisomerization for stilbene? The purpose in the present study is to answer this question, and actually, we found there are four enantiomer (totally eight) conical intersections between the ground-state S0 and the lowest excited-state S1. It is shown in Figure 1a that the dihedral angles labeled as D1 (CCCC) and D2 (HCC H) are involved as variables to describe the twist around the central CC double bond, whereas the other two dihedral angles D3 (CCCC) and D4 (CCCC) correspond to the rotation of phenyl rings, respectively. It was found that these dihedral angles have been proved to be significant by the directly realistic dynamic simulation.16 What is more important is that the dihedral angles D1 and D2 can coordinately form a mirror rotation coordinate, and the mirror

2. COMPUTATION AND THE MIRROR ROTATION In the present study, all computations have been carried out by employing the state average complete active space self9023



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consistent field (SA-CASSCF) method.32,33 The active space includes both 2 and 6 active electrons, and both 2 and 6 active orbitals, which correspond to HOMO and LUMO as assigned to π and π* type molecular orbitals, respectively.11 The average states correspond to the concerned ground-state S0 and the first-excited-state S1. The 6-31G basis set is used and no symmetry restriction is imposed on geometries and electronic wave functions calculation.34 These computational levels are consistent with the previous one as the geometrical optimizations have just been slightly improved by higher level methods.8 Besides, the main purpose in the present work is to find four-enantiomer conical intersections so that the present calculation levels are good enough to demonstrate the new concept. We have utilized the method of Bearpark et al.35 to optimize the conical intersection geometries as implemented in Molpro program package,36 and all other calculations reported in the present work have been completed by MOLCAS 7.0 program package.37 There are four dihedral angles (D1, D2, D3, and D4) as shown in Figure 1a that represents the main internal coordinates for conical intersections. They are defined as

shown in Table S1 of the Supporting Information, and we found SA-CASSCF(6,6) works well and can be good enough for running on-the-fly nonadiabatic photoisomerization dynamics in the future. Stilbene has 26 atoms with 72 dimensional internal coordinates that define molecular geometry uniquely, in which 58 dimensional internal coordinates are related to the two phenyl groups and the rest of 14 dimensional internal coordinates are related to the ethylene-like structure as well as the rotation around the vinyl−phenyl bond. In Tables 1 and 2, Table 2. Optimized Geometrical Parameters and Potential Energies for Conical Intersections and Middle-S0, Which Is the Geometry for the Mirror Planea indexb C1C2 C1C8 C1H9

D1 = C11C8C1C2

D2 = H10C8C1H9

C8H10

D3 = C8C1C2C3

C8C11

and D4 = C12C11C8C1

C8C1C2

(1) H9C1C8

where D1 and D2 form the mirror rotation coordinate defined as

H10C8C1

λ + 10 D1 = − D10 + D10 10

C11C8C1

and λ + 10 D2 = − D20 + D20 10

H9C1C8C2 D10

(2)

in which the mirror rotation coefficient λ ∈ [−10, +10]. At λ = −10 we have D1 = D10 and D2 = D20 defined as the left-hand general stationary point, and at λ = 10 we have D1 = −D10 and D2 = −D20 defined as the corresponding right-hand general stationary point. At λ = 0, D1 = 0 and D2 = 0 are defined as the corresponding mirror plane. Actually, the geometry of mirror plane (called middle-S0 throughout the present paper) is optimized at the fixed dihedral angles D1 = D2 = 0, and its energy is even lower than the energy of trans-S0. It should be noted that the left-hand stationary point and corresponding right-hand stationary point have opposite signs for all dihedral angles along the mirror rotation coordinate λ. Before investigating the four enantiomer conical intersections, we would like to recompute conventional stationary points on the ground-state S0 and excited-state S1 potential energy surfaces separately. These conventional stationary points include cis- and trans-equilibrium geometries and transition states on both S0 and S1 surfaces. Initial guesses of those stationary points can be found from the other computational results,7−9 and we have made calculations by both SACASSCF(6,6) and SA-CASSCF(2,2) methods. Calculated results are shown in Table 1 for the most important internal coordinates around the conical intersections, and all Cartesian coordinates are given in Supporting Information S4. We made a comparison with results calculated by SA-CASSCF(14,12),8 as

D20 D30 D40 DD10 DD20 ΔE (eV)c

HT1CI-L

HT2-CI-L

OBF1CI-L

OBF2CI-L

middle-S0

1.644 97 1.6241 1.410 35 1.418 37 1.083 79 1.084 74 1.093 84 1.093 90 1.459 77 1.459 53 82.48 83.35 118.43 117.26 124.27 124.13 124.42 124.45 101.20 102.88 114.41 114.57 39.25 41.29 −101.49 −98.12 −175.23 −175.24 76.83 77.93 37.58 36.64 4.974 5.138

1.493 50 1.494 89 1.395 00 1.400 80 1.088 60 1.089 29 1.071 56 1.073 02 1.668 49 1.653 33 132.46 132.71 117.98 118.00 132.40 130.32 80.06 81.19 −176.97 −178.05 −74.48 −72.69 −145.49 −143.47 −29.14 −30.63 95.28 96.63 −109.99 −108.08 35.50 35.39 5.534 5.725

1.467 28 1.462 19 1.390 92 1.390 87 1.200 56 1.192 57 1.095 23 1.096 47 1.456 17 1.452 95 128.56 128.02 73.79 75.48 122.31 122.29 125.15 125.27 −97.33 −99.02 −148.07 −146.64 −60.21 −60.02 48.70 50.30 −174.65 −174.66 −104.14 −103.33 −43.93 −43.31 4.847 5.054

1.435 32 1.439 66 1.381 24 1.382 27 1.201 51 1.199 15 1.086 71 1.087 26 1.483 45 1.481 85 140.21 138.62 71.45 71.94 118.24 118.04 130.19 130.62 −104.33 −103.69 36.59 36.79 106.78 107.51 63.41 57.79 174.61 173.47 71.69 72.15 -35.10 −35.36 5.151 5.341

1.482 42 1.471 07 1.350 97 1.336 15 1.076 53 1.076 78 1.076 94 1.077 38 1.491 25 1.489 90 131.39 131.95 115.59 115.21 116.37 116.41 129.14 129.24 −178.65 −178.53 0.00 0.00 0.00 0.00 11.06 11.68 79.36 87.45 0.00 0.00 0.00 0.00 −0.690 0.214

The first and second rows are data calculated from CASSCF(6,6) and CASSCF(2,2). Bond lengths are in angstroms (Å); bond and dihedral angles are in degrees. bDD10 = (D10 + D20)/2 and DD20 = (D10 − D20)/2. cΔE is all relative to trans-S0. a

we just list these 14 dimensional internal coordinates, which are most important degrees of freedom for photoisomerization of stilbene. Actually, only four dihedral angles D1, D2, D3, and D4 (eq 1) out of 14 degrees of freedom appear to have large changes and the other 10 internal coordinates show small changes among those stationary points. This is evidence from both SA-CASSCF(6,6) and SA-CASSCF(2,2) calculations that these four dihedral angles are actually responsible for 9024



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photoisomerization of stilbene. The cis-isomer and trans-isomer on the S0 surface listed in Table 1 are defined as left-hand isomers: cis-S0-L and trans-S0-L, where L stands for left-hand.

3. RESULTS AND DISCUSSION We first optimized geometry of mirror plane middle-S0 as shown in Figure 1a and then we optimized geometries of four left conical intersections as shown in Figure 1b for HT1-CI-L, in Figure 1c for HT2-CI-L, in Figure 1d for OBF1-CI-L, and in Figure 1e for OBF2-CI-L. Definition of the left and right conical intersection is relative in meaning, and we here define four CIs found in the first place as left conical intersections. The last column in Table 2 shows that all important internal coordinates are almost same calculated from both SACASSCF(6,6) and SA-CSSACF(2,2) methods, but energy optimized from the SA-CASSCF(6,6) method is 0.6 eV lower than the energy of trans-S0 whereas it is 0.21 eV higher from the SA-CASSCF(2,2) method. However, the CASPT2 correction confirms that the energy of trans-S0 is the lowest as the global minimum. From dihedral angles D1 and D2, we can see that the geometry of middle-S0 is close to the geometry of cis-S0. We utilize the geometry of middle-S0 as a mirror plane to investigate the enantiomers. We emphasize here that the left enantiomer and the corresponding right enantiomer have all the same bond lengths and bond angles, but opposite signs for all dihedral angles. We set up the energy of middle-S0 as the zero point in Figures 2−4, which is the lowest point in energy Figure 3. Same as in Figure 2 except from left-hand hula-twist-CI to right-hand hula-twist-CI: (a) HT1-CI-L to HT1-CI-R and (b) HT2CI-L to HT2-CI-R.

from the SA-CASSCF calculation. All important internal coordinates for the four left-hand conical intersections are listed in Table 2, and all Cartesian coordinates are given in Supporting Information S4. From now on, the following discussion is based on the results calculated from the SACASSCF(6,6) method. 3.1. Cis-Enantiomers and the Trans-Enantiomers. We insert 9 configuration points between the middle-S0 and lefthand cis-isomer (cis-S0-L) with respect to D1 and D2 from λ = 0 (D1 = D2 = 0) to λ = −10 (D1 = −5.79° and D2 = −3.74) in eq 2. On the basis of the S0 state, we did partial optimizations at fixed D1 and D2 for 6 points from λ = −1 to λ = −6 and then we did single-point calculations for 3 points (λ = −7, −8, −9) with geometries obtained from linear interpolations in all internal coordinate between λ = −6 and −10. Potential energy profiles for positive λ from λ = 1−10 are computed by single point calculation from corresponding geometries of negative λ with changing signs of all dihedral angles. We plot potential the energy curve along the mirror rotation coefficient λ from λ = −10 to λ = +10 as shown in Figure 2a, in which energy profiles of the S1 state are vertical excitation energies. We confirm the potential energy curves have a mirror image with respect to middle-S0 and potential energy curves on both S0 and S1 states are very flat except for a small barrier at λ = ±7. We insert 9 configuration points between middle-S0 and the left-hand trans-isomer (trans-S0-L) with respect to D1 and D2 from λ = 0 (D1 = D2 = 0) to λ = −10 (D1 = −179.47° and D2 = −175.87°) in eq 2. On the basis of the S0 state, we did partial optimizations at fixed D1 and D2 for 7 points from λ = −1 to λ

Figure 2. Potential energy curves of S0 and S1 along mirror rotation coordinate λ from the left-hand (λ = −10) isomer to the right-hand (λ = +10) isomer: (a) cis-S0-L to cis-S0-R and (b) trans-S0-L to trans-S0R. 9025



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= −7 and then we did single-point calculations for 2 points (λ = −8, −9) with linear interpolations in all internal coordinate between λ = −7 and −10. Potential energy profiles in Figure 2b show relatively big change in comparison with potential energy profiles in Figure 2a for the cis-isomer. This is because both D1 and D2 change from 0° to almost −180° for the trans-isomer whereas they change from 0° to about 5° for the cis-isomer. We confirm again the potential energy curves have a mirror image with respect to middle-S0. The potential energy curves have an energy barrier of about 2 eV on the S0 state and have an energy well about 1 eV on the S1 state at λ = ±5, as shown in Figure 2b. Potential energy profiles show that photoisomerization can be freely transferred from trans-S0-L to trans-S0-R, as shown in Figure 2b, but it needs a thermal energy of about 1.5 eV to transfer from cis-S0-L to cis-S0-R, as shown in Figure 2a. It should be very interesting to see the dynamic effect from onthe-fly trajectory simulation in future. 3.2. HT1-CI Enantiomers and the HT2-CI Enantiomers. Let us discuss hula-twist S1/S0 conical intersections first. We insert 9 configuration points between middle-S0 and left-hand HT1-CI-L with respect to D1 and D2 from λ = 0 (D1 = D2 = 0) to λ = −10 (D1 = 114.41° and D2 = 39.25°) in eq 2. On the basis of the S0 state, we did partial optimizations at fixed D1 and D2 for 4 points from λ = −1 to λ = −4 and then we did single-point calculations for 5 points (λ = −5, −6, −7, −8, −9) with the linear interpolations in all internal coordinate between λ = −4 and −10. Potential energy profiles show monotonically rising from middle-S0 to HT1-CI-L and HT1-CI-R on S0 state and show small wiggling on S1 state, as shown in Figure 3a. We insert 9 configuration points between middle-S0 and lefthand HT2-CI-L with respect to D1 and D2 from λ = 0 (D1 = D2 = 0) to λ = −10 (D1 = −74.48° and D2 = −145.49°) in eq

Figure 4. Same as in Figure 2 except from left-hand one-bond-flip-CI to right-hand one-bond-flip-CI: (a) OBF1-CI-L to OBF1-CI-R and (b) OBF2-CI-L to OBF2-CI-R.

Figure 5. Four two-dimensional potential energy surfaces around vicinity of (a) HT1-CI-L, (b) HT2-CI-L, (c) OBF1-CI-L, and (d) OBF2-CI-L, in terms of branching plane g−h coordinates. The corresponding CI is zero energy point for each potential energy surface. 9026



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Figure 6. Potential energy contour plot for the S0 state from Figure 5: (a) HT1-CI-L, (b) HT2-CI-L, (c) OBF1-CI-L, and (d) OBF2-CI-L.

Figure 7. Same as in Figure 6 except for potential energy contour plot of the S1 state.

3.3. OBF1-CI Enantiomers and the OBF2-CI Enantiomers. Next let us discuss one-bond-flip S1/S0 conical intersections. We insert 9 configuration points between middleS0 λ = 0 (D1 = D2 = 0) and left-hand OBF1-CI-L λ = −10 (D1 = −148.07° and D2 = −60.21°), and between middle-S0 λ = 0

2. On the basis of the S0 state, we did partial optimizations for 4 points from λ = −1 to λ = −4 and then we did single-point calculations for 5 points (λ = −5, −6, −7, −8, −9) with the linear interpolations. The potential energy profiles for HT2-CI in Figure 3b show a structure very similar to that for HT1-CI. 9027



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Figure 8. Energy levels for all stationary points and four left-hand conical intersection, all energies are referenced to the trans-S0 isomer: (a) CASSCF and (b) CASPT2 within active space CASSCF(6,6); (c) CASSCF and (d) CASPT2 within active space CASSCF(2,2).

(D1 = D2 = 0) and left-hand OBF2-CI-L λ = −10 (D10 = 36.59° and D20 = 106.78°). On the basis of the S0 state, we did partial optimizations for 4 points from λ = −1 to λ = −4 and then we did single-point calculations for 5 points (λ = −5, −6, −7, −8, −9) with the linear interpolations. Potential energy profiles for OBF1-CI in Figure 4a and OBF2-CI in Figure 4b show a structure very similar to that for HT1-CI and HT2-CI in Figure 3. It can be seen that except for the two dihedral angles D10 and D20, the other 12 internal coordinates shown in Table 2 are quite similar for OBF1-CI-L and OBF2-CI-L so that the two are related with each other, but the energy of OBF2-CI-L is 0.30 eV higher than the energy of OBF1-CI-L. The energy of OBF2-CI-L is just 0.015 eV higher than the energy of the transS1 isomer, and this means that both OBF2-CI-L and OBF1-CIL could be energetically open for photoisomerization. In conclusion, all left enantiomers can be exchangeable with all right enantiomers for cis- and trans-isomers as well as for the four conical intersections. Photoisomerization can be energetically accessible to all configuration space of both the left and right manifold of potential energy surfaces. 3.4. Two-Dimensional Potential Energy Surfaces. From now on, we focus on left-hand four conical intersections to investigate landscapes of the potential energy surfaces in the vicinity of each left-hand conical intersection because righthand ones are just mirror images. At each conical intersection, we construct two-dimensional potential energy surfaces in terms of branch plane g−h coordinates;30 the direction in the gcoordinate is defined by derivative difference of two adiabatic

potential energy surfaces as the tuning coordinate and the direction in the h-coordinate is defined by the nonadiabatic coupling vector as the coupling coordinate. Starting from the conical intersection point at g = h= 0 that corresponds to Cartesian coordinates (xi0, yi0, zi0) where i = 1, 2, ..., N (number of atoms in stilbene), we compute normalized (gix0, giy0, giz0) and (hix0, hiy0, hiz0) vectors and then xi(n ,m) = xi0 + ngix 0δν + mhix 0δu yi (n ,m) = yi0 + ngiy0δν + mhiy0δu zi(n ,m) = zi0 + ngiz 0δν + mhiz 0δu

(3)

where stepsize δv and δu for the g−h coordinates are set up to equal 0.01 bohr in length for HT1-CI-L, OBF1-CI-L, and OBF2-CI-L and 0.02 bohr for HT2-CI-L. Finally, integer’s n and m are converted to g−h coordinates in Figures 5−7 and 200 × 200 configuration points are computed for HT1-CI-L, OBF1-CI-L, and OBF2-CI-L, and 100 × 100 points for HT2CI-L. The topology of the four conical intersections shows a structure very similar to that shown in Figure 5 in which the potential energy surfaces for the S0 state are quite flat cones and are regular cones for the S1 state. The potential gap for HT1CI-L varies slowly in both g and d directions as shown in Figure 5a, it varies slowly in the h direction and fast in the d direction for HT2-CI-L as shown in Figure 5b, and it varies fast in both g and d directions for OBF1-CI-L as shown in Figure 5c and for OBF2-CI-L as shown in Figure 5d. These can be seen from contour maps for the S0 state (Figure 6) and for the S1 state 9028



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onto one of the phenyl groups, and thus electronic excitations occur on the phenyl group not on the CC bond. This is very different from the excitation from the HOMO to LUMO for the CASSCF(2,2) wave function. To quantitatively understanding decay lifetimes, branch ratios, and cooperative interaction dynamics of four left-hand conical intersections, it may be necessary to run full-dimensional trajectory surface hopping dynamics.

shown (Figure 7). There is a potential well for HT1-CI-L on contour maps for the S0 state, as shown in Figure 6a, whereas there is no potential well for the rest of three left CIs, as shown in Figure 6b−d. It is very interesting to note that there are potential wells in different sections of h−g plane for each conical intersection in the S1 state; the potential well for HT1CI-L is in the section −1 < h < 0 and −1 < g < 0 (Figure 7a), for HT2-CI-L it is in the section 0 < h < 1 and −1 < g < 0 (Figure 7b), for OBF1-CI-L it is in the section −1 < h < 0 and 0 < g < 1 (Figure 7c), and for OBF2-CI-L it is in the section 0 < h < 1 and 0 < g < 1 (Figure 7d). Four photoproducts that are separated by the h−g plane demonstrate four possible isomerization pathways related with four left-hand conical intersections. The four corresponding right-hand conical intersections can further complicate the dynamics. To see another aspect of conical intersection, namely a seam surface that might be in the perpendicular direction of h and g coordinates, we construct two-dimensional potential energy surfaces again in terms of two key dihedral angles D1 and D2 that define the mirror rotation coordinate in eq 2 and the remaining 70 internal coordinates are all fixed at geometry parameters for each conical intersection. For good visualization of potential energy surfaces in terms of conical intersections, we convert D1 and D2 into

4. CONCLUDING REMARKS We have investigated the photoisomerization mechanism of stilbene mainly in terms of four dihedral angles D1, D2, D3, and D4 around center CC double bonds. We have found four left-hand conical intersections and four right-hand conical intersections, namely HT1-CI-L and HT1-CI-R, HT2-CI-L and HT2-CI-R, OBF1-CI-L and OBF1-CI-R, and OBF2-CI-L and OBF2-CI-R. Four enantiomer conical intersections have the same mirror plane constructed by the mirror rotation coordinate at D1 = D2 = 0°, where we found the middle-S0 state by partial optimization with fixed D1 = D2 = 0°. Furthermore, we have found that cis-S0 enantiomers and transS0 enantiomers have the same mirror plane as at D1 = D2 = 0°. This mirror plane actually plays a bridge to connect left-hand manifold and right-hand manifold potential energy surfaces of stilbene, and these two dihedral angles play a crucial role for the photoisomerization. We have plotted one-dimensional potential energy curves along the mirror rotation coordinates and the energy profiles are a kind of minimum energy path obtained with partial optimization (at fixed D1 and D2). Therefore, we conclude that the left-hand isomer can be energetically transferred with the right-hand isomer. In the vicinity of each left-hand conical intersection, we have constructed two-dimensional potential energy surfaces in terms of branching plane g−h coordinates. The landscapes of the potential energy surfaces on the S1 state show that there are four potential wells separated in different sections of the g−h plane related to different conical intersections. This means that each conical intersection might be related to its own photoisomerization pathway. To quantitatively understanding decay lifetimes, branch ratios, and cooperative interaction dynamics of four left-hand conical intersections, it is necessary to run full-dimensional trajectory surface hopping dynamics, which is currently underway, and we have utilized SACASSCF(6,6) for the dynamic simulation.

DD1 = (D1 + D2)/2 and

DD2 = (D1 − D2)/2

(4)

All discussion of landscapes of four conical intersections in terms of DD1 and DD2 coordinates are given in Supporting Information S3. Finally, we summarize energy levels obtained from Tables 1 and 2 in Figure 8 to make a clear energetic comparison. It should be noted that the energy levels of cis-S1 minimum and vertical excitation and of trans-S1 minimum and vertical excitation are all plotted in Figure 8. From the SA-CASSCF(6,6) calculation, it appears that the lowest energy is at middleS0 on the S0-state potential energy surface that is lower in energy than both cis-S0 and trans-S0, as shown in Figure 8a. However, it appears that the highest energy is at middle-S0 on the S1-state potential energy surface, which is higher in energy than both cis-S1 and trans-S1. Then we did a CASPT2 correction and results are summarized in Figure 8b in which the energy of trans-S0 is the lowest, but still the energy of middle-S0 is 0.01 eV lower than the energy of cis-S0. Besides, the energy of OBF1-CI is lower than the energy of HT1-CI under SACASSCF(6,6) but it reverses under CASPT2 correction. From SA-CASSCF(2,2) calculation, however, the energy of trans-S0 is the lowest in both CASSCF and CASPT2, as shown in Figures 8c,d, respectively. The energy of OBF1-CI is lower than the energy of HT1-CI under SA-CASSCF(2,2) but it reverses under CASPT2 correction. The energies for the all other stationary points do not change order under CASPT2 correction. All four conical intersections are energetically accessible for photoisomerization from both CASSCF and CASPT2 calculations. The main reason for the middle-S0 being lower in energy than the trans and cis geometries is due to the dramatic change of CASSCF(6,6) wave functions. The dominant configuration for the S0 state still corresponds to no excitation, but two dominant configurations for S1 arise from the excitations from both HOMO−1 to LUMO+1 and HOMO−2 to LUMO+2. These four orbitals are delocalized

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ASSOCIATED CONTENT

* Supporting Information S

Cartesian coordinates of the all optimized stationary points with comparison to higher level CASSCF calculations. Geometry structures of cis- and trans-isomers and conical intersections. Two potential energy profiles are plotted in terms of two important dihedral angles around vicinity of each left conical intersection and seam surfaces are clearly shown up. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*C. Zhu. Phone: +886-3-5131224. Fax: +886-3-5723764. Email: [email protected]. Notes

The authors declare no competing financial interest. 9029



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ACKNOWLEDGMENTS This work is supported by Ministry of Science and Technology of the Republic of China under Grant No. 100-2113-M-009005-MY3. Y. Lei thanks Postdoctoral Fellowship supported by Ministry of Science and Technology of the Republic of China under Grant No. 100-2811-M-009-004 and the project of Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JQ2060) for support. C. Zhu thanks the MOE-ATU project of the National Chiao Tung University for support. L. Yu thanks Postdoctoral Fellowship supported by Ministry of Science and Technology of the Republic of China under Grant No. 102-2811-M-009-044.

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