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Theoretical Studies of the Ground and Excited State Structures of Stilbene Rajat K. Chaudhuri† Indian Institute of Astrophysics, Bangalore 560034, India

Karl F. Freed‡ The James Franck Institute and Department of Chemistry, University of Chicago, Chicago, Illinois 60637, United States

Sudip Chattopadhyay* Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India

Uttam Sinha Mahapatra§ Department of Physics, Maulana Azad College, 8 Rafi Ahmed Kidwai Road, Kolkata 700013, India S Supporting Information *

ABSTRACT: Optimized geometries are evaluated for the ground and low lying excited states of cis-stilbene, trans-stilbene, and 4a,4b-dihydrophenanthrene (DHP) from calculations performed with the improved virtual orbital, complete active space configuration interaction (IVO-CASCI) method. The calculations indicate that a nonplanar conformer of trans-stilbene is the most stable among the isomers. The calculated ground and low lying excited state geometries agree well with experiment and with prior theoretical estimates where available. Our IVO-CASCI based multireference Möller−Plesset (MRMP) computations place the 1Bu state of trans stilbene to be ∼4.0 eV above the ground X1Ag state, which is in accord with experiment and with earlier theoretical estimates. The 11Bu state of trans-stilbene can be represented by the highest occupied molecular orbital (HOMO) → lowest unoccupied molecular orbital (LUMO) transition (ionic type) from the ground state, whereas its 21Bu state is dominated by the HOMO → LUMO+1 and HOMO-1 → LUMO transitions (covalent type). Likewise, the 11B and 21B states of cis-stilbene and DHP are also found to be of ionic and covalent types, respectively.

I. INTRODUCTION

derivatives have been of interest for more than half a century because of their important role in chemistry and in various photochemical and photophysical processes.2−14 Stilbene has two principal conformers, cis and trans. Room temperature experiments indicate that 35% of photoexcited cis-stilbene molecules isomerize to trans-stilbene, 10% cyclize to 4a,4bdihydrophenanthrene, and the remaining 55% return to the original reactant15,16 (see Figure 2). Stilbene isomers continue to be investigated experimentally17−52 and theoretically53−74 because the nature of some of its ground and excited state

Stilbene is widely used in manufacturing dyes, optical brighteners, scintillators, etc. Many stilbene derivatives occur naturally in plants, and several of them, e.g., resveratrol, a hydroxylated stilbene found in grapes, are reported to provide a variety of health benefits.1 Stilbene (see Figure 1) and stilbene

Special Issue: Oka Festschrift: Celebrating 45 Years of Astrochemistry Received: November 21, 2012 Revised: March 26, 2013

Figure 1. Structure and atom labeling of trans-stilbene. © XXXX American Chemical Society

A

dx.doi.org/10.1021/jp311493w | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

Article

very limited for the cis-stilbene. The first theoretical characterization of the ground and excited states structures of cis- and trans-stilbene by Warshel55 includes a description of the vibrational frequencies. Extensive calculations by Choi et al.71 for the ground state geometries and normal frequencies of cisand trans-stilbene apply HF, B3LYP, and MP2 theories with various basis sets. MP2 and B3LYP probes of the minimum energy structure by Kwasniewski et al.66 and Improta et al.,69 respectively, concur with other calculations66,69,71 and with experimental data47 indicating that the nonplanarity of the ground state of cis-stilbene arises due to strong van der Waals interactions between the H8 and H8′ atoms (see Figure 1). Compared to the cis isomer, information is even more scarce concerning the ground and excited states of 4a,4b-dihydrophenanthrene (DHP). The lack of stability, perhaps, renders this system difficult to access experimentally. To our knowledge, only a HF/4-21G calculation16,85 is available for the ground state geometry and vibrational frequencies of DHP, even though this polyene plays an important role in the photoisomerization of cis-stilbene. Controversy surrounds the assignments of the lowest two singlet excited states of Bu symmetry. The S1 and S2 states of trans-stilbene belong to the 1Bu representation under C2h pointgroup symmetry. The configuration of one of these states, to a large extent, is dominated by a HOMO → LUMO excitation [indicated as 1Bu (HL)] from the ground state,65 whereas the other [assigned as 1Bu(-)] is dominated by the HOMO → LUMO-1 and HOMO-1 → LUMO transitions [Ψ(1Bu(-)) = L−1 CL−1 − CLH−1 ΦLH−1, where Ci and Φi are the mixing H ΦH coefficients and the zeroth order wave functions, respectively]. Thus, the oscillator strength (f) for the X1Ag → 1Bu(-) transition is expected to be vanishingly small compared to that for the X1Ag → 1Bu(HL) transition. Since the intensity for excitation to the 1Bu(-) state is very low, experiments have been pursued33 for similar molecules to trans-stilbene which demonstrate that the 1Bu(-) state is higher in energy than the 1 Bu(HL) state. Using the complete active space second order perturbation theory (CASPT2),86,87 Molina et al.64 assign the 1 Bu(-) state [E(1Bu(-)) − E(X1Ag) = 4.03 eV, f = 0.04] to be lying 0.01 eV below the allowed 1Bu(HL) state [E(1Bu(HL) − E(X1Ag) = 4.04 eV, f = 0.7]. The predicted presence of a weak transition below the allowed, strong transition disagrees with experimental evidence17−19,22−25,33 and earlier theoretical predictions.62,63,72−74 Roos and co-workers65 have reexamined these two 1Bu states using multistate CASPT2 (MS-CASPT275) methods and obtain decent agreement with the experimental finding that the 1Bu(HL) state lies below the 1Bu(-) state. They further conclude that an accurate description of systems having quasi-degenerate states (like S1 and S2 in trans-stilbene), requires the inclusion of mixing between these states. In other words, independent CASPT2 calculations must be replaced by multistate CASPT2 procedures when energy states of the same symmetry are close in energy. Subsequent theoretical studies with time-dependent DFT (TD-DFT)68 and n-electron valence state second order perturbation theory (NEVPT2)67,88−91 agree that the allowed 1Bu(HL) state of trans-stilbene is more stable than the mixed 1Bu(-) state. The oscillator strength for the X1Ag → 1Bu(HL) transition is also an order of magnitude higher than the corresponding X1Ag → 1Bu(-) transition. In this article, we investigate some of the above-mentioned controversies, viz., the global minimum energy structure of stilbene and the character of its low-lying 1B states, by calculating the ground and excited state geometries of trans-

Figure 2. IVO-MRMP ground state potential energy versus reaction coordinate. The reaction coordinates α and β are C2−C1−C′1−C′2 (dihedral angle) and ∠C1−C1′ −C2′ of Figure 1, respectively.

structures, vibrational frequencies, excited state properties, etc. remain unresolved and controversial. The degree of nonplanarity ascribed to trans-stilbene depends on the experimental technique used and on the experimental conditions. For instance, fluorescence spectroscopy and ultra cold (6 K) collision-free molecular beam experiments34−36 suggest a planar structure for the ground state (S0) of trans-stilbene. On the other hand, a nonplanar structure is implicated from gas phase electron diffraction experiments.37−39 Collision-free molecular beam experiments concur with the presence of a planar excited state structure of transstilbene. In addition, the torsional angle between the phenyl groups appears to increase with temperature. For example, the phenyl groups are deduced to be twisted by 15−20° from UV photoelectron spectra40 at 393−483 K, whereas the gas phase electron diffraction experiment (at 473−553 K) imply a larger 30° twist.37 In summary, trans-stilbene is planar in the crystal phase, but may loose planarity in the gas phase. Despite extensive theoretical attempts to obtain a reliable and accurate description of the ground and excited state properties of trans-stilbene from a variety of semiempirical and ab initio many-body methods, some of these properties are inconclusive and depend on the method used in the calculations. Notably, the consistent force field (CFF) method,55 complete active space self-consistent field (CASSCF)75 calculations,65,70 density functional calculations66,71,76,77 based on the hybrid three parameter Lee−Yang−Parr (B3LYP) functional,78−80 and a 631G* coupled cluster81−83 calculations wth double excitations66 all yield a planar structure for the ground state of trans-stilbene. In contrast, geometry optimization using Hartree−Fock (HF) and second order Mö ller−Plesset (MP2) perturbation theories71 predict a nonplanar (C2) ground state. Molecular mechanics73 studies (using the CHARMM force field84) also favor a nonplanar conformation (see ref 66. for further details). Disagreements also exist in the assignment and in the magnitude of the normal-mode frequencies of the ground and excited states of trans-stilbene predicted by various theoretical and experimental methods. While a plethora of information is available for the ground state of trans-stilbene, theoretical and experimental data are B



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Table 1. Optimized Ground State Geometry of trans-Stilbene from IVO-CASCI and CASSCF Calculations with a (14/14) CAS that Comprises 7π and 7π* Molecular Orbitals (Full π Space)a parameters

b

point group C1−C1′ C1−C2 C2−C3 C3−C4 C4−C5 C5−C6 C6−C7 C7−C2 C1′ −C1−C2 C1−C2−C3 C2−C3−C4 C3−C4−C5 C4−C5−C6 C5−C6−C7 C6−C7−C2 C3−C2−C7 C1−C2−C7 C′1−C1−C2−C7 C2−C1−C′1−C′2

CASSCFc

CCDd

B3LYPe

B3LYPf

MP2f

CASSCF

IVO-CASCI

ANO

6-31G*

cc-pVDZ

6-31G*

6-31G*

cc-pVDZ

cc-pVDZ

C2h 1.350 1.476 1.404 1.399 1.396 1.402 1.394 1.409 126.7 118.5

C2 1.345 1.476 1.403 1.394 1.395 1.396 1.393 1.404 125.1 119.4 120.9 120.1 119.5 120.3 120.7 118.4 122.2 27.7 179.3

C2 1.351 1.468 1.410 1.396 1.397 1.401 1.393 1.411 127.1 118.7 121.4 120.1 119.3 120.5 121.0 117.3 123.6 0.0 180.0

C2 1.349 1.466 1.408 1.393 1.395 1.399 1.391 1.409 127.2

C2h 1.357 1.471 1.411 1.398 1.403 1.399 1.400 1.409 126.5

0.1 179.9

0.0 180.0

C2h (C2) 1.350 (1.328) 1.479 (1.478) 1.400 (1.386) 1.393 (1.391) 1.380 (1.390) 1.397 (1.396) 1.391 (1.389) 1.407 (1.404) 126.8 (126.3) 118.6 (119.0) 121.3 (121.4) 120.3 (120.0) 119.5 (119.3) 120.4 (120.3) 120.8 (120.7) 117.8 (118.2) 123.6 (122.7) 0.0 (21.3) 180.0 (179.9)

C2h (C2) 1.349 (1.328) 1.480 (1.478) 1.398 (1.386) 1.390 (1.390) 1.390 (1.389) 1.393 (1.396) 1.388 (1.389) 1.400 (1.404) 126.8 (126.3) 118.5 (119.0) 121.3 (121.7) 120.1 (120.0) 119.3 (119.3) 120.5 (120.3) 120.9 (120.7) 117.9 (118.2) 123.6 (122.7) 0.0 (20.4) 180.0 (179.9)

123.6 0.0 180.0

expt.g 1.326 1.471 1.392 1.384 1.381 1.383 1.381 1.397 126.4 119.0

123.2 0.0−30.0c

expt.h 1.329 1.481 1.398

127.7

32.5

a

The optimized geometries for the non-planar conformation from (10/10) CASSCF/6-31G* and (10/10) IVO-CASCI/6-31G* are shown in the parentheses. bBond lengths and bond angles are given in Å and degrees, respectively. cReferences 36, 37, and 42. dReference 65. eReference 66. f Reference 71. gFrom X-ray study. Reference 43. hFrom electron diffraction experiment. Reference 37.

state 0.31 eV above the 11Bu state. All these theoretical calculations concur with experimental findings33 in unambiguously assigning the lowest 1Bu state of the trans isomer as the 1 Bu(HL) state and the other one as the 1Bu(-) state. Interestingly, the 1B states of cis-stilbene also exhibit a similar trend,69 i.e., (a) E(11B) − E(X1A) ≈ 4.0 eV, (b) E(21B) − E(11B) ≈ 0.6 eV, (c) the 11B and 21B excited states are ionic and covalent types, respectively. The 11B and 21B states of DHP and the nonplanar trans-stilbene are found to be of B(HL) and B(-) types, respectively. The geometries obtained using IVO-CASCI method agree well with experiment and with earlier theoretical calculations. The ground and excited states normal-mode frequencies emerge from the IVO-CASCI calculations as all being real (see Tables S1−S6 in the Supporting Information), and hence, the geometries are stable. The IVO-CASCI vibrational frequencies are computed without empirical scaling, whereas the frequencies reported by Arenas et al.74 and Zhou et al.85 for the ground state of cis-stilbene and DHP apply the scaled quantum mechanical force field methodology.111 To our knowledge, no prior theoretical or experimental results are available for the normal-mode frequencies for the excited 3Ag state of trans-stilbene and the 3B state of DHP. In addition, we report the ground state vibrational frequencies of the nonplanar conformation of trans-stilbene, which is a stable isomer according to B3LYP71 and IVO-CASCI (as well as CASSCF) calculations.

stilbene, cis-stilbene, and DHP using the IVO-CASCI method, which has emerged as a computationally efficient and powerful tool that provides an efficient alternative to replace the more laborious CASSCF method by eliminating the multiconfigurational orbital optimization step altogether.92−95 The IVOCASCI orbitals (other than the occupied SCF orbitals) are simply obtained via an unitary transformation of unoccupied HF orbitals, with no further refinement required. Several prior applications95−105 demonstrate that these orbitals yield good nondynamical correlation energies with small reference spaces, a balanced description of ground and excited states, and geometries almost identical to CASSCF geometries (at a fraction of the computational cost). Our study includes the determination of (a) geometries and normal-mode frequencies of the ground and excited states of trans-stilbene, cis-stilbene, and DHP, (b) low-lying vertical excitation energies of these systems, and (c) the investigation of the potential energy curve of stilbene as a function of the C2− C1−C1′ −C2′ torsional angle (α) and the C1C2−phenyl angle (β) [see Figure 2]. The calculations are performed with 631G*,106,107 correlation consistent polarized double-ζ (ccpVDZ),108 and Karlsruhe polarized valence triple-ζ (KpVTZ)109,110 basis sets for various active space to analyze the influence of basis set and active space. The IVO-MRMP calculations place the lowest 1Bu states of trans- stilbene to be ∼3.9 eV above the ground X1Ag state, which is comparable to the experimental value of 4.0 eV.36 The X1Ag → 11Bu transition energy yielded by this method is also in good agreement with multistate CASPT2 (3.86 eV), statespecific (SS)-NEVPT2 (3.84 eV) and with the TD-DFT (3.91−4.10 eV) estimates. The IVO-MRMP predicts the 21Bu of trans-stilbene to be ∼0.6 eV higher than the 11Bu, which is also in accord with TD-DFT estimate69 (∼0.7 eV). The multistate CASPT2 method, on the other hand, places the 21Bu

II. COMPUTATIONAL DETAILS The geometry optimizations are performed using a (14/14) complete active space (CAS) with Pople’s 6-31G* basis106,107 and with a correlation consistent polarized valence double-ζ (cc-pVDZ) basis.108 Here, the first entry of the CAS designation (m/n) denotes the number of active electrons, C



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Table 2. Optimized Ground State Geometry of cis-Stilbenea

parameters

b

C1′ −C1 C1−C2 C2−C3 C3−C4 C4−C5 C5−C6 C6−C7 C7−C2 C1′ −C1−C2 C1′ −C1−C2−C7 C2−C1−C′1−C′2 a

MP266

B3LYP66

MP271

CASSCF

IVO-CASCI

IVO-CASCI

6-31G**

cc-pVDZ

6-31G*

6-31G*

6-31G*

cc-pVDZ

(C2)

(C2)

(C2)

(C2)

(C2)

(C2)

expt.47

1.350 1.473 1.404 1.395 1.396 1.397 1.393 1.405 126.7 43.1 5.6

1.352 1.477 1.409 1.396 1.398 1.400 1.395 1.409 131.1 34.2 6.7

1.356 1.478 1.408 1.398 1.402 1.400 1.401 1.407 126.8 42.4

1.327 1.486 1.396 1.380 1.393 1.392 1.393 1.399 129.2 44.6 4.2

1.327 1.487 1.397 1.387 1.387 1.390 1.390 1.398 129.2 44.9 3.7

1.349 1.488 1.400 1.393 1.384 1.392 1.393 1.402 129.0 44.5 4.4

1.334 1.489 1.398 1.398 1.398 1.398 1.398 1.398 129.5 43.2

The point-group symmetry is noted in parentheses. bBond lengths and bond angles are given in Å and degrees, respectively.

and the second entry indicates the number of active orbitals. This (14/14) complete active space encompasses the full π manifold which comprises seven π orbitals and seven π* orbitals. The computation of normal-mode frequencies, on the other hand, is performed with a smaller (10/10) active space generated by removing two lowest occupied π and two highest unoccupied π* orbitals from the (14/14) CAS. Vertical excitation energies are calculated for trans-stilbene using three different active spaces and two different basis sets to analyze the influence of electron correlation and basis set. The largest (14/14) active space is described above. The second (8/ 8) CAS is constructed by deleting the three lowest occupied π orbitals and three highest unoccupied π orbitals. The correlation consistent polarized double-ζ basis108 (256 molecular orbitals) and Karlsruhe valence triple-ζ basis with a set of single polarization functions (K-pVTZ)109,110 (338 molecular orbitals) are used to compute the excitation energies of trans-stilbene. The excitation energies of cis-stilbene and DHP are, however, determined from (14/14) IVO-MRMP/KpVTZ calculations. All these calculations (except for the nonplanar trans-stilbene) are performed at the (14/14)IVOCASCI/cc-pVDZ optimized geometry. The vertical excitation energies for the nonplanar conformation are carried out at the (10/10)IVO-CASCI/6-31G* optimized geometry. All calculations employ the GAMESS quantum chemistry software112 which has been interfaced with our IVO-CASCI module for generating improved virtual orbitals.

(a) The IVO-CASCI and CASSCF optimized geometries are almost identical to each other, a feature consistent with earlier findings by Chaudhuri et al.101−105 (b) While coupled cluster66 and B3LYP71 calculations with a 6-31G* basis set predict a nonplanar conformation for the ground state of trans-stilbene, the CASSCF65 with ANO basis,113 B3LYP66/cc-pVDZ, MP2/6-31G*71 and the present CASSCF and IVO-CASCI calculations with a cc-pVDZ (6-31G*) basis set favor a planar conformation. (c) Almost all the theoretical methods very slightly overestimate the C1C1′ and phenyl bonds of trans-stilbene by 0.002 and 0.001 Å with respect to experiment (see Table 1). (d) The dihedral angle and vinyl bond length predicted by the CASSCF and IVO-CASCI methods for the nonplanar trans-stilbene ground state are well within the experimental range (0−30)36,37,42 (see Table 1). (e) The IVO-CASCI/cc-pVDZ treatment overestimates the C1C1′ vinyl bond by 0.015 Å, but reproduces the C1− C2 single bond to within 0.001 Å (see Table 2). Table 1 indicates that the torsional angle C′1-C1−C2-C7 is accurately described by the MP2, CASSCF and IVO-CASCI methods, with the MP2/6-31G**66 torsional angle the most accurate (deviations of 0.1°). B3LYP66/cc-pVDZ calculations also favor a nonplanar conformation, but the torsional angle predicted by the B3LYP method is 10° less than experiment. Table 3 summarizes the optimized geometry of DHP as determined from (14/14) IVO-CASCI and B3LYP calculations with a cc-pVDZ basis, along with HF/4-21G predictions by Negri et al.16 for completeness. The agreement of the IVOCASCI/cc-pVDZ and HF/4-21G geometries is most likely a coincidence. This molecule must adopt a nonplanar conformation with similar C2−C′7 and C7−C′2 torsional angles to allow the cyclization of cis-stilbene to DHP. B. Excited State Geometries of trans- and cisStilbenes. Using the consistent force field (CFF) method, Warshel55 deduces a planar structure for the lowest 1Bu excited state of trans-stilbene which is in accord with CASSCF predictions using an ANO basis.65 On the other hand, Warshel55 and Improta et al.69 obtain a nonplanar geometry for the lowest 1B state of the cis isomer from the CFF and TDDFT procedures, respectively. Table 4 presents the IVOCASCI geometry for the first excited 1Bu and 1B states of the

III. RESULTS AND DISCUSSION A. Ground State Geometries of trans- and cis-Stilbene and 4a,4b-Dihydrophenanthrene (DHP). Tables 1 and 2 present a comparison of the ground state optimized geometry of trans- and cis-stilbene from (14,14) IVO-CASCI calculations with available theoretical and experimental (X-ray43,44 and electron diffraction37) data. The IVO-CASCI geometry optimizations for planar (nonplanar) trans-stilbene are pursued assuming C2h (C2) point-group symmetry. Electron diffraction data are unavailable for the phenyl bonds of trans-stilbene because electron diffraction is incapable of resolving the small bond length differences within the ring. Tables 1 and 2 exhibit several general trends in the optimized geometries which are summarized as follows: D



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Table 3. Optimized Ground X1A State Geometry of 4a,4bDihydraphenanthrene (DHP) from (14/14) IVO-CASCI calculations IVO-CASCI

B3LYP

HF85

parametersa

cc-pVDZ

cc-pVDZ

4-21G

C1′ −C1 C1−C2 C2−C3 C3−C4 C4−C5 C5−C6 C6−C7 C7−C2 C1′−C1−C2 C1−C2−C3 C2−C3−C4 C3−C4−C5 C4−C5−C6 C5−C6−C7 C3−C2−C7 C2−C7−C7′ −C2′

1.454 1.360 1.463 1.352 1.467 1.345 1.517 1.526 121.1 121.6 121.4 120.2 121.3 122.6 119.0 46.8

1.433 1.369 1.443 1.360 1.456 1.349 1.514 1.531 121.7 122.1 121.5 120.4 121.2 122.0 118.6 44.3

1.438 1.360 1.448 1.353 1.453 1.345 1.510 1.524 121.2 121.8 121.4 120.3 121.2 122.4 118.4 46.8

is close (within 0.01 Å) to that for a typical CC double bond, whereas the TD-DFT estimates are close to C−C single bond. To illustrate the nature of the conjugation, Table 5 presents the deviation of the optimized C1−C2, C1−C1′ , and C2−C3 bond lengths from those for standard single, double, and phenyl bonds, along with the differences of these C−C bond lengths between the cis and DHP isomers and the trans isomer. Inspection of Table 5 reveals that (a) the C1−C′1 and C2−C3 bonds in the ground state of the cis and trans isomers remain the same, (b) the trans → cis transition is accompanied by an increase in the C1−C2 bond length (the excited states also exhibit a similar increase), and (c) the variation in the C1−C2 and C1−C′1 bond lengths is most significant (∼0.1 Å) for the cis → DHP transition. C. Relative Stability. The normal-mode frequencies from the present calculations are all real (see the Supporting Information). Hence, the CASSCF and IVO-CASCI predicted nonplanar trans-stilbene ground state is stable and energetically lower (∼ 2 kcal/mol) than the planar conformer. However, some residual uncertainties still remain in the prediction of the global minimum energy structure because the (10/10) IVOCASCI/cc-pVDZ geometry optimization for the nonplanar conformation yields an imaginary frequency, whereas those from the IVO-CASCI/6-31G* procedure are all real. The relaxation pathways from cis-stilbene to DHP and transstilbene are depicted in Figure 2, where the ground state energy of cis-stilbene from (10/10) IVO-MRMP/cc-pVDZ calculations is plotted against the dihedral angle C2−C1−C′1−C′2 (α) and the angle ∠C2−C1−C′1= ∠C1−C′1−C′2 (β). The cis−trans and cis−DHP barrier heights estimated from this potential energy curve are 18 and 22 kcal/mol, respectively, but the estimates are rough because the geometry is constrained to the same as the cis geometry except for the angles α for cis−trans isomerization and β for cis−DHP ring closure. D. Vertical Excitation Energies. Table 6 presents a comparison of the vertical excitation energies of trans-stilbene from IVO-CASCI and IVO-MRMP calculations with prior theoretical and experimental data. As mentioned in the introduction, the 11Bu state of trans-stilbene is represented by a HOMO → LUMO transition, whereas the 21Bu state is dominated by HOMO → LUMO+1 and HOMO-1 → LUMO

a

Bond lengths and bond angles are given in Å and degrees, respectively.

trans and cis isomers, along with the geometry of the lowest trans-stilbene excited triplet state of Ag symmetry. The bond lengths and bond angles, summarized in Table 4 from IVO-CASCI calculations with a cc-pVDZ basis, agree favorably (maximum deviation of 0.002 Å for the C5−C6 phenyl bond) with those obtained using the CASSCF/ccpVDZ procedure. Indeed, the CASSCF/ANO and CASSCF/ cc-pVDZ optimized geometries for the lowest 1Bu of transstilbene are almost identical. The IVO-CASCI estimated C1 C′1 (C1−C2) bond is 0.05 Å (0.1 Å) longer than a typical double (single) bond. The IVO-CASCI and CASSCF procedures predict the C1C1′ bond length to be of vinyl type. On the other hand, the C1−C2 optimized bond length is found to be the average of the lengths for typical single and double bonds. Similar trends emerge for the lowest 1B state of cis-stilbene, where the C1C1′ vinyl bond from the IVO-CASCI treatment

Table 4. Optimized Excited State Geometries of trans- and cis-Stilbenea trans-stilbene

cis-stilbene

1

3

Bu

CASSCF

a

1

Ag

IVO-CASCI

B

CASSCF

IVO-CASCI

IVO-CASCI

parameters

ANO65

cc-pVDZ

6-31G*

6-31G*

6-31G*

6-31G*

C′1−C1 C1−C2 C2−C3 C3−C4 C4−C5 C5−C6 C6−C7 C7−C2 C′1−C1−C2 C1−C2−C3 C1′ −C1−C2−C7 C2−C1−C1′ −C2′

1.383 1.435 1.434 1.414 1.410 1.413 1.417 1.430 126.5 118.6 0.0 180.0

1.384 1.435 1.435 1.413 1.409 1.411 1.416 1.431 126.6 118.8 0.0 180.0

1.381 1.434 1.433 1.412 1.407 1.410 1.414 1.429 126.7 118.8 0.0 180.0

1.359 1.465 1.448 1.343 1.442 1.418 1.382 1.436 126.7 118.9 0.0 180.0

1.359 1.465 1.448 1.343 1.441 1.418 1.381 1.436 126.7 118.9 0.0 180.0

1.337 1.467 1.426 1.412 1.411 1.411 1.415 1.425 132.8 117.6 27.7

TD-DFT69 1.417 1.412 1.428

1.447 124.8 120.9 11.6

Bond lengths and bond angles are given Å and degrees, respectively. E



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Table 5. Variation in the Computed/Experimental C1−C1′, C1−C2, and C2−C3 Bond Lengths of the cis and DHP Isomers with Respect to Planar trans Conformer ground state cis standard C−C bond length C1−C1′ 1.34 (sp2−sp2)

C1−C2 1.54 (sp3−sp3)

C2−C3 1.40 (phenyl)

a

trans Ag

Δa

1.334 1.349 1.352 1.356

1.329 1.349 1.351 1.357

experiment IVO-CASCI B3LYP MP2

1.489 1.488 1.477 1.478

experiment IVO-CASCI MP2 B3LYP

1.398 1.400 1.411 1.409

method

X1A

experiment IVO-CASCI B3LYP MP2

. DHP

cis

A

Δb

11B

0.005 0.000 0.001 −0.001

1.454 1.433

−0.105 −0.081

1.471 1.480 1.468 1.471

0.018 0.008 0.009 0.006

1.360 1.369

1.398 1.398 1.408 1.410

0.000 0.002 0.003 −0.001

1.463 1.443

1

trans Bu

Δa

1.337

1.381

−0.044

0.128 0.108

1.467

1.434

0.033

−0.063 −0.032

1.426

1.433

−0.007

1

1

Δ(R) = R(cis) − R(trans). bΔ(R) = R(cis) − R(DHP).

Table 6. Vertical Excitation Energies (in eV) of trans-Stilbene and Nonplanar trans-Stilbene Calculated by IVO-CASCI and IVO-MRMP Methods Using Karlsruhe Polarized Valence Triple Zeta Basis Set (K-pVTZ)a trans-stilbene method

11Bu(HL)

21Bu(-)

IVO-CASCI IVO-MRMP CASSCF67 SS-NEVPT2(PC)67

5.65 (5.79) 3.69 (3.75) 5.95 4.01

6.28 (6.43) 4.30 (4.29) 6.10 4.76

IVO-CASCI IVO-MRMP CASSCF67 MS-CASPT265 SS-NEVPT2(PC)67 CASPT2b65 TD-DFT68 TD-DFT(PBE0)69 TD-DFT(cc-pVDZ) TD-DFT(K-pVTZ) expt.

5.50 (4.76) 3.87 (3.81) 6.15 3.86 3.84 4.04 3.91 3.94 3.94 3.86 4.036

6.10 (5.82) 4.43 (4.98) 4.82 4.17 4.65 4.03 4.58 4.60 4.54 4.53

nonplanar trans-stilbene 13Bu

11B(HL)

21B(-)

13B

CAS (8/8) 3.67 (3.71) 2.16 (2.10)

5.80 (6.01) 3.77 (3.86)

6.36 (6.59) 4.36 (4.45)

3.84 (4.00) 2.20 (2.00)

CAS (14/14) 3.68 (2.95) 2.27 (2.38)

5.65 (5.91) 3.93 (4.01)

6.20 (6.41) 4.47 (4.52)

3.86 (3.98) 2.29 (2.45)

4.00 3.91

4.57 4.51

2.35 2.33

2.27 2.34 2.13,452.1546

a

Results from cc-pVDZ basis are shown in the parentheses. The 1Bu states are assigned according to its dominant configuration(s). Here, the 1Bu states dominated by HOMO → LUMO and HOMO-1 → LUMO/HOMO → LUMO+1 are designated as 1Bu(HL) and 1Bu(-), respectively. bFrom a (10/12) CAS calculations with ANO basis set.

The relative positions of the low lying 1Bu states are assigned on the basis of the transition amplitudes of the dominant configuration(s) because they principally determine the oscillator strength for the transition between the ground X1Ag and the excited 1Bu states. The (8/8) IVO-CASCI/K-pVTZ calculations indicate that the HOMO → LUMO transition amplitude contributes 0.87 to the lowest excited 1Bu state, whereas the overall contribution from the HOMO → LUMO+1 and HOMO-1 → LUMO transition amplitudes to the 2 1 B u state is 0.75. The contributions from the HOMO → LUMO+1 and HOMO-1 → LUMO amplitudes to the 21Bu state are 0.32 and 0.43, respectively. Calculations for the nonplanar conformation find that the HOMO → LUMO transition amplitude to the lowest

transitions. The (10/12) CASPT2 calculations of Molina et al.64 predict the 11Bu state to be of lower intensity than the 21Bu state. Thus, in order to reproduce experimental observations, the contribution from the HOMO → LUMO transition amplitude should be dominant for the 11Bu state wave function. On the other hand, the major contribution to the 21Bu state should come from the HOMO → LUMO+1 and HOMO-1 → LUMO transition amplitudes. The excitation energies and oscillator strengths from MS-CASPT2,65 NEVPT2,67 and TDDFT68,69 (and from our TD-DFT estimates) are, however, in accord with the experiment. Comparison with experimental data is precluded because experimental values refer to absorption maxima, while our computations are for vertical excitation energies. F



Article

excited 1B state is 0.87, and the overall contribution of the HOMO → LUMO+1 and HOMO-1 → LUMO transitions to the 21Bu state wave function is 0.75. The (8/8) IVO-CASCI calculations assign the 1Bu and 21Bu states as 1Bu (HL) and 1 Bu(-), respectively, and this character persists in IVO-MRMP computations. For instance, the HOMO → LUMO, HOMO-1 → LUMO and HOMO → LUMO-1 transition amplitudes contribute 0.55 (0.34), 0.14 (0.19), and 0.16 (0.28) to the first (second) excited 1B u state. The IVO-MRMP/K-pVTZ calculations predict the 1Bu (1B) and 21Bu (21B) states of the planar (nonplanar) conformer to be 3.69 (3.77) and 4.30 (4.36) eV above the ground 1Ag state, which are ∼0.3 eV off from the SS-NEVPT2 estimates of 4.01 and 4.76 eV, respectively. The calculations further show (a) the excited state energies obtained using cc-pVDZ and K-pVTZ basis set are almost identical and (b) the excitation energy from the ground to the lowest excited state of B-symmetry is more accurate for the nonplanar conformer than the planar conformer. The wave functions from the (14/14) IVO-CASCI calculations indicate that the HOMO → LUMO transition amplitude has a weight of 0.85 in the first excited 1Bu state, while the HOMO → LUMO+1 and HOMO-1 → LUMO amplitudes contribute 0.42 and 0.30, respectively, to the second excited 1Bu state. The wave function from the (14/14) CAS IVO-MRMP calculations, on the other hand, assign maximum weight (0.44) to the HOMO → LUMO transition for the 11Bu state, while the 21Bu state is dominated by the HOMO → LUMO+1 (0.44) and HOMO-1 → LUMO (0.14) transitions. The 1Bu(HL) state lies at a lower energy (3.87 eV) than the covalent 1Bu(-) state (4.43 eV) in the (14/14) IVO-MRMP calculations. Both (8/8) and (14/14) IVO-MRMP calculations place the ionic 1Bu(HL) state ∼0.6 eV below the covalent 1Bu() state, which is consistent with experiments and with TD-DFT and CAS based results.65,67 The IVO-CASCI calculations further exhibit that (a) the dynamical electron correlation contribution is roughly the same (∼1.7 eV) for the ionic 1 Bu(HL) state and the covalent 1Bu(-) state, (b) the excitation energies computed from the (14/14) IVO-MRMP method are of comparable accuracy to other (14/14) CAS space estimates, and (d) the excitation energies obtained using the K-pVTZ and cc-pVDZ basis sets are almost identical. The excitation energies of the lowest triplet state of transstilbene from IVO-CASCI and IVO-MRMP calculations are compared with experiment and with other correlated calculations in Table 6. The experimental study by Evans114 places the origin at 2.20 eV for the T1 → S0 transition of transstilbene in chloroform at 76 atm in O2. Dyck et al.7 observe the T1 → S0 absorption band at 2.15 eV in ethyl iodide. Aime et al.115 find the absorption band at 2.15 eV in single crystals, while Saltiel et al.116 and Ikeyama et al.46 observe this band at 2.14 and 2.13 eV, respectively. The T1 → S0 transition energy computed by Molina et al.64 with CASSCF and CASPT2 methods, an atomic natural basis (ANO), and a (10/12) complete active space is 2.56 eV, which deviates by 0.4 eV from the observed value. The IVO-MRMP reproduces the experimental triplet energy within 0.1 eV. To our knowledge, experimental data are unavailable for the ground to excited state transition energies of cis-stilbene [see Table (7)]. Thus, we benchmark our predictions against TDDFT calculations of Improta et al.69 and additional TD-DFT estimates (with cc-pVDZ and K-pVTZ basis sets) generated by us using GAMESS112 (see Table 7). Interestingly, the TD-DFT

Table 7. Vertical Excitation Energies (in eV) of cis and DHP Calculated by IVO-CASCI and IVO-MRMP Methods Using (14/14) CAS with Karlsruhe Valence Triple Zeta Basisa method

symmetry 1

cis-stilbene

1 B(HL)

IVO-CASCI IVO-MRMP TD-DFT(K-pVTZ) TD-DFT(cc-pVDZ) TD-DFT(PBE0)69 DHP 16

CNDO/S QCFF16 IVO-CASCI IVO-MRMP TD-DFT(cc-pVDZ) TD-DFT(K-pVTZ) expt.41

6.14 4.04 4.02 4.09 4.09 11B(HL) 3.15 3.77 3.93 2.87 2.40 2.36 2.76

21B(-)

13B

6.32 4.69 4.51 4.58 4.61

5.02 2.81 2.60 2.61

21B(-)

21A

31A

4.18 3.28 5.32 3.83 4.13 4.02

3.59 3.68 4.48 3.06 3.26 3.23

4.63 5.15 5.79 4.09 3.92 3.88 4.00

a The 1Bu states are assigned according to its dominant configuration(s).

calculations predict that the oscillator strength of the 11B state is larger than that of the 21B state, a feature which is similar to the behavior of the trans conformer. Moreover, the TD-DFT estimated 1B and 3B state transition energies for the cis isomer are pretty close (0.1 eV for 1B and 0.01 eV for 3B) to those of the 1Bu and 3Bu states of the trans conformer. The excitation energies of these states from IVO-MRMP calculations with a (14/14) CAS agree well (to 0.05 eV) with the TD-DFT determinations (see Table 7) and both these calculations place the 1B(HL) below the 1B(-). Table 7 compares the IVO-CASCI and IVO-MRMP geometries of the low lying S0 (1A) excited electronic states of DHP with experiment41 and with those computed using the CNDO/S [Complete Neglect of Differential Overlap (Second Version)117−119] and QCFF/PI (semiempirical π−electron method120,121) by Negri et al.16 The vertical energies estimated from TD-DFT with cc-pVDZ and K-pVTZ bases are also listed in Table 7. The transition energies provided by the IVOMRMP approach for the 11B(HL) and 31A states are 0.11 and 0.09 eV higher than the observed values. The corresponding CNDO/S, QCFF, and TD-DFT/K-pVTZ (TD-DFT/ccpVDZ) estimates are off by 0.39, 1.01, and 0.36 (0.40) eV and 0.63, 1.15, and 0.08 (0.12) eV, respectively. The IVO-based calculations further indicate that the 11B and 21B excited states are of B(HL) and B(-) types, respectively. E. Potential Energy Curve. The relaxation pathways of cisstilbene to DHP and trans-stilbene is depicted in Figure 2, where the ground state energy of cis-stilbene from (10/10) IVO-MRMP/cc-pVDZ calculations is plotted against the dihedral angle C2−C1−C1′ −C2′ (α) and the angle ∠C2−C1− C1′ = ∠C1−C1′ −C2′ (β). The cis−trans and cis−DHP barrier heights estimated from this potential energy curve are found to be 18 and 22 kcal/mol, respectively. This is, however, a rough estimate of the barrier heights because the geometry is constrained to the cis geometry except for the angles α for cis−trans isomerization and β for cis−DHP ring closure.

IV. CONCLUSION Extensive comparisons with experiment and prior high level computations demonstrate the accuracy of the IVO-CASCI G



Article ‡

methods for geometries and vibrations and the IVO-MRPT methods for the excitation energies and character of the low lying excited states. Of particular note is the fact that the geometries represent stable minima. These comparisons instill confidence in the accuracy of the predictions for cases in which experimental and/or theoretical data are absent. The ground state geometries and vibrational frequencies of cis-stilbene, trans-stilbene, and DHP are evaluated using the improved virtual orbital complete active space configuration interaction (IVO-CASCI) method. The vibrational frequencies of the ground state of these molecules are all found to be real, thus confirming the stability of these structures. The calculated geometries and vibrational frequencies agree favorably well with experiment and with prior estimates where available. Similar IVO-CASCI calculations for the excited states of these system yield real normal-mode frequencies for the 11Bu and 3Ag states of trans-stilbene, the 1B state of cis-stilbene, and the 3B state of DHP The unscaled IVO-CASCI vibrational frequencies for the 11Bu state agree well with observed low frequency modes. This agreement suggests that the vibrational frequencies predicted for other excited states should be reliable. Some residual uncertainties remain in the prediction of the global minimum energy structure because the (10/10) IVO-CASCI/cc-pVDZ geometry optimization for the nonplanar conformation yields an imaginary frequency, whereas those from the IVO-CASCI/ 6-31G* procedure are all real. The TD-DFT and CAS calculations concur that the lowest excited 1Bu state of trans-stilbene lies ∼4.0 eV above the ground X1Ag state. These calculations further show that the oscillator strength for the X1Ag → 11Bu transition is higher than that for the X1Ag → 21Bu transition, as the 11Bu state is primarily dominated by a HOMO → LUMO transition. Theoretical calculations find the lowest 1B state of cis-stilbene to be ∼4.0 eV above the ground state and of B(HL) type. Interestingly, the lowest excited 1B state of DHP also exhibits similar trends, except for the X1A → 1B(HL) transition energy that in DHP is 2.75 eV (less than the X1A−11B(HL) energy gap in cis- and trans-stilbene). In closing, let us mention that although the presented results are promising and can be compared very well to the many-body methods from the literature, a much more extensive testing must be performed in the context of global minimum energy structure for nonplanar trans-stilbene ground state. This will be attempted in the near future.

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E-mail: [email protected]. E-mail: [email protected].

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ACKNOWLEDGMENTS This research is supported, in part, by the Department of Science and Technology (DST), India (Grant SR/S1/PC-61/ 2009) and NSF Grant No. CHE-1111918.

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(1) Baur, J. A.; Sinclair, D. A. Therapeutic Potential of Resveratrol: The in vivo Evidence. Nat. Rev. Drug Discovery 2006, 5, 493−506. (2) Malkin, S.; Fischer, E. Temperature Dependence of Photoisomerization. III.1Direct and Sensitized Photoisomerization of Stilbenes. J. Phys. Chem. 1964, 68, 1153−1163. (3) Hammond, G. S.; Saltiel, J.; Lamola, A. A.; Turro, N. J.; Bradshaw, J. S.; Cowan, D. O.; Counsell, R. C.; Vogt, V.; Dalton, C. Mechanisms of Photochemical Reactions in Solution. XXII. Photochemical cis-trans Isomerization. J. Am. Chem. Soc. 1964, 86, 3197− 3217. (4) Lomola, A. A.; Hammond, G. S.; Mallory, F. B. The Blue Emission from cis-Stilbenes. Photochem. Photobiol. 1965, 4, 259−263. (5) Muszkat, K. A.; Gegiou, D.; Fischer, E. Temperature Dependence of Photoisomerization. IV. Evidence for the Involvement of Triplet States in the Direct Photoisomerization of Stilbenes. J. Am. Chem. Soc. 1967, 89, 4814−4815. (6) Phillips, D.; Lemaire, J.; Burton, C. S.; Noyes, W. A., Jr. Advances in Photochemistry; Noyes, W. A., Jr., Hammond, G. S., Pits, J. N., Jr., Eds.; Interscience: New York, 1968; Vol. V. (7) Dyck, R. H.; McClure, D. S. Ultraviolet Spectra of Stilbene, pMonohalogen Stilbenes, and Azobenzene and the trans to cis Photoisomerization Process. J. Chem. Phys. 1962, 36, 2326−2345. (8) Berveridge, D. I.; Jaffe, H. H. The Electronic Structure and Spectra of cis- and trans-Stibene. J. Am. Chem. Soc. 1965, 87, 5340− 5346. (9) Gelbart, W. M.; Freed, K. F.; Rice, S. A. Internal Rotation and the Breakdown of the Adiabatic Approximation: Many-Phonon Radiationless Transitions. J. Chem. Phys. 1970, 52, 2460−2473. (10) Borrell, P.; Greenwood, H. H. The Photochemistry of Stilbene: Some s.c.f. Molecular Orbital Calculations. Proc. R. Soc. London A 1967, 298, 453−466. (11) Bromberg, A.; Muszkat, K. A. The Geometry of Hindered Stilbenes in Their Ground and Excited States. Tetrahedron 1972, 28, 1265−1274. (12) Dou, Y.; Allen, R. E. Another Important Coordinate in The Photoisomerization of cis-Stilbene. Chem. Phys. Lett. 2003, 378, 323− 329. (13) Todd, D. C.; Jean, J. M.; Rosenthal, S. J.; Ruggiero, A. J.; Yang, D.; Fleming, G. R. Fluorescence Upconversion Study of Cis-Stilbene Isomerization. J. Chem. Phys. 1990, 93, 8658−8668. (14) Todd, D. C.; Fleming, G. R.; Jean, J. M. Calculations of Absorption and Emission Spectra: A Study of cis-Stilbene. J. Chem. Phys. 1992, 97, 8915−8925. (15) Petek, H.; Yoshihara, K.; Fujiwara, Y.; Lin, Z.; Penn, J. H.; Fedrick, J. Is the Nonradiative Decay of S1 cis-Stilbene Due to the Dihydrophenanthrene Isomerization Channel? Suggestive Evidence from Photophysical Measurements on 1,2-diphenylcycloalkenes. J. Phys. Chem. 1990, 94, 7539−7543. (16) Negri, F.; Orlandi, G. Theoretical Analysis of the Force Field and Resonance Raman Spectrum of 4a,4b-Dihydrophenanthrene. Chem. Phys. Lett. 1992, 195, 523−530. (17) Saltiel, J.; Megarity, E. D. Mechanism of Direct cis−trans Photoisomerization of the Stilbenes. Solvent Viscosity and the Azulene Effect. J. Am. Chem. Soc. 1972, 94, 2742−2749. (18) Sun, Y. P.; Saltiel, J. Application of the Kramers Equation to Stilbene Photoisomerization in n-Alkanes using Translational Diffusion Coefficients to Define Microviscosity. J. Phys. Chem. 1989, 93, 8310−8316.

ASSOCIATED CONTENT

S Supporting Information *

Normal mode frequencies (in cm−1) for the ground state of planar and nonplanar trans-stilbene. Vibrational frequencies (cm−1) for the ground state of cis-stilbene and 1A from DHP IVO-CASCI calculations. Vibrational frequencies (cm−1) for the first excited 1Bu and 3Ag states of trans-stilbene. Vibrational frequencies (cm−1) for the first excited states of cis-stilbene and DHP from IVO-CASCI calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: sudip_chattopadhyay@rediffmail.com. Notes

The authors declare no competing financial interest. † E-mail: [email protected]. H



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