are azobenzenophanes rotation-restricted?

THE JOURNAL OF CHEMICAL PHYSICS 123, 174317 �2005�

Are azobenzenophanes rotation-restricted?Cosimo Ciminelli, Giovanni Granucci, and Maurizio PersicoDipartimento di Chimica e Chimica Industriale, via Risorgimento 35, 56126 Pisa, Italy

�Received 4 August 2005; accepted 7 September 2005; published online 3 November 2005�

We simulated the photoisomerization dynamics of an azobenzenophane with a semiclassical surfacehopping approach and a semiempirical reparametrized quantum mechanics/molecular mechanicsHamiltonian. Only one of the two azobenzene chromophores in the molecule is taken into accountquantum mechanically: the other one is treated by molecular mechanics. Both n→�* and �→�* excitations are considered. Our results show that the photoisomerization reaction mainlyinvolves the rotation around the NvN double bond. The excited state relaxation features are inqualitative agreement with experimental time-resolved fluorescence results. © 2005 AmericanInstitute of Physics. �DOI: 10.1063/1.2098628�

I. INTRODUCTION

The photoisomerization dynamics of azobenzene and itsderivatives has been widely studied because of their impor-tance as photochromic compounds.1–16 The mechanism ofphotoisomerization has been for a long timecontroversial.1,17–21 The most recent experimental and theo-retical studies of the azobenzene photoisomerization in non-viscous solvents point to a rotational mechanism with out-of-plane torsion of the NvN double bond.22–25

Less clear is what happens when azobenzene is in a“constraining” environment induced by a chemical modifica-tion or by a viscous solvent. More than 20 years ago, Rauand Lüddecke17 measured the trans→cis photoisomeriza-tion quantum yields for the 2 ,19-dithia�3.3��4,4’�-diphenildiazeno�2�phane �2S-ABP, see Fig. 1�. In thiscompound, two -CH2-S-CH2- bridges keep two azobenzenemoieties in a face-to-face position. The quantum yield withn→�* excitation �436 nm� was found 0.24, close to that ofazobenzene in the same solvent �benzene�. Instead, with �→�* excitation �366 nm� the quantum yield was 0.21, twiceas that of azobenzene: rather surprisingly, the structural con-straints make this compound “more isomerizable.”

Recently it has been found that the rigidity of theazobenzenophanes increases the S1 relaxation times: in par-ticular, from time resolved fluorescence measurements Luet al.24 obtained S1 lifetimes of 5.7 ps for 2S-ABP and 56 psfor 4S-ABP �with four sulphur bridges�, versus about 1 ps fortrans-azobenzene �TAB�. Similar results have been reportedby Pancur et al.26 for an azobenzene derivative capped witha crown ether. It is worth to note that solvent viscosity hassimilar but more complex effects. Tahara and co-workers21,27

performed time-resolved Raman measurements on azoben-zene after S2 excitation in both hexane and ethylene glycolsolvents. The observed S1 state was found to have a planarstructure around the NvN double bond and a lifetimestrongly dependent on the solvent: �1 ps in hexane and�12 ps in ethylene glycol. In the time-resolved fluorescenceanisotropy measurements of azobenzene upon S1 excitation,performed by Diau and co-workers,23 the fluorescence depo-
larization rate measured in hexane turned out to be equal to
0021-9606/2005/123�17�/174317/10/$22.50 123, 1743

Downloaded 05 Nov 2005 to 131.114.75.86. Redistribution subject to

the slow component of the fluorescence transients �1–2 ps�.In ethylene glycol the fluorescence anisotropy exhibited nodiscernible decay feature, while the decay rate of the corre-sponding transient was 3–4 ps. These results show that, in aviscous solvent, the S1 lifetime is longer and, since the tor-sional motion is considered responsible for the observed de-polarization, the deactivation of the S1 state should mainlyinvolve a nonrotational mechanism.

Torsion of the double bond is thought to be blocked, orat least severely hindered, in the azobenzenophanes. How-ever, even in free azobenzene our previous simulations of thephotodynamics25 show that the out-of-plane rotation aroundthe NvN bond occurs without displacing to a large extentthe phenyl rings, nor changing the relative orientation oftheir planes: this requires the simultaneous torsional motionaround the N-C bonds, a sort of “hula-twist.” In thispaper we perform a simulation of the nonadiabatic dynamicsof the 2 ,19-dithia�3.3��4,4’�-trans-diphenildiazeno�2�phane �2S-TTABP� after n→�* and �→�* excitation.Our aim is to investigate the photoisomerization dynamics of2S-TTABP with emphasis on the mechanism and on thecomparison with free azobenzene.

II. METHOD AND POTENTIAL ENERGY SURFACES

The molecular system considered in this study is shownin Fig. 1. All the calculations have been done with our de-velopment version of the MOPAC package.28 The quantummechanical treatment of two interacting azobenzene chro-mophores poses serious problems of computational effi-

FIG. 1. The 2S-TTABP isomer. Balls represent the QM atoms, sticks the
MM atoms. The connection atoms are shown in black.
© 2005 American Institute of Physics17-1

AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

http://dx.doi.org/10.1063/1.2098628

http://dx.doi.org/10.1063/1.2098628

174317-2 Ciminelli, Granucci, and Persico J. Chem. Phys. 123, 174317 �2005�

ciency and consistency. In fact the molecular orbital �MO�active space should contain the same number of n ,�, and �*

orbitals for each of the azobenzene units, and it would bedifficult to avoid including also the lone pairs of the S atoms,thus making the calculations totally impractical. Therefore,to describe the electronic structure of 2S-ABP we resort to aquantum mechanical/molecular mechanical �QM/MM�strategy.29,30 One azobenzene moiety is treated quantum me-chanically, with the same semiempirical method used in ourstudy25,31 of the photoisomerization of the bare azobenzene:reparametrized AM1 Hamiltonian, MO from SCF with float-ing occupation numbers and MRCI �CAS-CI with four elec-trons in six MOs �Ref. 35� and single excitation involvingthe last five occupied and the first five virtual MOs, for atotal of 259 determinants�. With this Hamiltonian we wereable to nicely reproduce very accurate ab initio results.22,32–34

Since to our knowledge only very few data are known about2S-ABP �see Table I� we did not attempt any further rep-arametrization. The QM part also contains the two carbonatoms directly bound to the azobenzene, treated as connec-tion atoms30 with the MM part and labeled as CCA. The semi-empirical parameters of the CCA have been taken fromAntes and Thiel.36 The rest of the molecule is treatedat the molecular mechanics level. For the MM moiety theGAFF-AMBER force field has been employed, withatomic charges obtained from the AM1 electrostatic potentialof 2S-TTABP, using the stabilization method of Lévy andEnescu.37,38 In particular we have used �=0.02 bohr−2, lead-ing to the elimination of two eigenvalues of the least-squaresmatrix, and the Mulliken atomic charges were taken as thereference set q0. Because of symmetry restrictions, only 10independent atomic charges were determined. According toAntes and Thiel’s prescription, the two CCA had atomiccharges equal to 1, while the charges of the four MM hydro-gens directly bound to them were set to zero: to ensure theneutrality of the MM moiety the charges of the two sulphuratoms were reset from −0.0291 to −0.0146.

In a previous paper30 we used a different QM/MM par-tition for 2S-ABP, in which the connection atoms were the S

TABLE I. Geometrical parameters �in Å� and vertical excitation energies �ineV� for 2S-TTABP. Dazo is the distance between the two azo groups. Dring isthe average distance between the two phenyl rings. The transition energyshifts are referred to TAB. A positive value indicates a blue shift from TABto 2S-TTABP.

QM/MM Exp.a

DistancesDazo 3.63 3.86Dring 3.41 3.57

CCA-CQM 1.468CCA-SMM 1.829

Vertical transition energiesS1 3.09 2.79S2 4.16 3.65

S1 shift 0.15 �0S2 shift −0.12 −0.25

aReferences 17 and 24.

atoms. Such a kind of partition has been abandoned because


of spurious interaction of SCA with the QM part. In fact, theconnection atom is only provided with one electron in s-typeorbital; for SCA that orbital has to be sufficiently diffuse toyield a SCA-CQM bond of the right length �about 1.8 Å, seeTable I�, and this results in a spurious attractive interactionwith other QM atoms that may approach SCA during the dy-namics or the geometry optimizations �for instance the CH2

hydrogens, see Fig. 1�.Analytical gradients of the CI energy were evaluated

with our implementation of the Z-vector formalism put for-ward by Patchkovskii and Thiel for semiempiricalmethods,39,40 extended to the treatment of floating occupa-tion numbers MOs as shown in the Appendix. This advance-ment greatly improved the efficiency of geometry optimiza-tions and trajectory calculations �an order of magnitudefaster than with the previous technique29,31,41,42�.

In Table I we show some geometrical parameters of2S-TTABP at its ground state equilibrium geometry, alongwith the vertical excitation energies n→�* and �→�*.Overall, our results compare reasonably well with the avail-able experimental data. The main drawback of our QM/MMcalculations is the complete neglect of the interaction be-tween the two chromophores. As expected on the basis of theexciton coupling model �see Granucci et al.43 for a recentcase study�, around the ground state equilibrium geometrysuch interaction is especially strong for the �→�* state �S2

in TAB�, because of its large transition dipole. In fact, afeature at 380 nm �3.26 eV� in the experimental absorptionspectrum24 of 2S-ABP indicates a non-negligible interactionbetween the �→�* states, while the two n→�* states seemto absorb independently.17

Figures 3 and 4 show the potential energy curves of thefirst two singlets along the torsion and inversion isomeriza-tion pathways. We found two different forms for the2S-CTABP isomer, produced through either the torsion or theinversion pathways from 2S-TTABP, labeled 2S-CTABPt

i i

FIG. 2. The 2S-CTABP isomer. Top panel: the 2S-CTABPi form; bottompanel: the 2S-CTABPt form �see text�.

and 2S-CTABP , respectively �see Fig. 2�. 2S-CTABP is


174317-3 Are azobenzenophanes rotation-restricted? J. Chem. Phys. 123, 174317 �2005�

more stable than 2S-CTABPt by about 0.3 eV �Ref. 44� �seeTable II�. Both forms are non-negligibly strained, as one cansee from Table II comparing energies and NNC angles withthe free cis-azobenzene �CAB�: note also that the NNCangles in 2S-TTABP and TAB are almost identical. However,the potential energy curves shown in Figs. 3 and 4 look verysimilar to the corresponding ones for azobenzene �see Ref.25�, the main differences being: �a� the cis isomer is higherin energy; �b� following the inversion minimum energy pathin S1, a surface crossing with S0 is encountered well beforethe midpoint of the reaction coordinate. Due to the latterfeature, the inversion mechanism should be even less effec-tive in 2S-TTABP than in TAB. Moreover, our results showthat in no way 2S-ABP can be considered rotation-lockedwith respect to azobenzene. In fact, for azobenzene, goingfrom TAB to the rotamer we found an energy increase of2.05 eV in S0 and an energy decrease of 0.28 eV in S1 �0.74eV starting from the Franck–Condon point�. Note that thedifferences between the potential energy surfaces of azoben-zene and of 2S-ABP, due to the constraints of the ring struc-ture and to steric hindrances, are only marginally influencedby the details of the semiempirical calculations.

III. PHOTOISOMERIZATION DYNAMICS

The simulation of the photoisomerization dynamics of2S-ABP has been carried out with a semiclassical trajectory

FIG. 3. Potential energy curves for the torsion around the NvN bond. Allinternal coordinates are optimized as the CNNC angle of the QM part is

TABLE II. Energies and geometries of ground state stationary points forazobenzene �from Ref. 25� and 2S-ABP �this work�. Energies with respect toTAB for azobenzene and to 2S-TTABP for 2S-ABP in eV. Distances in Å,angles in degree. In parentheses, values for the MM part.

Structure Energy RNN �NNCa�CNNC

2S-TTABP 0.00 1.239�1.235� 116�118� 178�176�2S-CTABPt 1.21 1.215�1.234� 132�118� 10�176�2S-CTABPi44 0.92 1.220�1.234� 130�119� 1�160�

131�116�

TAB 0.00 1.239 117 180CAB 0.40 1.221 124 4

aIn case of asymmetric geometries we show two different values.

varied. Solid lines: S0 is optimized. Dashed lines: S1 is optimized.


surface hopping approach with “on the fly” semiempiricalQM/MM calculation of energies and wavefunctions. Themethod has been put forward by our group29–31 and is imple-mented in our development version of the MOPACpackage.28

The initial coordinates and momenta were sampled withthe following procedure. First, a Monte Carlo calculationwas run �2�105 steps, T=298 K� on S0, starting from theequilibrium geometry of 2S-TTABP. For each set of geom-etries generated by the Monte Carlo procedure, a set ofnuclear velocities was also produced according to the equi-partition principle, with random directions. Second, a Lange-vin dynamics during 50 fs was run starting from each set ofcoordinates and velocities previously obtained. Two swarmsof 164 and 241 trajectories were run with two differentranges of initial excitation energies: 2.5–3.5 eV and 3.7–4.7eV in order to reproduce the n→�* and �→�* excitations,respectively. The vertical excitation procedure is described inAppendix C of Ref. 25. The integration time step was 0.1 fs.Each trajectory was terminated according to the followingstop conditions: �a� the time spent on the S0 PES exceeded0.5 ps; or, �b� the total time exceeded 5 ps.

The quantum yield � of the photoisomerization reactionis obtained as the fraction of reactive trajectories �i.e., thoseending up at the 2S-CTABP isomer� with respect to the totalnumber of trajectories. The results are shown in Table IIIalong with the experimental quantum yields obtained by Rau

TABLE III. Quantum yields for the trans→cis photoisomerization reaction.The standard deviation for the quantum yield � computed over a total of NT

trajectories is �=��1−�� /NT.

n→�* excitation �→�* excitation Ref.

Computed2S-TTABP 0.067±0.019 0.070±0.016 This work

TAB 0.33±0.03 0.15±0.02 25

Experimental2S-TTABP 0.24 0.21 17

TAB 0.20-0.36 0.09-0.20 1

FIG. 4. Potential energy curves for the N inversion. All internal coordinatesare optimized as one NNC angle of the QM part is varied. Solid lines: S0 isoptimized. Dashed lines: S1 is optimized.



and Lüddecke17 and the corresponding values for TAB pho-toisomerization. Probably the most striking difference be-tween the 2S-TTABP and TAB photoisomerization reactionsis the lack of wavelength dependence of the quantum yieldfor 2S-TTABP. Our simulations are able to reproduce thisexperimental result:17 we found a non-negligible quantumyield for the 2S-TTABP photoisomerization and the twokinds of excitation �n→�* and �→�*� result in very similarvalues of �. However, our computed quantum yields aremuch lower than the experimental ones.

Such disagreement can be explained at least in part withthe fact that in our simulations only one of the two azogroups of 2S-ABP is photoreactive �the one being treated atthe QM level�. The QM/MM scheme does not allow for anyeffect arising from the interaction between the excited statesof the two chromophores, such as excitation transfer andmodifications of the PES. According to the exciton couplingmodel, at non-symmetric geometries the excitation localizeson the azobenzene moiety with lower S1 energy. A distorsionof one of the two chromophores along the reaction coordi-nate, lowering the S1 energy, concentrates the excitation en-ergy on that chromophore, thus increasing the quantumyield: in other words, each chromophore may act as an an-tenna for the other one. The absence of this effect in oursimulations may contribute to the underevaluation of thequantum yields. Another source of disagreement can derivefrom the overestimation of the rate of excited state decay bythe surface hopping method45 �see below for more details�.

All the reactive trajectories are shown in Ref. 46. In Fig.
5 we show the values of the CNNC and NNC angles of the

QM part, as functions of the time elapsed after excitation.We averaged the angles separately for the reactive and theunreactive trajectories. Again, as already pointed out abovefor the potential energy curves, these results are very similarto the corresponding ones for azobenzene �see Ref. 25�, inparticular for the NNC angles. As in azobenzene, the isomer-ization mechanism mainly involves the rotation around theNvN bond, both for the n→�* and for the �→�* excita-tion. This point can be further elucidated considering Fig. 6,where all the reactive trajectories are shown in two polar r ,�plots �one for n→�* excitation, 11 trajectories, and the otherfor �→�* excitation, 17 trajectories�. The r coordinate isthe complement of the largest �NNC to 180°, while �=�CNNC. All the trajectories start on the left-hand side ofthe plot, corresponding to a neighborhood of the Franck–Condon point, then proceed toward the 2S-CTABP isomer�right-hand side�. A trajectory showing a “pure” inversion-type mechanism should go from the left to the right along astraight line, while a “pure” rotational trajectory should de-scribe a semicircle. A remark, valid for all the trajectories,can be drawn by inspecting Fig. 6: during the first half of theisomerization process, that takes place on the S1 surface, theNNC angles just oscillate around their equilibrium value.Thus, at least the first half of the isomerization process defi-nitely involves a rotational mechanism. Once on the groundstate PES, the reactive trajectories proceed more directly to-ward the cis minimum, sometimes approaching the inver-tomer geometry ��NNC=180° �. We arbitrarily count a re-active trajectory as representing an inversion-type pathway if

FIG. 5. Average CNNC and NNCangles as function of time, for the QMpart. Reactive and unreactive trajecto-ries are shown separately. Two aver-ages are shown for NNC angles: onefor the smaller and one for the largerangle.

one NNC angle overcomes 160° at any time �and a rotational



pathway otherwise�: in this way, we have an inversion torotation ratio �inv /�rot=0.18 for n→�* excitation and�inv /�rot=0.35 for �→�* excitation.

Further insight in the isomerization mechanism can begained by inspection of Fig. 7, where we show the behaviorof some angles specifying the orientation of the four phenylrings as functions of the reaction coordinate �the CNNC di-hedral�, averaged on reactive trajectories only. In particular,we monitored the deviation from parallelism of the stackedphenyl rings, one belonging to the QM and the other to the

FIG. 7. Average of and CCNN angles �defined in the text� for reactive
trajectories.

MM subsystem. The plane of each phenyl was defined asbeing perpendicular to the principal axis of inertia with thelargest eigenvalue for the six C atoms. In Fig. 7 we report theaverage of the two dihedral angles for the two pairs ofstacked rings, further averaged over all reactive trajectories�this angle with be labeled as and corresponds to the thickline�. We also show the CCNN dihedrals �only for the QMpart�, evaluated in this way:

FIG. 6. Behavior of the CNNC dihe-dral and the largest NNC angle for re-active trajectories, in polar �r ,�� coor-dinates: r=180°−�NNC, �=�CNNC. The most populated elec-tronic states are indicated by differentcolors �or shades of grey� and the la-bels S0 ,S1, and S2.

FIG. 8. Time-dependent populations of the electronic states.



�CCNN = 90 ° − ��CaCNN + �CbCNN

2� , �1�

where Ca and Cb label two carbon atoms belonging to thesame phenyl ring, and the dihedrals CaCNN and CbCNNhave values between −180° and 180°. When �CCNN=0 thefive atoms are coplanar, otherwise a torsion around theCuN bond is present. At transoid geometries, the phenylrings of each azobenzene moiety are approximately coplanarand parallel to the rings of the other azobenzene moiety.Therefore, both and the CCNN angles have small values.At cisoid geometries �structures in Fig. 2�, the two phenylrings of the QM moiety cannot be coplanar because of thesteric hindrance of the hydrogens �note that this feature isshared with the bare azobenzene�. Therefore both and�CCNN must increase when the CNNC torsion angle goesfrom 180° to 0°, but Fig. 7 shows that such changes onlyoccur in the second half of the reaction path. Most of theincrease of the angle takes place after �CNNC drops be-low 40°, both for n→�* and �→�* excitations. The CCNNangles start to increase earlier, when the CNNC torsion angleis about 80°. Apparently, the stacked rings have a strongtendency to remain parallel because of steric hindrance, untilthey are pulled away at cisoid geometries. Even when theCCNN angles start to change, the deviation from parallelism remains small, thanks to a corresponding torsion of the CNbonds of the nonisomerizing azobenzene moiety. The mainconstraint introduced in 2S-ABP seems therefore to be a ten-
dency to keep the stacked rings parallel. This does not imply

restrictions for the rotation around the NvN double bond,except for low values of the CNNC angle �below 40°, seealso the potential energy curves, Fig. 3�. It should be notedthat in ring structures obtained by meta substitution of theazobenzene phenyl rings �see, e.g., Pancur et al.26 and Luet al.24�, much stronger constraints are expected to correlatethe motions along the �CNNC and �CCNN coordinates.

The time-dependent populations of the electronic statesare shown in Fig. 8. In the range 0–300 fs the pattern isalmost identical to azobenzene, both for the n→�* and the�→�* excitations. In the case of �→�* excitation, thesimilarity also holds for the decay of the S2 state. At longertimes, the pattern is still similar to that of TAB, but2S-TTABP shows a slower S1 decay. To bring out the differ-ences between 2S-TTABP and TAB in the S1 decay mecha-nism, in Fig. 9 we report dot density plots showing theCNNC and the largest NNC angles versus the S1−S0 energydifference at the S1→S0 surface hopping, in the case of n→�* excitation. Bottom panels show results for t�300 fs,top panels for t�300 fs. The larger density of dots for TABis due to the larger number of trajectories that were run �215for TAB, 164 for 2S-TTABP�. The overall pattern is verysimilar, except that for 2S-TTABP at t�300 fs the hops oc-cur predominantly at quite large energy S0-S1 differences andtransoid geometries, with �CNNC spread in the range 130°–180° and �NNCmax�140°. This is an indication that torsion�but also inversion� is somewhat hindered in 2S-TTABPsince the fraction of trajectories that “linger” at transoid ge-

FIG. 9. S1→S0 surface hoppings forn→�* excitation. S1−S0 energy dif-ference �eV� versus CNNC dihedral�cross� or the largest NNC angle �opencircles�. Upper panels: t�300 fs;lower panels: t�300 fs. Left panels:2S-TTABP; right panels: TAB.

ometries in S1 for more than 300 fs is larger than in TAB. For



these trajectories a significant occurrence of “frustrated”hops is expected because of the large S0-S1 energy gap, i.e.,many hops from S0 to S1, that should take place according tothe computed probabilities, are given up because the kineticenergy is too small to compensate for the increase in poten-tial energy. This artifact of the surface hopping treatmentaccelerates the S1 to S0 conversion and lowers the quantumyields. A similar behavior �long lifetime component of S1 attransoid geometries� is observed after �→�* excitation.

We now turn to the analysis of femtosecond fluorescencetransients, recently observed experimentally by Diau andco-workers24 for 2S-TTABP. Both the n→�* and the �→�* excitations yield, after deconvolution of the experi-

TABLE IV. Time-resolved fluorescence experiments fexcitation and fluorescence wavelengths. Transient lreport relative weights in parentheses.

Molecule � �pump� nm � �fluor.

2S-TTABP 360 450500550

440 550600650600

TAB 432 550-75440 520-68480 600-68280 550-75


mental signal, transients with lifetimes in two distinct ranges:0.1–0.6 ps and 2.5–6.0 ps. These experimental results aresummarized in Table IV, together with the correspondingdata available for TAB, for comparison purposes. In the toppanels of Fig. 10 we show our simulated total �i.e., frequencyintegrated� emission transients for 2S-TTABP and TAB. Inthe case of n→�* excitation we find pronounced oscillationsat short times �t�300�, for both 2S-TTABP and TAB. Theseoscillations are closely linked to those of the CNNC angle inS1, since both the S0−S1 transition dipole moment and theenergy gap strongly depend on that angle. Since theexperiments24 were conducted with a pulse length compa-rable with the period of the oscillations in the fluorescence

-TTABP �in CH2Cl2� and TAB �in hexane� at variouses �1 and 2� are shown. For 2S-TTABP we also

1 fs 2 fs Ref.

110�1.00� 24100�0.98�100�0.89� 2500�0.10�110�0.91� 1400�0.09�340�0.81� 3500�0.18�500�0.77� 5500�0.23�590�0.72� 6000�0.27�

160-280 780-1560 47150-330 600-1700 23120-240 1300-2000�500 48

FIG. 10. Fluorescence transients. Toppanels: total fluorescence intensity, for2S-TTABP and TAB, irrespective ofthe emission wavelength. Bottom pan-els: fluorescence intensity in various�fluor ranges, for 2S-TTABP. Insets for�→�* excitation enlarge the regionof lower emission intensity.

or 2Sifetim

� nm

0000



intensity �about 200 fs�, the latter exhibits a much smoothertime profile than in our simulations. At longer times, theTAB fluorescence decays abruptly while 2S-TTABP shows alonger tail. This is in qualitative agreement with experimen-tal results.24 In the case of �→�* excitation �top left panelof Figure 10� the fluorescence intensity is about two ordersof magnitude stronger at short times, but the behavior atlonger times is similar to that of n→�* excitation.

For a better comparison with the experimental results,we split the fluorescence transients in low, medium and highenergy components, according to the emission wavelength�see the bottom panels of Fig. 10�. Decay times 1 and 2

were extracted from each component assuming the same ki-netic model used by Lu et al.24 in the analysis of their time-resolved fluorescence experiments:

A→1

B→2

C. �2�

This leads to the decay law X�t�=w1exp�−t /1�+w2

��exp�−t /2�−exp�−t /1��. The results of the fit of our low,medium, and high energy fluorescence transient with thiskind of decay law are shown in Table V. We consider firstthe case of n→�* excitation. The time coefficient 1 shows abehavior in reasonably good agreement with the experience:it increases with �fluor, while the relative weight of the fastdecay component w1 / �w1+w2� decreases. The 2 time coef-ficient extracted from our simulations is shorter than in allexperiments, especially for the long wavelengths.

Also in the case of �→�* excitation the results of our fitare in qualitative agreement with the experimental findingsof Lu et al.,24 for the short and medium wavelengths, whereboth the S2→S0 and the S1→S0 transitions do contribute. Inparticular, we have an ultrafast decay, almost completely de-scribed by a single exponental and mainly due to the S2

→S1 relaxation. The long wavelength component, almost ex-clusively due to the S1→S0 transition, shows a raise anddecay in time, at variance with the experimental transient.

As we have seen, the discrepancies of the simulated andexperimental fluorescence decay profiles mainly concern theslow, long wavelength, components. The main reason isprobably that the electronic relaxation rate is overestimated,because of the surface hopping artifacts and of the neglect ofsolvent effects in our simulations, as in the case ofazobenzene25 �see above�. The accuracy of the transition di-pole functions, determined at the semiempirical level, is alsoquestionable, especially at distorted geometries. Finally, we

TABLE V. Fitted time coefficients for 2S-TTABP time-resolved fluores-cence. Relative weights �see text� are given in parentheses.

Excitation � �fluor.� nm 1 fs 2 fs

n→�* ��520 13 �0.995� 811 �0.005�520��650 74 �0.60� 793 �0.40�

��650 239 �0.34� 850 �0.66�

�→�* ��400 8 �0.92� 97 �0.08�400��550 34 �0.87� 304 �0.13�

��550 418 �0.0008� 424 �0.9992�

recall the complete neglect of the interaction between the


two chromophores, as a further feature of our model thatmay influence in a negative way the quality of the computedfluorescence transients.

IV. CONCLUSIONS

The analysis of the potential energy surfaces of 2S-ABPand of its photoisomerization dynamics show that in ourmodelization this molecule is not rotation-locked: theisomerization, in particular, proceeds mainly via torsionaround the NvN double bond. This mechanism is also themain responsible for the S1 state decay, although a non-negligible fraction of the S1→S0 radiationless transitions oc-cur at transoid geometries.

Our simulations of time-resolved fluorescence experi-ments suffer from several drawbacks: oversimplifications ofthe model �only one active chromophore, no solvent effects�,artifacts of the surface hopping method, and inaccuracy ofthe semiempirical calculations. For these reasons, the com-puted decay times are shorter than the experimental ones.Nevertheless, our results are in overall qualitative agreementwith the experiments, and reproduce some important trends:in particular, the S1→S0 relaxation time in 2S-TTABP islonger than in TAB, and it increases with the fluorescencewavelength.

ACKNOWLEDGMENTS

We are grateful to E. W.-G. Diau for helpful discussionsand for communicating to us the results of his experimentson the azobenzenophanes prior to publication. This work wassupported by grants of the Italian MIUR and of the Univer-sity of Pisa.

APPENDIX: SEMIEMPIRICAL CI ENERGY GRADIENTWITH FLOATING OCCUPATION MOS

The use of the Z-vector technique for the evaluation ofanalytical CI gradients is mandatory for a large system evenwith semiempirical methods since it allows to reduce thescaling from O�N3Nat� to O�N3�, where N is the number ofbasis functions and Nat the number of atoms.40 Of course thisis particularly important for QM/MM calculations. In thisappendix we will use i , j ,k ,… to label CI-active orbitals anda ,b ,c ,… for any kind of MOs. For computational conve-nience, only the CI-active orbitals are allowed to have float-ing occupation numbers. The occupation number Oi of theactive orbital �i is spread on the energy axis according to aGaussian function f i� � in such a way that �see Ref. 42�

Oi = −�

F

f i� �d �A1�

where F is the Fermi level, set by imposing that the sum ofthe occupation numbers is equal to the total number of activeelectrons. The CI energy of the state n can be profitably

49
written as


ECIn = EHF − Eg

+ + ij

�ijn ij

+ + ijkl

�ijkln �ij�kl . �A2�

Here EHF is the floating occupation SCF �see Ref. 42� energyand

ij+ = i�ij −

1

2k

�ij�kk Ok, �A3�

Eg+ =

1

2i

� ii+ + i�Oi, �A4�

where i is the energy of the orbital i, �ij�kl =2�ij�kl − �ik�jl , and �ij�kl =��i�r1�� j�r1�r12

−1�k�r2��l�r2�dr1dr2. Notethat EHF−Eg

+ represents the contribution of the inactive elec-trons to the CI energy, and ij

+ are the matrix elements of themonoelectronic Hamiltonian. Therefore �n and �n are theone-electron and two-electron CI density matrix for the staten. The derivative of ECI

n with respect to a nuclear coordinateR� can be separated in a static and in a response part.40 Weare only interested here in the response part, containing thederivatives of the SCF density matrix �i.e., of the MO coef-ficients and of the occupation numbers� and the derivativesof the orbital energies. The response part of the derivative ofECI

n can be evaluated making use of the followingrelations:40–42

� �i

�R�

j�kl� = a

Bia� �aj�kl , B� =

�C†

�R�

C , �A5�

�� EHF

�R��

response=

i

i�Oi

�R�

, �A6�

�Oi

�R�

= j

�ij� j

�R�

, �A7�

�ij = � f j� F�kfk� F�

− �ij� f i� F� , �A8�

where C is the matrix of the orbital coefficients. We define amatrix X� collecting the CPHF unknown

X� = B� +�

�R�

. �A9�

Note that B� is antisymmetric and is the diagonal matrix ofthe MO energies. In analogy with Ref. 40, the response partof the derivative is then given by �from now on, we drop theindex n of the state�

�� ECI

�R��

response=

a

i

Xia� qia �A10�

with

qii = �ii − Oi −1

2 �ki�lj��kk ��lj − �ljOl� , �A11�
jkl

qia = 4jkl

�ijkl�aj�kl − jk

�ijOk�kk�aj

− jk

� jkOi�ai�jk + j

OiOj�ai�j j �i � a� .

�A12�

Equation �A10� can then be recast in this form:

�� ECI

�R��

response=

a�b

Xab� Qab �A13�

with

Qii = qii, �A14�

Qij = qij − qji, �A15�

Qdi = − qid d inactive, �A16�

Qiv = qvi v virtual. �A17�

In all the other cases Qab is zero.It is useful to distinguish between redundant and nonre-

dundant CPHF variables: in particular, only the nonredun-dant variables contribute to the derivative of the SCF densitymatrix.39,42 They are Xab

� with Oa�Ob, a�b and, in the caseof floating occupation numbers, Xii

�. Using subscripts R andNR to label the redundant and nonredundant part, respec-tively, Eq. �A13� becomes

�� ECI

�R��

response= QNR

+ XNR� + QR

+XR� , �A18�

where Q and X� are treated as vectors with the compoundindex ab.

The quantity X� is the solution of the CPHF equations,written explicitly for the case of floating occupation orbitalsin Ref. 42 �see Eq. �20��. Following Ref. 39, these equationscan be profitably rearranged in this matrix form:

�AM + G�NX� = − F�, �A19�

where matrices and vectors have compound indices ab andcd. F� is the static derivative of the Fock matrix in the MObasis �rhs of Eq. �22�, Ref. 42�, and

A = �ANR 0

AR 0� , �A20�

M = �MNR 0

0 1� , �A21�

G = �GNR 0

0 GR� , �A22�

N = �NNR 0

0 NR� . �A23�

Explicitly we have

�A�ab,cd = 1 �4�ab�cd − �ac�bd − �ad�bc � , �A24�
2


�NNR�ab,cd = �ac�bd�Oc − Od + �ab� , �A25�

�NR�ab,cd = �ac�bd, �A26�

�GNR�ab,cd = �ac�bd b − a − �ab

Oa − Ob + �ab, �A27�

�GR�ab,cd = �ac�bd� b − a − �ab� , �A28�

�MNR�ab,cd = ��ab�cd�ac if a,c � active,

�ac�bd otherwise.� �A29�

Note that all the above matrices are symmetric in the indicesab ,cd. From Eq. �A19�, the nonredundant and redundantcomponents of vector X� are

XNR� = − NNR

−1 �ANRMNR + GNR�−1FNR� , �A30�

XR� = − GR

−1�FR� − ARMNR�ANRMNR + GNR�−1FNR

� � .

�A31�

Substituting Eqs. �A30� and �A31� in Eq. �A18� we obtainthe final expression for the energy derivative:

�� ECI

�R��

response= Y+FNR

� − QR+GR

−1FR� , �A32�

where the vector Y is found by solving the following linearsystem:

�MNRANR + GNR�Y = MNRAR+GR

−1QR − NNR−1 QNR. �A33�

For the system considered in this paper �66 basis func-tions, 10 active orbitals, 34 MM atoms� the calculation of theCI gradient is 19 times faster using the Z-vector method withrespect to the conventional technique.29,31,41,42 Overall, ourdynamics calculations are 11 times faster. The gain factor isexpected to increase with the size of the molecular system.

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are azobenzenophanes rotation-restricted?

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