Vibronic Coupling Explains the Different Shape of Electronic

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Vibronic coupling explains the different shape of electronic circular dichroism and of circularly polarized luminescence spectra of hexahelicenes Yanli Liu, Javier Cerezo, Giuseppe Mazzeo, Na Lin, Xian Zhao, Giovanna Longhi, Sergio Abbate, and Fabrizio Santoro J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00109 • Publication Date (Web): 27 Apr 2016 Downloaded from http://pubs.acs.org on April 27, 2016

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Journal of Chemical Theory and Computation

Vibronic coupling explains the different shape of electronic circular dichroism and of circularly polarized luminescence spectra of hexahelicenes

Yanli Liu1,2, Javier Cerezo2, Giuseppe Mazzeo3, Na Lin1, Xian Zhao1, Giovanna Longhi3,*, Sergio Abbate3, Fabrizio Santoro2,* 1

State Key Laboratory of Crystal Materials, Shandong University, 250100 Jinan, Shandong, People’s Republic of China

2

CNR – Consiglio Nazionale delle Ricerche, Istituto di Chimica dei Composti Organo Metallici (ICCOM-CNR), UOS

di Pisa, Area della Ricerca, via G. Moruzzi 1, I-56124 Pisa, Italy 3

Dipartimento di Medicina Molecolare e Traslazionale, Università di Brescia, Viale Europa 11, 25123 Brescia, Italy

* Corresponding authors: [email protected]; [email protected];

Abstract We present the simulation of the absorption (ABS), electronic circular dichroism (ECD), emission (EMI) and circularly polarized luminescence (CPL) spectra for the weak electronic transition between the ground (S0) and the lowest excited state (S1) of hexahelicene, 2-methyl-hexahelicene, 2-bromo-hexahelicene and 5-aza-hexahelicene. Vibronic contributions have been computed at zero Kelvin and at room temperature in harmonic approximation including Duschinsky effects and accounting for both Franck-Condon and Herzberg-Teller contributions. Our results nicely capture the effects of the different substituents on the experimental spectra. They also show that HT effects dominate the shape of ECD and CPL spectra where they even induce changes of signs; HT effects are also relevant in ABS and EMI, tuning the relative intensities of the different vibronic bands. HT effects are the main reason for the differences in the lineshapes of ABS and ECD and of EMI and CPL spectra and for the mirror-symmetry breaking between ABS and EMI and between ECD and CPL spectra. In order to check the robustness of our results, given also that few examples of calculations of vibronic CPL spectra exist, we adopted both adiabatic and vertical approaches to define the model potential energy surfaces of the (S0) and the (S1) states; moreover we expanded the electric and magnetic dipole transition moments around both the S0 and S1 equilibrium geometries.

1

ACS Paragon Plus Environment


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1 Introduction Helicenes are polyaromatics endowed with planar chirality, consisting of ortho-fused benzene rings, with large optical rotation, intense circular dichroism spectra and several enhanced physical-organic properties.1-3 They have been a target of intensive theoretical and experimental studies2,4-10 in various areas of science and technology. The most remarkable feature of helicenes is their inherent chirality7 which is due to the existence of a stereogenic axis. The possible left- and right-handed helical structures (configurations M and P respectively), are characterized in most of the cases by negative and positive optical rotation, respectively.4,7,11 Thanks to their intrinsic chiral character, helicenes have promising applications in biological chemistry for their capability to bind selectively to DNA and are employed for molecular chiral recognition in polymers chemistry.12 In addition, hexahelicene and larger [n]helicenes with high optical stability have been regarded as promising candidates as chiral catalysts and ligands in asymmetric syntheses.2,4,13,14 Hexahelicene has been adopted for one of the first meaningful calculations of optical rotation (OR) through the celebrated Rosenfeld formula.15,16 Furthermore, it served as a prototypical molecule to formulate the theory encompassing Rayleigh optical activity and optical rotation17. Recently,

various

synthetic

and

spectroscopic

studies

have

been

conducted

on

hexahelicenes.4,18,19 The assignment of the absolute conﬁguration (AC) of hexahelicene was unequivocally established on the basis of combined OR, electronic circular dichroism (ECD) and X-ray diffraction methods.11 Substituted hexahelicenes were also very recently investigated for their two-photon absorption and circular dichroism response.20 Among the different chiroptical spectroscopies, ECD has been employed in various helicenes6,9,21-23. Besides ECD, circularly polarized luminescence (CPL) has also been employed for the assignment of AC of helicene21,24. CPL can be viewed as the emission analogue of ECD25. Therefore, the combination of ECD and CPL techniques is expected to provide more complete information on both the ground and the excited electronic states. If the same electronic states are involved, the transition is Franck-Condon allowed, and the spectra are dominated by the effect of the geometry displacements, ECD and CPL are expected to be mirror symmetric. Therefore any deviation from this behavior indicates that some of the above conditions are not fulfilled providing 2


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a valuable piece of information about the system. Very recently, some of us21 studied four different types of hexahelicenes shown in Fig. 1: hexahelicene, 2-methyl-hexahelicene, 5-aza-hexahelicene, 2-bromo-hexahelicene. One photon absorption (ABS), spontaneous emission (EMI), ECD and CPL -reported here for completeness in Figure 1- were measured and analyzed also in terms of the helical sense-responsive (H) and substituent-sensitive (S) characterization proposed by Inoue et al.6,22 Helicity of helicenes is strictly connected to the delocalization of the π electrons, therefore in ref. 21 the H category was extended to include those features that are actually sensitive to changes in the π conjugation. The picture arising from the analysis of the experimental spectra is that the strong ECD signals mainly exhibit H signatures since they are invariant with respect to addition of the diﬀerent substituents while CPL spectra manifest mainly S features.21 Electronic calculations presented in ref. 21, could not account for the rich patterns of peaks observed experimentally for the S0/S1 transition. This is not surprising since, already long time ago, Weingang and his co-workers24 suggested that vibronic coupling may play an important role for such transition in helicenes. These considerations prompted us to undertake a detailed vibronic simulation of the shapes of ABS, ECD, EMI and CPL spectra associated to the S0/S1 transition of the above mentioned helicenes by employing first principles calculations. The goal is to reproduce the fine structures observed in the experimental spectra, to assign them to specific vibronic contributions and to relate the different responses of the four molecular systems to differences in their potential energy surfaces (PESs) or in their transition dipoles. Moreover we aim at investigating if the useful H and S categorization, first introduced for the interpretation of ECD spectra of helicene-based systems and then extended in ref. 21 to vibrational spectroscopy, can still be extended and provide a useful framework to analyse in detail vibronic spectra. In this paper, the vibronic spectra have been computed by applying density functional theory (DFT) and its time-dependent (TD) extension (TD-DFT) combined with AH (Adiabatic Hessian) and VH (Vertical Hessian) models.26-29 Both Duschinsky30 and Herzberg-Teller effects31 were considered.

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2 Theoretical background Let us consider an electronic transition from the initial electronic state i

to the final state

f . In a time-independent (TI) approach it is possible to write a general equation for one-photon

ABS, EMI, ECD and CPL as a weighted sum of the transitions between the vibrational states and ν f

associated to the initial and final electronic state, ν i

σ χ (ω ) = Cχ ω

nχ

∑ν∑ν Z f

−1 − β hωvi i

e

∑ α

g (ω, ωiν i f ν f )

i, f

= x, y ,z

respectively. 29

Aiαν i f ν f Bαf ν f iν i

(1)

where χ stands for ABS, EMI, ECD or CPL, T is the temperature, β = (k BT )−1 and k B is the Boltzmann constant. Moreover, Z i

is the vibrational partition function of the initial

electronic state, Z i−1 exp( −hωivi / kbT ) the thermal population of the initial vibrational state

(

)

ν i , ωivi fv f = ω fv f − ωivi , and ω is the circular frequency of the incident/emitted photon. The exponent n χ is 1 for ABS and ECD and 3 for EMI and CPL (assuming that the number of α emitted photons is detected). Aiανi f ν f = ivi µ fv f

is the matrix element of the Cartesian

component α of the electric dipole moment operator µˆ . Bαf ν f iν i is

ˆ α ivi EMI and Im fv f m

fv f µˆ α ivi

for ABS and

ˆ is the magnetic dipole moment for ECD and CPL, where m

operator. Equation (1) is valid for ECD and CPL in case of randomly oriented molecules. The constant C χ is different for each of the four spectra. ABS and ECD are usually reported as the molar absorptivity (ABS) and its anisotropy (ECD) and the expression of C χ in terms of fundamental quantities has been reported many times (see e.g. ref. 3,29). Here we only recall that expressing all the terms in Eq. 1 in atomic units, in order to obtain the molar absorptivity and its anisotropy in dm3mol-1cm-1, C χ must have the numerical values 703.300 and 20.5288, respectively3. Experimental EMI and CPL on the contrary are usually given in arbitrary units. However their ratio, the so called dissymmetry factor glum is well defined, since the relative 4


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values of EMI and CPL can be measured. Assuming that molecules are randomly oriented and that their rotational (or tumbling) relaxation time is small with respect to the lifetime of ES, for each vibronic transition iν i → f ν f

g iν f ν = lum i

4 Im  ivi µ α fv f c0 ivi µ α fv f

f

glum corresponds in SI units to32,33

fv f mˆ α ivi  fv f µˆ α ivi

(2)

where c0 is the speed of light. While we will report EMI and CPL calculated spectra in arbitrary units, the ratio given by Eq. (2) is directly comparable with experiment. The line shape g (ω , ωivi fv f ) is chosen here to be a Gaussian function:

(

)

g ω , ωiν i f ν f =

− (|ωiν f ν |− ω )2 /2 Γ 2 1 e i f 2π Γ

(3)

where Γ is the standard deviation of the Gaussian function. The matrix elements of the electric and magnetic dipole transition moment can be explicitly evaluated as

µiαν f ν = vi i µˆ α f v f = vi µeα,if (Qκ ) v f

(4a)

miαν i f ν f = Im vi i mˆ α f v f = vi meα,if (Qκ ) v f

(4b)

i

f

Here the vector Qκ represents respectively the full set of normal coordinates {Qκa} of either the initial or the final electronic state ( κ = i, f ) and µeα,if ( Qκ ) and meα,if (Qκ ) are usually called electronic transition dipole moments. They can be Taylor expanded around the equilibrium geometry of state

κ , Q0κ  ∂µeα,if ( Qκ )  α (0κ ) α (1κ )  Qκ a + ..... = µe,if + ∑ µe,if (a )Qκ a + ..... ∂ Q a κa  Q0κ

µeα,if (Q) = µeα,if (Q0κ ) + ∑ 

(5a)

 ∂meα,if ( Qκ )  α (0κ ) α (1κ ) me,if (Q) = me,if (Q ) + ∑   Qκ a + ..... = me,if + ∑ me,if (a )Qκ a + ..... ∂Qκ a Q0κ a  a

(5b)

a

α

α

0κ

µeα,if(1κ ) and meα,if(1κ ) are the first derivative of electronic and magnetic transition dipole moments with respect to normal coordinate Qκ a , respectively, and the sum if extended over all the normal modes. 5



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The label

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κ in the parentheses is used to recall that the expansion is performed around the

equilibrium geometry of state electronic state

κ and the sum is extended over all normal modes Qκ of

κ . Insertion of the first two terms of these expansions into Eqs. (4) yields:

µiαν f ν = µeα,if(0κ ) vi v f + ∑ µeα,if(1κ ) ( a ) vi Qκ a v f i

(6a)

f

a

miαν i f ν f = meα,if(0κ ) vi v f + ∑ meα,if(1κ ) (a ) vi Qκ a v f

(6b)

a

The first term of the rhs of Eqs 6 represents the Franck-Condon (FC) term and the second term the Herzberg-Teller (HT) term. FC, HT and FCHT (FC+HT) calculations of the spectra include respectively only the first, only the second, and both terms in the rhs of Eqs. 6. Due to vibronic couplings, small displacements along the normal coordinates can mix electronic states involved in the transition with close lying states. The linear terms in Eqs. 6 introduce the effects of such couplings in the spectra, within the limit of first-order perturbation theory.31 We define the total intensity

I

tot

( χ ) of a given transition χ as the integral over the frequency −n

domain of the lineshape of the corresponding spectrum defined as Cχ−1ω χ σ χ (ω) . Let us only consider the FC and HT term in the expansion of the transition dipole moments (Eqs. (5a) and (5b)). Then

I

tot

( χ ) can be obtained analytically from the Boltzmann weighted sum of the Aiανi f ν f Bαf ν f iν i

terms in Eqs. (1), integrating over the normal coordinates of the initial electronic state3,34:

 ∂µeα,if ( Qκ )   β hωκ a  ( µe,if (Q ) ) + ∑ 2ω coth  2  ∑  ∂Q  I tot( ABS / EMI ) = α =∑ α = x, y ,z  x, y ,z a κa κa Q0κ 2

α

=I

FC tot

( ABS / EMI ) + ∑ I

HT tot

0κ

2

( ABS / EMI ) = I

h

FC tot

( ABS / EMI ) + I

HT tot

(7a)

( ABS / EMI )

a

I

tot

( ECD / CPL ) =

∑ α

= x, y ,z

Im  µeα,if ( Q0κ ) meα,if ( Q0κ )  +

 ∂µeα,if ( Qκ )   ∂meα,if ( Qκ )   β hωκ a  coth Im     ∑a 2ω   ∑  2 α = x , y , z  ∂Qκ a Q0κ  ∂Qκ a  Q0κ κa h

(7b)

FC FC HT = I tot ( ECD / CPL ) + ∑ I aHT ( ECD / CPL ) =I tot ( ECD / CPL) + I tot ( ECD / CPL ) a

where ωκ a is the harmonic frequency of mode a in the initial state, 6


κ =0 for ABS and ECD and 1

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for EMI and CPL. Therefore

I

tot

( χ ) is the sum of two terms. The first term corresponds to the

FC total FC intensity ( I tot ) that is the square modulus of the transition dipole at the geometry of the

initial state for ABS and EMI, and the rotatory strength at the geometry of the initial state for ECD HT ) and is the sum of individual and CPL. The second term represents the HT contribution ( I tot

contributions ( I aHT ( ECD / CPL ) ) of the normal modes of the initial state. It is noteworthy that for HT ECD (or CPL) I tot arises from the algebraic sum of terms (one for each mode) which may have

different signs, so that the HT contribution to the total intensity may be very small even in cases where there are strong HT effects. It is therefore useful for analysis to define also a sum of their HT

absolute values ( I a ( ECD / CPL) ).

I tot ( ECD / CPL) = ∑ I a ( ECD / CPL ) HT

HT

a

=∑ a

 ∂µeα,if ( Qκ )   ∂meα,if ( Qκ )   β hωκ a  coth  Im      ∑ 2ωκ a  2  α = x, y ,z  ∂Qκ a Q0κ  ∂Qκ a Q0κ

h

(7c)

Notice that for analysis in the following we also used normalized lineshapes defined as −n

Lχ (ω) = N χω χ σ χ (ω) , where N χ is a normalization factor so that the integral over the frequency domain is 1. The calculation of the spectra can be recast in a TD formalism as

ε χ (ω )=

Cχ ω

nχ

2π

×Z

∞ −1 g −∞

∫

dt [ χ FC (t , T ) + χ FC / HT (t , T ) + χ HT / HT (t , T )]e

± iωt −

Γ2 2 t 2

(8)

where the positive sign should be selected for EMI and CPL and the negative one for ABS and ECD. The quantities

r

r

χFC (t, T ) = Im  µe(0,ifκ ) ⋅ me(0,ifκ )  ∫ dQ Q e

−itH f / h − ( β −it ) Hi / h

e

χ FC / HT (t , T ) = ∑ α ,a Im  µeα,if(1κ ) (a )meα,if(0κ )  ∫ dQ Q Qκ a e

∑α ,a Im  µeα,if(0κ )meα,if(1κ ) (a ) ∫ dQ Q e

− itH f / h

Q ,

(9a)

− itH f / h − ( β − it ) H i / h

e

Q +

Qκ a e − ( β −it ) H i / h Q ,

7


(9b)


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χHT / HT (t, T ) = ∑α ,a,b Im  µeα,if(1κ ) (a)meα,if(1κ ) (b) ∫ dQ Q Qκae

−itH f / h

Qκbe−( β −it ) Hi /h Q ,

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(9c)

are correlation functions whose expression is analytical for harmonic PESs and is given, for example, in ref. 35. In Eqs. 9, H i and H f are the vibrational Hamiltonians associated to electronic states i

and

f .

3 Computational Details In order to compute equilibrium geometries and Hessians, we adopted DFT for the ground state (GS) and TD-DFT for the lowest-energy excited-state (ES). We employed the long-range corrected CAM-B3LYP functional36 with standard parametrization (α=0.190, β=0.460, µ=0.330) that has been shown to improve considerably on standard hybrid B3LYP especially for Rydberg-like and charge-transfer states36,37 and has been already adopted for CPL calculations26. Computations of CPL intensities at highly-accurate equation-of-motion coupled cluster level have been presented very recently,38 but the employment of this method to retrieve all the information needed for a vibronic calculation still appears very challenging. In the following, vibronic spectra have been computed in harmonic approximation, accounting for Duschinsky30 and HT effects31, exploiting both the TI pre-screening method34,39,40 and the TD approach implemented in our code FCclasses.41 We considered ABS, EMI, ECD and CPL spectra.42 Geometry optimizations, as well as the calculations of the vibrational normal modes of the electronic GS and ES were carried out using Gaussian 09 program.43 TZVP basis set has been applied throughout all the calculations, according to previous studies on helicenes. 21,44-47 AH and VH models have been used to build the harmonic PESs26-29. In AH both the initial and the final-state PESs are quadratically expanded around their own equilibrium geometries. As a consequence the same GS and ES PES models are adopted for both the absorptive (ABS and ECD) and the emissive (EMI and CPL) processes. In VH, on the contrary, both the initial and final state PESs are expanded around the equilibrium geometry of the initial state (GS for ABS/ECD and ES for EMI/CPL). Therefore the PES models adopted for absorptive (ABS and ECD) and emissive (EMI and CPL) processes are in general different and they become identical only if the PESs are 8


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exactly harmonic. When the harmonic approximation is accurate, AH and VH deliver similar FC spectra; if significant differences exist, VH is expected to outperform AH in the description of the region of the spectral maximum, while AH is expected to better describe the region close to the 0-0 transition27. Concerning HT approximation, in Eqs 4 we have specified the geometry adopted for the linear expansion of the transition dipoles, i.e the one for the initial or for the final state κ = i, f . If the linear approximation is correct this choice is immaterial and the

κ label can be dropped. However

in all practical cases the transition dipole moments are not perfectly linear functions of the normal coordinates and the two expansions lead in principle to different results. Remarkable differences between the results obtained with the two approaches highlight the inadequacy of the linear approximation implicit in HT treatment. There is a very limited experience in the calculation of vibronic CPL spectra and in the comparison of the line shapes of the full set of the ABS, ECD, EMI and CPL spectroscopies.32,33 To the best of our knowledge, the only recent calculations of vibronic CPL were reported by Pritchard and Autschbach in 201033 and by Barone et al. in 201448. The former authors investigated d-camphorquinone and (S,S)-trans-b-hydrindanone,33 performing FC|AH calculations with the TI method implemented in a distributed version of our code FCclasses.41 Barone et al. studied the spectrum of camphorquinone, adopting the analogous TI method implemented in a development version of the Gaussian code in combination with both FC|AH and FCHT|AH models.48 For the systems studied in this contribution, convergence problems challenge the TI strategy and we performed TD calculations to ensure full convergence of the spectral lineshapes. Moreover, in order to check the robustness of our predictions, we decided to adopt both the AH and the VH model. For the same reason, we also employed the two possible models to describe HT effects, named HT0 and HT1 respectively, depending on whether the derivatives of the transition electric and magnetic dipole moments are taken at the equilibrium geometry of the GS or ES. For FC calculations the transition dipoles are taken at the initial state geometry (GS for ABS and ECD, and ES for EMI and CPL). In the remaining we use the following labels to specify the different kinds of calculation: FC|AH, FCHT0|AH, FCHT1|AH, FC|VH, FCHT0|VH (for ABS/ECD), FCHT1|VH (for EMI/CPL). 9



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In current implementations of TD-DFT level the derivatives of the electric and magnetic dipoles come at no additional cost as a side-product of an ES frequency calculation. The ES Hessian is computed in Gaussian 09 by numerical differentiation of the analytical energy gradients, adopting positive and negative steps of ∆ =10-3 Å along each of the 3 N a Cartesian coordinates qi (i=1,3 N a ). In these steps the transition dipoles µe,if and me ,if are calculated as well, and therefore the derivatives

of

their

Cartesian

component

α

can

be

computed

as

α ∂µeα,if ∂qi = (2 ∆ ) −1  µeα,if ( qi + ∆ ) − µeα,if ( qi − ∆ )  and the analogous expression for me ,if .

Differences in FCHT0|AH and FCHT1|AH models depend only on different HT expansions of the transition dipole while the differences in FCHTx|AH and FCHTx|VH (x=0,1) are due to different PESs. Among the adopted models, FCHT1|AH (for ABS and ECD) or FCHT0|AH (for EMI and CPL) are the most demanding for electronic calculations, since they require the equivalent of two Hessian calculations on the ES. However, they are introduced to better dissect the effect of PES and HT expansions differences. All the vibronic spectra reported in this work refer to the M enantiomers and have been phenomenologically broadened with a Gaussian function with full width at half maximum (FWHM) of 0.08 eV. They have been computed with a development version of our code FCclasses41.

4. Results and Discussion 4.1 Molecular structures Let us first briefly discuss the main geometrical differences between the GS and ES states. The structures of the four hexahelicenes with labels are sketched in Fig. 1, and the values of some key structural parameters are collected in Table 1. Hereafter we use the following abbreviation for the investigated compounds: c1 (hexahelicene), c2 (2-methyl-hexahelicene), c3 (5-aza-hexahelicene), c4 (2-bromo-hexahelicene). The structural features listed in Table 1 are similar for all the systems, whose two edge benzene rings form a dihedral angle C1-C3-C9-C2 (labeled as τ5, see Figure 1) close to 0 degree. The substitution of H with a methyl enhances the co-planarity of the edge rings, with GS and ES torsion 10


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angles τ5 varying from 3.4 and 3.6 to 2.3 and 2.5 degrees, respectively. The largest torsion angle τ5 among the four systems is exhibited by c3 but it is still quite small (5.6 deg). On the same footing, the distance H1-H2 (labeled as d1, see Figure 1) is maximum for c3. The torsions about inner CC bonds τx (x=1,5, see Table 1 for definitions) are very similar for the four molecules, increasing or decreasing very slightly changing the substituent. For example, moving from c1 to c2, c3, and c4, τ1 decreases by ∼0.2º, ∼1.6º, and ∼0.2º, respectively. C3 exhibits the maximum changes among the four molecules. According to our calculations, for each molecule, the dihedral angle τ and d parameters are very similar in ES and GS. More in detail, in ES τ and d increase and decrease slightly, respectively. For instance, for c1, in GS τ1−τ5 increase (in absolute value) respectively by ∼1.5º, ∼0.5º, ∼0.5º, ∼1.5º and ∼0.25º in ES, while d1 is shortened by 0.12 Å, and d2 is increased by 0.09Å (see Table 1).

4.2 The S0/S1 electronic transition Table 2 reports the calculated FC and HT total intensities at the GS and ES equilibrium FC geometries. I tot ( ABS / EMI ) increases upon substitution of H with methyl or Br and is largest for

c3. The rotatory strength I totFC ( ECD / CPL ) exhibits a markedly different behavior for the different compounds which reflects the different deviation from 90 degrees of the angle formed by the FC electric and magnetic transition dipole moments.21 I tot ( ECD ) is the strongest (in absolute terms)

and positive for c3, it is reduced by almost a factor 2 and negative for c4, and is very small for c1 FC (positive) and c2 (negative). I tot (CPL) is positive for c1-c3, being the strongest for c3 and the FC weakest for c2. c4 exhibits a very small and negative I tot (CPL) .

For all the systems, total HT intensities are comparable to the FC ones except for c3 where they are weaker. Accordingly, c1, c2 and c4 spectra are expected to exhibit remarkable HT |HT | effects.46 Considering ECD and CPL, the I tot values are significantly larger than | I totHT | ones,

indicating that the HT contributions induced by different modes do not always have the same sign, and therefore likely involve more than one higher electronic state.

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4.2.1. Nature of the excited state To get further insight into the nature of the S0/S1 electronic transition, Table 2 also lists the most important contributions to S1 state for each molecule. The corresponding molecular orbitals for c1 are shown in Fig. S1 in the Supporting Information. The first excited state (S1) of c1, corresponds to a combination of the transitions from the second highest occupied molecular orbital (HOMO-1) to the lowest unoccupied molecular orbital (LUMO) and from HOMO to LUMO+1. According to the plots in the SI such π Molecular Orbitals (MOs) are largely delocalized above the skeleton of the whole molecule. Table 2 shows that the first excited state for c2 is dominated by a combination of

HOMO-1→LUMO+1

and

HOMO→LUMO+2. For c3 S1 arises from HOMO-1→LUMO+1 and HOMO→LUMO transitions. Finally, for c4 S1 mainly derives from four combined transitions: HOMO-1→LUMO, HOMO→LUMO, HOMO→LUMO+1 and HOMO-1→LUMO+1. Although the MOs involved in the S0→ S1 excitation have different labels in the different species, the nature of the transition is indeed very similar, as evidenced by the difference of the electronic density of the S1 and S0 states, plotted Figure 2 for the four compounds. Additionally, these plots show that the transition mainly involves the four central rings of the hexahelicenes. Among the different substitutions, the one that has the most recognizable (although mild) effect is the substitution of a carbon with a nitrogen in c3. As a matter of fact, this species is the one showing the strongest FC intensities (see Table 2) and consequently, as we will show in the following, the less evident HT effects. 4.2.2. A brief discussion of the transition energies Table 2 also reports the vertical absorption EVA , emission EVE and adiabatic energies Ead . They correspond respectively to the excitation frequency at the S0 ( EVA ) and S1 ( EVE ) geometry and to the difference between S1 and S0 minimum energies ( Ead ). These three quantities are extremely similar for the four molecules, highlighting that the effect of the substituents and of the nitrogen is small; c3 exhibits the largest Stokes shift, EVA - EVE =0.36 eV, and c4 the smallest one, 0.33 eV. For all compounds, Ead is very close to ( EVA + EVE )/2 indicating that the absorption and emission reorganization energies are pretty similar. 12


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Since S1 rotatory strength is small, it can be expected that the S0/S1 transition borrows intensity from higher-energy states. Table 3 reports the total FC intensity for absorption and ECD of the first three electronic states of the four compounds. For all the four compounds, the lowest-energy state exhibiting strong ABS and ECD signals is S3 which has a negative rotatory strength. This is in line with the fact that in Table 2 I totHT ( ECD / CPL) for S0/S1 transition is always negative.

4.3 Computed spectra In Figure 3 we superimpose the four experimental ABS, ECD, EMI and CPL spectra for c1-c4. In the subsequent Figures 4-7 we compare the computed spectra with the experimental ones, focusing onto the energy region pertinent to the S0-S1 transition, for c1 (Figure 4), c2 (Figure 5), c3 (Figure 6) and c4 (Figure 7). In order to better compare the shapes of computed and experimental spectra, the computed spectra have been shifted along the energy axis by ∼0.4 eV (see the Figures’ captions for details). Both computed and experimental EMI and CPL are reported in relative intensities and scaled to properly fit in the same figures, therefore the comparison focus on their shapes only. However, as mentioned in Section 2, the absolute values obtained for the experimental and computed CPL/EMI ratios can be directly compared (the measured ratios may be directly evaluated from Fig. 3). Before coming to the detailed analysis of the spectra it is useful to make some general remarks. According to the adopted theory, in FC approximation ABS and ECD, and EMI and CPL have the same shape. Moreover if the GS and ES states share the same Hessian, the normalized FC line shapes of ABS and EMI, LABS (ω ) and LEMI (ω ) (see Section 2) are mirror-symmetric, i.e. they are the same, except that they are in reverse energy order; the same is true for LECD (ω ) and LCPL (ω ) . Such symmetry relation does not hold anymore taking into account differences in GS and

ES vibrational frequencies and Duschinsky mixings or introducing HT effects. Therefore the fact that, in Figure S2 in SI the FC shapes LABS (ω ) = LECD (ω ) and LEMI (ω ) = LCPL (ω ) are close to be mirror-symmetric is an indication that Duschinsky mixing and frequency changes are only moderate. 13



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Inspection of Figures 4-7 shows that at FC level or when the same HT expansion is adopted, AH and VH models deliver very similar spectra both at 0 K and at room temperature. This finding indicates that the harmonic approximation is accurate enough for treating the S0-S1 transition of these helicenes, supporting the reliability of our predictions. Figures 4-7 highlight the existence of remarkable HT contributions that determine the shape of the spectra of the investigated species. In fact, they strongly modify the FC spectral shapes, introducing differences between the shapes of ABS and ECD, and EMI and CPL and breaking the mirror symmetry between ABS and EMI and between ECD and CPL. Moreover, they give rise to the sign inversion observed for ECD spectra of some of the investigated species. All these features improve in general the agreement with experiments moving from FC to FCHT models. However, results also show that the description of HT effects is more challenging than a pure FC approach. In fact, as we will discuss in the following, in some cases the shapes of the spectra depend remarkably on whether the transition dipole are expanded around the GS or ES equilibrium geometry (compare FCHT1|AH and FCHT0|AH results), and the difference becomes even more remarkable considering temperature effects. This fact suggests that the inclusion of higher-order terms, beyond the linear one considered in HT expansion, might have some impact on the computed spectra. FC and HT terms can interfere leading to a complex redistribution of the intensities among the different vibronic transitions.34 Notwithstanding this, some simple statements are possible on the total intensity (i.e. the integral of the spectral lineshapes, see Section 2). In fact, according to our level of theory, the FCHT intensity is the sum of a FC and a HT contribution (Eqs. 7). However, while for ABS and EMI they are both positive so that FCHT intensity is always larger than FC one, this is not true for ECD and EMI where FC and HT contributions can have opposite signs.3 4.3.1. Absorption spectra As it was anticipated in our previous studies,3,21,49,50 the results here reported show that HT effects are significant in the absorption spectra of the investigated species. In particular, with respect to FC predictions, in all systems the introduction of HT effects reverses the order of the relative intensity of the first and second vibronic bands improving agreement with the experiment. It is likely that in experiment the relative intensity of these two bands is also the result of the effect of 14


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the red edge of S3 absorption (the same is true for ECD signals, see below), since the latter is surely the responsible for the steep increase of the intensity at frequencies larger than ∼3.3 eV. A further contribution by the very weak S2 is also possible. Notice in fact that, according to Table 3, the difference of the vertical transitions ( EVA ) of S2 and S1 is 0.21-0.23 eV, not too much larger than the spacing of the two bands (∼0.17 eV). Therefore it is possible that the lowest vibronic band of S2 may reinforce the second band in the spectra (at ∼ 3.2 eV). The agreement of FCHT calculations with experiment is excellent for c3 and very good for c4 apart from a moderate overestimation of the total intensity, which is especially marked when the HT0 expansion is considered. The agreement is still satisfactory for c1 and for c2 even if the overestimation of the intensity with both HT0 and HT1 calculations increases, particularly for this last case. In summary, the computational results strongly suggest that the two small lowest-energy bands seen in the experiment for c1-c4 derive from a vibronic progression and that HT effects have a remarkable impact on their relative intensity. However the possible direct effect of S2 and S3 on the second band suggests that our calculations may have overestimated the HT contributions. As far as the total intensity is concerned, HT1 expansion provides spectra in slightly better agreement with experiment as compared to HT0. 4.3.2. ECD spectra HT effects are more significant in ECD than in ABS and cause a clear difference between the ECD and ABS lineshapes. As already shown in previous studies,51-53 the HT contribution may induce a sign inversion on ECD response of an electronic excited state. In the present cases, a change of sign within vibronic bands of the S0/S1 electronic transition is actually observed in experiment for both c1 and c2 and it cannot be explained at FC level which predicts a weak positive signal for c1 and a weak negative signal for c2. Introduction of HT effects allows us to correctly reproduce the sign inversion at 0 K for c1 (all three levels of calculations) and for c2 (only FCHT1|AH) level. Thermal excitation washes out such sign inversion worsening the agreement with experiment. This scenario indicates that HT effects are responsible for the experimental finding but also suggests that the experimental spectral shape is the result of a subtle balance of many factors that cannot be quantitatively captured by our calculations, probably due to both computational inaccuracies and limitations of the model, inherent in the linear HT expansion and in harmonic 15



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approximation for low-frequency modes. The HT effects are definitively weaker but still noticeable for c4 (Figure 7) where they make the second vibronic band more intense than the lowest one. This phenomenon is qualitatively similar to what happens in ABS (apart from the fact that the ECD signal is negative). For c4 the agreement with experiment is excellent with FCHT1|AH calculations. The adoption of the HT0 expansion both in combination with AH and VH PES models deliver spectra that follow the same correct qualitative trend. However, in this case the HT effect appears to be slightly overestimated on both total intensities and relative heights of the vibronic bands. In c3 HT effects do not appear strong enough to induce a sign inversion between the first and the second lowest energy peaks, which are both positive in the experiment. This finding is very nicely reproduced by our calculations. However, HT contributions (both at FCHT1 and FCHT0 level) improve the results. In fact they lead to a remarkable quenching of the second lowest-energy peak with respect to FC predictions, approximatively matching the intensity ratio of the experimental peaks. In the high energy wing the computed spectrum acquires a negative sign more evident when an HT0 expansion is considered. As for all the other compounds the negative intensity in the blue-wing of the spectrum is, in any case, much smaller than what observed in experiment21 where most of the intensity arises from the negative contribution of S3 (see above) not considered in the present vibronic calculations. HT effects appear stronger for c3 than for c4, although in absolute terms the FC rotatory strength is larger for the former system. This is likely a simple consequence FC of the fact that for c3 I tot ( ECD ) of S1 is positive and the influence of S3 leads to a change of sign.

This is a more spectacular phenomenon than the mere increase of the intensity of the blue-bands observed for c4. As we commented for ABS spectra, an effect of the weak S2 state on the height of the second ECD band (at ∼ 3.2 eV) is also possible. However for all systems except c3, at least at FC level, S2 should give a positive contribution to such band (and no contribution to the first band at ∼ 3.0 eV). On the contrary, the difference of the intensities of the second and first peak is always negative in the experiment. Therefore if a contribution from S2 exists it should not be dominant. Clearly S2 may exhibit negative HT contributions induced by the proximity with S3. In summary we can conclude that the big part of the experimental finding, i.e. that starting from the second band, the ECD signal 16


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decreases and turns negative at high frequencies (or steeply increases its negative intensity for c4) can be attributed to S3. Its effect is direct, through contribution of its vibronic states, and/ or indirect through intensity-lending to S1 (and S2).

4.3.3. Spontaneous emission spectra. At variance with what happens for ABS and ECD, EMI (and afterward CPL) spectra are compared with experiment in relative intensities. More precisely, the computed spectra were scaled so to approximately match the intensity of the highest experimental peak for c1. Afterward, the same scaling was adopted for c2-c4, so that the relative computed intensities of the different species are comparable (the same is true for CPL). Moreover, the same scaling was adopted for EMI and CPL and therefore also measured and computed ratios are directly comparable. Experimental EMI and CPL spectra Φ λ (λ ) were measured in the wavelength domain and then mapped to the frequency domain ( Φ ω (ω ) = 2π c0ω −2Φ λ (λ ) ),54,55,56 in order to be directly comparable to computed spectra. Please notice that not only absolute values of experimental emission cannot be given with our data, but different gain factors were used for the different systems. Actually they are all comparable except for c4 where it was about 30 times larger. Accordingly, the total emitted intensity measured for c4 is much less than for the other species, a fact that is not reproduced by our calculations (notice the y-axis scales in Figures 4-7). This suggests that a physical process, not accounted for by our model, occurs only for c4. A possibility is an intersystem crossing triggered by the heavy-atom effect. The computed EMI spectra are similar for all the investigated systems and appear in good agreement with the experiment. In general the relative intensity of the second highest-energy band (1) is underestimated at FC level with respect to the highest-energy (0) band, while it improves introducing HT effects. The latter however appear too strong, resulting in an opposite overestimation of the relative intensity of the band 1. HT effects on the total intensity are higher in EMI than in ABS at least for c1 and c2. This observation can be put on more quantitative grounds, FC FC HT by inspecting the FC ( I tot ) and FC+HT ( I tot = I tot + I tot ) total intensities reported in Table 2.

FC Considering for example c1, I tot / I tot is seen to be ∼3 for ABS and ∼5.4 for EMI. In comparison,

17



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the HT effect on the spectral shape is less evident. The computations seem to qualitatively capture the moderate breaking of the mirror symmetry observed in experiment, reproducing the fact that the relative intensity of the 0 band is higher in ABS than in EMI. Introduction of thermal effects leads to a general loss of resolution of the single vibronic peaks and, for c2, to a moderate increase of the total intensity. Within the theory we adopted, such an increase can only be due to HT effects.26

4.3.4. Circularly Polarized Luminescence spectra Among the four considered spectroscopies, CPL measurements are the most challenging as evidenced by the largest noise of the signal. c3 exhibits the simplest CPL spectrum with a clear positive sign and a shape that shows slight differences with respect to the EMI spectrum, like the fact that the highest-energy band (0) is weakly enhanced in CPL. Computed spectra for c3 are similar with each of the three adopted models and nicely reproduce the experimental shapes. HT effects introduce changes between the EMI and CPL line shapes in qualitative agreement with experiment: consider in fact the relative intensity of the two experimental shoulders at 2.90 and 2.79 eV. However, from the quantitative point of view, CPL/EMI differences appear larger in simulation than in experiment suggesting an overestimation of HT contributions. With respect to EMI, the experimental CPL spectrum of c4 is dominated by noise which is due to the very low emissive intensity of this molecule. Notwithstanding this, the experimental spectrum suggests a negative signal which is correctly predicted by our computations (both at FC and FCHT levels). As far as the modulation of the line shape is concerned, any detailed analysis is made difficult by the experimental noise. The computed spectra show some clear vibronic pattern that is similar to what observed in EMI and may tentatively be identified also in the experimental CPL signal. The experimental CPL spectra of c1 and c2 exhibit interesting differences. As far as the line shape is concerned, the c2 spectrum shows a sign inversion characterized by a positive highest-energy band followed by a sequence of equally spaced less intense (in absolute terms) negative bands. At variance with this behavior, both the highest and the second highest peaks of c1 are positive. As far as the total intensities are considered, in comparison with EMI, the experimental CPL appears to be weaker for c2 than for c1. This feature is not reproduced by our calculations, since they predict very similar spectra for the two species that, on the balance, better match the c2 18


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experimental signals. In particular, the sign change after the highest energy band is qualitatively reproduced by all the three adopted models at 0 K. Unfortunately, the positive peak is washed out if temperature effects are included, a phenomenon analogous to what already described of the ECD spectra of c1 and c2. As for ECD, also for CPL HT1 expansion seems to perform slightly better than HT0, if one compares the relative intensities of the negative and positive peaks. Overall, the computed peaks appear less resolved than the experimental ones, a problem not met for the other spectroscopies although the same phenomenological broadening was used. The less satisfactory agreement with experiment is observed for the CPL of the unsubstituted species c1. Since in this case FC spectra appear to match better the experimental band-shapes, it is tempting to conclude that our calculations strongly overestimate HT effects. For c1 an overestimation of HT effects was also found possible for EMI, and also for ABS and ECD simulations (recall the discussion on the relative height of the first and second band in sections 4.3.1 and 4.3.2). Its impact on CPL is much more drastic, probably due to the overall weakness of the signal.

4.4. Dissymmetry factors at the 0-0 and maximum frequencies In order to further analyze our results, in Table 4 we compare the computed and experimental dissymmetry factors g (ECD/ABS) and g lum (CPL/EMI). We considered both values at the 0-0 lum energy E00 ( g00 and g 00 respectively) and at the spectral maxima of absorption E max ( g max )

lum m and of emission E max ( g mluax ). In the experiment, E00 is estimated from the point of intersection of

the line shapes of absorption and emission bands (absorption-fluorescence crossing point, AFCP),57,58 normalized so that the lowest-energy peak of absorption and the highest-energy peak of emission have the same height. As noticed above, introduction of thermal effects generally leads to a worsening of the agreement with experiment, signaling a specific problem when treating low-frequency modes at harmonic level. Therefore, to get rid of this bias, in the present analysis we consider spectra computed at 0 K. We focus on AH results, since this model allows a more straightforward comparison of ABS and EMI data (the same PESs are adopted in both cases). Moreover it offers the most reliable data for the 00 transition which, as discussed below, is the band for which a more robust comparison with experiment is possible. 19



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For all systems the computational results show a very similar overestimation of E00, by ∼0.4 eV. lum values the agreement is reasonably good for all four molecules. In Both for g00 and g 00

particularly, computed data perfectly match their experimental counterparts for c3 (both adopting FCHT1 and FCHT0 expansions). For c1, g 00 is very nicely reproduced at FCHT0 level and lum overestimated by 60-70 % at FCHT1 level and the opposite is true for g 00 . For c2 g 00 is

lum underestimated (more at FCHT1 level) and g 00 overestimated (more at FCHT0 level). Finally for

lum c4 the agreement is good for g 00 ; as far as g 00 is concerned, only its sign can be determined in

experiment and is correctly reproduced by our calculations. The computed dissymmetry factors at the spectra maxima are still in satisfactory agreement with experiment for ECD/ABS ( g max ) while the weakness of CPL signals makes more difficult an accurate comparison for g mluma x . A detailed analysis is reported in the SI

4.5 Assignment of the main vibronic bands While TD calculations allow one to obtain fully converged spectra with little computational time, TI calculations, based on the sum over state approaches in Eq. (1), permit an in depth analysis of the spectra in terms of the most intense vibronic stick bands. Such an analysis is reported in Figure 8 for the spectra of the M enantiomers of c3, c1, and c4 computed at 0 K with the FCHT1|AH model. The calculated features are reported as bars, whose height is proportional to the calculated intensity. The most intense fundamental and overtone bands are labelled as “nx”, where “n” indicates the normal mode of the final state (S1 for ABS and EMI and S0 for EMI and CPL) and “x” its number of quanta. In a similar way, combination bands with “x” and “y” quanta on modes “n” and “m” are specified with the symbol “nxmy”. The main properties of the modes corresponding to the strongest bands are reported in Tables 5-7. Moreover for c1 and c3 such modes are sketched in Figures 9-10 (for ABS/ECD) and in Figs S4-S5 of SI for EMI/CPL where more illustrative movies are also reported. Assignments of the corresponding FC spectra are reported in Figure S3 in the SI. We first focus on c3 for which HT effects are less important. As a matter of fact ABS and ECD show similar shapes and the same is true for EMI and CPL. The non-perfect mirror symmetry 20


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between ABS and EMI and between ECD and CPL is largely due to the HT contributions. In fact, Figure S2 in the SI shows that at FC level, and considering normalized line shapes (see Section 2 for definition) and not intensities, mirror-symmetry is approximately respected. Qualitatively speaking, both ABS/ECD and EMI/CPL spectra are dominated by a single effective progression with a spacing of 0.17 eV. In ABS/ECD the lowest peak of the convoluted spectrum at 3.0 eV (band 0) is due to the 0-0 transition and to progressions along S1 low-frequency modes 3 (66.7 cm-1) and 9 (233.3 cm-1), both corresponding to collective distortions of the helical structure (see Figure 10). The peak at 3.18 eV (hereafter, band 1, observed also at FC level), is mainly due to the fundamentals of S1 modes 83 and 85 (1385.65 and 1410.95 cm-1), and to a lower extent of 84. They are all combinations of CC stretchings and CCH in-plane bendings and are remarkably displaced during the electronic transition (see Figure 10 and data in Table 6). These fundamentals are dressed with progressions along mode 3 and 9, as it happens in band 0. The higher-energy bands at 3.35 and 3.52 eV arise from the overtones and combination bands of modes 83 and 85, convoluted with progressions along low-frequency modes. EMI and CPL show similar patterns as ABS/ECD (we now name the peaks 0, 1, 2 in order of decreasing frequency). The assignments of the most intense vibronic features are only seemingly different. In fact, as shown in Table 6, modes 82 and 83 of S0 are mainly projected on modes 83-85 of S1 (while modes 3 and 9 are extremely similar in S0 and in S1). Comparison with FC spectra in Figure S3 of the SI shows that HT effects enhance peaks 1-3 in ABS and EMI and de-enhance them in ECD and CPL. This is not surprising since the borrowed intensity is positive in ABS and EMI and is negative in ECD and CPL (see Table 2). Moreover, HT effects make the relative intensity of peaks 1-3 (with respect to 0) larger in ABS than in EMI, and larger in ECD than in CPL. A detailed analysis for peak 1 shows that this is a clear consequence of a constructive interferential FC/HT effects. The latter increase the intensity of ABS/ECD bands 831, 841 and 851 and decrease the intensity of the corresponding bands 821 and 831 in EMI/CPL. Let us now consider the spectra of the M enantiomer of c1. In this case intensities are smaller, and the spectral shapes are dominated by HT effects, as already evidenced long time ago by Weigang et al for ECD.24 Such phenomenon is manifested with the fact that the shapes of ABS and ECD differ remarkably, and the same is true for those of EMI and CPL. Moreover ABS and EMI significantly deviate from mirror symmetry and strong deviations from this rule are also observed 21



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for ECD and CPL. Further information can be gained by the inspection of the most intense stick bands. The lowest-energy band (band 0) in ABS and ECD mainly arises from combination of 0-0 transitions with the fundamentals of mode 3 and mode 10. The band 101 has a significant HT contribution (4% of the total HT intensity) which is clearly manifested in the fact that in ECD it exhibits a sign opposite to the 0-0 band. The main vibronic progression, with a spacing of 1370 cm-1, is observed also in experiment and can be attributed to mode 85 (1391.55 cm-1). It corresponds to a combination of CC stretchings and CCH bendings, similar to those observed for c3 which is remarkably displaced (see Table 5). Mode 10 of S1 is mainly projected on mode 11 of S0 (sketched in Figure S4). Therefore it is not surprising that in CPL band 111 has a strong HT component, and its sign is opposite with respect to the 0-0 band. Mode 85 in the S1 state is mainly projected on modes 84 and 86 in S0 (Figure S4) which show significant displacements (see Table 5). Fundamentals 841 and 861 give indeed remarkable contributions to the peak at ∼2.8 eV in EMI and to the CPL spectrum at the same energy. These bands are FC-active (see δ in Table 5), nonetheless they also exhibit clear HT effects. This is evidenced by the fact that 841 is stronger than 861 in FC approximation (see Figure S3 of SI) while it becomes weaker than 841 at FCHT level. A more in depth analysis shows that, actually, the derivative of the electronic transition dipole is stronger along mode 84 than along mode 86. However, the FC and HT contributions are out of phase, and this determines a destructive interference and a remarkable decrease of 841 intensity at FCHT level (by 33% in CPL). In the same frequency region, the fundamentals of other modes contribute to determine the spectral shape. These are 891 in ABS/ECD, and 901 and 911 in EMI/CPL (notice that S0 mode 90 is projected on S1 mode 89, Table 5). Their intensity mainly derive from HT contributions as may be appreciated by noticing that they are not distinguishable in the FC spectra (see Figure S3 in the SI and their negligible dimensionless shift in Table 5). Very interestingly, the minimum (in absolute terms) in the ECD spectrum, roughly corresponding to peak 1 (at 3.19 eV) in the ABS spectrum, arises from the compensation of the opposite contributions of the FC-active 851 band and the HT-active 891 band. At variance, the lowest-negative band calculated at 3.12 eV in ECD is apparently determined by the contribution of a multitude of stick bands. Among them a significant (but not dominant) contribution is given by the fundamental of HT-active mode 39, (with frequency 758.92 cm-1). 22


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HT effects have a remarkable impact on the intensity of other modes also. An example is given by mode 9 whose progression is clearly seen in FC spectra both in in EMI/CPL and in ABS/ECD (Figure S3 in SI). Introducing the HT effect, the fundamental 91 is strongly quenched in ABS (by a factor 10) and in ECD (by a factor 3) while it is enhanced in EMI (by a factor 3) and finally is almost unaltered in CPL. The results of this complicated phenomenon is that 91 is very intense (similar to the 0-0 one) in EMI, is still clearly distinguishable in CPL, and it is very weak in ECD and ABS. Finally we focus on c4. In this case the relevance of HT effects is intermediate between what observed for c1 and c3. In fact, ECD and CPL vibronic stick bands with sign opposite to that of the convoluted spectrum are more intense in c4 than in c3. However, HT effects in c4 are not as intense as in c1 where they are able to cause a sign inversion in the convoluted spectrum. The pattern of vibrational progressions shows clear analogies with what seen for c3 and c1. It is characterized by a strong FC-active progression along mode 86 (visible in all the 4 spectra) which is a combination of CC stretchings and in-plane CCH bendings and clearly resembles modes 85 (on S1) and 84/86 (on S0) of c1 and modes 83/85 in S1 and 82, 83 in S0 of c3. These similarities were actually expected from the fact that the electronic transition in the different compounds does show similar characteristics (see Figure 2). Even the most active low frequency modes show some resemblance in all the molecules, although mode 14 of c4 (see Figure 11) has not an exact analogue in c2-c4 because it has some component of the C-Br stretching. HT-active modes 91 (S1) and 92 (S0, see Figure 11) in c4 also carry clear similarities with modes 89 (S1) and 90, 91 (S0) of c1. However, in c1 they look more symmetric and concentrated either in central or in the peripheral rings while in c4 they exhibit some asymmetry due to the presence of the Br substituent. A noteworthy difference of c4 is the strong contribution in the CPL of a HT band at ∼1700 cm-1. It has no analogue in the spectra of the other species and is actually responsible for the most intense and negative peak of the computed convoluted spectrum. Even if such band cannot be distinguished in experiment where, as shown in Figure 7, the signal is extremely weak, it is interesting to notice that is due to the fundamental of mode 104 of S0, a collective CC stretching partially localized on the rings closer to the Br substituent (see Figure 11). We finally make few purely empirical comments on the significance of the normal modes mostly involved in vibronic activities in ABS/ECD and in EMI/CPL. As far as band 1 (and possibly 23



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2) is concerned, we see from Figures 9-10 that modes 82-86 are antisymmetric with respect to a central pseudo-symmetry plane. Those modes, that are in the range 1350-1410 cm-1, have been called D-modes in the PAH/graphene literature and are responsible for the strongest ROA band observed for c4 (Johannesen et al)17 in agreement with the predictions by Liegeois and Champagne (2010)59,60. Instead, the normal modes responsible for the strongest IR and VCD bands at cal. 1490-1510 cm-1(observed for both c4 and c3) do not seem to be relevant here21. As regards to the normal modes 3 and 9 which give important contributions to band 0, we could not find their signature in other chiroptical spectroscopies; the lowest wavenumber band observed for c4 in ROA is at ca. 400 cm-1 and is due to delocalized out-of-plane HCC bendings (normal mode 19).

5. Conclusions In this study, we have applied first principles calculations to systematically study the vibronic structure associated to the ABS, EMI, ECD and CPL spectra of four hexahelicene molecules. Simple helicene systems, like the ones considered here, appear ideal for these studies, since CPL contributions is quite weak and originates mostly from vibronic activity (this fact had been pointed out earlier, by Weigang24). Instead helicenes carrying several hetero-atoms, like the triaryl-amines of ref. 61 or the systems studied by Crassous et al.62, are less interesting for vibronic studies, while having high CPL activity, related to helical sensitive states. The vibronic spectra have been computed exploiting TI and TD approaches in harmonic approximation considering both Duschinsky and HT effects. The similarity of the predictions of VH and AH models in FC approximation, indicated that harmonic approximation is reliable for these systems. Different HT expansions have been considered as well, by computing the derivatives of the transition electric and magnetic dipole moments at the equilibrium geometry of GS and ES. The results of ABS or EMI spectra calculations are only slightly influenced by the choice of the HT models. Moderate but more significant differences arise in ECD and CPL spectra where, for example, the ratio of the positive and negative parts of the intensity is significantly modified by HT effects. The nice general agreement with experiment witnesses that first principles calculations including vibronic contributions at simple harmonic level, and the effect of state couplings with a 24


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perturbative HT approach, can allow a reliable simulation even of rather complicated signals like those exhibited by the investigated helicenes, providing a firm framework for their interpretation. Somewhat larger discrepancies are observed for CPL, where however the agreement is still satisfactory. This finding shows that, at least for these species, CPL signal arises from a subtle balance of many different factors that cannot be quantitatively thoroughly reproduced at the adopted level of theory.

We wish to remind that the present approach neglects, for example,

anharmonicites and/or the contribution of terms beyond the linear one in the Taylor expansion of transition dipoles. It is likely that, the worsening of the results at room temperature is largely connected with the combined effect of these two approximations. Beyond the limitations of the adopted vibronic models, it is also possible that part of the discrepancies with respect to experiment (like the general over-estimation of the HT effects) is due to inaccuracies in the electronic method. It has been shown for example that while DFT functionals with different amount of Hartree-Fock exchange predict similar spectra for the S0/S1 transition, they show some differences for higher-energy states, which in turn, may be involved in HT intensity-lending mechanisms.6,63 Our results clearly show that HT effects dominate the shape of ECD and CPL spectra and have a remarkable impact also on ABS and EMI. They are the responsible for the sign inversion observed in ECD and CPL, for the differences between EMI and CPL and between ABS and ECD spectral shapes and, finally, for the breaking of the mirror symmetry between ABS and EMI and between ECD and CPL. The results obtained in this contribution allow us also to reconsider the characteristics of ECD and CPL of the S0/S1 transition with reference to the H and S categories. Vibronic analysis clearly confirms the conclusions of ref. 21, since both ECD and CPL S0/S1 vibronic spectra are very dependent on the specific systems so that they are both S features. Notwithstanding this, apart from some exceptions, the vibronic patterns distinguished in experiment seem to be determined by modes that can be labeled as H. In fact they represent collective CC stretchings coupled with CCH bendings of the rings which, additionally, are rather similar in all the investigate hexahelicenes. In summary, our results suggest that the S character of the ECD and CPL is basically an electronic feature, since differences among c1-c4 can be traced back to the different angle of the 25



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Page 26 of 52

electronic transition dipoles,21 which determines the strength of their rotatory strengths. On the contrary, FC vibronic features mainly arise from H vibrations. Displacements along HT modes induce intensity borrowing from other electronic states, changing in principle the aforementioned angle. Our results however indicate that the tempting deduction that they exhibit S features would be too simplistic. In fact, what really rules the HT activity of a given mode is its ability to couple the electronic states involved in the transition with close-lying states. In the investigated systems they all correspond to excitations of the π system delocalized on the helical structure (see the description of the S3 state in Table 3 and the molecular orbitals in Figure S1 in the SI). Therefore, it is not surprising that HT modes, i.e. modes able to perturb π conjugation by mixing with other states, actually exhibit H features. Clearly, substituents can have an impact on HT contributions, for example by tuning the energy differences of close-lying states (and therefore the effect of their couplings). In the hexahelicenes we studied here, however, in most of the cases this is not reflected into a substituent effect on the coupling mode. While this is the general trend, some exceptions exist. For instance, mode 104 in c4 shows both H and S characters, since it is delocalized on the helix (H), but it is also asymmetric and concentrated on the rings closer to the substituent (S). Another example is provided by mode 14 (S0) of the same molecule which shows some pronounced effect of the presence of the substituent, although it is not localized on the latter.

Acknowledgment The support of MIUR (PRIN 2010- 2011 prot. 2010ERFKXL) is acknowledged. The National Nature Science Foundation of China (Grant No. 21573129), the National Nature Science Foundation of Shandong Province (Grant No. ZR2015BQ001), and the General Financial Grant from the China Postdoctoral Science Foundation (Grant No.2013M531595) are also acknowledged. Y.

L.

acknowledges

the

financial

support

from

China

Scholarship

Council

(CSC,

No.201506220064) and J.C. acknowledges “Fundación Ramón Areces” for funding his postdoctoral position. Y. L. and J.C. thank the Pisa Unit of ICCOM-CNR Pisa for hospitality. We also acknowledge a generous grant of computer time from the Norwegian Programme for Supercomputing. 26


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Supporting Information Analysis of the dissymmetry factors at the maxima of ABS and EMI spectra, plots of the molecular orbitals of c1 involved in S0/S1 transition, FC spectral line shapes, assignments of FC spectra. Movies of the modes reported in Figures 9-11 and sketches of the S0 modes involved in strong vibronic transitions in EMI/CPL spectra of c1 and c3. This information is available free of charge via the Internet at http://pubs.acs.org

27



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Page 28 of 52

Table 1: H1-H2 (d1, Å), C1-C2 (d2, Å), torsion about inner CC bonds (C3-C4-C5-C6:τ1, C4-C5-C6-C7:τ2, C5-C6-C7-C8:τ3, C6-C7-C8-C9:τ4, degree) and dihedral angle C1-C3-C9-C2 (τ5, degree) optimized at CAM-B3LYP/tzvp level for in gas-phase hexahelicene (c1), 2-methyl-hexahelicene (c2), 5-aza-hexahelicene (c3), 2-bromo-hexahelicene (c4).

c1 GS

c2

ES

GS

c3

ES

GS

c4

ES

GS

ES

d1

6.30

6.18

6.27

6.16

6.40

6.26

6.28

6.16

d2

3.15

3.24

3.15

3.23

3.13

3.22

3.15

3.22

τ1

-13.72 -15.17 -13.56 -15.22 -12.13 -12.98 -13.65 -15.01

τ2

-28.80 -29.32 -28.76 -29.08 -27.75 -28.82 -28.82 -29.00

τ3

-28.80 -29.32 -28.70 -29.36 -29.33 -29.12 -28.71 -29.13

τ4

-13.72 -15.17 -13.80 -15.12 -13.58 -16.01 -13.79 -15.41

τ5

3.41

3.65

2.28

2.53

5.29

5.63

28


4.13

4.52

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: Vertical absorption energy EVA , vertical emission energy EVE (in eV), adiabatic energy FC HT |HT | Ead (in eV), total FC ( I tot ) and HT ( I tot ) intensity together with sum I tot (all in atomic units),

as well as dominating transition nature for the lowest excited state of the M enantiomers of all the four compounds in gas phase obtained at CAM-B3LYP/TZVP level of theory. c1

c2

c3

c4

3.67

3.66

3.68

3.65

3.33

3.32

3.32

3.32

3.50

3.49

3.50

3.49

ItotFC ( ABS )

0.025

0.043

0.10

0.061

ItotFC ( EMI )

0.0097

0.022

0.16

0.035

ItotHT ( ABS )

0.050

0.052

0.070

0.056

ItotHT ( EMI )

0.043

0.045

0.055

0.048

I totFC ( ECD)

0.0017

-0.00028

0.015

-0.0084

ItotFC ( CPL )

0.0021

0.00027

0.038

-0.00032

I totHT ( ECD )

-0.0067

-0.0082

-0.012

-0.010

ItotHT ( CPL )

-0.0031

-0.0020

-0.0084

-0.0053

0.014

0.013

0.015

0.016

0.011

0.010

0.011

0.011

H-1→L (0.49)

H-1→L+1 (0.42)

H-1→L+1 (-0.30)

H-1→L (0.37)

H→L+1 (-0.46)

H→L+2 (-0.36)

H→L (0.48)

H-1→L+1 (-0.30)

EVA EVE Ead

HT I tot ( ECD ) HT I tot ( CPL )

Transition (Coefficient)

H→L (-0.35) H→L+1 (-0.30)

H, the highest occupied molecular orbital (HOMO); L, the lowest unoccupied molecular orbital (LUMO)

29



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Page 30 of 52

Table3: Vertical transition energies, EVA (eV), total FC intensity for absorption and ECD (in the length representation) in atomic units of the first three excited states of the four compounds (M enantiomers) in gas phase obtained at CAM-B3LYP/TZVP level of theory.

ItotFC ( ABS ) S1 c1 0.025

I totFC ( ECD)

S2

S3

S1

S2

0.008

4.2

0.0017

0.0025

EVA S3

S1

S2

S3

-1.8 3.67 3.88 4.15

c2 0.043 0.0038 3.96 -0.0003 0.00095 -1.9 3.66 3.88 4.13 0.10

0.055

3.67

c4 0.061

0.005

3.71 -0.0084

c3

0.015

-0.0019 -1.6 3.68 3.91 4.20 0.0006

-2.0 3.65 3.87 4.12

30


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Table 4: Comparison of the dissymmetry factors between ECD and ABS ( g00 ) and between CPL lum and EMI ( g 00 ) at the 00 frequency for the M enantiomer of all the considered compounds

Theoretical data are taken from spectra computed at 0 K with the AH model. molecule

c1 c2 c3 c4

E00 (eV)

-5 g 00 (x10 )

Exp AH exp FCHT1 FCHT0 3.00 3.40 92 170 101 2.98 3.39 23 19 39 2.96 3.40 516 514 492 2.99 3.39 -226 -314 -361

lum (x10-5) g 00

exp 139 51 590 -(*)

(*) only the sign is defined from experiment

31


FCHT1 166 150 628 -115

FCHT0 262 184 626 -130


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 52

Table 5: Characteristics of selected vibrational modes relevant for the vibronic structures of the spectra of c1*

I aHT HT I tot

I aHT HT I tot

(ABS,%)

(ECD,%)

(ECD,%)

3

2.34

-0.71

0.39

66.59

-1.07

10

4.31

0.18

0.097

246.40

39

3.05

1.16

0.64

85

0.31

-0.12

0.067

S1 mode (a)

ABS/ECD

89

S0 mode (b)

8.25

I aHT HT

I tot

0.29

I bHT HT I tot

I bHT HT I tot

0.16

I bHT HT

I tot

S0 mode

ωb

(b)

(cm-1)

1

3

69.25

-0.34

0.93

11

263.42

758.92

0.00062

0.84

37

764.76

0.14

39

771.87

1391.55

-0.94

0.45

84

1398.79

0.35

86

1407.04

0.84

90

1472.12

0.07

91

1492.09

S1 mode

ωa

(a)

(cm-1)

ωa

δ

-1

(cm )

1466.72

ωb

-0.0058

δ

-1

(cm )

2

J ab

2

J ab

(EMI,%)

(CPL,%)

(CPL,%)

3

2.05

1.6

0.62

69.25

1.08

1

3

66.59

9

1.57

0.57

0.22

250.46

0.88

0.93

9

234.62

11

5.95

1.46

0.57

263.42

-0.21

0.93

10

246.40

39

5.77

8.73

3.37

771.87

0.017

EMI/CPL 84

2.10

11.04

4.27

1398.79

-0.87

0.77

38

753.83

0.14

39

758.92

0.45

85

1391.55

0.33

86

1403.78

0.61

86

1403.78

86

0.04

4.47

1.73

1407.04

0.71

0.35

85

1391.55

90

5.11

0.011

0.0043

1472.12

0.047

0.84

89

1466.72

91

4.37

4.43

1.87

1492.09

-0.1

0.85

91

1475.70

0.07

89

1466.72

*

The contribution of each mode to the HT intensities is defined in Eqs. 7; δ is the displacement in

dimensionless units;

2

J ab

is the square of the Duschinsky matrix element representing the

projection of mode a of state S1 on mode b of the initial state S0, see detailed definition in ref. 3.

32


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I aHT HT I tot

I aHT HT I tot

(ABS,%)

(ECD,%)

(ECD,%)

3

1.29

2.81

2.13

66.69

1.19

0.97

3

70.04

9

2.26

0.94

0.71

233.31

-0.77

0.94

9

249.85

83

0.29

0.51

0.39

1385.65

-0.65

0.32

82

1402.41

84

0.99

1.69

1.27

1402.77

0.48

0.72

83

1407.44

85

0.43

0.24

0.18

1410.95

0.64

0.47

86

1443.58

0.33

82

1402.41

S1 mode

ωa (cm-1)

S1 mode (a)

ABS/ECD

S0 mode (b)

EMI/CPL

I bHT HT I tot

I bHT HT I tot

I aHT I

HT tot

I bHT HT

I tot

ωa (cm-1)

ωb

δ

δ

2

J ab

2

J ab

S0 mode

ωb (cm-1)

(b)

(a)

-1

(EMI,%)

(CPL,%)

(CPL,%)

(cm )

3

1.00

2.28

1.72

70.04

9

1.97

1.17

0.88

249.85

0.83

0.94

9

233.31

82

0.55

0.94

0.71

1402.41

-0.89

0.33

85

1410.95

0.32

83

1385.65

0.72

84

1402.77

0.16

83

1385.65

83

0.30

0.16

0.12

1407.44

-1.11

0.55

0.97

3

66.69

*

The contribution of each mode to the HT intensities is defined in Eqs. 7; δ is the displacement in

dimensionless units;

2

J ab

is the square of the Duschinsky matrix element representing the

projection of mode a of state S1 on mode b of the initial state S0, see detailed definition in ref. 3.

33



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Page 34 of 52


I aHT HT I tot

I aHT HT I tot

(ABS,%)

(ECD,%)

(ECD,%)

3

0.99

3.09

1.98

53.26

-0.78

0.86

3

55.76

5

0.21

0.68

0.44

92.53

-0.70

0.98

5

95.12

10

2.91

7.10

4.55

221.95

-0.22

0.91

10

227.53

11

2.37

6.27

4.03

246.51

-0.23

0.92

11

260.01

86

0.53

-0.24

-0.15

1390.50

-0.85

0.43

86

1402.59

0.27

85

1397.53

0.50

91

1471.67

0.32

92

1478.99

S1 mode

ωa (cm-1)

S1 mode (a)

ABS/ECD

91

7.11

I bHT HT I tot

S0 mode (b)

EMI/CPL

15.60

I bHT HT I tot

I aHT I

HT tot

10.01

I bHT I

HT tot

ωa (cm-1)

1467.10

δ

0.04

ωb

δ

2

J ab

2

J ab

S0 mode

ωb (cm-1)

(b)

(a)

-1

(EMI,%)

(CPL,%)

(CPL,%)

(cm )

3

0.40

2.21

1.12

55.76

-0.63

0.86

3

53.26

5

0.035

0.31

0.16

95.12

0.72

0.98

5

92.53

8

0.85

1.71

0.87

172.95

-0.33

0.98

8

165.06

13

3.14

3.98

2.03

281.09

-0.45

0.51

12

260.34

0.42

13

269.47

14

2.89

5.00

2.54

291.81

-0.30

0.90

14

283.82

36

1.52

8.57

4.36

697.07

0.05

0.84

37

698.72

0.15

36

685.76

86

0.93

91

3.24

0.79

9.02

0.40

4.59

1402.59

1471.67

0.93

-0.04

92

5.71

4.55

2.31

1478.99

-0.09

104

1.54

7.83

3.98

1703.07

-0.20

0.43

86

1390.50

0.36

87

1397.42

0.50

91

1467.10

0.32

92

1474.10

0.36

91

1467.10

0.29

92

1474.10

0.63

102

1634.86

0.12

101

1626.56

*

The contribution of each mode to the HT intensities is defined in Eqs. 7; δ is the displacement in 2

dimensionless units; Jab is the square of the Duschinsky matrix element representing the projection of mode a of state S1 on mode b of the initial state S0, see detailed definition in ref. 3.

34


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Figure captions:

Figure 1 Sketches of the M enantiomers of the four compounds investigated in this work: hexahelicene (c1), 2-methyl-hexahelicene (c2), 5-aza-hexahelicene (c3), 2-bromo-hexahelicene (c4).

Figure 2 Difference of the electronic density of the first excited state and ground state of c1, c2, c3, and c4 optimized at CAM-B3LYP/tzvp level in gas phase

Figure 3 Experimental ABS, ECD, EMI, CPL spectra, the plots have been mapped in the frequency domain as explained in the text. The ABS and ECD bands at low energy have been multiplied for different factors, as indicated in the legend. The emissive spectra are in arbitrary units and the relative intensity is not comparable: in particular c4 data have been acquired with very high gain. Comparison of EMI and CPL intensity of the same molecule can be instead correctly done and used to report the dissymmetry factor.

Figure 4 Vibronically resolved ABS, ECD, EMI, CPL spectra at CAM-B3LYP/tzvp level for c1 in gas phase convoluted with a Gaussian with full width at half maximum (FWHM) of 0.08 eV. For a clear comparison with the experimental spectra (also been given), the vibronic spectra of AH and VH have been shifted by 0.41 and 0.42 eV, respectively.

Figure 5 Vibronically resolved ABS, ECD, EMI, CPL spectra at CAM-B3LYP/tzvp level for c2 in gas phase convoluted with a Gaussian with FWHM of 0.08 eV. The vibronic spectra have been shifted by 0.42 eV for a clearer comparison with the experimental spectra (also been given).

Figure 6 Vibronically resolved ABS, ECD, EMI, CPL spectra at CAM-B3LYP/tzvp level for c3 in gas phase convoluted with a Gaussian with FWHM of 0.08 eV. For a clearer comparison with the 35



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experimental spectra, the vibronic spectra of AH have been shifted by 0.42 eV, ABS/VH and ECD/VH have been red-shifted by 0.42 eV, EMI/VH and CPL/VH have been red-shifted by 0.46 eV.

Figure 7 Vibronically resolved ABS, ECD, EMI, CPL spectra at CAM-B3LYP/tzvp level for c4 in gas phase convoluted with a Gaussian with FWHM of 0.08 eV. The vibronic spectra have been shifted by 0.4 eV for a clearer comparison with the experimental spectra (also been given).

Figure 8 Assignments of the main stick bands of FCHT1|AH (FCHT is equivalent to FC+HT) spectra of c1, c3 and c4. The transitions with largest HT contributions are colored in red. Vibrational contributions are labeled as “nx”, where x indicates the quanta deposited on the excited-state normal mode n. When the x is not explicitly given, it is intended that the corresponding mode is in the ground state (x=0). The sticks bands are reported in arbitrary intensity to fit on the same scale of the convoluted spectra.

Figure 9 Selected vibrational modes calculated of the excited electronic state of hexahelicene c1 computed at CAM-B3LYP/tzvp level of theory.

Figure 10 Selected vibrational modes calculated of the excited electronic state of 5-aza-hexahelicene c3 computed at CAM-B3LYP/tzvp level of theory.

Figure 11 Selected vibrational modes calculated of the ground electronic state of Br-hexahelicene c4 computed at CAM-B3LYP/tzvp level of theory.

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Figure 1 Liu et, al

c1

c2

c3

c4

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Figure 2 Liu et, al

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Figure 3 Liu et, al

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Figure 4 Liu et, al

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Figure 5 Liu et. al

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Figure 6 Liu et. al

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Figure 7 Liu et, al

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Figure 8 Liu et, al

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Figure 9 Liu et, al

c1_mode 3_S1

c1_mode 10_S1

c1 _mode 85_S1

c1_mode 89_S1

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Figure 10 Liu et. al

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c3_mode 9_S1

c3_mode 84_S1

c3_mode 85_S1

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Figure 11 Liu et. al

c4_mode 14_S0

c4_mode 92_S0

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c4_mode 104_S0


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Vibronic Coupling Explains the Different Shape of Electronic

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