Anisotropic Dissymmetry Factor, g: Theoretical Investigation on Single

Article pubs.acs.org/JPCA

Anisotropic Dissymmetry Factor, g: Theoretical Investigation on Single Molecule Chiroptical Spectroscopy Masamitsu Wakabayashi,†,‡ Satoshi Yokojima,§,‡ Tuyoshi Fukaminato,∥,⊥ Ken-ichi Shiino,† Masahiro Irie,# and Shinichiro Nakamura*,‡ †

Department of Biomolecular Engineering, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama, Kanagawa 226-8501, Japan ‡ Nakamura Lab, RIKEN Research Cluster for Innovation, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan § Tokyo University of Pharmacy and Life Sciences, 1432-1 Horinouchi, Hachiouji-shi, Tokyo 192-0392, Japan ∥ Research Institute for Electronic Science, Hokkaido University, N20, W10, Kita-ku, Sapporo 001-0020, Japan ⊥ PRESTO, Japan Science and Technology Agency (JST), 5-3, Yonbancho, Chiyoda-ku, Tokyo, 102-8666 Japan # Department of Chemistry, Rikkyo University, Nishi-Ikebukuro 3-34-1, Toshima-ku Tokyo, 171-8501 Japan S Supporting Information *

ABSTRACT: A formula for an anisotropic dissymmetry factor g evaluating the chiroptical response of orientationally fixed molecules is derived. Incorporating zeroth- and first-order multipole expansion terms, it is applied to bridged triarylamine helicene molecules to examine the experimental results of singlemolecule chiroptical spectroscopy. The ground- and excited-state wave functions and a series of transition moments required for the evaluation of the anisotropic g value are calculated using time-dependent density functional theory (TDDFT). The probability histograms obtained for simulated g values, uniformly sampled in regard to the direction of light propagation toward the fixed molecule, show that even for a given diastereomer, the dissymmetry factors have positive and negative values and can deviate from their averages to a considerable extent when the angle between the electric dipole transition moment and the propagation vector of the incident light is near 0 or 180°.

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INTRODUCTION The natural chiroptical response of polymer materials or chemical compounds is an important signal that reflects their states and properties. This fascinating phenomenon is of interest in fundamental research and has practical applications in many different fields.1−8 Circular dichroism (CD) occurs when the absorption of left-handed circularly polarized light differs from that of right-handed circularly polarized light, and CD spectroscopy can help investigators identify the absolute configurations of natural compounds and artificially synthesized compounds. It is also used to quantify enantiomeric excess in chemical samples. CD can be observed in a variety of systems and in various phases (solution, crystal, liquid crystal, gas, liquid, and solid),9−16 and the interpretation of CD spectra is considerably facilitated by auxiliary theoretical predictions.17−40 The chiroptical response of an absorbing medium can be evaluated by the dissymmetry factor g defined as follows: g=

IL − IR 1 (I + IR ) 2 L

conditions, and excitation wavelengths. Dissymmetry factors are by definition within the range from −2 to +2. In the case of isotropic bulk samples whose CD signals arise from small organic molecules, the absolute values of the dissymmetry factors are normally small: |g| ∼ 10−3. This is reasonable theoretically because the molecular sizes are much smaller than the helical pitch of circularly polarized light, and the magnitude of the denominator in eq 1 (IL + IR divided by 2) is mainly due to electric dipole transition, whereas the magnitude of the numerator (IL − IR) is mainly due to the coupling between electric and magnetic dipole transitions. In the case of anisotropic samples, the coupling between electric dipole and electric quadrupole transitions also contributes. The contributions of magnetic dipole transition and electric quadrupole transition to light absorption are usually a few orders of magnitude smaller than that of electric dipole transition. Therefore, the extreme values of g near −2 or +2 can only happen when dipole absorption is not dominant. The theory on light absorption and CD is described in detail in the Theoretical Background section.

(1)

where IL and IR are the absorption intensities for left- and righthanded circularly polarized light. This factor is a dimensionless quantity useful when the signs and the magnitudes of CD signals are compared among various kinds of samples, © 2014 American Chemical Society

Received: September 25, 2013 Revised: June 10, 2014 Published: June 11, 2014 5046

dx.doi.org/10.1021/jp409559t | J. Phys. Chem. A 2014, 118, 5046−5057

The Journal of Physical Chemistry A

Article

molecules are significantly oriented at a certain direction, we have to take meticulous care to avoid artifacts arising from the linear biases.60−65 Nearly replicating the experiments of Hassey et al. but correcting the linear bias inherent to the dichroic mirror and not preferentially selecting for photostable molecules, they concluded that the g distributions for the two diastereomers did not differ significantly and thus were experimentally indistinguishable. Cyphersmith et al. performed single-molecule FDCD experiments with the M-2 helicene dimer,66 and the results showed that, despite the measurement imprecision due to intensity fluctuations and residual ellipticity, the dissymmetry factors for individual molecules are larger than those for bulk samples. That is, single-molecule g values with an absolute value ≥0.1 are readily observable.66 These results are consistent with the possibility that disagreement between reported experimental results might be due to the different subpopulations of molecules sampled for the histograms.67,68 The problem of the artifacts arising from linear bias, however, still remains. Motivated by those experiments, Furumaki et al. applied the FDCD measurements to single immobilized chlorosomes isolated from the photosynthetic green sulfur bacterium Chlorobaculum tepidum to investigate the excitonic structure of the light-harvesting complexes.69 In their report they suggested that even both the left- and right-handed circularly polarized beams had perfect circular polarization at the sample plane, slight differences in their spatial intensity distribution could give rise to false CD signals. After taking into account the effect on the measurement and correcting the calculation, the difference between two single-chlorosome g values became much smaller than that calculated using the raw data. They indicated still another point where we should be careful when we discuss the distribution of g values. For the purpose of investigating the instrumental noise effect, they used two largely different excitation powers to induce different levels of noise and showed that the breadth of g value histogram can be affected by the factor. As we can see in these arguments, the single-molecule chiral detection still has many ambiguities. In this study, we derived a dissymmetry factor g formula taking account of molecular orientation and we evaluated the g values of single molecules to get insight into experimental results of interest. Experimental interpretation can greatly benefit from theoretical analysis. Our theoretical formulation of the g values of single molecules allows the effect of their experimental dependence on the incident light directions to be evaluated in a systematic way.

Although the chiroptical response is sensitive to molecular structure, orientation with respect to incident light, temperature, and the surrounding environment,6,41−49 ensemble measurement of CD spectra and the dissymmetry factors obscures the information associated with the microscopic conditions of individual molecules. Single-molecule spectroscopy, on the other hand, has a potential to clarify the hidden heterogeneities and fluctuations in physical systems and chemical samples50−54 and has recently been used to investigate the chiroptical response of individual molecules or complexes embedded in a polymer matrix. The chiroptical signals obtained from those measurements are quite interesting and provide new sorts of information. The first experiment on single-molecule fluorescence detected circular dichroism (FDCD) was reported by Hassey et al.55−57 The M-type and P-type of bridged triarylamine helicene58 derivatives (Figure 1) immobilized on a poly-

Figure 1. Structure of bridged triarylamine helicene.

cycloolefin surface were irradiated with left- and right-handed circularly polarized light alternately, the amount of photons emitted from each molecule was monitored, and the dissymmetry factors of individual molecules were evaluated by using the fluorescence intensities averaged over time as an index of the absorption rates associated with corresponding circularly polarized light. Very surprisingly, the g values of individual molecules obtained from their experiment were widely distributed in the range from −2 to +2 and both positive and negative values were observed even for a given diastereomer, which cannot be seen in the case of isotropic bulk samples. Some of the absolute values of the dissymmetry factors of individual molecules were much larger than those of the dissymmetry factors of bulk samples. Hassey and her colleagues analyzed data only for molecules with a photochemical lifetime long enough to prevent short-time intensity fluctuations (blinking), and they remark that this sampling bias might have distorted the g distribution to larger absolute values.56 Because the histograms of g values made from single-molecule FDCD measurements for individual P- and M-type diastereomers were, for the most part, mirror images of each other, they concluded that their results were attributed to the chiral responses of single chiral molecules: information that in the conventional ensemble measurement is hidden by averaging. Cohen and co-workers, however, pointed out that Hassey’s experiments were not free from artifacts due to linear biases that arise from the imperfect polarization of illumination.59 In the CD measurement of anisotropic samples, where the

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THEORETICAL BACKGROUND Circular dichroism is defined as a difference between the absorption of left- and right-handed circularly polarized light. Here we focus our discussion on the anisotropic dissymmetry factor in the case of electronic circular dichroism (ECD) whose signals originate from electronic excitations of molecules. We discuss CD theory based on Fermi’s golden rule, which is a consequence of the time-dependent perturbation theory in quantum mechanics. 63,70 CGS units are used for the formulation. The detailed and basic definitions of the vector operations and the notations used here are shown in the Supporting Information. CD arises through coupling between electric dipole transition and either electric quadrupole or magnetic dipole transitions. In the case of anisotropic CD, both types of 5047



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couplings may contribute to rotatory strength. In the case of isotropic CD, in contrast, due to the orientational average the contribution of the coupling between electric dipole and electric quadrupole transitions is zero. The CD of a sample consisting of randomly oriented molecules is quantified by the Rosenfeld equation (eq 2 or 3), which defines rotatory strength R as the imaginary part of scalar product of the electric dipole transition moment μ0n and magnetic dipole transition moment mn0, which are characteristics of the excitation from the ground state 0 to the nth excited state. R = Im{μ0n ·m n0} ⎧ ieℏ = Im⎨ ⟨0|e ∑ rs|n⟩·⟨n| − ⎪ mc 2 ⎩ s ⎪

w0(λ→) n =

ηn0 = ⟨n| ∑ e ik·rsps|0⟩ s

⎪

s

⎭

w0(λ→) n =

4R D

eR =

(12)

Figure 2. (a) Unit vectors defining the coordinates of the physical system. u3 is the propagation direction of the circularly polarized light. (b) Spherical coordinate system. The molecule that is fixed orientationally is at the origin. The incident directions of light u3 are chosen without bias in regard to ϕ and θ when sampling g values for the histrogram simulations.

By substituting eq 11 or eq 12 into eq 10 and using the relation

(5)

|ηn0|2 = |ηn0 ·u1|2 + |ηn0 ·u 2|2 + |ηn0 ·u3|2

(13)

we can write the transition probability density per unit time for left- or right-handed circularly polarized light as w(L) =

2π 2e 2 ρ (ω) I(ω){|ηn0|2 − |ηn0 ·u3|2 m2cω 2ℏ2 n + iu3·(η * × η )}

(14)

2π 2e 2 ρ (ω) I(ω){|ηn0|2 − |ηn0 ·u3|2 m2cω 2ℏ2 n − iu3·(η * × η )}

(15)

n0

(6)

n0

or

where A(ω) is the real amplitude of the electromagnetic field, eλ is the polarization vector, k is the wavenumber vector, rs is the coordinates of the sth electron, and ∇s is the gradient del operator of the sth electron. The light intensity is defined as

ω2 I(ω) = |A(ω)|2 2πc

u1 − iu 2 2

(4)

where λ is a polarization index and Ĥ ′λ is the time-independent part of the interaction between charged particles and light. The ρn(ω) represents the density of states for the n-th electronic excited state. A fully retarded expression for the Hamiltonian is ieℏ A(ω)eλ ·∑ e ik·rs∇s Hλ̂ ′ = mc s

(10)

where the indices L and R correspond to the left- and righthanded circularly polarized states of light and where u1, u2, and u3 form a right-handed system of mutually orthogonal unit vectors as shown in Figure 2a.

where dipole strength D is the square of the absolute value of electric dipole transition moment, i.e., |μ0n|2. In theoretical treatment of orientational CD, the anisotropy of the rotatory strength is represented by a second-rank tensor called the rotatory strength tensor.63,71,74−78 There are reports on quantum chemical studies estimating the elements of the tensor of specific molecules.76,79−83 Here we present an expression for the dissymmetry factor of oriented molecules for the purpose of theoretical evaluation. We start with Fermi’s Golden Rule (eq 5), which is more fundamental than the approximation given in eq 3. Consider a transition from a ground state |0⟩ to an excited state |n⟩. The transition probability density per unit time at the angular frequency ω is given by w0(λ→) n(ω) = (2π /ℏ2)ρn (ω)|⟨n|Hλ̂ ′|0⟩|2

4π 2e 2 ρ (ω) I(ω)e Tλ ηn0ηn*0Te*λ m2cω 2ℏ2 n

By defining the propagation vector u3 as the unit vector along the direction of incident light, we can write the vectors for leftand right-handed circular polarization vectors as follows: u + iu 2 eL = 1 (11) 2

(3)

The subscript n0 (0n) expresses acting the operator on |0⟩ (| n⟩) and taking the scalar product with ⟨n| (⟨0|), where the |0⟩ and |n⟩ are respectively the electronic wave functions of the ground and excited states. The s is an index indicating the sth electron. The Rosenfeld equation can be derived by considering the contribution of only electric and magnetic dipole transition moments and taking the average over all orientations.71,72 The dissymmetry factor of an isotropic bulk sample is given approximately as follows:73 g=

(9)

and then eq 8 becomes

⎫ ⎪

(8)

Because the momentum operator of the sth electron ps is −iℏ∇s, we introduce

(2)

∑ rs × ∇|s 0⟩⎬

4π 2e 2 ρ (ω) I(ω)|⟨n| ∑ ei k·rseλ ·∇|s 0⟩|2 m2cω 2 n s

w(R) =

n0

n0

The exponential in eq 9 is expanded in a power series as

(7)

e ik·rs = 1 + ik·rs +

By substituting eq 6 into eq 5 and using eq 7, we obtain the following equation84 5048

(ik·rs)2 + ··· 2!

(16)



Article

The common dipole approximation incorporates the terms of this series up to the zeroth-order and, together with eq 8, enables us to evaluate the absorption of light in any polarization state if electric dipole transition is dominant. For the calculation of CD, the difference between w(L) and w(R) is required. In this case, the first and the second terms in eq 14 and eq 15 cancel out, and the first-order term in the series becomes influential (see the Supporting Information, S38−S39). From now on we deal with the zeroth-order and first-order terms in the series and ignore the higher-order terms, whose contributions rapidly decrease to zero as the order increases. This treatment is valid when the molecular sizes are much smaller than the wavelengths; that is, k·r ≪ 1 (small molecule limit).77 Representing k explicitly as (ω/c)u3, we use ηn0 ≈ pn0 + i(ω/c)(pr T)n0 u3

(pr T)n0 =

⎛0 −(mz )n0 (my)n0 ⎞ ⎜ ⎟ imωn mc ⎜ −(mx )n0 ⎟ (mz )n0 0 = Q n0 + ⎟ 2e e ⎜ ⎜ − (m ) (m ) ⎟ 0 y n0 x n0 ⎝ ⎠ (23)

Note that the electric dipole transition moment can be described in two different ways: the velocity form and the length form. One of them uses the linear momentum operator p and is related to the other that uses the operator μ (=er) as follows:

(17)

p0n = −

where

pn0 = ⟨n| ∑ ps|0⟩ T (pr T)n0 = ⟨n| ∑ pr |0⟩ s s s

(19)

For the UV−vis excitation wavelengths of small organic molecules, under the small molecule limit the magnitude of the first-order term is much smaller (∼×10−3) than that of the zeroth-order term. Because the rate of excitation is proportional to the absorption intensity, if the observation time is long enough, the dissymmetry factor can be written as g=

RM = −

(R)

+w )

RQ = −

16π 2ωn 3cℏ2ω

ρn (ω) I(ω)u 3TRu3

(21)

where ωn is the Bohr angular frequency of the transition and R is the rotatory strength tensor defined as R=−

3e 2 p0n × (pr T)n0 2 2m cωn

3e 2 {p × (pr T − rp T)n0 } 4m2cωn 0n

(25)

3e 2 {p0n × (pr T + rp T)n0 } 2 4m cωn

(26)

It should be noted that in the case of anisotropic circular dichroism the contribution of the coupling between electric dipole and electric quadrupole transitions can be as large as that of the coupling between electric dipole and magnetic dipole transitions because both of them originate from the first-order term in eq 16. On the other hand, the denominator of eq 20 becomes

(20)

The numerator in eq 20 is rewritten by subtracting eq 15 from eq 14 and using eq 17, as follows: w(L) − w(R) =

(24)

The other is coupling between electric dipole and electric quadrupole transitions:

w(L) − w(R) 1 (w(L) 2

imωn μ e 0n

This can be directly derived from the Heisenberg equation of motion. The equation holds true when the wave functions |0⟩ and |n⟩ are exact. Note that if |0⟩ and |n⟩ are obtained approximately, as is usually the case with ab initio calculations, eq 24 does not always hold true.85−87 We can divide R into two parts. One is coupling between electric dipole and magnetic dipole transitions:

(18)

s

1 1 (pr T + rp T)n0 + (pr T − rp T)n0 2 2

1 (L) 2π 2e 2 (w + w(R)) = 2 2 2 ρn (ω) I(ω)[|pn0|2 − |pn0 · u3|2 2 m cω ℏ + (ω /c)2 {|(pr T)n0 u3|2 − |u 3T(pr T)n0 u3|2 }] (27)

(22)

(The detailed derivation of the numerator and the denominator of eq 20 from eqs 14, 15, and 17 is given in the Supporting Information.) By substituting eqs 21 and 27 into eq 20, we obtain

(For a more detailed derivation of eq 21, see the Supporting Information.) The numerator in eq 20, w(L) − w(R), is the scalar quantity proportional to the CD intensity. The CD for the specific direction of light propagation u3 is obtained by using eq 21. The tensor R can be obtained by calculating the transition from |0⟩ to |n⟩ in a system under investigation. The quadratic form uT3 Ru3 results in a scalar quantity, which gives the rotatory strength in the anisotropic case. The factor on the right-hand side of eq 22 is selected so that the average of uT3 Ru3 over all orientations is equivalent to the rotatory strength defined by the Rosenfeld equation (see Proof S1 in the Supporting Information). The tensor (prT)n0 can be represented as the sum of the symmetric part and the antisymmetric part, which are related to the electric quadrupole transition moment tensor Qn0 and magnetic dipole transition moment mn0:

g=

8m2ωωn 3e 2

· u 3TRu3

[|pn0|2 − |pn0 · u3|2 + (ω /c)2 {|(pr T)n0 u3|2 − |u 3T(pr T)n0 u3|2 }] (28)

This equation gives the dissymmetry factor for a given propagation vector u3. The tensor (prT)n0 appearing in the denominator can be evaluated by using eq 23. Qn0 and mn0 are obtained approximately from the ground-state and excited-state wave functions |0⟩ and |n⟩ by quantum chemical calculations. If for the denominator we consider only the zeroth-order term of 5049



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the power-series expansion (eq 16), the third and fourth terms on the right-hand side of eq 27 do not appear. Due to the inconsistent approximation, the estimated g values deviate from the range from −2 to +2 or diverge to infinity when pn0 and u3 are nearly parallel. In contrast, as shown in the following section, such improper behavior is avoided in the numerical evaluation of eq 28. Thus, it is important to consider the power-series expansion consistently as shown in eq 16 where the terms up to the first-order expansion are included for both the numerator and denominator. It is known that for some variables there is a gauge origin problem. That is, the magnetic dipole transition moment mn0, the electric quadrupole transition moment Qn0, and the length form rotatory strength Im{μn0·mn0} are dependent on the choice of the origin for calculation. On the other hand, both the velocity and length form electric dipole transition moments pn0 and μn0; the velocity form rotatory strength and the rotatory strength tensor R (eq 22) are independent of the choice of origin. The origin dependence depends on the approximation we use. As for the anisotropic g (eq 28), the numerator uT3 Ru3 and the first and second terms in the denominator (i.e., |pn0|2 − |pn0·u3|2) are independent of the choice of origin, whereas the last term in the denominator (i.e., (ω/c)2{|(prT)n0u3|2 − | uT3 (prT)n0u3|2}) is origin-dependent. In spite of the origin dependence of the g given by eq 28, the formula is useful because its origin dependence is small. This is because the origin-dependent term results from the first term in the multipole expansion (eq 16) and its contribution is small in comparison with the contributions of the first and second terms in the denominator unless pn0 is parallel to u3. When the vectors pn0 and u3 are nearly parallel, the first and the second terms in the denominator are small and the last term in the denominator dominates the result. We found, however, that in this case the origin dependence disappears from the last term in the denominator, as shown in the Supporting Information. (The gauge origin dependence of some variables appearing in this paper is discussed both theoretically and numerically in the Supporting Information.) Apart from the velocity gauge we use here, the “gauge-including atomic orbitals” (GIAOs, London orbitals) is another option to avoid the gauge origin problem originates from basis set incompleteness.83 We should be careful not to confuse the dissymmetry factor of an isotropic bulk sample with the average of dissymmetry factors observed in orientational CD. In the former case, the set of IL and IR obtained is ensemble averaged over all the contributions of individual molecules, as is the set of IL − IR and 1/2(IL + IR). In the latter case, on the other hand, singlemolecule measurements are made and the average of dissymmetry factor is calculated after getting the g value for each incident light direction. This is shown by ⟨f (x)⟩ ≠ ⟨g (x)⟩

f (x ) g (x )

Article

COMPUTATIONAL DETAILS

The geometry of the bridged triarylamine helicene molecule (Figure 1) was optimized using density functional theory (DFT)88 with the hybrid functional B3LYP89 and the 631+G(d,p) basis set.90−99 Each optimized geometry was confirmed by frequency analysis to be a minimum on the potential energy surface. Franck−Condon excited-state wave functions and the transition moments at the optimized geometries were calculated using time-dependent density functional theory (TDDFT)100 with the 6-31+G(d,p) basis set. To validate and confirm our method, several basis sets for the excited-state calculations were tested, and the results are presented in the Supporting Information (Tables S2−S9). The calculations were carried out using the Gaussian09 program.101 Bridged triarylamine helicenes have attracted the attention of researchers who have made theoretical investigations from various standpoints.55,102,103 Here we focus on the properties of the rotatory strength tensor and the anisotropy of the dissymmetry factor. The formulas described in the previous section concern absorption CD. In applying them to the single-molecule FDCD measurement case, we assumed that the rate of fluorescence emission is proportional to the rate of excitation. Fluorescence emission has indeed been used to monitor absorption differences in FDCD experiments.56,59 The tensor (prT)n0 and the rotatory strength tensors R, RM, and RQ are constructed by three transition moments (electric dipole, magnetic dipole, and electric quadrupole transition moments) as shown in eqs 22−26. The probability histograms of the dissymmetry factor for each diastereomer were obtained by using eq 28 for each light beam direction u3. The incident directions were chosen without bias in terms of the relative angle to the orientationally fixed molecule. These samplings over the light beam direction were performed uniformly over sets of zenith angle θ and azimuth angle ϕ (Figure 2b). For each angle the step size of each angle is 1°. The geometry of the molecule was fixed during the sampling calculation.

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RESULTS AND DISCUSSION Electronic Structures. We optimized the geometry of the M-type atropisomer of bridged triarylamine helicene (Figure 1) and obtained four conformations with different orientations of the camphanate group. The structure with the lowest energy among them is shown in Figure 3a,b together with the canonical Kohn−Sham orbitals for the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Other conformers are higher in energy by at least 2.46 kcal/mol. Optimizing the geometry of the P-type atropisomer, we obtained five conformations. The optimized geometries and their energies are shown in the Supporting Information (Figures S1 and S2). The electronic excitation wavelength with the lowest excitation energy was calculated to be 432 nm, which agrees with the absorption spectrum. The experimental absorption band of bulk samples is broad, extending from about 400 nm to about 480 nm,104 possibly because at room temperature the molecules are flexible and have many conformational variations. The calculated wavelength of the second-lowest electronic excited state was 374 nm, which is energetically quite separated from the former absorption band. The excitation wavelengths within the absorption band with the lowest excitation energy were

(29)

where the functions f(x) and g(x) are dependent on a variable x at the same time. In the actual single-molecule CD experiments, the set of IL and IR is obtained by the alternate radiation of the left- and right-circularly polarized light. Thus, our results are the idealistic case where the time scale of the change of the molecular orientation is much slower than the duration of the left- and right-circularly polarized light alternation cycle. 5050



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S3). From these calculated components the dissymmetry factor for the isotropic bulk case can be estimated by using eq 4 to be −0.0009, which is in agreement with the one obtained experimentally: −0.0010.104 Furche et al. systematically investigated the CD spectra of [n]-helicene (n = 4−7,12) by using TDDFT calculations and showed that the calculations reproduce important spectral features and facilitate the assignment of the low-energy bands.105 As far as hexahelicene (n = 6) is concerned, the CD signal of the lowest energy electronic excitation is very small and its sign is not easily reproduced correctly. This also holds true for the bridged triarylamine helicenes. We used diffuse functions (the 6-31+G(d,p) basis set) because they are required to reproduce the sign of rotatory strength reliably.105 The calculated value of the rotatory strength is shown in Table 2. Although there is no experimental report on the rotatory strength for this molecule, we can compare one-third of the trace of the rotatory strength tensor with the experimental CD intensity; the CD intensity is proportional to the trace of the rotatory strength tensor. From Table 2, the one-third of the trace of the rotatory strength tensor for the lowest excited state is −7.19 (10−40·erg·esu·cm)/ Gauss. Additionally, we calculated the one-third of the trace of the rotatory strength tensor for the second lowest excited state and found that it was 20.53 (10−40·erg·esu·cm)/Gauss. The ratio of the experimental CD intensity between the lowest and second lowest excited states is about −1:+2.104 This ratio agrees with the calculated ratio using the rotatory strength tensor of −1:+2.86. 3D-Representation of Numerator and Denominator in Eq 28. Three-dimensional representations of the R, RQ, and RM of an M-type diastereomer are shown in the upper half of Figure 4, and the corresponding representations of a P-type diastereomer are shown in the lower half. (As discussed in the Theoretical Background section, R appears in the numerator in eq 28 for g, and R = RM + RQ.) This useful display method was introduced by Pedersen et al.74 Consider an incident light propagating toward the origin of the molecular coordinate system: when the light traverses the red surface the sign of the rotatory strength is minus, and the distance from the surface to the origin corresponds to the magnitude of the rotatory strength. That is, the color of the surface indicates the sign, and the distance from a surface point to the origin indicates the magnitude of the rotatory strength for a given light beam propagating along that direction. The pictures in Figure 4 show the magnitude and the sign of CD as a function of direction of incident light, indicating that even one given diastereomer molecule can show both positive and negative CD signals when molecules are fixed. Although both positive and negative surfaces appear in the 3D representations of RM and RQ, averaging RQ over all orientations always turns out to be zero (Proof S2 in the Supporting Information by showing that (1/ 3)TrRQ = 0), whereas averaging RM over all orientations turns out to be a nonzero value but fairly small as for the case under consideration. Instead of using eq 3, we started with Fermi’s Golden Rule and incorporated terms up to the first order in multipole expansion (eq 16) for the desctiption of anisotropic CD. There are two features unique to the orientational CD measurements, including the single-molecule cases where no orientation randamization is included. One is that both RM and RQ can contribute to the total rotatory strength tensor R to the same extent. If RQ is not 0, both positive and negative surfaces should

Figure 3. Canonical Kohn−Sham orbital isosurfaces (isoval = 0.02 au) of (a) HOMO and (b) LUMO. The HOMO-to-LUMO transition accounts for more than 97% of the electron rearrangement at the lowest-energy electronic excitation.

aimed at in the single-molecule FDCD measurements. Hence we hereafter consider the first excited state. The excitation is characterized by π → π* transition, in which one electron is excited from the HOMO to the LUMO, which accounts for more than 97% of electron rearrangement through the excitation (i.e., the corresponding CI coefficient of the TDDFT calculation is 0.69803). As shown in Figure 3a,b, the HOMO and the LUMO are extending mainly over the helical π-conjugated system in the bridged triarylamine helicene molecule. Its CD properties must reflect their nature. The calculated components of the electric and magnetic dipole transition moments and the quadrupole transition moment of the M-type diastereomer are shown in Table 1. The x-, y-, and z-axes in the molecular coordinate system and the directions of electric and magnetic dipole transition moments are shown in the Supporting Information (Figure Table 1. Electric Dipole Transition Moment (Length),a Electric Dipole Transition Moment (Velocity),a Magnetic Dipole Transition Moment,b,d and Electric Quadrupole Transition Moment (Velocity)c of M-Type Diastereomer Whose Configuration Has the Lowest Energy among the Optimized Structurese Electric Dipole Transition Moment μn0 (length) x

y

−0.78

(e/imωn)pn0 (velocity) z

x

y

z

−2.40 1.55 −0.89 −2.49 Magnetic Dipole Transition Moment

1.63

mn0 x

y

z

3.38

−0.75 Electric Quadrupole Transition Moment

0.26

(e/imωn)(prT + rpT)n0 (velocity) xx

yy

zz

xy

xz

yz

−2.80

−1.98

1.62

−5.26

1.74

0.40

Unit: 10−18 esu·cm. bUnit: 10−21 erg/Gauss. cUnit: 10−26 esu·cm2. These values are pure imaginary and only the coefficients are shown. e The transition is from the ground state to the energetically lowest excited state, i.e., n = 1. a

d

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Table 2. Rotatory Strength Tensor and Its (Electric Dipole) − (Magnetic Dipole) Contribution Part and (Electric Dipole) − (Electric Quadrupole) Contribution Parta R RM RQ a

Rxx

Ryy

Rzz

Rxy

Rxz

Ryx

Ryz

Rzx

Rzy

−11.94 34.37 −46.31

50.74 −38.74 89.48

−60.39 −17.21 −43.18

22.96 −10.00 32.97

28.51 3.50 25.00

101.82 126.18 −24.36

132.97 9.79 123.18

−31.66 −82.73 51.07

−28.27 18.32 −46.59

Unit: 10−40 (erg·esu·cm)/Gauss.

Small Rotatory Strength in Isotropic Bulk Sample. The rotatory strength of an isotropic bulk sample is defined by the Rosenfeld equation (eq 3) as the imaginary part of the scalar product of the electric and magnetic dipole transition moments. The rotatory strength is zero if the angle formed by the electric dipole transition moment and the magnetic dipole transition moment is exactly 90°. With increasing the angle between the electric and magnetic dipole transition moments, the rotatory strength changes sign from positive to negative at the angle of 90°. Its absolute value approaches the maximum, if the angle approaches 0 or 180°. In the case of the M-type diastereomer the calculated angle is 93.8°, which is only slightly larger than 90°. It explains the fact that the experimentally measured CD intensity of this molecule is quite small. Moreover, the influence of surroundings and even small structural fluctuations can affect the sign and the magnitude of the rotatory strength. As Figure 4 shows, averaging over all orientations leads to the complete cancellation of the contribution of RQ and partial but considerable cancellation of the contribution of RM, which results in small rotatory strength. Rotatory Strength in Single-Molecule Measurement. In the case of the orientational CD, including single-molecule measurements where the molecules are immobilized on the polymer surface, the rotatory strengths can be very different from the rotatory strength of the bulk sample, −7.19 (10−40· erg·esu·cm)/Gauss (1/3Tr R, Table 2). If the direction of the incident radiation is along the z-axis, the value becomes Rzz, −60.39 (10−40·erg·esu·cm)/Gauss (Table 2). The maximum and minimum values of the rotatory strength are +101.45 and −91.31 (10−40·erg·esu·cm)/Gauss. These values correspond to the length of the two principal axes, which are obtained by solving the eigenvalues of the symmetric part of the total rotatory strength tensor R, i.e., (RT + R)/2. The diagonalized symmetric part of the total rotatory strength tensor becomes

Figure 4. Three-dimensional representation of rotatory strength tensors. The distance from the origin to the surface indicates the magnitude of rotatory strength along that direction of light propagation. R, total rotatory strength tensor (left); RM, electric dipole-magnetic dipole contribution (center); RQ, electric dipoleelectric quadrupole contribution (right). In the picture of R of M-type, the longest distance from the origin to a blue surface point, is 101.45 × 10−40 erg·esu·cm/Gauss, and the scale is the same in each of these pictures.

appear in the three-dimensional representation of RQ because it turns out to be 0 on isotropic average. The other is that a helix generally has both clockwise and counterclockwise parts. We will explain this peculiar fact by using the pentahelicene molecule (Figure S4, Supporting Information). For example, seen along the helical axis, the helix approaches us when it is traced clockwise (Figure S4, left, Supporting Information). If we take a look from another direction, perpendicular to the axis, we will find a part that approaches us when it is traced counterclockwise (Figure S4, right, Supporting Information). As a matter of fact, we can see the parts of the left-handed coil in the actual right-handed coil. The sign of the isotropic optical activity of a helicene is determined by the handedness of the dominant helix. In contrast to what we can see in the 3D-representation of the numerator (Figure 4), the sign of the denominator never becomes negative, which is a natural consequence of eq 1 or eq 27. (The three-dimensional representation of the denominator in eq 28 is shown in Figure S5 (Supporting Information). The surface is shown in one color (orange) due to the positivity of the denominator.) The first term in the denominator is usually dominant compared with the second one, and therefore, the denominator is approximately proportional to sin2 θ, where θ is the angle between the electric dipole transition moment, i.e., μn0 or pn0, and the propagation vector of the light beam, i.e., u3. The denominator becomes large when θ is around 90°, and it becomes smaller as the θ approaches 0 or 180°. At 90°, the electric field of circularly polarized light can be parallel to μn0, leading to large dipole absorption. At 0 and 180°, the field is perpendicular to μn0, and therefore, there is no dipole transition. At this point the magnetic dipole and electric quadrupole terms dominate, leading to the large g values but very low absorption.

R diagonalized

⎛101.45 0 ⎞ 0 ⎜ ⎟ = ⎜0 −31.73 0 ⎟⎟ ⎜ ⎝0 −91.31⎠ 0

(30)

By considering the orientations of individual molecules, we can, at least partly disentangle their contributions to the experimental results, which are not seen in bulk-sample, ensemble-averaged, measurements. Theoretically Calculated g Value and Simulated Histogram. To construct the simulated histograms, we tried sampling g values for every direction of light propagation without any bias. Equation 28 was used to calculate g values at 1° intervals of ϕ and θ in the spherical coordinate system (Figure 2b). The simulated probability histograms are shown in Figure 5. It is theoretically predicted that the dissymmetry factors can take positive and negative values, as expected from the character of the three-dimensional representations in Figures 4 and S5 (Supporting Information). 5052



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Figure 7. Graph of g as a function of IL and IR (i.e., eq 1).

function of the two variables, IL ≥ 0, IR ≥ 0. The absolute value of g becomes large when at least one of the two variables is small. g deviates far from zero when both are small. That corresponds to the situation where the angle is near 0 or 180° and the theoretical prediction is consistent with this behavior. Note that this situation is rare in terms of molecular orientation. We compare the results of single molecules to those of bulk samples. There are two reasons that the g value is small in bulk samples from the orientationally averaging viewpoint. (i) The first reason is a purely geometrical one. When the direction of light propagation is chosen at random, the probability of the angle between the propagation vector u3 and the electric dipole transition moment μn0 being near 90° is higher than that of the angle being near 0 or 180°. Plainly, it can be understood if we think of the cases where the angle is just 90° and the cases where the angle is just 0 or 180°. The former ones occur when u3 is on the equatrorial plane with respect to μn0. On the other hand, the latter ones only occur when the direction of u3 is just parallel or antiparallel to μn0. This first reason is illustrated in Figure 8. When the light

Figure 5. Simulated probability histograms of dissymmetry factor, g. The blue and red lines respectively show the histograms for the P-type and M-type diastereomers.

The distribution of the dissymmetry factor (Figure 5) is narrower than that of the experimental results, as is also expected from the small molecule limit.77 The size of the molecule is much smaller than the helical pitch of circularly polarized light, so that the light response does not show significant molecular dissymmetry. In other words, the UV−vis circularly polarized light shows that the dissymmetry of small organic molecules is quite small in general. If the theoretical evaluation is seen in detail, however, the value can be large under specific conditions. In Figure 6 we show the correlation between the g value and the angle formed by the electric dipole transition moment and the propagation vector of the light beam.

Figure 8. Comparison between the two possibilities of the different situations. The probability of the angle between the light propagation vector and the electric dipole transition moment being 80−100° is greater than that of its being 170−180°, 0−10° even though the range widths are identical (20°). Those probabilities are respectively proportional to the areas of the surfaces, S∼90 (deep red) and S∼180 (green).

Figure 6. Relation between dissymmetry factor and the angle between the electric dipole transition moment and the propagation vector of the light.

The absolute value of the dissymmetry factor becomes small when the angle is around 90°. This is because under those conditions the denominator of g is large due to strong dipole absorption and, therefore, g values become very small. For purely geometrical reasons, the angles around 90° correspond to very high probabilities of occurrence (vide infra). As a result, the majority of the calculated values are in the range −0.01 ≤ g ≤ +0.01. Interestingly, on the other hand, it is possible that g values are distributed in the wide range from −2 to +2 when the angle is close to 0 or 180°. For clarity, a figure focusing on the behavior in the vicinity of 0° is inserted in Figure 6. For more detail, Figure 7 shows the graph of g = 2(IL − IR)/(IL + IR) as a

reaches the origin O going through one of the points on the surface within the range 80−110°, i.e., S∼90 (near 90°), the surface is shown by deep red, whereas when the light reaches the origin O going through one of the points within the range 170−180° or 0−10°, i.e., S∼180, the surface is shown by green. The deep red area of the surface S∼90 is much larger than to find the direction of light that of the green area of the surface S∼180. Therefore, the probability to find the direction of light in S∼90 is much larger than that in S∼180. In this case the values of the denominator are large and, therefore, g values are small. (ii) The second reason comes from the fact that the g value near 90° turns out to be small because the absorption rate of 5053



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vector is near 0 or 180°. Finally, it is noteworthy that, as explained by eq 29, a dissymmetry factor obtained by measurement of an isotropic bulk sample is by definition essentially different from the statistically averaged g value obtained by orientational CD in the single-molecule case.

allowed electronic transition is approximately proportional to sin2 θ (see Proof S3 in the Supporting Information). These two reasons do not apply to single-molecule measurements unless each molecule is oriented at random. In general, however, there is bias in the orientation of molecules immobilized on a polymer surface. Experimental vs Theoretical Value and the Width of Histogram. The simulated histograms of M- and P-type diastereomers are almost mirror images of each other as the experimental results show. The M- and P-type diastereomers have asymmetric carbon in common within the camphanate group, and therefore, the two molecules are not completely mirror images in terms of structure. Both experimental and simulated histograms split into two peaks. The theoretically estimated g values are distributed from −2 to +2, but almost all of them fall into the range from −0.01 to +0.01. The full width at half-maximum (fwhm) is ∼0.002. The g values obtained experimentally, in contrast, are much more widely distributed: fwhm is ∼0.762. There may be a sampling problem. In the simulations, the orientation of molecules at surface is assumed to be at random in order not to impose artificial bias. In reality, however, their random orientation may not be plausible; that can cause the difference between the experimental results and our theoretical results. Actually, it is conceivable that a certain range of molecular orientation would be favored on a polymer surface that is spatially and electronically inhomogeneous. Besides, it is known that fluorescence lifetime often depends on the molecular orientation as well as the geometry of the environment.106,107 It is also possible that the photodegradation rate depends on molecular orientation. If the photochemical lifetime tends to be short when the propagation direction of light and electric dipole transition moment is near 90° due to high absorption rate, to sample only the molecules with long photochemical lifetime is to leave out the majority of situations where the magnitude of g is small. There is a possibility that the experimental results are affected by artifacts due to imperfection of the circular polarization and the unstabilized fluctuation of photon emission. In our simulation, in contrast, the interaction between light and molecules occurred in the idealized situation. That is, we took into account neither the effects of conformational variation brought about by the thermal fluctuation nor the influence of the environment, both of which could significantly influence the distributions of the histograms.

■

ASSOCIATED CONTENT

* Supporting Information S

The calculated excitation wavelengths and oscillator strengths up to 20th lowest excitation energy, the lowest electronic excitation energy in wavelength calculated by a series of basis sets, electric dipole, magnetic dipole, and electric quadrupole transition moments, and the components of rotatory strength tensors calculated by a series of basis sets, the optimized structures and Cartesian coordinates of M-type and P-type, the directions of electronic and magnetic transition dipole moments of the bridged triarylamine helicene, an example of one coil seen as left- and right-handed by changing the angle, three-dimensional representation of the denominator of eq 28, definitions of symbols and operations used in the Theoretical Background section, proofs on the relations between R (RQ) and the Rosenfeld equation, sin2 θ dependence of the absorption rate of circularly polarized light in zeroth-order of multipole expansion, detailed derivation of the numerator and denominator of eq 28, and gauge origin dependence for some variables appeared in this paper. This material is available free of charge via the Internet at http://pubs.acs.org

■

AUTHOR INFORMATION

Corresponding Author

*S. Nakamura: E-mail: [email protected]. Phone: +81-48467-9477. Fax: +81-48-467-8503. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The calculations were performed by using the RIKEN Integrated Cluster of Clusters (RICC), Research Center for Computational Science, Okazaki Japan, and supercomputer TSUBAME at the Tokyo Institute of Technology. We thank the KAITEKI institute (Mitsubishi Chemical holdings) for support. This work was partly supported by JSPS KAKENHI Grant Number 24540443. This work was supported by RIKEN Junior Research Associate Program.

■

■

CONCLUSION A formula for anisotropic g values that is based on Fermi’s Golden Rule and that incorporates the zeroth- and first-order terms of multipole expansion is presented together with a quantum chemical evaluation. It is applied to bridged triarylamine helicene molecules to explain reported experimental single-molecule measurements. Compared from theoretical and geometrical points of view with the values measured in bulk samples, single-molecule g values can have a certain distribution width that originates from the orientational difference, which means g values can differ distinctly in both magnitude and sign. Simulated probability histograms without any orientational bias show a narrower distribution than that of the g values measured experimentally, which can be expected in consideration of small molecule limits. The single-molecule g values can, however, vary greatly when the angle between the electric dipole transition moment and the light propagation

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