#
Chiroptical Symmetry Analysis: Exciton Chirality-Based Formulae to Understand the Chiroptical Responses of C_{n} and D_{n} Symmetric Systems

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}or D

_{2}symmetry, one can get the same results by applying the selection rules. In the present article, the analysis of the selection rules for systems with symmetries C

_{n}and D

_{n}with n = 3 and 4 is used to uncover the origin of their chiroptical responses. We foresee that the use of the Chiroptical Symmetry Analysis (CSA) for systems presenting the symmetries explored herein, as well as for systems presenting higher symmetries will serve as a useful tool for the development of chiroptical applications.

## 1. Introduction

_{12}, emerges from the excitation energy splitting between the in-phase and out-of-phase simultaneous two-chromophore transitions, strongly dependent on their relative orientation (Scheme 1).

_{12}, is given (in au) by Equation (1), where ${\overrightarrow{R}}_{12}$ is the vector connecting the origin of ${\overrightarrow{\mu}}_{1}^{t}$ and ${\overrightarrow{\mu}}_{2}^{t}$. When these chromophores display also permanent dipoles in the excited or ground state, represented respectively by ${\overrightarrow{\mu}}_{i}^{0}$ and ${\overrightarrow{\mu}}_{i}^{a}$ additional terms have to be considered within V

_{12}, as will be shown below.

_{2}and D

_{2}symmetric systems in terms of the selection rules. In recent years, we have contributed with a number of highly symmetric systems such as linear [12] and cyclic oligomers [17], molecular containers [18,19], or even all-carbon double helices [20], presenting remarkable chiroptical responses in the search for valuable materials for everyday chiroptical applications [21]. However, analysis of the chiroptical responses is often tedious [22,23]. In this regard, in 1937, Condon [24] revised the work from Kirkwood [25], where he demonstrated the possibility to determine the optical activity of a molecule by considering the relative orientation of the different constituent groups. In 1974, Schellman developed the theory of optical activity with simple electric and magnetic dipole interaction with radiation capable of resampling rather accurately the experimental results [26]. In order to have a more accessible understanding of the behavior of highly symmetric systems, herein we perform the Chiroptical Symmetry Analysis (CSA) and apply it to systems with C

_{n}and D

_{n}symmetries for n = 2, 3, and 4. Even when this approach is more specific, CSA could be applied to any other symmetry, and therefore, we hope that it will serve as a design tool for the development of everyday chiroptical applications in the near future. Additionally, since an exciton coupling approach in Vibrational CD (VCD), known as the coupled oscillator model, has also been employed successfully by Monde [27] and further evaluated by Polavarapu [28], in analogy to electronic transition, the CSA could also be applied for the prediction of the VCD of highly symmetric systems.

## 2. Results and Discussion

#### 2.1. Theoretical Background (Binary Systems)

_{i}

^{0}, to an empty (virtual) one, Φ

_{j}

^{a}. This produces a rearrangement of the electron density of the molecule, basically described by the EDTM as a linear displacement of electron density during the transition (Scheme 3). Moreover, the intensity of the transition between the two states is proportional to the square of the modulus of the EDTM, and forbidden electronic transitions yield null values of EDTM. The selection rules establish that for a transition to be allowed, at least one of the components of the EDTM must contain the totally-symmetric representation of the system point group, and consequently, the direct product for the representations of the involved wave functions must contain at least one of the representations of the electric dipole moment components.

^{a}. In fact, when considering that the electronic transition only affects the chromophores, it is assumed that the rest of the molecule remains unchanged by the transition. We consider it to be worth reviewing this model with a certain level of detail before expanding it to three and four chromophores.

#### 2.1.1. Approximated Wave Functions and Energy Levels

^{0(0)}, can be represented by the product of two ground state wave functions for the isolated chromophore (3), Ψ

^{0}. In the same vein, the zero-order degenerate excited electronic state, Ψ

^{a(0)}, can be expressed by the product of ground and excited states, Ψ

^{a}, of single chromophores, with two equivalent possibilities: Ψ

_{1}

^{a(0)}(4) and Ψ

_{2}

^{a(0)}(5). The former refers to the chromophore 1 in the ground state and the chromophore 2 in the excited state. This assumes that the electronic excitation only modifies the electron density distribution of the chromophore where it takes place.

^{0(1)}, the usual expression given by Equation (6). Considering only the dipole-dipole interaction between the chromophores, the first-order correction for the energy of the ground state is that shown by Equation (7), where ${\overrightarrow{\mu}}_{i}^{0}$ are the dipole moments of the chromophores in the ground state, named ${V}_{12}^{00}$.

^{a(1)}is the first-order correction to the energy of the excited state for the two-chromophore system. Condition (8) is only fulfilled by the two E

^{a(1)}values contained in Expression (9). We observe that these values involve the dipole–dipole interaction between one chromophore in the ground state and another in the excited state, ${V}_{12}^{0a}$, and a term where the interaction applies over the electronic transition, ${V}_{12}^{tt}$, shown in (10).

^{a}, can be expressed by Equation (11). Obviously, the reliability of the formula can be improved including further corrections, but the most significant fact is the splitting of the excited level into two different states (usually named α and β), whose energy differs in $2{V}_{12}^{tt}$. Moreover, replacing the two solutions for energy correction (9) in the system of homogeneous equations and normalizing, it is found that the zero-order wave functions for α and β states follow, respectively, Expressions (12) and (13).

_{2}arrangement. In this case, energies for α and β states (named, respectively, A and B according to symmetry) can be calculated as average values (Scheme 4) (14).

#### 2.1.2. Rotatory Strength

_{jk}and p

_{jk}are, respectively, the position and linear momentum vectors for electron k in chromophore j. Both are the summation of one-electron operators. Within the context of the independent exciton model, the summation only involves the N electrons provided by the chromophores, and these electrons can be assigned to a certain chromophore j.

#### 2.1.3. Electric Dipole Transition Moment

#### 2.1.4. Magnetic Dipole Transition Moment

#### 2.1.5. Expressions for Rotatory Strength

_{2}, a further step can be done, and the values of rotatory strength can be predicted in terms of simple geometry parameters. Thus, for C

_{2}symmetry, a convenient reference system for the vectorial operations indicated above has its origin at the center of mass; the z axis is given by the C

_{2}axis; and x axis connects the molecular center of mass with that of Chromophore 1. We notice that the orientation of the EDTMs is given by two angles, which are formed with the C

_{2}axis, coincident with the z axis, θ, and the angle between the projection of EDTM for each chromophore on the zx plane with the x axis, ω. Thus, simple calculations lead to Expression (35), where rotatory strength is a function of the distance between chromophores, R

_{12}or R as the distance from the center of the chromophores to the C

_{2}

#### 2.2. Extending the Exciton-Independent Model to Systems with Three and Four Chromophores

_{3}(the system presents one C

_{3}axis) and D

_{3}(the system presents three C

_{2}axes perpendicular to a C

_{3}axis). Our aims are: (i) obtaining expressions for the splitting between those first electronically-excited states attainable from the ground state by absorption of electromagnetic radiation allowed by electric dipole transition; and (ii) predicting the corresponding rotatory strengths.

#### 2.2.1. C_{3} Geometries

_{3}Symmetry-Adapted Linear Combinations (SALC) for the first electronically-excited states. In this case, the reducible representation decomposes into the symmetry irreducible species A and the pseudodegenerate reducible species E, as shown in (38). Thus, one A excited state and two E excited electronic states can be predicted. They are represented in the zero-order approach, respectively, by Equations (39)–(41). Transitions A and E (E1 and E2) are both orbitally allowed.

_{3}axis, and the x axis connects the molecular center of mass with that of Chromophore 1. We notice that the orientation of the EDTMs is given by two angles, which are formed with the C

_{3}axis, θ, and the angle between the projection of EDTM for each chromophore on the xy plane with the x axis, ω for Chromophore 1. Thus, the total EDTM for the transition to the A state displays only one non-zero component. It corresponds to the symmetry axis (z axis by convention) as Equation (49) shows. Total EDTMs to E1 and E2 states are orthogonal to that of the A state and contain imaginary parts, as indicated in Expressions (50) and (51).

_{3}symmetry, results in (56), where R is the distance from the center of the chromophores to the C

_{3}axis,

_{j}and μ

_{j}vectors are applied.

#### 2.2.2. D_{3} Geometries

_{3}arrangement, the totally-symmetric species is represented as A

_{1}(ground electronic state). There are again three equivalent monoexcitations given by (37). They form a suitable basis set for constructing D

_{3}SALCs for the first electronically-excited states. The reducible representation of this basis, ${\mathsf{\Gamma}}^{a}$, decomposes into different symmetry-irreducible species depending on how the rotation around any of the three perpendicular C

_{2}axes affects the first electronically-excited state of the isolated chromophore, ${\psi}^{a}$. When ${\widehat{C}}_{2}{\psi}^{a}={\psi}^{a}$, the typical situation when all the MOs describing the excited state are σ ${\mathsf{\Gamma}}^{a}$ decomposes into A

_{1}and the truly degenerate reducible species E. In contrast, when the excited state of the chromophore contains one π MO: ${\widehat{C}}_{2}{\psi}^{a}=-{\psi}^{a}$, and ${\mathsf{\Gamma}}^{a}$ contains A

_{2}and E species. Thus, the two degenerate E excited electronic states, represented in the zero-order approach by any linear combination of Equations (61) and (62), can be accompanied by one A

_{1}or A

_{2}state, represented by a common expression, equivalent to that shown above for C

_{3}systems (39).

_{2}rotations (e.g., π* states). In this case, the application of the first-order perturbation theory leads to the same splitting of the excited level obtained for C

_{3}geometries: $3{V}_{12}^{tt}$. On the other hand, when C

_{2}rotations are symmetric (this is the case of σ* states), the transition to the A

_{1}excited state is orbitally forbidden, and there is no splitting.

_{2}axes orthogonal to C

_{3}restrains the orientations for EDTMs in D

_{3}systems with regard to C

_{3}ones to those where ω = π/2. As a consequence, the EDTM for the transition to the A

_{2}state follows the same expression obtained for C

_{3}symmetry (49), standing along the main symmetry axis, whereas the EDTMs for transitions to true degenerate E states are contained within the orthogonal plane, being linear combinations of the vectors represented by (61) and (62).

_{2}state follows the same expression as that to the A excited state in C

_{3}symmetry (52). Consequently, the expression for the rotatory strength for this transition is common for C

_{3}and D

_{3}symmetries (56). Transitions to E states display, in contrast, different expressions for MDTMs in D

_{3}systems (linear combinations of (63) and (64)) and rotatory strength ((65) and (66)).

#### 2.2.3. C_{4} and D_{4} Geometries

_{4}structure including four chromophores (Scheme 6), a pairwise approach for the interaction among them, where V

_{12}and V

_{13}are not equivalent, (27) the same approximations, and a similar treatment to that shown above (see Supporting Information) leads to the following conclusions: (i) there are four monoexcited states, two of them pseudodegenerate, belonging to the E symmetry species, the other with A and B symmetry; (ii) the electronic transition from the ground state to the B one is orbitally forbidden, whereas those to A and E are allowed; and (iii) the energy of state A differs from those of E (equivalent within the first-order approach) in $2\left({V}_{12}^{tt}+{V}_{13}^{tt}\right)$; which is the splitting in this case.

_{n}symmetry by D

_{n}splits some symmetry species (A into A

_{1}and A

_{2}and B into B

_{1}and B

_{2}), while pseudodegenerate E states become degenerate. A

_{1}and B

_{1}species are symmetric for C

_{2}rotations over any perpendicular axis and correspond to excited states with forbidden transitions from the ground state. In contrast, A

_{2}and B

_{2}states are antisymmetric for these perpendicular rotations, and the transitions between the ground state and A

_{2}are orbitally-allowed. As a consequence, the excited state splitting is only observed for excited states involving the excitation of one chromophore to an antisymmetric state with regard to a perpendicular rotation. In this case, the energy of state A

_{2}differs from the E ones again by $2\left({V}_{12}^{tt}+{V}_{13}^{tt}\right)$ (see the Supporting Information).

_{4}systems ((68) and (69)) show that the absorption for 0→A and 0→E transitions depends on the molecular geometry. This can be described in terms of θ and ω angles, defined as in C

_{3}systems, but now identifying the z axis with C

_{4}and considering the recursive formula ω

_{j+1}= ω

_{j}+ π/2 for the series of ω angles of the chromophores.

_{4}system follow, respectively, Equations (70) and (71). Thus, we arrive at Expressions (72) and (73) for the corresponding rotatory strength.

_{4}symmetry (ω = π/2 for Chromophore 1) does not alter any of the expressions shown for transition A. Nevertheless, it allows obtaining vectors with real components for EDTMs and imaginary ones for MDTMs to E degenerate states. These vectors are linear combinations of those shown respectively in (74) and (75). The former is always contained in the orthogonal plane to the C

_{4}axis, whereas the latter are parallel to it.

#### 2.3. Comparison with DFT Calculations

_{2}, D

_{3}, and D

_{4}symmetry, respectively (Scheme 7).

## 3. Conclusions

_{ij}can be calculated according to the Davydov equation; (ii) the allowed transitions can be predicted by applying the selection rules; (iii) ΔE between those transitions can be calculated according to theory; (iv) the symmetry of the allowed transitions enables simple calculation of the total EDTM, MDTM, and rotatory strength associated.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Scheme 1.**Representation of the splitting of the energy excitation of two interacting chromophores due to the in-phase and out-of-phase interactions. Arrows represent the electronic transition for two chromophores, blue and magenta. Curves represent the absorption band in the UV-Vis spectra. Different torsion angles between the two chromophores lead to different relative intensities and ΔE between the two possible transitions.

**Scheme 2.**Representation of the bisignated CD signal arising from the interaction between two independent chromophores with a relative positive (

**left**) or negative (

**right**) twist.

**Scheme 3.**Representation of the o→a electronic excitation of a model compound. The electron density redistribution from the ground state o to the excited state a results in the Electric Dipole Transition Moment (EDTM). EDTM has a defined direction and two equally-probable orientations. The blue spheres represent the center of electron density for the electronic states.

**Scheme 4.**

**Top**: Representation of angles to define the position of the chromophores.

**Bottom**: Representation of a system presenting D

_{2}symmetry. According to the selection rules, the two equivalent chromophores 1 (blue) and 2 (magenta) can undergo a simultaneous excitation following A or B

_{3}symmetry leading to total EDTMs (black) parallel to y or z, respectively, as a result of the summation of the EDTM of each chromophore. A circle with a dot in the middle represents an arrow perpendicular to the plane pointing to the reader.

**Scheme 5.**Representation of a system presenting D

_{3}symmetry. According to the selection rules, the three equivalent chromophores 1 (blue), 2 (magenta), and 3 (green) can undergo a simultaneous excitation following A

_{2}or E symmetry, leading to total EDTMs (black) parallel to z or y, respectively, as a result of the summation of the EDTM of each chromophore. A circle with a dot in the middle represents an arrow perpendicular to the plane pointing to the reader.

**Scheme 6.**Representation of a system presenting D

_{4}symmetry. According to the selection rules, the four equivalent chromophores 1 (blue), 2 (magenta), 3 (green), and 4 (red) can undergo a simultaneous excitation following A

_{2}or E symmetry leading to total EDTMs (black) parallel to z or y, respectively, as a result of the summation of the EDTM of each chromophore. A circle with a dot in the middle represents an arrow perpendicular to the plane pointing to the reader.

**Scheme 7.**Top: DFT predicted (graph bottom) and CSA predicted (graph top) rotatory strengths for D

_{2}, D

_{3}, and D

_{4}(from left to right) symmetries. The bottom axis represents the relative energy for each symmetry (from left to right: higher to lower). Bottom: D

_{n}systems.

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**MDPI and ACS Style**

Castro-Fernández, S.; Peña-Gallego, Á.; Mosquera, R.A.; Alonso-Gómez, J.L.
Chiroptical Symmetry Analysis: Exciton Chirality-Based Formulae to Understand the Chiroptical Responses of *C*_{n} and *D*_{n} Symmetric Systems. *Molecules* **2019**, *24*, 141.
https://doi.org/10.3390/molecules24010141

**AMA Style**

Castro-Fernández S, Peña-Gallego Á, Mosquera RA, Alonso-Gómez JL.
Chiroptical Symmetry Analysis: Exciton Chirality-Based Formulae to Understand the Chiroptical Responses of *C*_{n} and *D*_{n} Symmetric Systems. *Molecules*. 2019; 24(1):141.
https://doi.org/10.3390/molecules24010141

**Chicago/Turabian Style**

Castro-Fernández, Silvia, Ángeles Peña-Gallego, Ricardo A. Mosquera, and José Lorenzo Alonso-Gómez.
2019. "Chiroptical Symmetry Analysis: Exciton Chirality-Based Formulae to Understand the Chiroptical Responses of *C*_{n} and *D*_{n} Symmetric Systems" *Molecules* 24, no. 1: 141.
https://doi.org/10.3390/molecules24010141