# Plasmonic Circular Dichroism in Chiral Gold Nanowire Dimers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Discussion

_{2}symmetry z-axis: we will refer to this system in the following as the parallel geometry.

_{2}z-axis: we will refer to this system as the rotated geometry. In Figure 1 we have reported the same system viewed from two different perspectives: in the side view (b), the chemical bonds connecting the two nanowires are visible, while in top view (c) it is possible to appreciate the relative rotation of the two nanowires. In this case, the relative orientation itself is chiral; in fact, in the present relative orientation, we do not have any symmetry plane and a chiral system is obtained. We may therefore distinguish between structural chirality (of the single nanowire) and induced chirality (by the relative orientation between the two objects). In order to distinguish even more clearly the two effects we have also considered a chiral relative orientation of two achiral nanowires, like in Figure 1d,e. In this case we first built an achiral nanowire with the same size of the chiral one (152 gold atoms), whose structure has been generated starting from the Au

_{12}icosahedral cluster with a gold–gold interatomic distance of 2.88 Å, adding 14 equatorial ribbons of Au

_{10}units, obtaining finally the Au

_{152}cluster with D

_{5d}symmetry. Then two of such clusters were paired in the same way as the rotated geometry of the previous Figure 1b,c. This new configuration was reported in Figure 1d,e and will be referred to as rotated achiral; note that, in this case, only induced plasmonic CD is expected.

^{−40}esu

^{2}·cm

^{2}, i.e., an increase of four orders of magnitude of the CD signal with respect to the single-nanowire system. This value is of the same order of magnitude as that obtained for the helical nanowires [23], which reached a maximum around 40,000 × 10

^{−40}esu

^{2}·cm

^{2}. In the Figure 3 we have reported the ICM-RS plots of both present pair of interacting nanowires (boxes (a) and (c)) as well as those of the single chiral nanowire (boxes (b) and (d)) taken from our previous work [23]. Moreover, we have generated both 2D (boxes (a) and (b)) as well as 3D plots (boxes (c) and (d)) in order to have a more direct visualization of the effects. All the details regarding the definition and calculation of the ICM-RS plots have been reported in Appendix A.2 of the Appendix A of the present work. Such plots consist of decomposing the rotator strength (R) of a given transition in its components in terms of occupied-virtual pairs; on the x and y axis, the occupied and virtual orbital energies are considered. The presence of a ‘spot’ indicates that the orbital pair that had the corresponding energy is involved in the transition. In 2D, the ‘intensity’ of the involvement is given by a colour scale; for 3D plots, the ‘intensity’ corresponds to the scale of the z axis. As observed previously and considering the present Figure 3b,d, the negligible CD of the single chiral nanowire is a consequence of a destructive interference of two opposite and large contributions. These opposite (positive and negative) contributions in the ICM-RS spectrum of the (5,3) nanowire are individually very large but are practically equal in absolute value. They thus cancel each other almost perfectly, producing a nearly zero CD spectrum. This suggested that by perturbing the system with a proper coupling, it should be possible to remove, at least partially, such a destructive interference, allowing the manifestation of a plasmonic CD. The present results fully confirm this hypothesis: here we have demonstrated that the coupling between a pair of nanowires is sufficient to allow a partial suppression of the destructive interference phenomenon. Indeed, in Figure 3a,d, we report the ICM-RS plots of the parallel pair taken at the energy corresponding to the maximum dichroism. Only the y dipole component was considered (along the direction of maximum nanowire length), the other components being negligible. It is evident that there was still destructive interference, since there were regions with the opposite sign, but now the positive region was wider and more intense than the negative one, such that there was only a partial cancellation. However, this also shows that the destructive interference had been only partially removed, and that there was still wide room for further increasing the dichroism, suggesting a promising path for future work.

^{−40}esu

^{2}·cm

^{2}, the one rotated by 45° gave a pair of strong peaks with opposite sign, separated by only 0.14 eV, with rotator strength up to ±50,000 × 10

^{−40}esu

^{2}·cm

^{2}. This finding suggests that, in this case, induced plasmonic CD was stronger than structural plasmonic CD, analogously to what was reported in our previous work for the comparison between linear and helical nanowires [23]. It is hard to say if this is a general behavior or one specific to the present systems. In fact, it is worth noting that the structural plasmonic observed in the parallel situation kept a large amount of destructive interference; therefore, it is still possible that more effective coupling between nanowires may remove further the destructive character giving rise to much higher structural plasmonic CD. Induced densities of the rotated geometry at the two energies corresponding to the maximum and minimum CD, respectively 0.96 and 1.10 eV, are reported in Figure 5. In both cases, the induced density displays a clear dipolar shape for each individual nanowire, typical of a plasmon. However, at lower energy, the individual dipoles of the wires displayed opposite direction, corresponding to a negative scalar product; at higher energy, the dipoles had the same direction.

^{−40}and −40,000 × 10

^{−40}esu

^{2}·cm

^{2}are roughly a factor of three larger than the pure structural CD reported in the previous Figure 2 for the parallel geometry. We may conclude this analysis by saying that, at least for the system considered in the present study, the strength of the induced CD was roughly three times the structural one, and these effects sum up when both of them are present in the same system.

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Appendix A. Theoretical Method

#### Appendix A.1. The Complex Polarizability Method

_{r}+ iω

_{i}, where the real part ω

_{r}is the scanned photon frequency (energy) and ω

_{i}is the imaginary part that corresponds to a broadening of the discrete lines and can be interpreted as a pragmatic inclusion of the excited states’ finite lifetime. The complex dynamical polarizability is calculated by solving the following non-homogeneous linear system:

**S**is the overlap matrix between fitting functions,

**b**is the unknown vector with the expansion coefficients b

_{µ}(ω) of the induced density ρ

^{(1)}

_{z},

**d**is the frequency dependent vector corresponding to the known non-homogeneous term, and finally the elements of the frequency dependent matrix

**M**are:

_{KS}refers to the Kohn-Sham frequency-dependent dielectric function and K to the kernel.

**M**(ω) as a linear combination of frequency independent matrices

**G**

^{k}with frequency dependent coefficients s

_{k}(ω), with the following expression:

**G**

^{k}} was calculated and stored only once at the beginning. Then the matrix

**M**(ω) was calculated very rapidly at each photon energy ω, as a linear combination of the {

**G**

^{k}} matrices with the following coefficients:

**μ**and

**m**are the electric dipole and magnetic dipole moment operators and γ is a constant.

_{0n}, is therefore defined as follows:

_{0n}by the complex polarizability algorithm [30], it is convenient to consider the dipole moment induced by an electromagnetic field [34]:

_{v}and B

_{v}are the electric and magnetic field components, c is the speed of light, α is the dynamical polarizability tensor and β is the optical rotation tensor, which is related to the rotatory strength by the following sum over states (SOS) expression:

_{0n}corresponds to the $|0$→$|n$ excitation energy. Therefore, it is convenient to extract R

_{0n}from the β imaginary part as in conventional photoabsorption. From Equation (A9), β consists in the electric dipole moment induced by a time-dependent (TD) magnetic field and can be calculated by the following expression:

_{µ}and the product between the i-th occupied and the a-th virtual orbitals, ${\phi}_{i}\left|{\mu}_{z}\right|{\phi}_{a}$ and ${\phi}_{i}\left|{m}_{z}\right|{\phi}_{a}$ are the electric and magnetic dipole moment matrix elements respectively, between the same occupied-virtual (ia) orbitals pair, the matrix

**L**is defined by Equation (28) of Ref. [31], t

_{k}is given by:

**b**is the solution of the linear system (2).

#### Appendix A.2. Individual Component Maps of Rotatory Strength (ICM-RS) Analysis

_{i}) and virtual (ε

_{a}) orbitals is generated. ICM-RS (ω) plots allow one to visualize the source of chiral response in momentum space, including signed contributions, therefore highlighting cancellation terms that are ubiquitous and critical in chiral phenomena.

#### Appendix A.3. Computational Details

_{2}point group, the z axis being the binary rotation axis, such symmetry has been exploited in the calculations. Core electrons have been kept frozen up to the Au 4f level. Relativistic effects have been considered at scalar level employing the zero order regular approximation (ZORA) level [38].

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**Figure 1.**(

**a**) Structure of a pair of interacting gold chiral nanowires with parallel axis so their arrangement is not chiral. (

**b**) side view and (

**c**) top view of a pair of interacting gold chiral nanowires with axis rotated by 45 degrees so their relative orientation is chiral. (

**d**) side view and (

**e**) top view of a pair of interacting gold achiral nanowires with axis rotated by 45 degrees so their relative orientation is chiral.

**Figure 2.**Photoabsorption (upper panel) and CD (lower panel) for the single chiral nanowire (red lines) and the pair of interacting gold chiral nanowires with parallel axis (blue line). Oscillator strengths are given in atomic units, while R is given in 10

^{−40}esu

^{2}·cm

^{2}.

**Figure 3.**ICM-RS plots relative to the y component taken at the energy corresponding to the maximum CD: 1.04 eV for the single nanowire in panels (

**a**,

**c**); 1.24 eV for the pair of interacting gold chiral nanowires with parallel axis in panels (

**b**,

**d**) as in Figure 1. ε

_{i}and ε

_{a}are energies of occupied and virtual orbitals, respectively.

**Figure 4.**Photoabsorption (upper panel) and CD (lower panel) for the pair of interacting gold chiral nanowires with parallel axis (blue line) and rotated axis by 45° (black line) and 30° (red line). Oscillator strengths are given in atomic units, while R is given in 10

^{−40}esu

^{2}·cm

^{2}.

**Figure 5.**Induced density plots relative to the x and y components taken at the energy corresponding to the maximum CD (0.96 eV) and the minimum CD (1.10 eV) of a pair of interacting gold chiral nanowires with axis rotated by 45 degrees as in Figure 1.

**Figure 6.**ICM-RS plot relative to the x and y components taken at the energy corresponding to the maximum CD (0.96 eV) and the minimum CD (1.10 eV) of a pair of interacting gold chiral nanowires with axis rotated by 45 degrees as in Figure 2. ε

_{i}and ε

_{a}are energies of occupied and virtual orbitals, respectively.

**Figure 7.**Dipole components partial contributions of photoabsorption (left panels (

**a**,

**c**)) and CD (right panels (

**b**,

**d**)) for the pair of interacting gold chiral nanowires with parallel axis (upper panels (

**a**,

**b**)) and rotated axis (lower panels (

**c**,

**d**)). Oscillator strengths are given in atomic units, while R is given in 10

^{−40}esu

^{2}·cm

^{2}, x, y, z and total contributions are in red, green, blue and black lines, respectively.

**Figure 8.**Photoabsorption (upper panel) and CD (lower panel) for the pair of interacting gold chiral nanowires with rotated axis (black line) and achiral nanowires with rotated axis (red line). Oscillator strengths are given in atomic units, while R is given in 10

^{−40}esu

^{2}·cm

^{2}.

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**MDPI and ACS Style**

Toffoli, D.; Medves, M.; Fronzoni, G.; Coccia, E.; Stener, M.; Sementa, L.; Fortunelli, A.
Plasmonic Circular Dichroism in Chiral Gold Nanowire Dimers. *Molecules* **2022**, *27*, 93.
https://doi.org/10.3390/molecules27010093

**AMA Style**

Toffoli D, Medves M, Fronzoni G, Coccia E, Stener M, Sementa L, Fortunelli A.
Plasmonic Circular Dichroism in Chiral Gold Nanowire Dimers. *Molecules*. 2022; 27(1):93.
https://doi.org/10.3390/molecules27010093

**Chicago/Turabian Style**

Toffoli, Daniele, Marco Medves, Giovanna Fronzoni, Emanuele Coccia, Mauro Stener, Luca Sementa, and Alessandro Fortunelli.
2022. "Plasmonic Circular Dichroism in Chiral Gold Nanowire Dimers" *Molecules* 27, no. 1: 93.
https://doi.org/10.3390/molecules27010093