Efficient first-principles method for calculating the circular dichroism of nanostructures

A first-principles method is developed to study the natural optical activity of nanostructures, making large-scale calculations of electronic circular dichroism feasible. Expressions to calculate circular dichroism using density-functional theory for finite and periodic systems are obtained and implemented within the SIESTA program package. To show the versatility and applicability of the method, the circular dichroism of the high fullerenes ${\text{C}}_{76}$ and ${\text{C}}_{78}$ is investigated. The results for these fullerenes show good consistency with previous semiempirical calculations, and a very good agreement with experiments. The method is generalized to treat periodic structures such as nanotubes, and the circular dichroism of the carbon single-wall nanotube (4,2) is studied, and the spectrum is interpreted in terms of its electronic density of states. It is found that the calculated circular dichroism spectra can be used to discriminate among different nanostructures through optical activity experiments. It is concluded that this methodology provides theoretical support for the quantification, understanding, and prediction of chirality and its measurement in nanostructures. It is expected that this information would be useful to motivate further experimental studies and the interpretation of natural optical activity in terms of the electronic circular dichroism in nanostructures.

Figure 1

(Color online) Schematic model of a linearly polarized electromagnetic field $\left(E\right)$ composed by the sum of left $\left(E_{\text{L}}\right)$ and right $\left(E_{\text{R}}\right)$ circularly polarized light that is traveling in an absorbing medium made of a collection of randomly oriented chiral nanoparticles. The system absorbs differently L and R circularly polarized light, presenting the circular dichroism phenomenon, and resulting in an elliptical polarized beam.

Figure 2

(Color online) Atomic models of the two different isomers of ${\text{C}}_{76}$ and the three isomers of ${\text{C}}_{78}$. The isomers in the right-hand side for both sizes are chiral.

Figure 3

(Color online) CD spectra of the $D_2$ isomer of ${\text{C}}_{76}$. (a) Semiempirical SCF calculations from Ref 41. The spectrum was shifted $-0.6\text{eV}$. (b) Experimental data also taken from Ref. 41. (c) Our DFT calculated CD spectrum with a Gaussian broadening of 0.4 eV. The spectrum was shifted $+0.9\text{eV}$. The arrows are used to indicate the correspondence between the main peaks of both calculated CDs with the experimental curve.

Figure 4

(Color online) CD spectra of the $D_3$ isomer of ${\text{C}}_{78}$. (a) Experimental data taken from Ref. 39. (b) Our DFT calculated CD spectrum with a Gaussian broadening of 0.2 eV. The calculated spectrum was shifted $+0.4\text{eV}$ and multiplied by a factor of 1/30 for a better comparison with the experimental curve. The arrows are used to indicate the correspondence of the main peaks between both CDs.

Figure 5

(Color online) DFT calculated CD spectra of one of the $D_2$ isomers of (a) ${\text{C}}_{92}$ and (b) ${\text{C}}_{100}$, with a Gaussian broadening of 0.2 eV. The atomic models of both fullerenes are also shown.

Figure 6

(Color online) CD spectrum of the zigzag, then, achiral SWNT (5,0), where numerical errors smaller than 0.1% are observed. The atomic model of SWNT (5,0) is also shown.

Figure 7

(Color online) DFT calculated CD spectra of the SWNT (4,2) and its enantiomer, the SWNT (2,4). The spectra have a Gaussian broadening of 0.2 eV. The atomic models of both SWNTs are also shown.

Figure 8

(Color online) (a) Electronic density of states of nanotube (4,2) as a function of energy. The main parallel optical transition energies $E_{nn}$ are indicated. (b) CD spectrum of nanotube (4,2). The energies of the allowed optical transitions for parallel polarization of light are also indicated.