# What software are used to model the optical behavior of metallic nanoparticles?

It's fairly routine to calculate absorption of single molecules, essentially this can be done by calculating the excited-state wavefunctions and the oscillator strengths between the states. For example, using TD-DFT or configuration interaction methods.

However nanoparticles behave much differently, from what I understand the electrons behave more like a particle in a box. The size and shape of the particle directly controls the optical properties.

What type of software can be used to model optical properties of nanoparticles and how do these approaches differ from molecules?

There are several DFT based commercial or open source softwares that could be used to simulate the optical properties of nanomaterials, particularly metallic nanoparticles. Basically, what you are looking for is calculating the susceptibility tensor from Kubo-Greenwood relation:

$$\chi_{ij}(\omega) = \frac{e^{2}}{\hbar m_{e}^{2} V} \sum_{n,m,\mathbf{k}} \frac{f_{m,\mathbf{k}} - f_{n,\mathbf{k}}}{\omega_{nm}^{2}(\mathbf{k}) (\omega_{nm}(\mathbf{k})-\omega-i\frac{\Gamma}{\hbar})}p_{nm}^{i}(\mathbf{k})p_{mn}^{j}(\mathbf{k})$$

Where $$p_{nm}^{i}(\mathbf{k}) = \langle n\mathbf{k} | \mathbf{p}^{i} | m\mathbf{k}\rangle$$ is i-th component of the momentum operator between states n and m. $$m_{e}$$ is the electron mass, $$e$$ is electron charge, $$V$$ is the volume, $$\Gamma$$ the energy broadening, $$\hbar\omega_{nm}(\mathbf{k})= E_{n}(\mathbf{k}) - E_{m}(\mathbf{k})$$, and finally $$f_{n,\mathbf{k}}$$ is the Fermi function evaluated at the band energy of $$E_{n}(\mathbf{k})$$.

Relative dielectric constant ($$\epsilon_{r}$$), polarizability ($$\alpha$$), and optical conductivity ($$\sigma$$) are related to susceptibility as:

$$\epsilon_{r}(\omega) = 1 + \chi(\omega)$$

$$\alpha(\omega) = V \epsilon_{0} \chi(\omega)$$

$$\sigma(\omega) = -i \omega \epsilon_{0} \chi(\omega)$$

Finally, the refractve indices are derived from relative dielectric constant ($$\epsilon_{r}$$) as:

$$n(\omega) + i\kappa(\omega) = \sqrt{\epsilon_{r}(\omega)}$$

# QuantumATK

An example of commercial software that could be used to extract these optical properties for metallic nanoparticles is QuantumATK.

• @PeterMorgan QuantumWise is the old name for QuantumATK before synopsys acquire it from University of Copenhagen. May 4, 2020 at 4:49
• Ah, ok. That explains it. Thanks for changing. May 4, 2020 at 4:51

# DDSCAT

You can use the DDSCAT (Discrete Dipole Scattering) package for nanoparticles >5 nm. Many of the optical properties of nanoparticles (absorption, scattering, extinction coefficient) can be solved via classic electrodynamics, i.e. solving the Maxwell Equations when light interacts with a nano-sized object with no quantum packages necessary. To determine the optical properties of nanomaterials classically, the most rigorous approach is solving Maxwell's Equations analytically via Mie Theory (see section 2.2 of Bertens). However, exact analytical solutions to the scattering problem only exist for a limited number of geometries (sphere, cylinder, etc.). The DDSCAT package approximates a continuous object utilizing a series of point dipoles on a 3D lattice, of which a direct solution to Maxwell's Equations can readily be solved.

The image above shows an example of point dipoles constructed for input into DDSCAT (Draine, 1988). In addition to the point geometry, to run DDSCAT, the complex polarizability of the material and an 'effective' radius must be provided. The complex polarizability of materials like gold, silver, etc. can be found readily online. The effective radius of an object is solved by determining the total volume of your object, then solving for radius as if the object were spherical (Vobj = 4/3𝜋reff3).