\label{sec:plasmonic}
The scope of real-space real-time approaches is not confined to the atomistic description of matter. For instance, finite-difference time-domain \cite{Taflove_1980} (FDTD) is a standard numerical tool of computational electromagnetism, while lattice Boltzmann methods \cite{Benzi_1992} (LBM) are widely used in computational fluid dynamics. Indeed, real-space real-time approaches can be used to model physical processes on rather different space and time scales. This observation also bears an important suggestion: numerical methods based on real-space grids can be used to bridge between these different space and time scales.
Numerical nanoplasmonics is a paradigmatic case for multiscale electronic-structure calculations. A nanoplasmonic system – e.g., made up of metal nanoparticles (MNPs) – can be a few tens of nanometers across, while the region of strong field enhancement – e.g., in the gap between two MNPs – can be less than 1 nm across \cite{Savage_2012}. The field enhancement, \(h\left({\bf r}\right)\), is essentially a classical observable, defined as \[h\left({\bf r}\right)=\sqrt{\frac{\left\langle {\bf E}_{\rm tot}^{2}\left({\bf r}\right)\right\rangle }{\left\langle {\bf E}_{\rm ext}^{2}\left({\bf r}\right)\right\rangle }}\;,\] where \({\bf E}_{\rm tot}\) is the total electric field, \({\bf E}_{\rm ext}\) is the external (or driving) electric field, and \(\left\langle \cdots\right\rangle \) indicates a time average. Large field enhancements are the key to single molecule surface-enhanced Raman spectroscopy (SERS) and values as large as \(h>100\) (the intensity of the SERS signal scales as \(h^{4}\)) are predicted by classical electromagnetic calculations \cite{Kneipp_2002}.
In classical calculations, the electronic response is modeled by the macroscopic permittivity of the material. The classical Drude model gives the following simple and robust approximation of the metal (complex) permittivity: \[\label{eq:local-optics} \epsilon_{r}\left(\omega\right)=\epsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega\left(\omega+i\gamma\right)}\ .\] For gold, typical values of the high-frequency permittivity \(\epsilon_{\infty}\), the plasma frequency \(\omega_{p}\), and the relaxation rate \(\gamma\), are: \(\epsilon_{\infty}=9.5\), \(\hbar\omega=8.95\) eV and \(\hbar\gamma=69.1\) meV \cite{Grady_2004}. A non-local correction to the Drude model can also be included by considering the plasmon dispersion \cite{Dobson_2000,Raza_2011}. The metal (complex) permittivity then reads \[\label{eq:non-local-optics} \epsilon_{r}\left({\bf k},\omega\right)=\epsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega\left(\omega+i\gamma\right)-\beta^{2}{k}^{2}}\;.\] The parameter \(\beta\) can be fitted to model the experimental data, although the value \(\beta=\sqrt{\frac{3}{5}}v_{F}\), where \(v_{F}\) is the Fermi velocity, is suggested by the Thomas-Fermi approximation.\cite{Boardman1982}
Regardless of the level of sophistication of the permittivity model, all classical calculations assume that electrons are strictly confined inside the metal surfaces. This is a safe approximation for microscopic plasmonic structures. However, at the nanoscale the electronic delocalization (or spillout) outside the metal surfaces becomes comparable to the smallest features of the plasmonic nanostructure, e.g., to the gap between two MNPs. In this scale, the very definition of a macroscopic permittivity is inappropriate and the electronic response must be obtained directly from the quantum dynamics of the electrons.
TDDFT is currently the method of choice to model the plasmonic response of MNPs , via the simplified jellium model, in which the nuclei and core electrons are described as a uniform positive charge density, and only the valence electrons are described explicitly. Early calculations – especially nanospheres \cite{Marinica_2012,Esteban_2012} – have suggested the existence of new charge-transfer plasmonic modes, which have been also demonstrated by pioneering experiments \cite{Savage_2012}. In the future, as the field of quantum plasmonics \cite{Tame_2013} – i.e., the investigation and control of the quantum properties of plasmons – will further develop, the demand for accurate, yet scalable, numerical simulations to complement the experimental findings is expected to grow. This demand represents both a challenge and an opportunity for computational physics.
Scaling up the TDDFT@jellium method to model larger and more complex plasmonic nanostructures is a challenge which can be addressed by high-performance real-space real-time codes, like Octopus. The code has been initially applied to investigate the plasmonic response of single gold nanospheres (Wigner-Seitz radius, \(r_{s}=3.0\) bohr) \cite{Townsend_2012}. A clear plasmonic resonance appears in the absorption cross section – computed by real-time propagation – for spheres containing a large enough number of electrons (\(N_{e}>100\)). A new plasmonic mode, deemed the “quantum core plasmon”, has been also suggested from the analysis of the absorption cross-section. This new mode has been further characterized by probing the sphere at its resonance frequency. Within a real-time propagation scheme, this is simply done by including an external electric field, the “laser pulse”, oscillating at a given frequency.
As versatility is a major strength of real-space real-time approaches, other jellium geometries can be easily modeled by Octopus, including periodic structures. For instance, a pair of interacting sodium nanowires (with periodicity along their longitudinal direction) has been investigated to assess the accuracy of classical methods based on the model permittivity in eq. (\ref{eq:local-optics}) and eq. (\ref{eq:non-local-optics}). Compared to pairs of nanospheres, nanowires display a stronger inductive interaction due to their extended geometry \cite{Stella_2013,Teperik_2013}. This is manifest in the absorption cross-section which already shows a large split of the plasmonic peak for a small gap between the wires (see Fig. \ref{fig:FIGURE_plasmonics}(a)). Due to the electronic spillout and the symmetry of the system, it also turns out that the largest field enhancement is reached at the center of the gap, not on the opposing surfaces of the nanowires as predicted by the classical methods (see Fig. \ref{fig:FIGURE_plasmonics}(b)). The maximum field enhancement estimated by the TDDFT@jellium method is also smaller than the classical estimates. Once again, the quantum delocalization ignored by the classical methods plays a crucial role in “smearing” the singularities of the induced field, effectively curbing the local field enhancement.
Simple jellium geometries have been implemented in Octopus and they can be used as effective “superatomic pseudopotentials”. The similarity between the jellium potential and atomic pseudopotentials can be further exploited to develop an external “jellium pseudopotential” generator to be used with Octopus. In this way, a larger selection of jellium geometries will be made available along with refined, yet scalable, jellium approaches to include \(d\) electron screening in noble metals \cite{Rubio_1993}. Efforts in this direction are being currently made.
Finally, a word of caution about the domain of applicability of the TDDFT@jellium method is in order. The non-uniformity of the atomic lattice is expected to affect the absorption cross-section of small MNPs. A careful assessment of the lattice contributions – including the lattice symmetry – on the main plasmon modes of a pair of nanosphere is available \cite{Zhang_2014}. This last investigation further demonstrates the possibility to bridge between atomistic and coarse-grained electronic calculations by means of a real-space real-time approach.