Have you ever thought something would likely not work, but in the end, it does? That's the story of this work. When I showed the comparison between the first few calculations with the available experiments, my supervisor jokingly said, "Stop fitting the data to the experiments!" But this is, of course, not what we’ve been doing. And in the end, we now have a paper we both are happy about.

Who are we?

Let's start with who we are and what we do. We are computational physicists, focusing on studying the optical properties of molecules and clusters: how light interacts with a bunch of atoms bonded to one another. For our analysis, we use a method called density-functional theory (DFT), which is a way of calculating the total electronic charge density distribution for a given arrangement of atoms. To calculate optical absorption, we use real-time time-dependent density-functional theory (RT-TDDFT), a method to simulate the dynamics of electrons. In both the above-mentioned methods, instead of taking into account interactions between electrons individually like in Schrödinger's equation, approximate interactions are calculated from the total electronic charge density: the Hartree term for the electron-electron Coulomb interaction, and exchange-correlation functional (xc-functional) to approximate other complex effects. Consequently, the quality of the result depends on the xc-functional approximation used. 

Challenges when modeling silver clusters

When working with structures containing noble-metal atoms like silver, extra caution is required to approximate the electronic interactions because of the localized 4d electrons. The Hartree term, which approximates the Coulombic interactions from the total electronic density, contains a significant amount of contribution coming from the 4d electrons interacting with themselves (self-interaction error), making them spatially spread more than they should be to minimize electrostatic repulsion. And this error becomes significant when simple approximations to the xc-functional such as the local-density approximation (LDA) and generalized-gradient approximations (GGA), are utilized. One way to resolve this issue is to use other accurate xc-functionals at the expense of computational cost, or find a simpler way to correct the description of only the problematic d electrons. And indeed there is a method called DFT+U, in which U is the empirical correction that could be applied locally to the d electrons at each atomic site, correcting their energetic position with respect to the highest occupied molecular orbital (HOMO), to roughly 4 eV below HOMO, and increasing the localization of their spatial distribution, thereby changing the screening of the s electrons responsible for the LSPR. Consequently, the use of RT-TDDFT+U, improves the quality of spectra significantly, at a computational cost only slightly higher than for LDA or GGA calculations.

Our important findings

Using this method for three different sizes of clusters: small molecular like 4 to 8 atoms, intermediate sized 20 to 92 atom clusters, and all the way to 923 atoms (approximately 3 nm diameter), we found the method to work well for a unique U value of 4 eV. For the smaller clusters, we correctly describe electron-hole-like transitions, while for the intermediate sized clusters, LSPR peaks clearly start to emerge, at the same time showing intricate changes in the peak position with cluster size caused by the superatom shell-closing effect. For larger sizes, there is a red shift with increasing cluster sizes. The method seems to work very well in describing optical absorption for silver clusters of all sizes.

Why are our findings crucial?

To unlock the full potential of the predictive power of computational modeling, it's essential to know what approach works and what does not. Our work highlights a method that's been known for a while, works well for modelling silver clusters, while at the same time being computationally efficient (hence, energy efficient). The lower computational cost of the approach makes it suitable for extending this method to investigate systems relevant to real-life applications, such as sensors for detecting changes in localized surface-plasmon resonance driven by the local environment around the silver particle, as photocatalysts for harnessing light energy to drive chemical reactions, in energy harvesting to enhance the efficiency of solar cells by incorporating silver nanoparticles into solar cells, among many more. Our work could help improve the accuracy and reliability of the calculations in describing the complicated interactions for silver clusters. Ultimately, our research not only contributes to the scientific community but also has the potential to impact real-world technologies.