structure. All of the discussion in this manuscript is focused on bare silver clusters,
but there is also an extensive literature related to clusters with ligands, especially for
gold clusters [1], but we will not consider this possibility.
In this paper we study the spectra of bare silver clusters in more detail than in
past work, with the goal of determining sensitivity of the results to the electronic
structure model used and to cluster structure. We also study the nature of the
excited states that are produced in plasmon-like intraband excitation, and contrast
that with results for interband excitation that is also found for the clusters we have
studied. Much of our focus is on the tetrahedral structures of Ag 20 and Ag 84 ,
corresponding to particles between 0.9 and 1.5 nm in size, so that we can study
trends in the results with cluster size. However we also consider a broader range of
clusters going from Ag 10 to Ag 220 to establish trends in the results with cluster size.
In all calculations we use time-dependent density functional theory (TDDFT), but
here we consider both the frequency domain (FD) version that is contained in
NWChem and Amsterdam Density functional theory (ADF) and the real-time (RT)
version that is available in CP2K. These three codes provide access to a variety of
density functionals and basis sets, so it is important to determine how the results
differ as there have been many recent calculations based on these codes [1, 12–14],
using the Ag 20 cluster as a reference for understanding SERS. Both NWChem and
CP2K rely on pseudopotentials for describing the core electrons, and we also
investigate a jellium model for a pseudopotential and a dual-space separable model.
In a RT-TDDFT study, Chen et al. [12] developed a method for calculating
absorption spectra of molecules interacting with metal particles by integrating the
time-dependent Kohn-Sham equations subject to a short (1 fs) pulse of light that
couples to the molecule and particle through the dipole μ · E interaction. The field
E is obtained either for the field in vacuum or from an electrodynamics calculation
if one wants to include a nearby particle. In the latter case, this approach makes it
possible to describe plasmon enhancement classically, and then the molecular
response with RT-TDDFT (and thus the method is called QM/ED). In this procedure, the induced dipole resulting from the μ · E perturbation is Fourier transformed
to determine the polarizability. The absorption is proportional to the imaginary part
of the polarizability. A key component of this approach is that the dipole is assumed
to be damped in determining the Fourier transform, using a damping energy
(0.1 eV) that is derived from the plasmon width for bulk silver. Except for the
electrodynamics description of the metal particle, the theory is the time domain
version of a frequency domain approach that included damping that was developed
by Jensen et al. [15] and used with the ADF code. In either the frequency or time
domain, the inclusion of a cluster of atoms from the particle in the TDDFT calculation makes it possible to describe chemical effects in SERS. However this
procedure was not done in the past work, and no silver cluster extinction spectra
were generated using TDDFT.
A similar QM/ED theory, but now in the frequency domain, was recently presented by Mullin and coworkers [13, 14], who used an approach similar to Jensen
and coworkers, but with the NWChem code. In describing SERS, the particle is
Understanding the Electronic Structure Properties …
39
but there is also an extensive literature related to clusters with ligands, especially for
gold clusters [1], but we will not consider this possibility.
In this paper we study the spectra of bare silver clusters in more detail than in
past work, with the goal of determining sensitivity of the results to the electronic
structure model used and to cluster structure. We also study the nature of the
excited states that are produced in plasmon-like intraband excitation, and contrast
that with results for interband excitation that is also found for the clusters we have
studied. Much of our focus is on the tetrahedral structures of Ag 20 and Ag 84 ,
corresponding to particles between 0.9 and 1.5 nm in size, so that we can study
trends in the results with cluster size. However we also consider a broader range of
clusters going from Ag 10 to Ag 220 to establish trends in the results with cluster size.
In all calculations we use time-dependent density functional theory (TDDFT), but
here we consider both the frequency domain (FD) version that is contained in
NWChem and Amsterdam Density functional theory (ADF) and the real-time (RT)
version that is available in CP2K. These three codes provide access to a variety of
density functionals and basis sets, so it is important to determine how the results
differ as there have been many recent calculations based on these codes [1, 12–14],
using the Ag 20 cluster as a reference for understanding SERS. Both NWChem and
CP2K rely on pseudopotentials for describing the core electrons, and we also
investigate a jellium model for a pseudopotential and a dual-space separable model.
In a RT-TDDFT study, Chen et al. [12] developed a method for calculating
absorption spectra of molecules interacting with metal particles by integrating the
time-dependent Kohn-Sham equations subject to a short (1 fs) pulse of light that
couples to the molecule and particle through the dipole μ · E interaction. The field
E is obtained either for the field in vacuum or from an electrodynamics calculation
if one wants to include a nearby particle. In the latter case, this approach makes it
possible to describe plasmon enhancement classically, and then the molecular
response with RT-TDDFT (and thus the method is called QM/ED). In this procedure, the induced dipole resulting from the μ · E perturbation is Fourier transformed
to determine the polarizability. The absorption is proportional to the imaginary part
of the polarizability. A key component of this approach is that the dipole is assumed
to be damped in determining the Fourier transform, using a damping energy
(0.1 eV) that is derived from the plasmon width for bulk silver. Except for the
electrodynamics description of the metal particle, the theory is the time domain
version of a frequency domain approach that included damping that was developed
by Jensen et al. [15] and used with the ADF code. In either the frequency or time
domain, the inclusion of a cluster of atoms from the particle in the TDDFT calculation makes it possible to describe chemical effects in SERS. However this
procedure was not done in the past work, and no silver cluster extinction spectra
were generated using TDDFT.
A similar QM/ED theory, but now in the frequency domain, was recently presented by Mullin and coworkers [13, 14], who used an approach similar to Jensen
and coworkers, but with the NWChem code. In describing SERS, the particle is
Understanding the Electronic Structure Properties …
39