# DESI constraints on varying electron mass model and axion-like early dark
energy

Osamu Seto<EMAIL_ADDRESS>Yo Toda
<EMAIL_ADDRESS>Department of Physics, Hokkaido University,
Sapporo 060-0810, Japan

###### Abstract

Baryon acoustic oscillation (BAO) is one of the important standard ruler in
cosmology. The results of the latest BAO measurements by Dark Energy
Spectroscopic Instrument (DESI) survey has been reported. Cosmology with the
varying electron mass model and the early dark energy (EDE) models are
regraded as interesting models to resolve the Hubble tension. We present
constraints on the varying electron mass model and EDE models by including new
DESI data as well as cosmic microwave background by Planck and the
conventional BAO data from 6dF, MGS, and DR12 and supernovae light curve data
into analysis. Since new DESI BAO data indicates slightly longer sound horizon
$r_{d}h$ than the other BAO observations, for the varying electron mass model,
the larger $H_{0}=69.44\pm 0.84$ km$/$s$/$Mpc is indicated.

††preprint: EPHOU-24-006

##  I Introduction

The $\Lambda$CDM model has been successful in explaining observations of the
universe at various redshifts. However, as observations have become more
precise, a discrepancy has emerged between the Hubble constants from distant
(high redshifts) observations and local (low redshifts) observations. The
spectrum of temperature anisotropy of the cosmic microwave background (CMB)
infers the present Hubble parameter as $H_{0}=67.36\pm
0.54\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ Aghanim  _et al._ (2020a). By
combining CMB, baryon acoustic oscillations (BAO) and Big Bang nucleosynthesis
(BBN) data, we find $H_{0}=67.6\pm 1.0\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$
Schöneberg  _et al._ (2022a). Other distant observations Dutcher  _et al._
(2021); Aiola  _et al._ (2020); Schöneberg  _et al._ (2019) also report
consistent values of $H_{0}$ with the Planck measurement. On the other hand,
local measurements of the Hubble constant have reported as $H_{0}=73.04\pm
1.04\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ by the SH0ES Riess  _et al._ (2022)
and $H_{0}=73.30^{+1.7}_{-1.8}\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ by the
H0LiCOW Wong  _et al._ (2020). The method of SNe Ia and the Tip of the Red
Giant Branch (TRGB) observations reported a slightly lower Hubble constant,
$H_{0}=69.6\pm 2.5\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ Freedman  _et al._
(2020). This discrepancy is called the Hubble tension.

Among various ideas to solve the Hubble tension Di Valentino  _et al._ (2021);
Abdalla  _et al._ (2022); Perivolaropoulos and Skara (2022); Freedman (2021);
Schöneberg  _et al._ (2022b); Vagnozzi (2023), the varying electron mass
(varying $m_{e}$) model and the early dark energy (EDE) models are regarded as
the most promising solutions Schöneberg  _et al._ (2022b). If we assume that
the electron mass was larger by a few percent than the present value
$m_{e0}=511$ keV before the recombination epoch $z\gtrsim 1000$, then the
varying electron mass model seems a potential solution to the Hubble tension
Ade  _et al._ (2015); Sekiguchi and Takahashi (2021); Seto and Toda (2023);
Hoshiya and Toda (2023); Solomon  _et al._ (2022), because shorter sound
horizon scale at the recombination epoch indicates smaller $H_{0}$.

The EDE model Poulin  _et al._ (2018, 2019); Braglia  _et al._ (2020); Agrawal
_et al._ (2023); Ye and Piao (2020); Smith  _et al._ (2020); Lin  _et al._
(2019); Niedermann and Sloth (2021, 2020) has been intensively also studied as
a possible solution to the $H_{0}$ tension. In the EDE model, temporal energy
of EDE before the recombination epoch makes the sound horizon shorter, which
results in smaller inferred $H_{0}$. The EDE models are constrained not to
increase the amplitude of matter fluctuation $\sigma_{8}$ Poulin  _et al._
(2018) or the baryon density $\Omega_{b}h^{2}$ Seto and Toda (2021); Takahashi
and Toda (2023).

Many models of EDE are proposed with different potentials, but in this paper,
we focus on the axion-like EDE Poulin  _et al._ (2018), which could explain
the cosmic birefringence Capparelli  _et al._ (2020); Murai  _et al._ (2023)
suggested by the recent analyses of the Planck data Minami and Komatsu (2020);
Diego-Palazuelos  _et al._ (2022).

Dark Energy Spectroscopic Instrument (DESI) survey has reported the results of
the latest BAO measurements Adame  _et al._ (2024a, b) and that the
cosmological constant dark energy $(w,w_{a})=(-1,0)$ is $2.1\sigma$ discrepant
with the combination analysis of DESI$+$SDSS$+$CMB$+$PantheonPlus Adame  _et
al._ (2024c), where $w_{a}$ is the derivative of the equation of the parameter
$w$ with respect to the scale factor $a$. To explain this result, many
extended models of dark energy have been discussed Tada and Terada (2024);
Cortês and Liddle (2024); Carloni  _et al._ (2024); Giarè  _et al._ (2024);
Christiansen  _et al._ (2024); Wang and Piao (2024); Qu  _et al._ (2024);
Poulot  _et al._ (2024); Andriot  _et al._ (2024); Pookkillath and Koyama
(2024). In this paper, we discuss implication of the DESI BAO results to the
varying $m_{e}$ model and the axion-like EDE model.

This paper is organized as follows. We present the explanation of the model in
Sec II, the method and the data sets of our analysis in Sec. III, the results
in Sec. IV, and the summary in Sec V.

##  II Models

###  II.1 Varying electron mass

In the varying electron mass model, we assume that the value of the electron
mass at early Universe differs from its present value and it dramatically
changes to the current value after the recombination is complete. As discussed
in the previous work Ade  _et al._ (2015), the primary contribution of an
increased electron mass is the increasing energy level of hydrogen and Lyman
alpha photon, both of which are directly proportional to the electron mass
$E\propto m_{e}$. Since the energy level of hydrogen is higher, a photon
energy at the standard recombination temperature is too small to excite the
hydrogen. In theory with large electron mass, the recombination takes place
earlier than the standard and the resultant sound horizon becomes shorter.
Thus, to reproduce the same measured multiple moment of acoustic peaks of CMB
power spectrum, the larger electron mass results in a shorter diameter
distance and a higher $H_{0}$.

Another consequence of the increased electron mass is Thomson scattering rate
$\sigma_{T}$ which is proportional to $m_{e}^{-2}$ and suppresses the peak of
high-$l$ CMB power spectrum. The other minor contributions include the
photoionization cross-sections, the recombination coefficient, K-factors,
Einstein A coefficients, and the two-photon decay rates (see Ref. Ade  _et
al._ (2015) for the detail).

The varying electron mass model can significantly relieve the Hubble tension
without spoiling the CMB fitting because there is the degeneracy between cold
dark matter density $\omega_{\mathrm{c}}=\Omega_{\mathrm{c}}h^{2}$, baryon
density $\omega_{\mathrm{b}}=\Omega_{\mathrm{b}}h^{2}$, and the sound horizon
scale in the CMB spectrum, which transformed into the degeneracy between
$m_{e}$ and $H_{0}$. On the other hand, the BAO bound can resolve this
degeneracy, a negative $\Omega_{K}$ model with positive spatial curvature
$K>0$ weakens the BAO constraints on variation of $m_{e}$ nevertheless.
Therefore, it is also important to consider the combination of varying
$m_{e}+\Omega_{K}$ Sekiguchi and Takahashi (2021).

To examine the varying electron mass model, we make all the above mentioned
modifications to the recombination code recfast Scott and Moss (2009) and
perform the MCMC analysis sampling $m_{e}\in[0.9\,\,1.2]$ in addition to the
$6$ standard parameters.

###  II.2 Axion-like EDE

The potential of the axion-like EDE takes the form Poulin  _et al._ (2018)

$V(\phi)=\Lambda^{4}\left(1-\cos\left(\frac{\phi}{f}\right)\right)^{n}\,,$ (1)

where $\Lambda$ is the energy scale of the potential, $f$ is the breaking
scale of shift symmetry, and $n$ is the power index of the cosine function. As
in the previous paper by Poulin et al Poulin  _et al._ (2019), we use the
three phenomenological parameters: $z_{c}$, $\Theta_{i}$, and
$f_{\mathrm{de}}(z_{c})\equiv\frac{\rho_{\mathrm{de}}(z_{c})}{\rho_{\mathrm{tot}}(z_{c})}$,
which stand for the redshift when $\phi$ starts to oscillate, the initial
value of the scalar field $\phi/f$, and the energy fraction of EDE to the
total energy density $\rho_{\mathrm{tot}}$ at $z_{c}$. After the transition
$z<z_{c}$, the energy density of EDE decreases as $\rho_{\mathrm{de}}\propto
a^{-4}$ for $n=2$, which is faster than the background energy density does.

To examine the axion-like EDE, we use the camb Howlett  _et al._ (2012) where
axion-like EDE is already implemented and perform the MCMC analysis sampling
$f_{\rm de}(z_{c})\in[0.00001,0.15]$, $z_{c}\in[1000,50000]$, and
$\Theta_{i}\equiv\phi_{\rm ini}/f\in[0.01,3.14]$ in addition to the 6 standard
parameters.

##  III Data sets and methodology

We perform a Markov-chain Monte Carlo (MCMC) analysis on the time-varying
electron mass model and the axion-like EDE, using the public MCMC code
CosmoMC-planck2018 Lewis and Bridle (2002). We analyze the models by referring
to the following cosmological observation and we call the above three data
$\mathcal{D}$ that are always included.

  * •

CMB from Planck Aghanim  _et al._ (2020a). We use the temperature and
polarization likelihoods for high $l$ plik ($l=30$ to $2508$ in TT and $l=30$
to $1997$ in EE and TE) and low$l$ Commander and lowE SimAll ($l=2$ to $29$).
We also include CMB lensing Aghanim  _et al._ (2020b).

  * •

BAO from 6dF Beutler  _et al._ (2011), MGS Ross  _et al._ (2015), and DR12
Alam  _et al._ (2017).

  * •

light curve of SNeIa from Pantheon Scolnic  _et al._ (2018).

  * •

DESI BAO Adame  _et al._ (2024c).

##  IV Results

We present the results of models we have examined. Those likelihood are shown
in Fig. 1 and parameter values in the best-fit points are listed in Tab. 1.

We find that the DESI BAO results disfavor lower $m_{e}$ and indicate slightly
larger electron mass $m_{e}/m_{e0}=1.0092\pm 0.0055$ (CMB+BAO+DESI BAO). This
indicates the shorter sound horizon at the recombination and larger $H_{0}$
than the analysis without DESI BAO. This is due to the measured sound horizon
$r_{d}$ relative to the diameter distance $r_{d}/D$ by DESI BAO. Here, we
define the sound horizon

$\displaystyle r_{d}=$
$\displaystyle\int_{z_{d}}^{\infty}\frac{c_{s}(z)}{H(z)}dz,$ (2)
$\displaystyle D_{M}=$
$\displaystyle\frac{c}{H_{0}\sqrt{\Omega_{K}}}\sinh\left[\sqrt{\Omega_{K}}\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})/H_{0}}\right],$
(3) $\displaystyle D_{H}=$ $\displaystyle\frac{c}{H(z)},$ (4)

and $D_{V}=(zD_{M}(z)^{2}D_{H}(z))^{1/3}$, where $z_{d}$ is the redshift of
the drag epoch. Diameter distance $D$ is inversely proportional to the Hubble
constant $H_{0}$. Therefore, in the BAO measurements, a product of the sound
horizon and the Hubble constant $r_{d}h$ is constrained. For the greater
electron mass, the Hubble constant $H_{0}$ increases while the sound horizon
$r_{d}$ slightly decreases, and the product $r_{d}h$ increases, which is
preferred from DESI BAO measurements as in Fig. 1.

We show the best-fit point of $(\Omega_{m},r_{d}h)=(0.3094,99.773$ km/s$)$ for
$m_{e}/m_{e0}=0.99$ and $(\Omega_{m},r_{d}h)=(0.2979,101.364$ km/s$)$ for
$m_{e}/m_{e0}=1.01$ for the data combination of CMB$+$BAO$+$JLA$+$DESI BAO, in
the Tab 2. As a result using only DESI BAO analysis, we quote
$(\Omega_{m},r_{d}h)=(0.295,101.8$ km/s$)$ for the best-fit point of
$\Lambda$CDM model from the DESI paper Adame  _et al._ (2024c). By comparing
those, for the smaller electron mass, the Hubble constant $H_{0}$ becomes too
small, and the sound horizon relative to the diameter distance $r_{d}/D\propto
r_{d}h$ becomes too short, which leads to poor DESI fitting. The difference in
the $\chi^{2}$ values accounts for
$\Delta\chi^{2}_{\mathrm{DESI,\,bestfit(m_{e}/m_{e0}=0.99)}}-\Delta\chi^{2}_{\mathrm{DESI,\,bestfit(m_{e}/m_{e0}=1.01)}}\simeq
4.7$ as in Tab 2. Figs. 2, 3, and 4 show the comoving diameter distance over
the sound horizon at the drag epoch $D_{M}/r_{d}(z)$, the Hubble distance over
the sound horizon at the drag epoch $D_{H}/r_{d}(z)$ and the angle-average
distance over the sound horizon at the drag epoch $D_{V}/r_{d}(z)$ of
$m_{e}/m_{e0}=1.01$ and $0.99$ models, respectively, as the function of
redshifts with the error bar of the DESI measurements.

Figure 1: Posterior distributions of some parameters for the varying $m_{e}$
model. Different colors of reddish and bluish contours stand for different
data sets.

Figure 2: The comoving diameter distance over the sound horizon at the drag
epoch $D_{M}/r_{d}(z)$ as a function of redshift. We use the best-fit value at
$m_{e}=1.01$ for the orange curve and $m_{e}=0.99$ for the blue curve. The
green error bars represent the data from the DESI BAO measurements. In the
right panel, we illustrate the difference from the $D_{M}(z)/r_{d}$ assuming
$(\Omega_{M},r_{d}h)=(0.295,101.8)$ which is the mean value from the only DESI
BAO constraint Adame  _et al._ (2024c).

Figure 3: Same as Fig. 2, for the Hubble distance over the sound horizon at
the drag epoch $D_{H}/r_{d}(z)$ as a function of redshift.

Figure 4: Same as Fig. 2, for the angle-average distance over the sound
horizon at the drag epoch $D_{V}/r_{d}(z)$ as a function of redshift.

Next, we discuss the results of varying $m_{e}$ in a curved Universe (the
$m_{e}+\Omega_{K}$ model). As is shown in the DESI paper Adame  _et al._
(2024c), we also find that a positive curvature ($K>0$) model does not fit the
DESI BAO data and a spatially flat ($K=0$) Universe is preferred when DESI BAO
data are included. The central value of a Hubble constant shifts slightly when
DESI BAO data are incorporated. The results are summarized in Fig. 5 and Tab.
1.

Figure 5:  Posterior distributions of some parameters for the
$m_{e}+\Omega_{K}$ model. Different colors of reddish and bluish contours
stand for different data sets.

We present the results of axion-like early dark energy, which is given in Fig.
6 and Tab. 1. As in the varying $m_{e}$ model, we find a higher Hubble
constant and a lower $\Omega_{m}$ when DESI BAO data are included. This result
is in agreement with the previous paper Qu  _et al._ (2024), while authors of
Ref. Qu  _et al._ (2024) have considered the $n=3$ model and excluded other
BAO data.

Figure 6:  Posterior distributions of some parameters for axion-like EDE
($n=2$) model. Different colors of reddish and bluish contours stand for
different data sets.

Finally, we calculate the Gaussian tension in Tab. 1 to make a fair comparison
of the three models. The Gaussian tensions for $H_{0}$, $S_{8}$, and
$\Omega_{b}h^{2}$ are respectively calculated as

$T_{H_{0}}=\frac{H_{0\leavevmode\nobreak\
\mathrm{\mathcal{D}(w./w.o.DESI)}}-73.30}{\sqrt{\sigma_{\mathrm{\mathcal{D}(w./w.o.DESI)}}^{2}+1.04^{2}}}\,,$
(5)

for the Hubble tension with direct measurement Riess  _et al._ (2022),

$T_{S_{8}}=\frac{S_{8\leavevmode\nobreak\
\mathrm{\mathcal{D}(w./w.o.DESI)}}-0.776}{\sqrt{\sigma_{\mathrm{\mathcal{D}(w./w.o.DESI)}}^{2}+0.017^{2}}}\,,$
(6)

for the $S_{8}$ tension with DES Abbott  _et al._ (2022).

From the viewpoint of the Hubble tension, as in Tab. 1, including DESI BAO
data improves the tension in varying $m_{e}$ model and the EDE models because
DESI BAO indicate slightly shorter $r_{d}$. On the other hand, the tension
increases for the varying $m_{e}+\Omega_{K}$ model, because the center value
does not change but the error is reduced by breaking the degeneracy. In the
respect of the $S_{8}$ tension, including DESI BAO data improves the tension
in all three models.

Parameter | $m_{e}$ | $m_{e}+\Omega_{K}$ | EDE
---|---|---|---
$\Omega_{b}h^{2}$ | $0.02251\pm 0.00016$ | $0.02288_{-0.00050}^{+0.00045}$ | $0.02253\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.1205\pm 0.0019$ | $0.1216\pm 0.0022$ | $0.1222_{-0.0026}^{+0.0016}$
$m_{e}/m_{e0}$ | $1.0054\pm 0.0065$ | $1.019\pm 0.018$ | -
$\Omega_{K}$ | - | $-0.0041\pm 0.0049$ | -
$f_{de}(z_{c})$ | - | - | $0.0117_{-0.010}^{+0.0039}$
$H_{0}$ | $68.6\pm 1.1$ | $69.7\pm 1.7$ | $68.74_{-0.90}^{+0.62}$
$\Omega_{m}$ | $0.3054\pm 0.0078$ | $0.299\pm 0.011$ | $0.3078\pm 0.0057$
$S_{8}$ | $0.824\pm 0.010$ | $0.824\pm 0.010$ | $0.824\pm 0.011$
$T_{H_{0}}$ | 3.10$\sigma\ $ | 1.81$\sigma\ $ | 3.51$\sigma$
$T_{s_{8}}$ | 2.43$\sigma\ $ | 2.43$\sigma\ $ | 2.37$\sigma$

Parameter | $m_{e}$ | $m_{e}+\Omega_{K}$ | EDE
---|---|---|---
$\Omega_{b}h^{2}$ | $0.02260\pm 0.00014$ | $0.02269\pm 0.00036$ | $0.02259\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.1210\pm 0.0018$ | $0.1213\pm 0.0021$ | $0.1216_{-0.0026}^{+0.0016}$
$m_{e}/m_{e0}$ | $1.0092\pm 0.0055$ | $1.013\pm 0.014$ | -
$\Omega_{K}$ | - | -0.000$9_{-0.0036}^{+0.0032}$ | -
$f_{de}(z_{c})$ | - | - | $0.0123_{-0.0092}^{+0.0049}$
$H_{0}$ | $69.44\pm 0.84$ | $69.7\pm 1.4$ | $69.19_{-0.84}^{+0.60}$
$\Omega_{m}$ | $0.2993\pm 0.0053$ | $0.2978\pm 0.0085$ | $0.3026\pm 0.0046$
$S_{8}$ | $0.8205\pm 0.0097$ | $0.8201\pm 0.0096$ | $0.8178\pm 0.0096$
$T_{H_{0}}$ | 2.89$\sigma\ $ | 2.06$\sigma$ | 3.23$\sigma$
$T_{s_{8}}$ | 2.27$\sigma\ $ | 2.26$\sigma$ | 2.14$\sigma$

Table 1: 68% constraints and Gaussian tension to other measurements from the data set $\mathcal{D}$ (upper) and $\mathcal{D}+$ DESI (lower)  Parameter |  $m_{e}$=0.99 |  $m_{e}$=1.00 |  $m_{e}$=1.01
---|---|---|---
$\Omega_{m}$ | 0.3094 | 0.3045 | 0.2979
$r_{d}h$ | 99.773 | 100.448 | 101.364
$\chi_{\mathrm{CMB}}^{2}$ | 2782.28 | 2775.04 | 2774.53
$\chi_{\mathrm{6DF}}^{2}$ | 0.022 | 0.000 | 0.033
$\chi_{\mathrm{MGS}}^{2}$ | 1.279 | 1.677 | 2.271
$\chi_{\mathrm{DR12}}^{2}$ | 4.281 | 3.520 | 3.509
$\chi_{\mathrm{DESI}}^{2}$ | 17.739 | 15.051 | 13.060
$\chi_{\mathrm{JLA}}^{2}$ | 1034.959 | 1034.799 | 1034.736
$\chi_{\mathrm{prior}}^{2}$ | 2.059 | 2.052 | 1.874
$\chi_{\mathrm{Total}}^{2}$ | 3842.63 | 3832.14 | 3830.01
Table 2: Best-fit point using $\mathcal{D}$ \+ DESI

##  V Conclusions

In this paper, we examine the varying electron mass model and the axion-like
early dark energy model, considering the recent DESI BAO measurements. In the
analysis using a combination of CMB, conventional BAO and light curves
(Pantheon), as in the previous studies, we find that the electron mass before
the CMB era is in agreement with the current value in a $1\sigma$ confidence
level. However, when the DESI BAO data is included, we find that the larger
electron mass is preferred as,

$\displaystyle\begin{cases}m_{e}/m_{e0}=1.0054\pm 0.0065\\\ H_{0}=68.6\pm
1.1\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\\\ \Omega_{m}=0.3054\pm
0.0078\end{cases}(\mathrm{CMB+BAO+Pantheon}),$ (7)
$\displaystyle\begin{cases}m_{e}/m_{e0}=1.0092\pm 0.0055\\\ H_{0}=69.44\pm
0.84\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\\\ \Omega_{m}=0.2993\pm
0.0053\end{cases}(\mathrm{CMB+BAO+Pantheon+DESI\,BAO}).$ (8)

Although the electron mass in the BAO era is assumed to be the same as today,
the larger electron mass at CMB era leads to a higher Hubble constant $H_{0}$
and a product of a sound horizon and Hubble constant $r_{d}h$ to fit the CMB
power spectrum, bringing it closer to the DESI best-fit value. This is the
reason why DESI BAO data prefers the larger electron mass in the CMB era. It
is also known that both $m_{e}/m_{e0}$ and $H_{0}$ increase when considering a
closed universe, but we find that such the case is not preferred from DESI BAO
measurements.

In the axion-like EDE ($n=2$) model, we also find that a larger amount of the
EDE and a higher Hubble constant are preferred when the DESI BAO data is
included.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number 23K03402 (OS), and JST
SPRING, Grant Number JPMJSP2119 (YT).

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11institutetext: S. K. Giri 22institutetext: Department of Chemistry,
Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United
States 33institutetext: G. C. Schatz 44institutetext: Department of Chemistry,
Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United
States, 44email<EMAIL_ADDRESS>
# Plasmon Enhanced Spectroscopy and Photocatalysis

Sajal Kumar Giri and George C. Schatz

###### Abstract

This study examines the Raman scattering and charge transfer properties of
molecules adsorbed on the surface of a tetrahedral Au120 nanoparticle based on
the time-dependent density functional tight-binding (TD-DFTB) method. We study
Raman scattering (SERS) enhancements for pyridine where the molecule is
adsorbed either on the tip (V-complex) or surface (S-complex) of the
nanoparticle. The scattering intensity is enhanced by a factor of 3-15 due to
chemical effects while significantly larger enhancements (in the order of
$10^{2}-10^{4}$) are observed for plasmon resonance excitation at an energy of
2.5 eV depending on the adsorption site. Furthermore, we demonstrate charge
transfer between the nanoparticle and a fullerene-based molecule after pulsed
excitation of the plasmon resonance which shows how plasmon excitation can
lead to negative molecular ion formation. All of these results are consistent
with earlier studies using either TD-DFT theory or experimental measurements.

###### Keywords:

Plasmons, SERS, Photocatalysis, Charge Transfer

##  1 Introduction

Surface-enhanced Raman scattering (SERS) is a widely used phenomenon among
various research communities where the scattering intensity of molecules can
be increased by several orders of magnitude when they are adsorbed on the
surface of plasmonic nanoparticles.SERS-VanDuyne-1997 (1, 2, 3) The
enhancement factors are as large as $10^{10}-10^{11}$ and therefore SERS can
be studied on the single-molecule level.SERS-Emory-1997 (4, 5, 6, 7, 8) The
unexpected enhancement in the Raman signal from pyridine adsorbed on an
electrochemically roughened silver electrode was the first evidence of
SERS,SERS-McQuillan-1974 (9) and this was attributed to the extensive surface
coverage of molecules on the rough nanostructured surface, but later it was
identified as a plasmon-based phenomenon.SERS-VanDuyne-1997 (1, 2) SERS has
been successfully applied to several fields including biosensingBiosensing-
VanDuyne-2003 (10, 11), drug analysis,Drug-Schatz-2009 (12) and
electrochemistryElectrochemical-Zen-2015 (13, 14).

Raman scattering arises because the electric field of the incident
electromagnetic radiation induces polarization in irradiated molecules and the
component of the charge oscillation at the shifted Raman frequencies leads to
emission of scattered light. The intensity of the scattered light depends on
two components: incident light intensity and the polarizability of the
molecule. In SERS, the weak Raman intensities for isolated molecules are
increased in the presence of plasmonic nanoparticles due to amplification of
the oscillating electric field at the molecule that arises from plasmon
excitation. Thus the treatment of SERS includes the interaction of light with
both molecules and particles.

Broadly speaking, the two possible mechanisms that describe SERS involve
electromagnetic and chemical effects. Localized surface plasmon resonance
(LSPR) excitation of the conduction electrons in metal nanoparticles of Au,
Ag, and Cu with dimensions of 5-200 nanometers generates an enhanced
electromagnetic field at the interface of the particle.LSPR-VanDuyne-2007 (15)
This enhanced field can modulate several optical processes at the surface
including photocurrent,Photocurrent-Schatz-2009 (16) luminescence,Exciton-
Achermann-2010 (17) and Raman scattering.SERS-VanDuyne-1997 (1, 2, 3) Raman
scattering scales with the local field as
$|E_{loc}(\omega)/E_{0}(\omega)|^{4}$, where the numerator is the localized
electromagnetic field and the denominator is the incident field.SERS-
VanDuyne-2008 (3) SERS enhancement factors due to electromagnetic effects can
reach values up to $10^{9}$.SERS-Schatz-2008 (6)

On the other hand, the chemical enhancement includes the effect of orbital
overlap that occurs when molecules adsorb on the surface. Static charge
transfer associated with adsorption modifies the Raman polarizability of the
molecule, leading to a contribution to SERS. Charge transfer (CT) excitations
at optical frequency can also contribute to enhancements through a resonance
Raman mechanism.Electronic-Schatz-2008 (18) Enhancements due to static
chemical effects can be up to a factor of 3-20 as demonstrated theoretically
for pyridine adsorbed on the surface of a small tetrahedral Au particle.TDDFT-
Schatz-2006 (19). Resonant CT can lead to larger enhancements, but usually the
resonant CT and LSPR frequencies are sufficiently different, or the resonance
CT transition is sufficiently broad, that one only has significant SERS
enhancement from plasmon excitation and static CT.

Atomistic simulations based on quantum mechanics provide insight into the
possible SERS mechanisms and enhancements, in addition to providing Raman
shifts and adsorbate structures and reactions. Previous theoretical works have
addressed SERS for pyridine adsorbed on the surface of Ag and Au particles
with enhancement factors up to $10^{2}-10^{4}$ based on the time-dependent
density functional theory (TD-DFT).TDDFT-Schatz-2006 (19, 19) However, the
size of the nanoparticles that can be described with electronic structure
theory has been limited to a small number of atoms ($\leq 20$) due to the
large computational complexities associated with the calculations. It is
therefore a subject of significant importance to explore SERS for larger
nanoparticles. Typically this should involve hundreds or even thousands of
atoms as quantum size effects lead to noticeable blue shifts of plasmon
frequencies with decreasing particle size for particles with less than 100
atoms. Semi-empirical methods based on INDO have been demonstrated
successfully for the efficient simulation of SERS for Ag-pyridine
complexes.HyperRaman-Schatz-1992 (20, 21) As an alternative to the INDO
calculations, the density functional tight-binding (DFTB) method may also
provide efficient solutions for larger complexes.DFTB-Aradi-2020 (22) We
explore this possibility in this work.

Two recent papers provide details of the DFTB method.DFTB-Aradi-2020 (22, 23)
DFTB is a semi-empirical method which contains many features that are similar
to the earlier INDO method, but it is developed from a DFT perspective, and
with much greater functionality. As a consequence, this method provides
results that are often comparable to DFT but with significantly lower
computational costs and therefore the ground and excited state properties of
larger complexes can be simulated efficiently. Previous works have
demonstrated the success of the DFTB method in various applications including
biomolecules,DFTB-Bio-Wong-2019 (24) metals,DFTB-Metal-2019 (25, 26, 27)
semiconductors,DFTB-SemiConductor (28) plasmonic photocatalysis,PDC-
Sanchez-2022 (29, 30, 31), etc. However, the accuracy of the prediction is
limited by the quality of the parameters that describe the Hamiltonian ($H$)
and overlap ($S$) matrices. To elevate accuracy and enable applications to a
larger portion of the periodic table, new parameters need to be developed and
further optimized.

The real-time time-dependent density functional tight-binding (RT-TD-DFTB)
method allows us to study plasmon-driven charge and energy transfer processes
at the metal-molecule interface by simulating laser-induced electron
dynamics.RTDFTB-Sanchez-2020 (23, 29, 30) In these calculations, hot carriers
generated through the decay of LSPR states can be transferred to the adsorbed
molecules, initiating reaction dynamics on the surface of the metal
nanoparticles. This includes several processes like molecular bond
dissociation and formation,PDC-Schatz-2023 (30, 31) charge transfer (CT),Hot-
Carrier-Erhart-2022 (32) energy transfer,PDC-Sanchez-2022 (29) spectral
shift,SERS-PCBM-Schatz-2019 (33) etc. In the following, we study CT processes
between Au120 and a fullerene (C60) after exciting plasmons at 2.5 eV with an
external laser pulse. This work was inspired by a recent study from the Van
Duyne group, where the fullerene-based molecule [6,6]-phenyl-C61-butyric acid
methyl ester (PCBM) was found to form negative ions in the presence of gold
dimers when irradiated at 532 nm as revealed by SERS measurements. Given this,
the present study addresses the ability of the DFTB method to study both SERS
and plasmon-driven electron transfer processes at the metal-molecule
interface.

The paper is divided into two parts, in the first part, we explore Raman
scattering for Au120-pyridine complexes and the second part contains plasmon-
driven CT in a Au120-C60 complex. For Raman scattering, we discuss
enhancements due to chemical and plasmon excitation at an energy of 2.5 eV.
The molecular Raman spectrum obtained with the DFTB method is compared with
the B3LYP/TZVP level of DFT theory. For the CT study, we demonstrate results
with RT-TD-DFTB calculations. Thus these two separate studies each addresses
key benchmarks for using DFTB in future studies where the two calculations are
combined.

##  2 Computational Details

All calculations in this work have been carried out with the DFTB+ and ORCA
codes. To perform time-dependent density functional tight-binding (TD-DFTB)
calculations, we use auorg/auorg-1-1 Slater-Koster parameters for gold-
molecule complexes downloaded from the dftb.org site. To elucidate the
accuracy of this method for predicting Raman spectra, we compare TD-DFTB
results with conventional results obtained with the B3LYP/TZVP level of DFT
theory. The self-consistent-charge density functional tight-binding (SCC-DFTB)
approach has been used to compute the ground state properties of all the
complexes studied. To treat the dynamics induced by external driving fields,
we use the time-dependent DFTB scheme that is an extension of SCC-DFTB to the
time domain.

The Raman scattering activity $S_{p}$ for a vibrational mode $p$ is calculated
from the polarizability derivatives as $S_{p}=45\alpha_{p}^{\prime
2}+7\gamma_{p}^{\prime 2}$, where $\alpha_{p}^{\prime}$ and
$\gamma_{p}^{\prime}$ are derivatives of the isotropic and anisotropic
polarizabilities with respect to the normal mode $p$. The derivatives of the
frequency-dependent polarizability are determined from single-point
computations for small Cartesian coordinate displacements from the equilibrium
geometry along the normal modes for both DFT and DFTB calculautions. We
compute polarizabilities in the frequency domain with a value of the damping
parameter $\Gamma=0.15$ eV. To obtain a smooth Raman spectrum as a function of
Raman shifts, the scattering activity is broadened by a Lorentzian function
with a $20$ cm-1 width.

In real-time TD-DFTB (RT-TD-DFTB) the one-body reduced density matrix is
propagated according to the Liouville-von Neumann equation in a nonorthogonal
basisRTDFTB-Sanchez-2020 (23)

$\dot{\rho}=-{\rm i}S^{-1}[H,\rho]S^{-1}$ (1)

starting from an initial state defined by Hamiltonian ($H_{0}$), overlap
($S_{0}$), and density ($\rho_{0}$) matrices. Here $S^{-1}$ represents the
inverse of $S$. The Hamiltonian in the above equation includes a field-matter
coupling term $V_{A}(t)=-\boldsymbol{\mu}\cdot\mathbf{E}(t)$, where
$\mathbf{E}(t)$ is the external field and $\boldsymbol{\mu}=\sum_{A}\Delta
q_{A}\mathbf{R}_{A}(t)$ is the dipole with atomic coordinate $\mathbf{R}_{A}$
and charge $\Delta q_{A}$ for atom A at time $t$. We consider a time-dependent
pulse of the following form to drive the system

$E(t)=E_{0}{\rm exp}(-(t-t_{c})^{2}/T_{l}^{2})\sin(\omega_{l}t)$ (2)

with field amplitude $E_{0}$, center of the pulse $t_{c}$, pulse duration
$T_{l}$, and frequency $\omega_{l}$. For calculating spectra, the density
matrix evolves in time due to purely electronic dynamics where the nuclei are
“clamped” in space. This treatment has been extensively used to study optical
processes of material and molecules induced by external time-dependent
electric fields.RTDFTB-Sanchez-2016 (34, 35, 23) The Mulliken population is
calculated as

$q_{A}={\rm Tr}_{A}[\rho S]$ (3)

where ${\rm Tr}_{A}$ denotes trace over orbitals centered on atom A, and the
$S$ matrix depends on the coordinates that are parameterized for each pair of
orbitals.

We compute all absorption spectra reported here in the time domain. The
density matrix is propagated in time with an external Dirac $\delta$
perturbation of the form $E(t)=E_{0}\delta(t)\hat{e}$ with field amplitude
$E_{0}$ and direction vector $\hat{e}$. The system evolves in time freely and
the induced dipole contains information related to the transition energy and
couplings between the ground and excited states. The absorption spectrum is
obtained from the imaginary component of the Fourier-transformed dipole moment

$\sigma_{abs}(\omega)=\frac{4\pi\omega}{3c}{\rm Im}\\{{\rm
Tr}[(\mu(\omega)-\mu_{0})/E_{0}]\\}$ (4)

with an initial permanent dipole $\mu_{0}$. This method reproduces linear
absorption spectra that are conventionally obtained in the frequency domain in
the limit of a weak field intensity, $E_{0}\rightarrow 0$, that induces linear
photoabsorption. To broaden the absorption lines in energy, we damp the time-
dependent dipoles with a damping function ${\rm exp}(-t/\tau)$ with a decay
timescale of $\tau=6$ fs.

Figure 1: Optimized geometries of Au120-pyridine complexes where pyridine is
adsorbed on one of the tips (V-complex) and triangle surfaces (S-complex).
Figure 2: Absorption spectra of (a) isolated fullerene (C60) and pyridine
molecules, and (b) Au120 complexes. In (b) peak positions for the first two
lowest energy peaks (gray dotted lines) of the isolated C60 are compared with
the peaks that appear for the Au120-C60 complex (magenta dotted lines) near
3.3 and 4.4 eV. Absorption intensity for (a) and (b) are scaled differently.

##  3 Results and Discussion

### Raman Scattering

#### Au120-Pyridine Complexes:

We use a tetrahedral Au120 nanoparticle for all of these studies, which is a
similar structure (but larger) to what has been considered in earlier SERS
studies.TDDFT-Schatz-2006 (19, 19) This allows two distinct binding sites for
an adsorbing pyridine molecule, either on one of the tips (V-complex) or in
the middle of the triangle faces (S-complex). These two sites provide
significantly different chemical environments for pyridine, affecting how the
molecule interacts with the particle. It has been found that the S-complex
better represents the experimental SERS spectra, however, the V-complex
provides different SERS features due to a unique interaction between the
particle and molecule.

We optimize all geometries with a maximum force tolerance of 10-5 a.u. for the
movement of atoms. We attach pyridine on the tip (for V-complex) and surface
(for S-complex) gold atoms of a previously optimized Au120 and further
optimize, freezing the atoms of the particle. Optimized geometries are shown
in Figure 1. For both the geometries pyridine binds through the N atom and the
Au-N bond aligns with the Z-axis. The distance between N and the nearest Au is
2.21 $\AA$ for the V-complex and 2.32 $\AA$ for the S-complex. A smaller
distance between the particle and molecule was also observed in a
Au20-pyridine V-complex with TD-DFT calculations.TDDFT-Schatz-2006 (19) We
find a small amount of electron transfer from pyridine to the particle in the
ground state structures, 0.05e and 0.03e for the V- and S-complexes,
respectively.

#### Absorption Spectra:

The absorption spectrum of pyridine is shown in Figure 2(a). Excited states
between 5-6 eV are weakly absorbing while a strong absorption peak appears at
an energy of 7 eV. The strong absorption peak involves $\pi\rightarrow\pi^{*}$
transition and is associated with a large oscillator strength. These spectral
features are consistent with previous studies based on INDO/SCI and TD-DFT
methods.INDO-Schatz-2017 (21) The absorption spectrum of fullerene (Fig.2(a))
is discussed in the next section.

Absorption spectra for Au120 and Au120-pyridine complexes are shown in Figure
2(b). The spectrum for the gold particle shows multiple peaks that are
comparable to the spectrum of a tetrahedral Au20 investigated using the TD-DFT
method.TDDFT-Schatz-2006 (19) It contains contributions from several intraband
transitions at lower energies and interband transitions at relatively higher
energies. However, the overall spectral intensity is increased with the
particle size due to the large density of states available in Au120. A weak
LSPR peak appears at around 2.5 eV. We note that although a gold particle
involving 120 atoms is large compared to the previous study with Au20, it is
still too small sustain significant LSPR excitation. Nevertheless, excitation
near 2.5 eV with dominant intraband transitions can generate results better
resembling the LSPR characteristics that are usually observed experimentally
for much larger particles. Note that the participation of $d$-electrons in
photoexcitation causes decoherence of plasmon excitation, resulting in a broad
plasmon peak that overlaps with inter-band transitions at higher energy. This
makes it difficult to identify the LSPR peak for gold particles with a nearly
spherical shape with less than about 300 atoms in the cluster, but for
tetrahedral particles, the LSPR peak is apparent in our 120 atom results. The
spectrum for the S-complex is very similar to the isolated gold particle
spectrum, whereas for the V-complex the absorption intensity is slightly
increased due to CT excitations as is consistent with previous TD-DFT
results.TDDFT-Schatz-2006 (19)

#### Enhancement of Raman Scattering:

We have used DFTB to compute the SERS enhancement for pyridine using low
frequency (nominally static) calculations to determine the chemical
enhancement factor, and with calculations at the plasmon energy of 2.5 eV for
Au120-pyridine complexes to determine the electromagnetic enhancement factor.
We find that resonant CT excitations for the Au120-pyridine complexes involve
small transition strengths that overlap with the dominant $d$-band excitations
and therefore can have negligible contributions to SERS. Thus in the
following, we only consider enhancements due to static chemical and
electromagnetic effects.

Figure 3: Raman spectra of pyridine using B3LYP/TZVP and TD-DFTB methods.

Normal Raman scattering spectra (obtained through computation of the static
polarizability derivatives) for gas phase pyridine are shown in Figure 3, with
the TD-DFTB result compared with the spectrum obtained from an analogous
B3LYP/TZVP DFT calculations. Here we have scaled the normal mode frequencies
from DFTB by a factor of 0.9 to correct for a systematic overestimation of the
vibrational frequency obtained from this method. Overall the spectra agree
reasonably well both for the mode frequencies and relative peak intensities.
The Raman active modes from the TD-DFT calculation include a pair of ring
deformation modes at 620 and 672 cm-1, a pair of ring breathing modes at 997
and 1039 cm-1, a symmetric C-H wag mode at 1242 cm-1, and a pair of ring
stretching modes at 1604 and 1605 cm-1. The two noticeable discrepancies
between spectra are (i) unphysical modes that appear in the TD-DFTB results
near 794 cm-1 and (ii) for the lower energy ring breathing mode (that appears
at 941 cm-1 in the TD-DFTB result) the intensity is significantly suppressed
compared to the others. As will be discussed later the unphysical modes near
794 cm-1 are not enhanced in SERS, however, the lower energy ring breathing
mode can be significantly enhanced even though it is vanishingly small for the
isolated gas phase molecule (TD-DFTB result).

Figure 4: Surface enhanced Raman scattering for Au102-pyridine complexes using
the static polarizability of the molecule. Throughout the scaling factors in
the inset represent enhancements compared to the normal Raman intensities of
an isolated pyridine. Figure 5: Surface enhanced Raman scattering of
Au120-pyridine complexes for an incident laser energy of 2.5 eV.

The normal Raman scattering spectra of pyridine adsorbed on the surface of
Au120 are shown in Figure 4. These results show that the chemical enhancements
are around a factor of 3 for the S-complex and 15 for the V-complex. These
enhancement factors are comparable with the reported values (a factor 3 for
the S-complex and 20 for the V-complex) for a Au20-pyridine complex based on
earlier TD-DFT calculations.TDDFT-Schatz-2006 (19) The larger enhancement and
sensitivity for the V-complex is understandable as the chemical interaction
between particle and molecule is larger for that complex compared to the other
given that the pyridine interacts with a coordinately less saturated gold
atom.

Interestingly the lower energy ring breathing mode (944 cm-1) that has a small
intensity for the isolated molecule (Fig.3) is now enhanced by a factor of 15
for the V-complex, while the higher energy ring breathing mode (1056 cm-1)
shows a weak enhancement. Between these two ring breathing modes the higher
energy mode is enhanced more than that for the lower energy mode for the
S-complex. Again, this trend in enhancements is consistent with the TD-DFT
calculation as observed for Au20-pyridine complexes.TDDFT-Schatz-2006 (19).
Further, the unphysical modes that appear in the isolated molecule near 794
cm-1 have not been enhanced for either complex. Another mode with much higher
enhancements is near 1459 cm-1. This peak was also seen in the TD-DFT work,
and it appears in experimental SERS measurements on gold but not silver.TDDFT-
Schatz-2006 (19)

The mode frequencies show blue-shifts for both complexes as observed in the
TD-DFT calculations, however, the shifts are relatively smaller compared to
the TD-DFT results.TDDFT-Schatz-2006 (19) The V-complex shows higher blue
shifts than the S-complex for most of the Raman active modes, as also observed
in the TD-DFT calculations. The larger blue shifts in the V-complex can be
associated with stronger bonding interaction in the V-complex as already
mentioned.

The SERS scattering spectra of gold-pyridine for plasmon excitation at an
energy of 2.5 eV are shown in Figure 5 for both complexes. The resonant
excitation of metal complexes leads to enhancements due to electromagnetic
effects along with chemical enhancements. The overall enhancement factors are
on the order of $10^{1}-10^{2}$ and $10^{3}-10^{4}$ for the S- and
V-complexes, respectively. Similar enhancement factors were also observed for
Au20-pyridine complexes with the TD-DFT method.TDDFT-Schatz-2006 (19) For the
V-complex, the 941 and 1635 cm-1 modes are enhanced significantly, by a factor
of $8\times 10^{4}$ and $4\times 10^{4}$, respectively, while for the
S-complex multiple modes are enhanced, with the maximum enhancement for 1636
cm-1 mode. Further, a new Raman active mode appears in the S-complex at 1459
cm-1. Interestingly this mode was strongly enhanced in the static Raman
spectra and in the TD-DFT results noted above.

Figure 6:  (Top) Optimized geometry of a Au120-C60 complex, and (Bottom)
snapshots of charge density difference with respect to the initial charge at
$t=0$. The isosurface cutoff is set to $\pm$0.05e. Yellow color represents
positive and cyan color represents negative amplitudes. Figure 7: Time
evolution of charges for particle and molecule for interaction with a Gaussian
laser pulse shown by a light gray line in the background.

### Charge Transfer

#### Au120C60 Complex:

To generate a suitable geometry to study plasmon-driven charge transfer, we
place a fullerene (C60) molecule on one of the triangular faces (as in the
S-complex) of Au120 (previously optimized) and then the geometry is optimized
freezing atoms of the particle. The optimized geometry of Au120C60 is shown in
Figure 6. The minimum distance between C and Au atoms is around 3.5 $\AA$. We
observe a small electron transfer (around 0.2e) from the particle to the
molecule for the optimized geometry in the ground state.

#### Absorption Spectra:

Absorption spectra for C60 and Au120C60 are displayed in Figures 2 (a) and
(b), respectively. The molecular spectrum exhibits multiple peaks, with the
first three peaks at 3.3, 4.35, and 5.2 eV. These calculated peaks well
represent the experimental peaks found at 3.8, 4.9, and 6 eV, respectively for
the gas phase C60 molecule.C60-Shobatake-1996 (36) The spectrum of the
Au120C60 complex is dominated by the gold particle with two major peaks near
3.27 and 4.3 eV. These two peaks are slightly red-shifted compared to the
first two lowest energy peaks of the isolated molecule. Previous studies
demonstrated a shift in LSPR peaks due to the interaction between plasmon and
molecular resonances.Plasmon-Molecule-VanDuyne-2006 (37, 38) Moreover, a weak
plasmon peak near 2.5 eV is also observed as in the other complexes in the
figure.

#### Laser-Induced Charge Transfer Dynamics:

A linearly polarized (polarized along the Z-axis) laser pulse of the form
given in Eq.2 is used to drive the system. Both molecule and particle couple
with the external field. We use a driving field amplitude $E_{0}$, frequency
$\omega_{l}$, and pulse duration $T_{l}$ of 0.7 V/$\AA$, 2.5 eV and 14 fs,
respectively. We note that for this value of the pulse duration, the field
amplitude decays to zero at about 50 fs with a peak position at 25 fs. Figure
7 displays the time evolution of charges for the molecule and particle
separately. We note that the total charge is preserved (here charge neutral
complex) at all times, however, charge transfer can occur across the interface
between the molecule and particle during the dynamics. As seen in Fig.7, the
initial charge (before interacting with a laser pulse) of the molecule is
slightly negative, reflecting ground state charge transfer to the molecule.
The charges of the two subsystems oscillate with the laser frequency during
interaction with the laser pulse. There is an effective transfer of electrons
from the molecule to the particle during the early time of the dynamics
(Figure 6), and the charge of the particle drops to about +0.15e at around 18
fs. After that the electron transfer changes direction and an effective
transfer of electrons from the particle to the molecule is observed. At long
times (after 50 fs), the charge on the molecule oscillates around a value of
-0.75e. Figure 6 displays the negative charge on the molecule at long times
(after 50 fs). The transfer of electrons and the formation of negative
molecular ions for fullerene-based molecules have been experimentally
demonstrated using SERS measurements, where the fullerene pentagonal pinch
mode was found to change frequencies from that for the neutral fullerene to
that for the fullerene anion after excitation of the gold/fullerene complex at
532 nm.SERS-PCBM-Schatz-2019 (33) We note that since the fullerene is still in
contact with the gold nanoparticle at the end of the calculation the amount of
charge residing on the molecule is not necessarily a full electron.

##  4 Conclusions

In this study, we have presented two interrelated projects that test the
ability of TD-DFTB to describe the influence of plasmon excitation on two
physical phenomena: surface-enhanced Raman scattering (SERS) and plasmon-
enhanced charge transfer. In the first, we calculated SERS spectra for
Au120-pyridine complexes, where we found enhancements due to both chemical and
plasmon excitation effects that are consistent with earlier TD-DFT
studies.TDDFT-Schatz-2006 (19) The enhancement factors for chemical effect are
around 3-15 depending on the site of the adsorption. For the electromagnetic
effect, the enhancement is on the order of $10^{3}-10^{4}$ for the V-complex
(for adsorption on one of the tips) whereas $10^{1}-10^{2}$ for the S-complex
(for adsorption on one of the triangle faces). In the second project, we
studied plasmon-driven charge transfer across particle-molecule junctions
after laser excitation of the particle near the plasmon energy at 2.5 eV. We
found substantial electron transfer from particle to molecule for a Au120-C60
complex leading to the formation of a negative molecular ion, C${}_{60}^{-}$
on the particle. This observation is consistent with a previous experimental
result that shows evidence of electron transfer from particle to molecule and
the formation of negative molecular ions for a fullerene derivative.SERS-PCBM-
Schatz-2019 (33) Noteworthy in this result is that we did not see other
photoprocesses in these simulations of laser pulses with intensities that are
close to threshold, including C-C bond breakage, fragmentation of the
particle, or desorption of the fullerene from the particle. The ability of the
DFTB method to simulate SERS and charge transfer problems opens several new
possibilities in the plasmonics field.

## Acknowledgements

This research was supported by the Office of Basic Energy Science, Department
of Energy, through grant DE-SC0004752.

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