A charge-dipole interaction model for the frequency-dependent polarizability of silver clusters.

Abstract

We present a charge-dipole interaction model for the calculation of the frequency-dependent polarizability of silver clusters. The model relies on the representation of silver atoms by both a net electric charge and a dipole. Time variations of the atomic charges are related to the currents that flow through the bonds of the structures considered and the atomic charges and dipoles are eventually determined from the application of a least-action principle. After a generalization that enables the bonds of the bulk and surface atoms to have specific resistances, the model is parameterized on data obtained by the time-dependent density functional theory for tetrahedral Ag(20), Ag(84) and Ag(120) clusters. We then study the polarization properties of dimers of silver clusters. We compare in particular the polarizability of the dimers with that of the isolated clusters, for a range of gap distances and frequencies. We also consider the field enhancements one can achieve with these systems. The results are in good agreement with reference data and enable an extension of these data to a wider range of situations. They show that significant field enhancements are achieved at frequencies associated with resonant polarization along the axis of the dimer.

Full text

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IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 20 (2009) 195204 (10pp)
A charge–dipole interaction model for the
frequency-dependent polarizability of
silver clusters
doi:10.1088/0957-4484/20/19/195204
A Mayer1,4, A L Gonz´ alez2, C M Aikens3and G C Schatz2
1FUNDP—University of Namur, Laboratoire de Physique du Solide, Rue de Bruxelles 61,
B-5000 Namur, Belgium
2Northwestern University, Department of Chemistry, 2145 Sheridan Road, Evanston,
IL 60208-3113, USA
3Department of Chemistry, Kansas State University 111, Willard Hall Manhattan,
KS 66506-3701, USA
E-mail: alexandre.mayer@fundp.ac.be
Received 10 February 2009, in final form 17 March 2009
Published 20 April 2009
Online at stacks.iop.org/Nano/20/195204
Abstract
We present a charge–dipole interaction model for the calculation of the frequency-dependent
polarizability of silver clusters. The model relies on the representation of silver atoms by both a
net electric charge and a dipole. Time variations of the atomic charges are related to the currents
that flow through the bonds of the structures considered and the atomic charges and dipoles are
eventually determined from the application of a least-action principle. After a generalization
that enables the bonds of the bulk and surface atoms to have specific resistances, the model is
parameterized on data obtained by the time-dependent density functional theory for tetrahedral
Ag20, Ag84and Ag120clusters. We then study the polarization properties of dimers of silver
clusters. We compare in particular the polarizability of the dimers with that of the isolated
clusters, for a range of gap distances and frequencies. We also consider the field enhancements
one can achieve with these systems. The results are in good agreement with reference data and
enable an extension of these data to a wider range of situations. They show that significant field
enhancements are achieved at frequencies associated with resonant polarization along the axis
of the dimer.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The response of systems to external fields is an important issue
in science. Electronic polarization indeed plays a role in the
dynamics [1–3] or field-induced organization [4] of complex
systems. It contributes to the determination of preferential
adsorption sites [5, 6],macroscopic polarizations [7],
hyperpolarizabilities [8, 9], Raman intensities [10, 11], field-
emission properties [12–14], etc. In this context, molecular
modeling in terms of both quantum chemical and classical
electrostatics models plays a prominent role [15–17].
molecular mechanics or force field models, electrostatic
interactions are, for example, described using classical
In
4Author to whom any correspondence should be addressed.
models [1, 2].
still commonly used in molecular simulations, the trend
is, however, to use force fields that include polarization
explicitly [18–20].This enables the polarizability of the
structures considered, as well as properties that arise from this
polarizability, to be determined.
In recent publications, one of us proposed a model for the
calculation of molecular polarizabilities in which each atom
of the structure considered is represented by a net electric
charge and by a dipole [21–26]. This model is essentially an
extension of previous work, in which atoms are represented
only by dipoles [5, 27–32]. It is in the line of work presented
by others [4, 33–38], who also associate with each atom a
net electric charge. These models differ essentially by the
self-energy associated with these charges and dipoles and by
Although non-polarizable force fields are
0957-4484/09/195204+10$30.00
© 2009 IOP Publishing LtdPrinted in the UK
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Nanotechnology 20 (2009) 195204A Mayer et al
the way the interactions between different atomic sites are
normalized (these issues are bound to the model used for the
spatial distribution of the atomic charges).
an oscillating external field are usually described by giving
the ‘monopolar’ and ‘dipolar’ polarizabilities a Lorentzian
dependence onthe frequency [31, 39–41]. Our approach to this
problem was to relate the time variations of the atomic charges
to the currents that flow through the bonds of the structure
considered. Given appropriate expressions for the potential
and kinetic energy of these oscillating charges and dipoles,
we determine them by applying a least-action principle [26].
Numerical stability is achieved by normalizing the interactions
between different atomic sites in a way that is fully consistent
with the expression used for the self-energies [24].
The structures considered in our previous work were
fullerenes, carbon nanotubes [21–24], alkanes, alkenes and
aromatic molecules [25, 26]. Metallic clusters are systems
for which models that account for free charges should be
especially adapted [42].We hence focussed in this paper
on silver clusters.These structures have attracted a wide
interest, in particular because of their ability to magnify,
locally, the fields that are applied to them (formation of
‘hot spots’) and because of the enhancement they induce in
the Raman scattering of molecular structures placed in their
vicinity [43–49]. In order to simulate the polarizability of
silver clusters with our charge–dipole model, it turned out that
bulk and surface atoms have to be given specific parameters.
This is motivated by the often-discussed spill-out of the free
electron density that occurs at the surface atoms, leading to
a decoupling of the intra-band and inter-band contributions
to the optical response that becomes important in the small-
particle limit [50, 51]. The agreement with reference data was
improved substantially by giving the ‘bonds’ of these clusters
a resistance that depends on the atoms actually connected. In
order to describe structures like dimers in which the transfer
of charges is restricted to a part of the whole system (in this
case, the two clusters), it was finally necessary to implement
charge-specification conditions for these different parts of the
system.
This paper is organized along the following lines.
section 2, we present a charge–dipole interaction model for
the polarizability of silver clusters (we focus essentially on
the adaptations of this model).
the frequency-dependent polarizability of tetrahedral Ag20,
Ag84and Ag120clusters. Reference data for these structures
were achieved using the time-dependent density functional
theory [48], in a local version of the Amsterdam density
functional (ADF) program [52]. These reference data were
essentially used to parameterize the model.
we study polarization properties of dimers as well as the
field enhancements one can achieve with these systems. The
results are representative of the electromagnetic coupling that
characterizes these systems. They are in excellent agreement
with reference data and enable an extension of these data
to a wider range of situations. They show that significant
field enhancements are achieved at frequencies associated with
resonant polarization along the axis of the dimers. Conclusions
are finally presented in section 5.
The effects of
In
In section 3, we simulate
In section 4,
2. Methodology
We present in this section the methodology used to simulate
the polarizability of metallic clusters. We focus essentially on
the improvements of this methodology, whose details can be
found in [22, 24, 26]. These improvements consist essentially
in considering bonds with different resistances (in contrast to
bonds that were all equivalent in our previous work). We also
implement several charge-specification conditions in order to
deal with situations in which charge transfers are restricted to
parts of the whole system. This prevents charges from being
transferred between structures that are not in contact with each
other and we actually solve an issue that arises systematically
in this type of model.
We assume that a set of NSstructures (with a total of N
atoms) is subject to an external field, whose time dependence
is given by
Eext(t) = Re[Eextexp(−iωt)],
where Eextstands for the amplitude of this external field and ω
for its angular frequency (angular frequency ω = frequency ν
times 2π).We associate with each atom of the structures
considered a net electric charge qi(t) as well as a dipole pi(t).
These quantities are expressed as
(1)
qi(t) = q0
pi(t) = p0
irefer to the static part of qi(t) and pi(t). qi
and pi are complex quantities that provide the amplitude and
the phase of the oscillating part of qi(t) and pi(t). We express
the time dependence of Eext(t), qi(t) and pi(t) using a factor
exp(−iωt), incontrast toexp(iωt) in our previouswork. When
dissipation is considered, this indeed provides polarizabilities
with a positive imaginary part, which is the convention usually
adopted.
The potential energy of this distribution of charges and
dipoles is given, according to our previous work [26], by
i+ Re[qiexp(−iωt)],
i+ Re[piexp(−iωt)],
(2)
(3)
where q0
iand p0
Epot(t) =1
2
N
?
?
i=1
N
?
pi(t) ·Ti,j
j=1
qi(t)Ti,j
q−qqj(t)
−1
2
N
?
N
?
i=1
N
j=1
p−ppj(t) −
N
?
i=1
N
?
j=1
pi(t) ·Ti,j
p−qqj(t)
+
i=1
qi(t)[χi+ Vi(t)] −
N
?
i=1
pi(t) ·Eext(t).
(4)
In this expression, Ti,j
the charge–charge, dipole–dipole and charge–dipole interac-
tions. The on-site terms Ti,i
on the parameters Ri
(Ri
atomic charges and to the atomic polarizabilities). The ex-
pression of these coefficients can be found in our previous
work [25, 26]. χiis the absolute electronegativity of the atom
i. Vi(t) = −Eext(t) · ri is the potential associated with the
external field Eext(t) at the atomic position ri.
The main idea of our model consists in relating the time
variation of the atomic charges qi(t) to the currents Il(t) that
q−q, Ti,j
p−pand Ti,j
p−qexpress, respectively,
q−qand Ti,i
p−pdepend, respectively,
qand αi, which are specific to each atom
qand αi correspond, respectively, to the extension of the
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Nanotechnology 20 (2009) 195204A Mayer et al
flow through the bonds of the structures considered.
generalization of our previous work [26], the kinetic energy
associated with this flow of charges is expressed as
In a
Ekin,q(t) =
Nbonds
?
l=1
RlI2
l(t),
(5)
where the sum runs over the Nbonds bonds of the structures
considered. The coefficients Rlexpress the ‘resistance’ of each
bond(these Rl, however, haveunitsof?s). Theyaredeveloped
as
Rl= ckin,q
m
2e2
d2
Sl
l
,
(6)
where dl is the length of the bond l and ckin,q is a
proportionality coefficient to be determined. The coefficient
Sl is a new feature of this model.
‘1/n’ factor of our previous work, where n was associated
with the number of electrons in the bond that contribute
to the current [26]. If i and j refer to the two atoms
that are connected by the bond l, Sl is actually defined
by the overlap
Gaussian distributions ρi(r) =
and ρj(r) =
with these two atoms in order to express their contribution
to the resistance Rl(the extension parameters Ri
specific to each type of atom and riand rjrefer to the atomic
positions). This definition is consistent with our model for
the atomic charges, which are also represented by Gaussian
distributions. Performing the integration in order to get an
analytical expression for Sl, one obtains
It generalizes the
? ? ?ρi(r)ρj(r)dV between the normalized
1
π3/2(Rj
1
π3/2(Ric)3exp[−|r − ri|2/(Ri
c)3exp[−|r − rj|2/(Rj
c)2]
c)2] one can associate
cand Rj
c are
Sl=
1
π3/2(Ri,j
?
c−c)3exp[−|ri− rj|2/(Ri,j
c−c)2],
(7)
where Ri,j
express the contribution of each atom to the resistance Rl.
With this formulation, the bond one can associate with any
pair of atoms is characterized by a resistance that increases
exponentially with the separation between the two atoms.
This naturally cancels any current between atoms that are too
far away from each other. This expression of the overlap
parameter Sl turns out to be appropriate for the simulation
of silver clusters. Different expressions may need to be
considered for other structures.
In order to establish a relation between the atomic charges
qi(t) and the currents Il(t) that flow through the bonds of each
structure, we have to enforce two sets of conditions. The first
set of conditionsexpresses the fact that, for each atom, the time
variation of the atomic charge qi(t) must be equal to the net
current that arrives through the bonds of this atom. This first
set of conditions is written as
?
where sl
atom i and sl
atom i [26].
c−c=
(Ric)2+ (Rj
c)2. The parameters Ri
cand Rj
c
l∈bonds of atom i
sl
iIl(t) = ˙ qi(t),
(8)
i= +1 if the current Il(t) is defined as going to the
i= −1 if the current Il(t) is defined as leaving the
The second set of conditions deals with the fact that, for
any closed path in the structures considered, there should be no
potential difference when going along that path. We refer by
‘cycles’ to the closed paths obtained by following the bonds of
each structure. Given the fact these bonds have a resistance Rl,
this second set of conditions is actually expressed as
?
where tl
as the path taken along the cycle c and tl
Il(t) is opposite to that path. This expression generalizes that
given in our previous work [26].
The remaining part of the methodology is similar to that
presented previously [26]. The conditions (8) and (9) can be
written in a matricial form as MI = ˙ q, where I is a vector that
contains the Il(t) of equations (8) and (9), and the vector ˙ q
contains either the ˙ qi(t) or the 0 of the right-hand side of these
equations. Using a generalized inversion [53], one can express
I as I = A˙ q, where A = (MtM)−1Mtwith t standing for the
transpose [26]. This enables one to express the kinetic energy
of the free charges as
l∈cycle c
tl
cRlIl(t) = 0,
(9)
c= +1 if the current Il(t) is oriented in the same way
c= −1 if the current
Ekin,q(t) =
Nbonds
?
l=1
Rl
?N
?
i=1
Al,i˙ qi(t)
?2
,
(10)
which generalizes the expression of our previous work through
the presence of Rland the definition of Al,i.
The kinetic energy of the oscillating dipoles is given in a
similar form to that used for the free charges:
Ekin,p(t) =
m
2e2
N
?
i=1
ci
kin,p˙ p2
i(t),
(11)
where ci
determined [26].
If NSstructures are considered at the same time, charge-
specificationconditionsmustbeimplementedforeach ofthem.
According to our previous work [26], these NSconditions can
be written as
?
?
where Qk
1,..., NS). Thestructuresconsideredhereafter willbeneutral,
so that Qk
The Lagrangian is then given by L = (Ekin,q+ Ekin,p) −
Epot.Taking account of the constraints associated with
the charge-specification conditions [54], the action J to be
minimized is given by
? ?
NS
?
kin,pis a parameter specific to each type of atoms to be
i∈structure k
q0
i= Qk
tot,
(12)
i∈structure k
qi= 0,
(13)
totis the net charge carried by the structure k (k =
tot= 0.
J =
L[qi(t), ˙ qi(t),pi(t), ˙ pi(t)]
?N
−
k=1
λk
?
i=1
fi
kqi(t) − Qk
tot
??
dt,
(14)
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Nanotechnology 20 (2009) 195204A Mayer et al
Figure 1. Representation of Ag20(left), Ag84(center) and Ag120(right) tetrahedral clusters, which are the structures used for the
parameterization of our model.
where fi
fi
k
= 0 otherwise.
associated with the charge-specification conditions.
Using the fact ¨ qj(t)
¨ pj(t) = −ω2Re[pjexp(−iωt)], the formulation of the least-
action principle (δJ
= 0) finally leads to the matricial
equations [26]:
k= 1 if the atom i belongs to the structure k and
The λk are the Lagrange multipliers
=−ω2Re[qjexp(−iωt)] and
?Tq−q
−Tq−p
−Tp−p
0
F
0
0
−Tp−q
Ft
??q0
p0
λ
?
=
?−χ
Qtot
0
?
,
(15)
and
?Tq−q− ω2Kq−q
?−V
0
Intheseexpressions,Tq−qisan N×N matrixthatcontains
the Ti,j
Tq−p = Tt
the Ti,j
2AtRA, withR an Nbonds×Nbondsdiagonalmatrixthat contains
the Rl. Kp−pis a 3N × 3N diagonal matrix that contains the
m
e2ci
with the kinetic energy of, respectively, the free charges and
the oscillating dipoles. q0and p0are two vectors of length
N and 3N that contain the q0
respectively. q and p are similar vectors that contain the qiand
the components of the pi. λ and η are two vectors of length
NS that contain the Lagrange multipliers associated with the
charge-specification conditions (12) and (13) [26]. χ and V
are two vectors of length N that contain the χiand Viof each
atom. E is a vector of length 3N that contains the components
of the external field Eexton each atom. Qtotfinally is a vector
of length NSthat contains the Qk
The expressions (15) and (16) generalize those presented
in our previous work [26] essentially by the presence of F
(for taking account of structure-specific charge-specification
conditions) and by the definition of Kq−q(for describing the
kinetics of free charges in structures characterized by bond-
specific resistances Rl).Dissipation can finally be taken
into account by replacing ω2Kq−qby (ω2+ iγqω)Kq−qand
ω2Kp−pby a diagonal matrix with elements given by (ω2+
−Tq−p
F
0
0
−Tp−q
Ft
−Tp−p− ω2Kp−p
0
??q
p
η
?
=
E
?
.
(16)
q−q, Tp−qis a 3N × N matrix that contains the Ti,j
p−qand Tp−pis a 3N × 3N matrix that contains
p−p. F is an N × NSmatrix that contains the fi
p−q,
k. Kq−q=
kin,p. Kq−qand Kp−pactually gather the terms associated
iand the components of the p0
i,
tot.
iγi
parameters to be determined (γi
atom).
pω)m
e2ci
kin,p.
γq = 1/τq and γi
p
= 1/τi
pare damping
pis specific to each type of
3. Parameterization for the simulation of silver
clusters
The methodological developments presented in section 2 were
achieved for the purpose of calculating the polarization of
silver clusters. Bulk silver has a face-centered cubic cell, with
a cell parameter a of 0.409 nm [55]. The ‘bonds’ we consider
in order to apply our charge–dipole model are the segments
that join first-neighbor atoms. Each atom in the bulk hence has
twelve first-neighbor atoms to which it is connected through
a ‘bond’ whose length is a/√2. The ‘cycles’ one needs to
consider in order to relate the ˙ qito the currents Ilare made of
three bonds only (taking larger cycles would only introduce
redundancies in the equations one has to solve in order to
establish this relation). For finite structures, the bonds hence
correspond to pairs of atoms that are separated by a distance
of a/√2 (within 10%). The cycles are detected by following
systematically the bonds of every atom encountered, starting
from a given atom and considering a maximum of three steps.
The structures considered for the parameterization of our
model are the tetrahedral Ag20, Ag84 and Ag120 clusters.
These structures are depicted in figure 1 (the figures include
the bonds considered for the application of our model).
The reference data we considered for this parameterization
consist of the real and imaginary parts of the polarizability
of these clusters.These data were obtained for angular
frequencies ω ranging between 0 and 6.8 × 1015Hz (energy
¯ hω of 4.5 eV) by using the time-dependent density functional
theory (TDDFT) [48]. These density functional theory
calculations were actually performed with a local version of
the Amsterdam density functional (ADF) program [52]. The
molecular structures of the species studied were calculated
using the gradient-corrected Becke–Perdew (BP86) exchange–
correlation functional [56, 57].
type basis set was employed with a [1s2–4p6] frozen core
for Ag. The zeroth-order regular approximation (ZORA) was
utilized in the calculations to account for scalar relativistic
effects [58].The real and imaginary polarizabilities were
computed using the AORESPONSE module [59] and a short-
time approximation to evaluate the perturbed density matrix
(and from this the frequency-dependent polarizability) [60].
A double-zeta (DZ) Slater
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Nanotechnology 20 (2009) 195204A Mayer et al
Table 1. Parameters used to compute the polarizability of silver clusters. The first set of parameters was established for the Ag20, Ag84and
Ag120tetrahedral clusters, while the second set of parameters is specific to Ag20.
Rq(˚ A)
αiso/(4π?0) (˚ A
3)
Rc(˚ A)
ckin,p
γp(1015s−1)
Ag20, Ag84, Ag120
Ag (bulk)
Ag (surface)
ckin,q= 0.9831 nm−3
γq= 0.19805 × 1015s−1
Ag20
Ag
ckin,q= 2.5564 nm−3
γq= 0.3083 × 1015s−1
3.3626
1.1853
1.8789
1.0821
7.9080
0.9532
2.6789
3.5891
1.0293
1.1326
1.36750.10221.3143 51.20030.1066
The adiabatic local density approximation (ALDA) was used
in all polarizability calculations. A damping parameter ? of
0.0037 au (which is determined by the dephasing time of the
conduction electrons) was previously found to be reasonable
for modeling the absorption spectra of silver clusters [60, 61].
Inordertoreproduce thesereference data, itwasnecessary
to distinguish between bulk and surface atoms.
atoms are defined in this context as silver atoms that are not
surrounded by twelve first-neighbor atoms. Considering hence
the existence of two types of atoms and using isotropic atomic
polarizabilities, the parameterization consists in searching for
the parameters Ri
minimize the deviations between the cluster polarizabilities
obtained using TDDFT and those obtained using our model.
This adjustment was achieved using an implementation of the
simulated annealing method [62]5as well as a conjugated-
gradient method [63]. The parameters Ri
γi
ckin,qand γqare general parameters. Their values are given
in table 1. We also provide parameters that are specific to
the Ag20cluster and provide polarizabilities that are in better
agreement with reference data (they are not meant to be used
for other structures).
The frequency-dependent polarizability of the tetrahedral
Ag20, Ag84and Ag120clusters is represented in figure 2. The
representation includes the real and imaginary parts of the
cluster polarizabilities, as obtained using TDDFT and our
charge–dipole model6. The agreement between the results
achieved by these two techniques turns out to be excellent.
The relative error on the polarizabilities is indeed 23.8% for
Ag20(6.8% when using parameters that are specific to Ag20),
6.0% for Ag84 and 4.0% for Ag120.
are obtained, for a given structure and a given set of angular
frequencies ω for which the polarizabilities are calculated,
by ? = (?
polarizabilities obtained using TDDFT or our charge–dipole
model.The peak intensity in the imaginary part of the
polarizability is associated with the ‘resonance’ of the system.
Figure 2 shows that the resonance appears at an angular
Surface
q, αi
iso, Ri
c, ci
kin,p, γi
p, ckin,q and γq that
q, αi
iso, Ri
c, ci
kin,pand
pare specific to the two types of atoms considered, while
These relative errors
{ω}|αTDDFT(ω) − αQ−P(ω)|)/(?
{ω}|αTDDFT(ω)|),
where αTDDFT and αQ−P refer, respectively, to the cluster
5The algorithm used to get the set of parameters of the Ag20cluster was an
adaptation of the original code written by Goffe and collaborators.
6The software used for these simulations is available at http://perso.fundp.ac.
be/∼amayer/DCDA.
frequency ω of 5.451 × 1015Hz (λ = 346 nm), 4.720 ×
1015Hz (λ = 399 nm) and 4.631 × 1015Hz (λ = 407 nm)
for Ag20, Ag84 and Ag120 tetrahedral clusters, respectively.
The resonance frequency is redshifted when the size of the
cluster increases, a behavior well known in nanosystems.
The agreement between the results achieved by these two
techniques demonstrates the pertinence of our approach.
The charge–dipole interaction model presented in our
previous work [26] fails at reproducing appropriately the
reference data beyond the first resonance and at describing the
shifts of this resonance. Using this previous version of our
model with an optimized parameter cq
energy of the free charges, we obtain a relative error on the
reference data of 76.4% for Ag20, 22.1% for Ag84and 18.8%
for Ag120. The results obtained with this previous version of
our model are represented in figure 2. The model presented
in this paper, which accounts for bond-specific resistances, is
therefore better suited for the description of silver clusters in
which bulk and surface atoms must be differentiated.
kinof 1.319 for the kinetic
4. Application: polarization properties of dimers of
silver clusters
As an application of this model, we consider the polarization
properties of dimers of silver clusters. These dimers consist
of pairs of tetrahedral Ag20, Ag84 or Ag120 clusters, which
are placed ‘tip-to-tip’ along the z axis with a gap distance d
(see figure 3 for a representation of the Ag120–Ag120dimer).
In order to enable a comparison of our results with previous
work, we will first consider a gap distance d of 7.89 ˚ A. This
distance is indeed that considered by Zhao et al in their time-
dependent density functional theory (TDDFT) simulations of
Ag20–pyrazine–Ag20 systems [47].
results for a gap distance d ranging between 3 and 30˚ A.
A quantity of interest, in order to quantify the coupling
between the two clusters, is the ratio between the polarizability
of the dimer (αAg−Ag) and the polarizability of an isolated
cluster (αAg). This ratio αAg−Ag/αAgtakes the value of 2 when
there is no coupling and this is indeed the value obtained when
d → ∞. Another quantity of interest, in order to quantify
the electromagnetic enhancement achieved by the dimer, is
the ratio between the electric field in the middle between the
two clusters (Emiddle) and the electric field that is applied to
the dimer (Eext). These quantities will be investigated for
We will then establish
5