A discrete interaction model/quantum mechanical method to describe theinteraction of metal nanoparticles and molecular absorptionSeth Michael Morton and Lasse Jensen Citation: J. Chem. Phys. 135, 134103 (2011); doi: 10.1063/1.3643381 View online: http://dx.doi.org/10.1063/1.3643381 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i13 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 135, 134103 (2011)

A discrete interaction model/quantum mechanical method to describethe interaction of metal nanoparticles and molecular absorption

Seth Michael Morton and Lasse Jensena)

Department of Chemistry, The Pennsylvania State University, 104 Chemistry Building, University Park,Pennsylvania 16802-4615, USA

(Received 30 June 2011; accepted 7 September 2011; published online 3 October 2011)

A frequency-dependent quantum mechanics/molecular mechanics method for the calculation of re-sponse properties of molecules adsorbed on metal nanoparticles is presented. This discrete interac-tion model/quantum mechanics (DIM/QM) method represents the nanoparticle atomistically, thusaccounting for the local environment of the nanoparticle surface on the optical properties of the ad-sorbed molecule. Using the DIM/QM method, we investigate the coupling between the absorptionof a silver nanoparticle and of a substituted naphthoquinone. This system is chosen since it showsstrong coupling due to a molecular absorption peak that overlaps with the plasmon excitation inthe metal nanoparticle. We show that there is a strong dependence not only on the distance of themolecule from the metal nanoparticle but also on its orientation relative to the nanoparticle. We findthat when the transition dipole moment of an excitation is oriented towards the nanoparticle there isa significant increase in the molecular absorption as a result of coupling to the metal nanoparticle.In contrast, we find that the molecular absorption is decreased when the transition dipole momentis oriented parallel to the metal nanoparticle. The coupling between the molecule and the metalnanoparticle is found to be surprisingly long range and important on a length scale comparable tothe size of the metal nanoparticle. A simple analytical model that describes the molecule and themetal nanoparticle as two interacting point objects is found to be in excellent agreement with the fullDIM/QM calculations over the entire range studied. The results presented here are important for un-derstanding plasmon–exciton hybridization, plasmon enhanced photochemistry, and single-moleculesurface-enhanced Raman scattering. © 2011 American Institute of Physics. [doi:10.1063/1.3643381]

I. INTRODUCTION

Controlling the optical behavior of molecules near thevicinity of noble metal nanoparticles continues to be an ac-tive research area in nanoscience. A molecular level under-standing of the optical properties of such metal–moleculecomplexes is important for many applications such as en-ergy harvesting,1, 2 nanoscale optical circuits,3–5 and ultra-sensitive chemical and biological sensors.6–12 It is by nowwell known that the metal nanoparticles can strongly affectthe response properties of molecules,13 which is explored intechniques such as plasmon resonance sensing,11, 14 surface-enhanced Raman scattering (SERS),15 and metal-enhancedfluorescence.16, 17 Of particular interest is the recent discov-ery of mixed exciton–plasmon states in metal–molecule com-plexes. Mixed exciton–plasmon states arise when excitons ofmolecules (such as J-aggregates) resonate with the plasmonmodes of metal nanoparticles.18–25 As a result, new peaks (so-called hybridized states) appear in the extinction spectra; theyare governed by the strength of the coupling between excitonsand plasmons. Such control over hybridized states paves away to enhance the application of plasmonics in realizing tun-able nanophotonic devices,26 molecular sensing,20 and plas-monic resonance energy transfer methods.27 However, a de-

a)Electronic mail: jensen@chem.psu.edu.

tailed understanding of how plasmons interact with molecularexcited states is currently not available.

The optical properties of large nanoparticles have beenshown to be well described using classical electrodynamicsmethods.28–32 These methods solve Maxwell’s equations us-ing the frequency-dependent dielectric functions of the bulkmetal and treat the nanoparticle and molecule(s) as a con-tinuous system, thus neglecting their internal structure. Thislack of microscopic structural detail makes it particularlyproblematic to incorporate effects of adsorbed molecules andmolecule–plasmon coupling. To retain the atomic structureof the nanoparticle–molecule system, first-principles calcula-tions are ideal and can readily be performed for nanoparti-cles smaller than 2 nm. For example, time-dependent densityfunction theory (TDDFT) has been used recently to estab-lish a microscopic picture of SERS for molecules interactingwith small metal clusters.33–35 However, first-principles cal-culations cannot be carried out for large nanoparticles due tocomputational constraints and thus it is advantageous to de-velop efficient hybrid approaches.

A variety of different hybrid approaches to describe theoptical properties of nanoparticle–molecule systems havebeen presented. Corni and Tomasi36 combined a boundaryelement method for the metal particles based on the po-larizability continuum method with a quantum mechanicaldescription of the molecules. This method has been usedto study excitation energies,37 Raman enhancements,38 and

0021-9606/2011/135(13)/134103/12/$30.00 © 2011 American Institute of Physics135, 134103-1

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134103-2 S. Morton and L. Jensen J. Chem. Phys. 135, 134103 (2011)

fluorescence enhancements39 of molecules on nanoparticleaggregates. Jörgensen et al.40 used heterogeneous solvationresponse theory to investigate linear and nonlinear electricproperties of metal–molecule systems where the quantum me-chanical part of the system is described by a multiconfigura-tional self-consistent field method. Lopata and Neuhauser41, 42

combined a finite-difference time- domain (FDTD) electro-dynamics description of the metal particle with a two-statequantum description of the molecule. Arcisaukaite et al.43

used a polarizable quantum mechanical/molecular mechani-cal (QM/MM) method to study the charge-transfer excitationenergies of pyridine interacting with small silver clusters.Masiello and Schatz44, 45 presented a formal many-bodytheory combining electronic-structure theory with acontinuum-electrodynamics description of the metal andan implementation of the theory within TDDFT to study pyri-dine interacting with metal nanospheroids. Chen et al. alsoreported a combination of real-time time-dependent densityfunction theory (RT-TDDFT) and a classical electrodynamicsFDTD approach.46 They used this method to study enhancedabsorption spectrum of the N3 dye molecule and the Ramanspectrum of pyridine interacting with a 20 nm diameter silversphere.

Our group has recently developed an atomistic elec-trodynamics model47, 48 and combined it with TDDFT.49

This method, which we denote the discrete interactionmodel/quantum mechanics (DIM/QM) method, represents thenanoparticle atomistically which enables the modeling of theinfluence of the local environment of a nanoparticle surface onthe optical properties of a molecule. This DIM/QM methodcan be seen as an extension of the discrete reaction fieldmethod,50–52 which is a polarizable QM/MM method for de-scribing optical properties of molecules in solution. In theDIM/QM method, the nanoparticle is considered as a col-lection of N interacting atoms that when combined describethe total response. The specific interaction model used hereis the capacitance-polarizability interaction model, whereineach atom is characterized by an atomic polarizability andan atomic capacitance, and these intrinsic atomic proper-ties are optimized by parameterization against reference dataobtained from TDDFT calculations. In our previous work,we have shown that this model is computationally efficientand reported static polarizabilities for silver and gold nan-otubes, nanodisks, and nanospheres with diameters as largeas 4.5 nm.47 In contrast to most previous work in whichthe macroscopic dielectric constants are used to describe themetal response, here the metal nanoparticle is described byintrinsic atomic properties which enable us to retain the de-tailed atomistic structure of the nanoparticle. We have usedthe combined DIM/QM method to study excitation energiesof rhodamine-6G and crystal violet interacting with quasispherical silver and gold nanoparticles with diameters as largeas 7.5 nm (∼10 000 atoms), respectively.

In this work, we extend the DIM/QM method to ac-count for the frequency-dependent response of the metalnanoparticle48 as well as the near field interactions with themolecule; in this way, we may explore the interaction ofthe optical properties of the metal nanoparticle with that ofthe molecule. We will use the combined DIM/QM method to

study the interaction of a silver nanoparticle with a substitutednaphthoquinone that have overlapping absorption so that wemay understand the coupling between a nanoparticle responseand a molecular excitation.

II. THEORY

A. The frequency-dependent DIM/QM system

We seek to solve the time-dependent Kohn-Sham (TD-KS) equations53–57 of a molecule in proximity to a metalnanoparticle. For such a hybrid system, we can write the ef-fective TD-KS equations as

i∂

∂tφi(r, t) = hKS[ρ(r, t)]φi(r, t), (1)

where the time-dependent density is given by

ρ(r, t) =occ∑i=1

ni |φi(r, t)|2, (2)

and ni is the occupation number of the ith time-dependentorbital φi . The effective time-dependent Kohn-Sham operator,hKS[ρ(r, t)], is given by

hKS[ρ(r, t)] = −1

2∇2 −

∑I

ZI

|r − RI | +∫

ρ(r, t)|r − r ′ |dr ′

+ δEXC

δρ(r, t)+ V pert(r, t) + V DIM(r, t) (3)

with the individual terms being the kinetic energy, the nu-clear potential, the Coulomb potential, the XC-potential(exchange correlation), the external perturbation operator(V pert(r, t)), and the embedding operator V DIM(r, t) describ-ing the frequency-dependent molecule–metal interactions andwhich is complex due to frequency-dependent DIM response.Because we are interested in properties pertaining to thefrequency-domain, for convenience we will write the remain-ing equations in the frequency-domain, which can be obtainedby taking the Fourier transform of Eq. (3). The complex em-bedding operator is given by

V DIM(r, ω) =∑

j

V el(rj , ω) +∑

j

V pol(rj , ω), (4)

where the electrostatic operator (V el(rj , ω)) is given as

V el(rj , ω) =∑m

q indm (ω)

rjm

=∑m

q indm (ω)T (0)

jm , (5)

and the polarization operator (V pol(rj , ω)) is given as

V pol(rj , ω) = −∑m

μindm,α(ω)rjm,α

r3jm

=∑m

μindm,α(ω)T (1)

jm,α.

(6)The perturbation to the density due to the V DIM(rj , ω) oper-ator can be thought of as the image field, i.e., the field aris-ing from the dipoles (and charges) that are induced in thenanoparticle (DIM system) by the presence of the molecule(QM system). The external perturbation operator is given by

V pert(r, ω) =∑

j

V ext(rj , ω) +∑

j

V loc(rj , ω), (7)

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134103-3 Frequency-dependent DIM/QM J. Chem. Phys. 135, 134103 (2011)

where V ext(rj , ω) represents the applied external potential,and the local field operator (V loc(rj , ω)) is given as

V loc(rj , ω) =∑m

qextm (ω)T (0)

jm +∑m

μextm,α(ω)T (1)

jm,α. (8)

The local field operator V loc(rj , ω) describes the influence ofthe response of the nanoparticle due to the external appliedfield on the QM system, and thus accounts for the enhancednear field due to the metal nanoparticle.

In the above equations, the subscripts α, β denote Carte-sian coordinates, indicies i, j denote QM electrons, m, n de-note DIM atoms, and I, J will denote QM nuclei. For Greekindicies the Einstein summation convention is employed. Theterms q ind

m (ω) and μindm,α(ω) are the frequency-dependent com-

plex charges and dipoles of the DIM subsystem as induced bythe QM system, respectively, whereas qext

m (ω) and μextm,α(ω) are

the frequency-dependent complex charges and dipoles of theDIM subsystem as induced by the external field. The termsT

(0)jm and T

(1)jm,α are the zeroth- and first-order real interaction

tensors and are described in more detail in Ref. 49.The induced atomic dipoles μind

m,α(ω) and induced atomiccharges q ind

m (ω) are obtained by solving a set of 4N + 1 com-plex linear equations as47, 48, 58–60⎛

⎜⎝A(ω) −M 0

−MT −C(ω) 1

0 1 0

⎞⎟⎠

⎛⎜⎝

μind(ω)

q ind(ω)

λ

⎞⎟⎠ =

⎛⎜⎝

ESCF(ω)

V SCF(ω)

qcluster

⎞⎟⎠ ,

(9)which is easily solved through standard linear algebra tech-niques. The three matrices A(ω), M, and C(ω) are composedof the capacitance, polarizability, and interaction tensor terms,and are given by

Amn,αβ (ω) ={

α−1m,αβ(ω) m = n,

−T(2)mn,αβ m �= n,

(10)

Mmn,α ={

0 m = n,

T (1)mn,α m �= n,

(11)

and

Cmn(ω) ={

c−1m (ω) m = n,

T (0)mn m �= n,

(12)

where T(2)mn,αβ is the second-order interaction real tensor

as described in Ref. 49. Each atom m is characterizedby a frequency-dependent complex capacitance (cm(ω)) anda frequency-dependent complex polarizability (αm,αβ(ω))which is obtained by fitting to reference data obtainedfrom first-principles calculations.48 The vectors ESCF(ω) andV SCF(ω) contain the frequency-dependent complex electricfield and electric potential, respectively, generated after eachQM self-consistent field (SCF) cycle. Thus, the coupling be-tween the metal nanoparticle and the QM system is fully self-consistent. The vectors μind(ω) and q ind(ω) are the frequency-dependent complex induced dipoles and charges, respectively,of the DIM system. The term λ is the Lagrange multiplierwhich ensures that the sum of the induced charges is equal tothe overall charge (qcluster) of the nanoparticle. Equation (9)

can be seen as the complex extension to what was presentedin Ref. 49.

The QM subsystem polarizes the DIM subsystemthrough the electric field (ESCF

m,α ) and potential (V SCFm ) given

by

ESCFm,α (ω) = EQM,el

m,α (ω) + EQM,nucm,α , (13)

V SCFm (ω) = V QM,el

m (ω) + V QM,nucm , (14)

where EQM,elm,α (ω) and V QM,el

m (ω) arise from the QM electrons,

EQM,elm,α (ω) =

∫ρ(rj , ω)

rjm,α

r3jm

drj = −∫

ρ(rj , ω)T (1)jm,αdrj ,

(15)

V QM,elm (ω) =

∫ρ(rj , ω)

rjm

drj =∫

ρ(rj , ω)T (0)jmdrj , (16)

and EQM,nucm,α and V QM,nuc

m from the QM nuclei,

EQM,nucm,α =

∑J

ZJ rJm,α

r3Jm

= −∑

J

ZJ T(1)Jm,α, (17)

V QM,nucm =

∑J

ZJ

rJm

=∑

J

ZJ T(0)Jm. (18)

For each SCF cycle, the new ESCF(ω) and V SCF(ω) are usedto solve Eq. (9) for μind(ω) and q ind(ω). The new μind(ω) andq ind(ω) are then in turn used to create a new density throughthe V DIM(r, t) operator in Eq. (3). Thus, the induced chargesand dipoles are obtained self-consistently by solving the DIMlinear equations at each SCF iteration.

The influence of the external perturbation on the DIMsystem is given by the external field Eext(ω) and external po-tential V ext(ω). By replacing ESCF(ω) and V SCF(ω) in Eq.(9) with Eext(ω) and V ext(ω), one then solves for μext(ω)and qext(ω), the frequency-dependent complex dipoles andcharges created by the external perturbation. Because theexternal perturbation is independent of the QM subsystemμext(ω) and qext(ω) only need to be solved once per ω.

B. Linear response of the DIM/QM system

Since we are interested in polarizabilities, we will use lin-ear response theory to obtain the first-order change in the den-sity to a time-dependent perturbation. We will use the typicalconvention and identify indicies a and b with virtual orbitals,i and j with occupied orbitals, and s and t with general or-bitals. The first-order change in the density is given in termsof the first-order density matrix (P

′st (ω)) by

ρ ′(r, ω) =∑i,a

P′ia(ω)φi(r)φ∗

a (r) + P′ai(ω)φa(r)φ∗

i (r).

(19)If Eq. (1) is expanded to first-order and, then solved forP

′st (ω), we get

P′st (ω) = nst

ω − ωst + i�V

′effst (ω), (20)

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134103-4 S. Morton and L. Jensen J. Chem. Phys. 135, 134103 (2011)

where nst is the difference in occupation number and � isa phenomenological energy broadening term that is due todamping of the excited state, i.e., it is related to the effec-tive lifetime of the QM excited state. The change in effectivepotential V

′effst (ω) is given by

V′effst (ω) = V

′pertst (r, ω) + V

′SCFst (r, ω), (21)

where the SCF potential, V′SCFst (r, ω), is

V′SCFst (r, ω) = V

′Coulst (r, ω) + V

′XCst (r, ω) + V

′DIMst (r, ω)

(22)and is composed of the Coulomb, XC, and DIM potentials, re-spectively, and V

′extst is the external potential. The SCF poten-

tial can also be written in terms of the coupling matrix Kst,uv

as

V′SCFst (r, ω) =

∑u,v

Kst,uvP′uv(ω), (23)

where the coupling matrix is defined as

Kst,uv = KCoulst,uv + KXC

st,uv + KDIMst,uv

= ∂V′Coulst

∂P′uv

+ ∂V′XCst

∂P′uv

+ ∂V′DIMst

∂P′uv

(24)

and thus describes the linear response of the self-consistentfield due to changes in the density. Inserting Eq. (23) intoEq. (21) yields

V′effst (ω) = V

′pertst (ω) +

∑u,v

Kst,uvP′uv(ω), (25)

which allows us to rewrite the first-order density matrix inEq. (20) as

P′st (ω) = nst

ω − ωst + i�

[V

′pertst (ω) +

∑u,v

Kst,uvP′uv(ω)

].

(26)

C. The dipole matrix

The dipole matrix of the QM system, Hαst (ω) is calculated

as

Hαst (ω) = 〈s|μα + V loc

α (ω)|t〉, (27)

where μα is the QM dipole operator in the α direction andV loc

α (ω) is the complex local field operator in the α direction.The polarizability of the system, ααβ , is found from the traceof the products of the dipole and density matrices as

ααβ(ω) = −Tr[Hα(ω)Pβ(ω)]

= −Tr[(HαR(ω) + i HαI(ω))(PβR(ω) + i PβI(ω))],

(28)

where the superscripts R and I indicate real and imaginarycomponents, respectively. Therefore, one finds that the realand imaginary polarizabilities are given by

αRαβ(ω) = −Tr[HαR(ω)PβR(ω) − HαI(ω)PβI(ω)] (29)

and

αIαβ(ω) = −Tr[HαR(ω)PβI(ω) + HαI(ω)PβR(ω)]. (30)

The real component of the polarizability is related to therefractive index of the system, and the imaginary componentis directly related to the absorption cross section.61 Theseare typically quantified using the isotropic polarizability (thesymmetric polarizability invariant), which is given by

α(ω) = 1

3Tr[α(ω)]. (31)

D. Iterative solution of the response equations

Unfortunately, for large systems the coupling matrixfrom Eq. (24) becomes very large and therefore it is not fea-sible to construct it explicitly. It is more efficient to solvethe linear equations iteratively as has been demonstrated inRef. 61. Using this method, the first-order density may be ex-pressed as(

P′R

iP′I

)=

(P

0R

iP0I

)+

(WR(P

′)

iWI (P′)

), (32)

where P0

is a constant arising from the external perturbation,and W is the self-consistent part. In Ref. 61, the external per-turbation V

′pertst (ω) is real because it only includes the real

V′extst term, and thus the terms P

0R and iP0I each contain a

single term. However, because V locst (ω) is complex, the addi-

tion of local fields results in V′pertst (ω) being complex. Thus,

P0

st (ω) = nst

ω − ωst + i�V

′pertst (ω)

= nst

ω − ωst + i�

(V

′pertRst (ω) + iV

′pertIst (ω)

). (33)

This can be separated into a real component

P0Rst (ω) = nst

(ω−ωst )2+�2

[(ω−ωst )V

′pertRst (ω)+�V

′pertIst (ω)

](34)

and an imaginary component

P0Ist (ω) = nst

(ω − ωst )2+�2

[(ω−ωst )V

′pertIst (ω)−�V

′pertRst (ω)

].

(35)

III. COMPUTATIONAL DETAILS

We have implemented the DIM/QM method into a localversion of the ADF program package.62, 63 The statistical av-eraging of (model) orbital potentials (SAOP) functional64 hasbeen employed. A triple-ζ polarized slater type all-electronbasis set from the ADF basis set library is used. Frequency-dependent complex polarizabilities were calculated using theAORESPONSE module as described in Ref. 61. The finitelifetime of the electronic excited states is included phe-nomenologically using a damping parameter of 0.0037 a.u.(∼800 cm−1), which was previously found to be acceptable.61

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134103-5 Frequency-dependent DIM/QM J. Chem. Phys. 135, 134103 (2011)

FIG. 1. (a) The geometry of the quasi-spherical Ag2057 nanoparticle. (b) The frequency-dependent imaginary polarizability (αI (ω)) of the Ag2057 nanoparticle(2 nm radius) as a function of frequency.

We also use the RESPONSE (Refs. 65 and 66) module to cal-culate the 10 lowest excitations of the naphthoquinones as acomparison to the AORESPONSE results.

The frequency-dependent polarizability (αi,αβ(ω)) andcapacitance (ci(ω)) of the Ag atoms are taken from Ref. 48and have the form

αi,αβ(ω)=

⎧⎪⎨⎪⎩

α0i,αβ ω = 0,

α0i,αβ

(ω2

i,1

ω2i,1−ω2−i�i,1ω

+ ωi,2(N)2

ωi,2(N)2−ω2−i�i,2ω)

)ω > 0,

(36)

ci(ω) =⎧⎨⎩

c0i ω = 0,

c0i

(1 + ω2

i,1

ω2i,1−ω2−i�i,1ω

)ω > 0,

(37)

where c0i = 2.7529 a.u. and α0

i,αβ = 49.9843 a.u. and are theatomic capacitance and polarizability in the static limit, ωi,1

= 0.0747 a.u. and ωi,2 = 0.0545 a.u. are the plasma frequen-cies of the Lorentzians, γi,1 = 0.0604 a.u. and γi,2 = 0.0261a.u. are the lorentzian widths, and

ωi,2(N ) = ωi,2

(1 + A

N13

), (38)

where A = 2.7759 is a size-dependent parameter and N is thenumber of atoms in the system. Please note that the value forthe capacitance reported in Ref. 48 incorrectly included anextra factor of

√π/2 and, therefore, needed to be scaled and

omitted the “1+” term in Eq. (37).In this work, we do not include any metal atoms in the

QM part. As a consequence, certain chemical interactions be-tween the metal and the molecule are necessarily neglected,such as charge-transfer and covalent bonding. As mentionedpreviously,49 these effects can be introduced by extendingthe QM system to include a few metal atoms close to themolecule; however, in this work we focus on long-range in-teractions where the contributions from orbital overlap effectscan be neglected.

IV. ABSORPTION OF A MOLECULE COUPLED TO AMETAL NANOPARTICLE

In order to understand how the absorption properties of adye molecule is affected by coupling to a metal nanoparticle,we choose to examine a substituted naphthoquinone interact-ing with a Ag2057 (2 nm radius) cuboctahedral nanoparticle.See Figure 1(a) for a representation of the geometry of thisnanoparticle. The Ag2057 Ag–Ag bond length used is the ex-perimental value of 2.889 Å. In this work, the focus is onunderstanding the importance of the near field coupling aswell as the relative orientation and distance between the metalnanoparticle and the molecule. In the following we will, forsimplicity, equate the imaginary part of the isotropic polariz-ability (αI (ω)) with the absorption spectrum, even though thisis not strictly true since the absorption cross section is relatedto the imaginary polarizability as σ (ω) = (4πω/c)Im[α(ω)].In Figure 1(b), we plot αI (ω) for the nanoparticle as a functionof frequency. We see that the nanoparticle is characterized bya strong broad absorption around 3.7 eV due to the plasmonexcitation. The reason that the plasmon excitation is so broadis due to the small size of the nanoparticle.48

Naphthoquinones are chosen since they possess a moder-ately strong absorption that can easily be tuned via functionalgroup substitution, so that the absorption matches the plas-mon response of the nanoparticle.67 We find that the 2-OH-6-NH2-1,4-naphthoquinone (NQ) molecule has a strong ab-sorption close to the plasmon excitation of the nanoparticle. Aball-and-stick representation of the NQ molecule is presentedin Figure 2(a). The αI (ω) of the NQ molecule is plotted inFigure 2(b). The absorption spectrum of NQ is characterizedby a peak at 3.8 eV and a smaller peak at 4.2 eV, which wewill denote S1 and S2; the transition dipole moment of theseare illustrated by the blue and red arrows, respectively, inFigure 2(a). The S1 excitation arises due to a transition fromthe HOMO-4 to the LUMO of the molecule, while the S2 ex-citation is due largely to a HOMO to LUMO+1 transition.The orientation of the S1 excitation is in the plane of the NQring and parallel with the short axis, while the S2 excitation is

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134103-6 S. Morton and L. Jensen J. Chem. Phys. 135, 134103 (2011)

FIG. 2. (a) The structure of the substituted naphthaquinone molecule (NQ). The transition dipole moment of the S1 peak at 3.8 eV (blue), S2 peak at 4.2 eV(Red), and S3 peak at 4.4 eV (green, not plotted in (b)) are overlaid on top of the molecule. (b) The frequency-dependent imaginary polarizability (αI (ω)) ofNQ.

roughly perpendicular to the S1 excitation, being parallel withthe long axis. In addition to the S1 and S2 excitations, thereis another weak excitation (S3, green arrow in Figure 2(a)) at4.4 eV that is oriented on the same axis as the S1 excitation;we will see in Sects. IV A and IV B that this excition can be-come close to as strong as the S1 excitation, depending on themolecular orientation with respect to the metal nanoparticle.

To understand the relative orientation and distance de-pendence between the metal nanoparticle and the NQ, wechoose to study two different orientations of the moleculewith respect to the metal nanoparticle; this is depicted inFigure 3. In the FLAT orientation the plane of NQ is flushagainst the Ag2057 surface, whereas in the SIDE orientationthe plane of NQ is orthogonal to the Ag2057 surface. Thesetwo orientations have been chosen so that the transition dipolemoment of S1 is parallel and perpendicular to the surface ofthe nanoparticle, respectively. We consider NQ at the centerof the edge of the Ag2057 nanoparticle; this adsorption loca-tion was chosen as it was demonstrated to have a large ef-fect on the optical properties of an adsorbed molecule in a

previous study.49 While it is likely that the specific adsorp-tion location is important at small metal-molecule separa-tion distances (< 7 Å), we expect only small difference atlong range since the Ag2057 nanoparticle is quasi-spherical.Thus, in this work we limit our study to a single absorptionsite.

The interactions between the metal nanoparticle and themolecule are governed by two different kinds of mechanisms.The first kind is due to the interactions between the electronicdensity of the molecule and its image induced into the metalnanoparticle, i.e., image field effects. In the DIM/QM model,this interaction is taken into account self-consistently throughthe V DIM(ω) operator. The second kind of interaction is due tothe induced polarization in the metal nanoparticle due to theexternal electric field, i.e., the near field interactions. Withinthe DIM/QM model, this interaction is accounted for by theV loc(ω) operator. In the following we will systematically in-vestigate the effect of these two kinds of interactions as wellas a simple analytical model to gain additional insights intothe metal–molecule coupling.

FIG. 3. (a) The NQ − Ag2057 system in the FLAT orientation and (b) in the SIDE orientation. For both configurations, the distance from the Ag2057 surface toNQ center-of-mass is 10 Å. The green arrows indicate the direction in which the separation r will be varied.

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134103-7 Frequency-dependent DIM/QM J. Chem. Phys. 135, 134103 (2011)

FIG. 4. Image field effects on the frequency-dependent imaginary polarizability (αI (ω)) as a function of distance and frequency for the NQ–Ag2057 system. (a)FLAT orientation, (b) SIDE orientation, (c) αI (ω) for the FLAT orientation at r = 3.0 Å and the free NQ molecule, and (d) αI (ω) for the SIDE orientation atr = 4.5 Å and the free NQ molecule.

A. Image field effects on molecular absorption

To understand the image field effect on the molecular ab-sorption, we calculate αI (ω) of the NQ–Ag2057 system by in-cluding the V DIM(ω) operator. In Figure 4, we plot αI (ω) as afunction of distance and frequency for the two different con-figurations, as well as detailed plots of αI (ω) at r = 3.0 Å andr = 4.5 Å for the FLAT and SIDE orientations, respectively,compared against the free molecule. Included in αI (ω) calcu-lated using the DIM/QM method is the polarizability of themolecule under the influence of the image field and the polar-izability induced into the metal nanoparticle from the inter-actions with the molecule. The polarizability of the isolatedAg2057 nanoparticle has been excluded from the spectrum,since it is independent of metal–molecule distance and twoorders of magnitude greater than the molecular absorption.The closest distance investigated was 3 Å for the FLAT ori-entation and 4.5 Å for the SIDE orientation, which for bothorientations correspond to the molecule being at a bondingdistance from the metal surface.

For the FLAT orientation at the closest distance, we seea slight redshift in the S1 excitation as well as an increase inthe peak width due to coupling with the metal nanoparticle.The interaction with the nanoparticle drops off quickly withdistance as illustrated in Fig. 2(b). Already at around 6 Å theinteraction is negligible and at 15 Å the gas phase result isrecovered exactly. We see a similar trend for the SIDE orien-tation, although for the closest distance the interactions with

the nanoparticle causes the absorption spectrum of NQ to lookvery different from that of the gas phase. For this distance, theS1 excitation is slightly redshifted to 3.85 eV, but the S3 peakredshifts 0.75 eV to 3.65 eV, becoming similar in strength tothe S1 excitation. The S2 state is now at about 3.7 eV but haschanged little in strength. Additionally, the absorption spec-trum of the SIDE orientation is significantly broader than thegas phase spectrum. The main difference between the SIDEand FLAT orientations is that the transition dipole moment ofthe S1 excitation is perpendicular and parallel to the nanopar-ticle surface, respectively. This is also the case for the S3 ex-citation, whereas the S2 excitation is roughly perpendicularto the surface for both orientations. Thus, for the S1 and S3excitations, the SIDE orientation sees a larger effect than theFLAT orientation (even though it is further from the surface)because the transition dipole moment points in the directionof the nanoparticle and, therefore, is aligned with its imagedipole. This results in enhanced absorption due to construc-tive dipole–dipole interactions between the transition dipolemoment and its image induced in the nanoparticle. Likewise,because the S2 excitation is not aligned with its image it ex-periences no constructive interference and thus no enhancedabsorption.

B. Near field effects on molecular absorption

The near field effects on the molecular absorption areexamined by calculating αI (ω) for the NQ–Ag2057 system

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134103-8 S. Morton and L. Jensen J. Chem. Phys. 135, 134103 (2011)

FIG. 5. Near field effects on the frequency-dependent imaginary polarizability (αI (ω)) as a function of distance and frequency for the NQ–Ag2057 system.(a) FLAT orientation, (b) SIDE orientation, (c) αI (ω) for the FLAT orientation at r = 3.0, 10, 50 Å and the free NQ molecule, and (d) αI (ω) for the SIDEorientation at r = 4.5, 10, 50 Å and the free NQ molecule.

in the two different orientations with both the V DIM(ω) andV loc(ω) operator included in the simulation; this takes intoaccount the electric field from the metal nanoparticle due tothe polarization induced by the external electric perturbationas well as the image field. The resulting αI (ω) as a functionof frequency and distance is displayed in Fig. 5. Also shownare comparisons between the gas phase spectrum and NQ–Ag2057 spectra obtained at distances of 3.0, 10, 50 Å from thesurface for the FLAT orientation and 4.5, 10, 50 Å for theSIDE orientation.

From Fig. 5, we see the effects of the near field dependstrongly on the distance from the metal nanoparticle and thatthe gas phase result is not recovered until around 40–50 Åaway from the nanoparticle. This distance is comparable tothe diameter of the nanoparticle and thus illustrates that thenear field has a rather long range, although it is strongest nearthe surface of the metal nanoparticle. This is in strong contrastto the image field effects which are found to be negligiblebeyond 6 Å; this distance is on the order of the size of themolecule.

It is also clear from Fig. 5 that there is a significant de-pendence on the orientation of the molecule relative to thenanoparticle when the near field effects are taken into account.For the FLAT orientation at the closest distance αI (ω) is sig-nificantly broader and slightly reduced in strength comparedwith the gas phase result. Even more striking is the result ob-tained at 10 Å, where αI (ω) is only around half that of the

gas phase result. Thus, for the FLAT orientation the interac-tion with the nanoparticle is destructive, especially when thenear field is included in the simulations. The result of this isthat αI (ω) reaches a minimum at around 10 Å. However, atall distances the shape of the absorption spectrum resemblesthe shape of the gas phase spectrum although it is significantlybroader and the peaks are blueshifted at the shortest distances.

The near field effect on the SIDE orientation is funda-mentally different from that of the FLAT orientation. In thisorientation, the interaction with the near field is constructiveat all distances. For the closest distance αI (ω) is six timesstronger than in the gas phase; even at 10 Å αI (ω) is stillstronger than the gas phase results by a factor of three. Asis seen for the image field effect, the absorption spectrum atthis distance is significantly broader than the gas phase resultsand is characterized by two peaks: S1 at 3.85 eV and S3 at3.65. However, in the near field case the strongest peak is S1,whereas S1 and S3 are comparable in the image field case.

As discussed in Sec. IV A, the main difference betweenthe two orientations is the direction of the transition dipolemoment of S1 (the strongest excitation). For the FLAT orien-tation this leads to destructive interference with the inducedpolarization in the metal nanoparticle and thus a reduction inthe molecular absorption. In the SIDE orientation S1 pointstowards the metal nanoparticle and this leads to constructiveinterference with the metal nanoparticle induced polarizationand a significant increase in the molecular absorption.

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134103-9 Frequency-dependent DIM/QM J. Chem. Phys. 135, 134103 (2011)

C. Analytical model of absorption of a moleculecoupled to a metal nanoparticle

To understand further the distance and orientation de-pendence of the near field on the molecular absorption, weconsider a simple analytical model. In this model, we con-sider the total system to consist of two interacting polarizableobjects characterized by polarizabilities αM and αNP for themolecule and metal nanoparticle, respectively. This model hasalso previously been used to illustrate that the signal of SERSis enhanced by a factor proportional to the fourth-power ofthe electric field enhancement, |Eloc|4.35, 68, 69 For this system,the total polarizability is given by Silberstein’s equations70, 71

as

α⊥ = αM + αNP − 2αMαNP

r3

1 − αMαNP

r6

, (39)

α‖ = αM + αNP + 4αMαNP

r3

1 − 4αMαNP

r6

, (40)

where α⊥ and α‖ are the polarizability for the componentsperpendicular and parallel to the separation axis, respectively.Strictly speaking, Silberstein’s equations are only valid if αM

and αNP are isotropic. However, if the αM and αNP polariz-ability tensors are diagonally-dominant, as is the case here,then we can neglect the off-diagonal components and use Sil-berstein’s equations to calculate the total polarizability of themolecule–nanoparticle system for each diagonal componentindividually, i.e., αM,αα or αM,ββ . In the equations above, theαM and αNP polarizabilities are complex, so to compare withthe results in Fig. 5 we need to isolate the imaginary part ofthe total polarizability. To do this, we insert αR + iαI for eachα and rearrange the equations to give (here only the imaginarycomponents are shown)

αI⊥ = αI

M + αINP − 2

(αR

MαINP +αI

MαRNP

)r3 + αI

NP

(αR2

M +αI2M

)+αI

M

(αR2

NP +αI2NP

)r6

1 − 2(αR

MαRNP −αI

MαINP

)r6 +

(αR2

M +αI2M

)(αR2

NP +αI2NP

)r12

, (41)

αI‖ = αI

M + αINP + 4

(αR

MαINP +αI

MαRNP

)r3 + 4αI

NP

(αR2

M +αI2M

)+αI

M

(αR2

NP +αI2NP

)r6

1 − 8(αR

MαRNP −αI

MαINP

)r6 + 16

(αR2

M +αI2M

)(αR2

NP +αI2NP

)r12

, (42)

where again the total polarizability of the molecule–nanoparticle system can be obtained by solving for each com-ponent individually. For αI

⊥ (Eq. (41)) the polarizability com-ponents are αI

M,αα and αINP,αα and the axis of separation is β,

where α �= β; in other words, the polarizability componentsare perpendicular to the axis of separation. For αI

‖ (Eq. (42))the polarizability components are αI

M,αα and αINP,αα , and the

axis of separation is α, meaning the polarizability componentsare parallel to the axis of separation.

To test if this simple analytical model can describethe interactions between the molecule and nanoparticle wecalculated αI (ω) for the NQ–Ag2057 system. The calculations

were done using Eqs. (41) and (42) where the differenttensor components of αM and αNP were taken from the freemolecule and free nanoparticle. The αI (ω) for the NQ–Ag2057system obtained using the analytical model as a function ofdistance and frequency is shown in Fig. 6. In the figure, thepolarizability of the free nanoparticle has been subtracted tobe comparable with the result obtained from the DIM/QMcalculations. Comparing the results shown in Fig. 6 with thatobtained from the DIM/QM calculations shown in Fig. 5, wesee that there is excellent agreement. There is some discrep-ancy at short distances but this is to be expected since at thesedistances the actual shape of the molecule and nanoparticle as

FIG. 6. Near field effects on the frequency-dependent imaginary polarizability (αI (ω)) as a function of distance and frequency for the NQ–Ag2057 systemobtained using an analytical model for the (a) FLAT and (b) SIDE orientations.

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134103-10 S. Morton and L. Jensen J. Chem. Phys. 135, 134103 (2011)

well as quantum effects and the image field are important, butthese are not accounted for in the analytical model. However,the excellent agreement between the analytical model and thefull DIM/QM simulations enables us to understand in moredetail the distance and orientation dependence of the nearfield effect on the molecular absorption.

At large distances, the 1/r3 term in the numerator isgreater than the 1/r6 term. Additionally, at all distances ofphysical relevance, both terms in the denominator are ≈ 0.Thus, the polarizability can be approximated as

αI⊥ = αI

M + αINP − 2

(αR

MαINP + αI

MαRNP

)r3

, (43)

αI‖ = αI

M + αINP + 4

(αR

MαINP + αI

MαRNP

)r3

, (44)

which shows that the parallel component increases withdecreasing distance, whereas the perpendicular componentdecreases. The polarizability of the metal nanoparticle isisotropic which means that the differences between the FLATand SIDE orientation is due to the anisotropy of the molecu-lar polarizability which is determined by the direction of theS1 transition dipole moment. For the FLAT orientation,the largest component of the polarizability is perpendicularto the separation axis. Therefore, αI

⊥ decreases faster than αI‖

leading to a net reduction in the total polarizability. The oppo-site is true for the SIDE orientation where the largest compo-nent of the polarizability is now parallel with the separationaxis, meaning that αI

‖ will dominate; this will lead to an in-crease in the total polarizability.

As the distance is further reduced, the 1/r6 terms inthe numerator of Eqs. (41) and (42) become important. Theseterms are all purely constructive for both the parallel and theperpendicular components. This leads to a further increasein the polarizability for the SIDE orientation as well as aincrease for the FLAT orientation. Therefore, for the FLATorientation the balance between the importance of the 1/r6

terms and 1/r3 terms in the polarizability lead to the mini-mum found in the calculations around 10 Å.

V. DISCUSSION

The results presented above for the absorption of amolecule interacting with a metal nanoparticle show a strik-ing dependence on the orientation of the molecule relative tothe metal nanoparticle. This is a result of overlapping absorp-tion bands for the molecule and the metal nanoparticle. In thesituation where the transition dipole moment of an excitationis oriented towards the metal nanoparticle there is a large in-crease in the molecular absorption. In contrast, a reduction inthe molecular absorption was found in the case where the tran-sition dipole moment is oriented perpendicular to the metalnanoparticle. These results show that the optical properties ofmolecules can be altered significantly by not only placing it inthe vicinity of a metal nanoparticle but also properly orientingthe molecule relative to the metal nanoparticle. Understand-ing and controlling the molecular absorption in the vicinityof nanoparticles is essential for plasmon hybridization,18–25

plasmon enhanced photochemistry,72–75 and single-moleculeSERS.76–80 Our results show that not only is it important tounderstand the distance dependence but also that the orienta-tion plays a significant role in determining the optical prop-erties of molecular near metal nanoparticles. This is remi-niscent of the work by Chance and co-workers in the 1970s(Refs. 81–84), where it was found that the fluorescence of anoscillating dipole above a reflecting surface is dependent onrelative orientation. More recently, Vukovic et al. have founda similar strong orientation dependence on the fluorescenceenhancement of molecules close to metal nanoparticles.39

Quite intriguing is the fact that the analytical model based ona generalization of Silberstein’s equations gives almost iden-tical results to the full DIM/QM calculations. Therefore, itis likely that this simple model can be used as a simple toolto understand molecule–metal nanoparticle interactions. Weare currently investigating how general the analytical modelis and if it can be used to predict molecule–metal nanoparti-cle interactions.

Currently, the DIM/QM implementation is limited tonanoparticles with 10 000 atoms or less due to the linearsolver currently employed to solve Eq. (9). However, we arecurrently working on addressing these issues so that muchlarger nanoparticles as well as multiple nanoparticles can beconsidered. Also, we do not consider solvent or retardationeffects in the current implementation. Retardation effects arenot important due to the small size of the nanoparticles stud-ied here. Solvent effects could be included by embedding thetotal system in a continuum dielectric matrix or by explicitlyincluded solvent molecules in a QM/MM model.

VI. CONCLUSIONS

A frequency-dependent quantum mechanics/molecularmechanics method for the calculation of response proper-ties of molecules adsorbed on metal nanoparticles has beenpresented. This DIM/QM method represents the nanoparticleatomistically which allows us to account for the local envi-ronment of the nanoparticle surface on the optical propertiesof the adsorbed molecule. Using the DIM/QM method, weinvestigated the coupling between the absorption of a silvernanoparticle and of a substituted naphthoquinone. The sys-tem was chosen since it showed a strong coupling due to thatthe absorption peak in the molecule overlaps with the plas-mon excitation in the metal nanoparticle. We showed thatthere is a strong dependence not only on the distance of themolecule from the metal nanoparticle but also its orientationrelative to the nanoparticle. We find that when the transitiondipole moment of the molecule is oriented towards thenanoparticle that there is a significant increase in the molecu-lar absorption as a result of the coupling to the metal nanopar-ticle. In contrast, we find that the molecular absorption isdecreased when the transition dipole moment is oriented par-allel to the metal nanoparticle. The coupling between themolecular and the metal nanoparticle is also found to be sur-prisingly long range and important on a length scale compa-rable to the size of the metal nanoparticle. A simple analyticalmodel that describes the molecular and the metal nanoparti-cle as two interacting point objects was found to be in very

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134103-11 Frequency-dependent DIM/QM J. Chem. Phys. 135, 134103 (2011)

good agreement with the full DIM/QM calculations over theentire range studied. This work has implications for plasmon-exciton hybridization, plasmon enhanced photochemistry, andsingle-molecular surface-enhanced Raman scattering.

ACKNOWLEDGMENTS

L.J. acknowledges the CAREER program of the NationalScience Foundation (Grant No. CHE-0955689) for financialsupport, start-up funds from the Pennsylvania State Univer-sity, and support received from Research Computing and Cy-berinfrastructure, a unit of Information Technology Servicesat Penn State. S.M.M. acknowledges the Academic Comput-ing Fellowship from the Pennsylvania State University Grad-uate School.

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