A Hybrid Atomistic ElectrodynamicsQuantum Mechanical Approach for Simulating Surface-Enhanced Raman Scattering JOHN L. PAYTON, SETH M. MORTON, JUSTIN E. MOORE, AND LASSE JENSEN* Department of Chemistry, The Pennsylvania State University, 104 Chemistry Building, University Park, Pennsylvania 16802, United States RECEIVED ON MARCH 15, 2013
CONSPECTUS
S
urface-enhanced Raman scattering (SERS) is a technique that has broad implications for biological and chemical sensing applications by providing the ability to simultaneously detect and identify a single molecule. The Raman scattering of molecules adsorbed on metal nanoparticles can be enhanced by many orders of magnitude. These enhancements stem from a twofold mechanism: an electromagnetic mechanism (EM), which is due to the enhanced local field near the metal surface, and a chemical mechanism (CM), which is due to the adsorbate specific interactions between the metal surface and the molecules. The local field near the metal surface can be significantly enhanced due to the plasmon excitation, and therefore chemists generally accept that the EM provides the majority of the enhancements. While classical electrodynamics simulations can accurately simulate the local electric field around metal nanoparticles, they offer few insights into the spectral changes that occur in SERS. First-principles simulations can directly predict the Raman spectrum but are limited to small metal clusters and therefore are often used for understanding the CM. Thus, there is a need for developing new methods that bridge the electrodynamics simulations of the metal nanoparticle and the first-principles simulations of the molecule to facilitate direct simulations of SERS spectra. In this Account, we discuss our recent work on developing a hybrid atomistic electrodynamicsquantum mechanical approach to simulate SERS. This hybrid method is called the discrete interaction model/quantum mechanics (DIM/QM) method and consists of an atomistic electrodynamics model of the metal nanoparticle and a time-dependent density functional theory (TDDFT) description of the molecule. In contrast to most previous work, the DIM/QM method enables us to retain a detailed atomistic structure of the nanoparticle and provides a natural bridge between the electronic structure methods and the macroscopic electrodynamics description. Using the DIM/QM method, we have examined in detail the importance of the local environment on molecular excitation energies, enhanced molecular absorption, and SERS. Our results show that the molecular properties are strongly dependent not only on the distance of the molecule from the metal nanoparticle but also on its orientation relative to the nanoparticle and the specific local environment. Using DIM/QM to simulate SERS, we show that there is a significant dependence on the adsorption site. Furthermore, we present a detailed comparison between enhancements obtained from DIM/QM simulations and those from classical electrodynamics simulations of the local field. While we find qualitative agreement, there are significant differences due to the neglect of specific moleculemetal interactions in the classical electrodynamics simulations. Our results highlight the importance of explicitly considering the specific local environment in simulations of moleculeplasmon coupling.
1. Introduction
of optical phenomena such as surface-enhanced linear and
Metal nanoparticles exhibit unique optical properties due to their ability to support surface plasmons. The coupling between molecules and plasmons leads to a wide range
nonlinear vibrational spectroscopy,1,2 surface-enhanced
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fluorescence,3,4 and plasmonexciton hybridization.5 This has led to many applications of plasmons in optics, in Vol. XXX, No. XX
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catalysis, and as sensors for chemical and biological detection.6 Thus, it is of fundamental importance to establish at a molecular level an understanding of the coupling between molecules and plasmons. Surface-enhanced vibrational spectroscopies, such as surface-enhanced Raman scattering (SERS),710 rely directly on the coupling between the molecule and the plasmon excitation. In SERS, the Raman scattering of adsorbed molecules can be enhanced by more than 108 such that a single molecule can be detected and identified due to the unique Raman fingerprint.1115 The SERS enhancement arises predominantly from the strong local field due to the plasmon excitation, the so-called electromagnetic mechanism (EM), and it be can shown that the Raman scattering scales as ∼|E|4 where |E| is the local field at the position of the molecule.1618 In addition to the EM, there is also an enhancement due to the short-range interactions between the metal and the molecule called the chemical mechanism (CM). The overlap between the wave functions of the molecule and the metal results in a renormalization of the molecular orbitals as well as the introduction of new mixed charge-transfer states; both of these effects will contribute to CM enhancement of the Raman signal.8,1921 While all of these mechanisms contribute to the observed enhancements, it is possible in some situations to separate the CM and EM mechanisms.22 The different enhancement mechanisms can then be modeled separately and compared with experiments. Although we have a fairly well developed understanding of the enhancement mechanisms, we only have a rudimentary understanding of spectral changes that occur in SERS due to the complicated nature of the metalmolecule interface.18,20 It remains a formidable challenge to simulate realistic SERS spectra from first-principles due to the complexity of correctly treating the interactions of an electronically localized molecular system with the electronically delocalized structure of a metal particle that is many nanometers in dimension. Typically, the EM contribution to SERS is considered by simulating the local electric field due to the plasmon excitation using classical electrodynamics.23 Several efficient approaches are available to simulate the optical properties of metal nanoparticles and have been shown to correlate with experimental results. However, recent work has highlighted the importance of quantum effects in junctions between metal nanoparticles with small gaps due to electron tunneling effects.5 This work has shown that there is a quantum mechanical limit on the EM enhancements in SERS.24 Furthermore, any microscopic detail of the coupling is neglected since the nanoparticle and the molecular layer B ’ ACCOUNTS OF CHEMICAL RESEARCH
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are represented as continuous objects characterized by their frequency-dependent dielectric functions. Despite their success for modeling the optical properties of nanoparticles, this lack of microscopic detail prevents a realistic description of the moleculemetal interface and thus cannot provide a complete description of SERS. First-principles simulations offer a unique tool that can provide the detailed understanding of the complicated interface by directly simulating the SERS spectrum that can be compared with experiments.1820,25 However, due to large computational requirements, first-principles methods are limited to small systems and thus are typically used to provide information about the CM in SERS.1822,25 Therefore, it is necessary to develop new methods that bridge the quantum mechanical description of the molecule and the classical electrodynamics description of the metal nanoparticle to provide a comprehensive understanding of SERS. In recent years, several different hybrid methods that combine a quantum mechanical description of the molecule and a classical description of the metal nanoparticle have emerged.2637 While these methods offer an improved quantum mechanical description of the molecule, the description of the metal nanoparticle is often based on a continuum treatment and thus neglects the relevant specific interactions. To overcome these limitations, our group has been developing an atomistic electrodynamics model38,39 and combined it with time-dependent density functional theory (TDDFT).4042 This method, which we denote the discrete interaction model/quantum mechanics (DIM/QM) method, represents the nanoparticle atomistically, enabling the modeling of the influence of the local environment of a nanoparticle surface on the optical properties of a molecule. The DIM/QM method can be seen as an extension of traditional polarizable QM/MM methods used for describing optical properties of molecules in solution. In contrast to most previous work, the DIM/QM method enables us to retain the detailed atomistic structure of the nanoparticle and provides a natural bridge between the electronic structure methods and the macroscopic electrodynamics description. In this Account, we will highlight our recent work on understanding the coupling between molecules and plasmons using the DIM/QM method. We will illustrate the importance of the local environment on molecular excitation energies, enhanced molecular absorption, and SERS. As a direct test of the well-known |E|4 approximation for describing SERS enhancements, we will present a detailed comparison between enhancements obtained using the
DIM/QM Approach for Simulating SERS Payton et al.
DIM/QM method and those obtained using the EM approximation. Our results highlight the importance of explicitly considering the specific local environment in simulations of moleculeplasmon coupling and the need for directly simulating the SERS spectra.
2. An Atomistic ElectrodynamicsQuantum Mechanical Approach In the DIM/QM method, the nanoparticle is considered as a collection of N interacting atoms that describe the total optical response. For large metal nanoparticles, each atom is characterized by an atomic polarizability obtained from the experimental dielectric constant. For smaller metal nanoparticles, we describe each atom by an atomic polarizability and an atomic capacitance that is obtained by fitting against TDDFT results for small silver clusters (N < 68).38,39 This allows us to include size-dependent effects and thus correctly describe the saturation of the polarizability of the nanoparticle as the size increases. In the DIM/QM model, the total energy is given by4042
UTOT [F] ¼ Ts [F] þ Z þ
1 2
ZZ
F(r)F(r 0 ) dr dr 0 þ Uxc [F] jr r 0 j
Variational minimization of the total energy given by eq 1 leads to the following effective KohnSham operator, hKS[F(rj)], given by 1 hKS [F(rj )] ¼ r2 2
∑j
Zj þ jrj Rj j
Z
F(rj ) δEXC ^ dim dri þ þV (rj ) δF(rj ) jrj ri j
(4) with the individual terms being the kinetic energy, the nuclear potential, the Coulomb potential, the XC-potential, ^ DIM(rj)) describing the and the embedding DIM operator (V moleculemetal interactions, respectively. The embedding operator is given by ^ DIM (rj ) ¼ V
qind
m ∑ ∑ m jrjm j m
μind m, R rjm, R jrjm j3
3
(5)
^ DIM(rj) operator The perturbation to the density due to the V can be thought of as the image field, that is, the field arising from the dipoles and charges that are induced in the nanoparticle (DIM system) by the presence of the molecule (QM system). The interactions are damped at short dis-
F(r)Vnuc (r) dr þ U DIM=QM [F]
(1)
with the individual terms being the kinetic energy of a fictitious noninteracting system, the Coulomb energy, the XC-energy (exchange correlation), the electronnuclear interaction energy, and the interaction energy describing the moleculemetal interactions, respectively. The interaction energy is given by UDIM=QM [F] ¼ U POL [F] þ U VDW
(2)
tances to avoid over polarization. The induced dipoles and charges needed to calculate the polarization energy are found by solving a set of 4N þ 1 linear equations expressed in supermatrix notation as 0 1 0 10 1 A M 0 μind ESCF @ MT C 1 A@ qind A ¼ @ VSCF A (6) 0 1 0 λ qDIM where the matrix A describes the dipoledipole interactions, the matrix M describes interactions between dipoles and charges, and the matrix C describes charge charge interactions. Solving these linear equations
where U [F] is the polarization energy (the energy required to induce the dipoles and charges in the DIM
is identical to variational minimization of the classical energy for a collection of interacting atoms described
system) and UVDW accounts for the dispersion and repul-
by their atomic polarizability and capacitance, under the constraint that the total charge of the system is fixed
POL
sion energy between the DIM and the QM system. U is treated purely classically and thus does not depend on the VDW
ground state density. The polarization energy for a neutral nanoparticle is given by U POL [F] ¼
1 2
N
1
N
SCF SCF μind ∑m qind m [F]Vm [F] m, R [F]Em, R [F] 2∑ m
(3)
ind where μm and qind m are the dipoles and charges induced in SCF the DIM system by the QM system, and ESCF m,R and Vm are the electric field and potential arising from the QM system.
using the Lagrangian multiplier, λ.38 To obtain molecular response properties, we will use linear response theory to obtain the first-order change in the density due to a time-dependent perturbation. We will use the typical convention and identify indices a and b with virtual orbitals, i and j with occupied orbitals, and s and t with general orbitals. The first-order change in the density is F0 (r, ω) ¼
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in terms of the first-order density matrix (Pst(ω)) 0
Pst (ω) ¼
Δnst 0 Vsteff (ω) ω ωst þ iΓ
(8)
where Δnst is the difference in occupation number and Γ is a phenomenological energy broadening term that is due to damping of the excited state; that is, it is related to the effective lifetime of the QM excited state. The change 0
in effective potential Vsteff(ω) is given by 0
0
pert
Vsteff (ω) ¼ Vst
0
0
(r, ω) þ VstCoul (r, ω) þ VstXC (r, ω) 0
þ VstDIM (r, ω)
(9)
and is composed of the Coulomb, XC, and DIM potentials, 0
respectively, and Vstpert is the external perturbation given by ^ pert (r, ω) ¼ V ^ ext (rj , ω) þ V ^ loc (rj , ω) V
(10)
^ ext(rj, ω) represents the applied external potential, where V ^ loc(rj, ω)) is given as and the local field operator (V ^ loc (rj , ω) ¼ V
(0) (1) ext qext ∑ m (ω)Tjm þ ∑ μm, R (ω)Tjm, R m m
(11)
ext The qext m (ω) and μm,R(ω) are the charges and dipoles
induced in the DIM system due to the external perturbation and are found by solving a set of linear equations similar to what is done for the DIM operator.41 The dipole matrix of the QM system, HRst(ω), is calculated as ^ loc (ω)jtæ H R (ω) ¼ Æsjμ^ þ V (12) st
R
R
where μ^R is the QM dipole operator in the R direction ^ Rloc(ω) is the complex local field operator in the and V R direction. From the solution of the linear response equations, we get access to excitation energies,40 frequencydependent polarizabilities,41 and recently the frequencydependent first-hyperpolarizability using the (2n þ 1) rule.43 Two different DIM/QM models are implemented for treating the metal nanoparticle. In the capacitancepolarizability interaction model (CPIM), each atom is described by an atomic polarizability and an atomic capacitance and is well suited for small silver nanoparticle systems (
