Density matrix calculation of Surface Enhanced Raman Scattering

for p-mercaptoaniline on silver nanoshells

Joshua W. Gibsona and Bruce R. Johnson

Department of Chemistry, Rice Quantum Institute and Laboratory for Nanophotonics

Rice University, Houston, TX 77005

Abstract

A theoretical analysis is performed of recent experiments measuring 782 nm Surface

Enhanced Raman Scattering of p-mercaptoaniline bound to silver nanoshells of different core

and shell radii [J. B. Jackson and N. J. Halas, Proc. Natl. Acad. Sci. 101, 17930 (2004)].

Electronic structure Hartree-Fock and Density Functional Theory calculations for Ag salts of p-

mercaptoaniline are used to characterize observed vibrational modes and CIS calculations are

carried out to examine excited states. Multimode vibronic density-matrix calculations are then

made including one excited electronic state, using a classical description of the strong local fields

and a phenomenological treatment of relaxations. The spectral behavior as a function of both

nanoshell surface plasmon resonance position and molecular electronic spacing is examined.

aCurrent address: Oceaneering Space Systems, 16665 Space Center Blvd., Houston, TX 77058.

1

I. Introduction

The enormous intensity increases in Surface Enhanced Raman Scattering (SERS) make it

a valuable vibration-specific spectroscopic tool in spite of lingering questions about the SERS

mechanism. Enhancement factors of up to ~106 for molecules on noble metal surfaces were

obtained decades ago.1,2 With the recent push toward single-molecule SERS, enhancements of

1012-1014 have been estimated.3,4 At the same time, nanoparticle/nanoarray design has become

more sophisticated, and new platforms for SERS are being made available. One recent set of

SERS experiments by Jackson, et al.,5,6 uses small silica spheres coated with silver shells. These

metal nanoshells are of great particular interest since the SERS effect is mediated by the

nanoshell surface plasmon resonances, and these are tunable throughout the visible and near-IR

regions of the spectrum via appropriate choice of inner and outer radii. The current investigation

is focused on quantum mechanical calculation and modeling of the SERS spectra for p-

mercaptoaniline (PMA, p-aminobenzenthiol, p-aminothiophenol) molecules on these nanoshells.

It is well-known that SERS depends first and foremost on the strong electromagnetic near

fields around the metal surface.7 This electromagnetic enhancement in SERS depends on the

specific molecules only to the extent that Raman-shifted vibronic frequencies fall within the

spectral width of the substrate surface plasmons. However, for physisorbed or chemisorbed

molecules, further differences are observed in the relative intensities of enhanced and

unenhanced vibrational features and molecule-specific differences in enhancement patterns are

found.8 These differences are ascribed to a "chemical effect" enhancement beyond, but smaller

than, the better-understood electromagnetic enhancement. The chemical effect is frequently

described by models involving transitions between neutral and ionic states of the molecule. In

2

charge transfer (CT) models,9-11 this interaction depends on overlaps between molecular and

extended metal orbitals and the position of the Fermi energy of the metal particle.

SERS-related quantum chemistry calculations are increasingly made that focus on

detailed interactions of a molecule with up to a few metal atoms, thereby including at least some

aspects of the chemical effect. Nakai and Nakatsuji12 take such an approach for CO adsorbed on

Ag, explicitly considering molecular distance from and orientation with respect to the metal

surface. They focus on molecular orbitals (MO's) of Ag2CO and Ag10CO using time-dependent

Hartree-Fock (HF) theory and the Placzek polarizability approximation to calculate SERS

intensities. From this point of view, the interactions are interpreted as surface polarization

interactions with molecular vibrations in an overall neutral complex, and the Raman

spectroscopy is resonant when the laser frequency is close to transition energies of the complex.

The investigation of Ag2-pyrazine by Arenas, et al.,13 uses HF and Configuration-Interaction

Singles (CIS) calculations to interpret observed SERS vibrational features according to local

symmetry. Similar calculations have been made for pyridine on Cu, Ag and Au clusters. 14

Bjerneld, et al.,15,16 have measured SERS spectra for the amino acid tyrosine adsorbed on Ag

nanoclusters, finding that the SERS spectra resemble the ab initio Raman spectra for Ag+-Tyr

rather than those for free Tyr. This is in accord with the results of Aroco, et al., 17 who find that

SERS spectra of phthalamide on silver colloids and islands correspond much more closely to

those mesured for the Ag-phthalamide salt with N-Ag linkage than those measured for the free

phthalamide molecule.

In the present work, ab initio calculations are undertaken for PMA with S-H replaced by

S-Ag and S-Ag3. This allows vibrational modes, electronic levels and transition moments more

appropriate to adsorbed PMA to be approximately determined. Ground-state calculations at the

3

HF and Density Functional Theory (DFT) levels are performed for the neutral closed-shell

moieties Ag-PMA and Ag3-PMA, and CIS calculations are performed for low-lying excited

electronic states. The lowest unoccupied molecular orbitals of the silver salts are of most interest

to the experiments of Jackson, et al.,5,6 with 782 nm excitation, since this is not resonant with the

CT process identified by Osawa, et al.18

Such quantum calculations can provide useful input in modeling the SERS process, but

cannot reflect the delocalized surface plasmon modes (collective motions of the conduction

electrons). These are sensitive to the overall particle shape, and are the critical determiners of

local field enhancement. Embedding ab initio cluster calculations within a much larger metal

body is at present a challenge, though related research is being made for molecular electronic

devices with contacts described by metal clusters.19,20 Nevertheless, we may be guided here by

the experimental findings of Jackson, et al.,5,6 that the PMA-nanoshell SERS spectra closely

follow the enhancement patterns of generalized Mie theory21,22 with respect to variations in

nanoparticle core and shell radii. Thus, this initial effort to model the SERS spectra combines a

quantum treatment of the AgPMA molecule with a classical treatment of local fields to capture

aspects of both chemical bonding and electromagnetic effect enhancement.

The molecular evolution is calculated via the density matrix formalism for molecular

interaction with classical fields. This has been developed in detail in recent work of Xu, et al., 23

and Johansson, et al.24 for two metallic nanospheres. Their treatment includes the important

effects of population transfer and phase relaxation, treats Raman scattering and fluorescence in a

unified manner, and readily generalizes to the case of a metal nanoshell and multiple vibrational

modes. Five dominant SERS-active vibrational modes identified by Jackson, et al., are

simultaneously included in the present treatment for PMA, verifying that this approach can

4

produce multimode emission spectra agreeing well with those obtained experimentally.

Overtones and combination bands are included to ensure that they remain weak in accord with

most SERS observations. The model spectra are then examined for off-resonance versus

resonance behavior with respect to both the surface plasmon modes and the molecular electronic

transition interval.

Section II discusses surface-enhanced spectroscopic experiments for PMA on silver

substrates. The electronic and vibrational structure of AgPMA and Ag3PMA is examined in

Sect. III. Details of the density matrix calculations are provided in Sect. IV and model SERS

spectra are examined in Sect. V. Section VI provides conclusions.

II. SERS for PMA absorbed on silver

Ordinary and surface-enhanced spectroscopies of PMA have been investigated by Osawa,

et al.,18 on silver island films and roughened silver electrodes, with the primary modes observed

classified as having a1 and b2 local symmetry with respect to the benzene ring. The Surface

Enhanced Infrared Absorption (SEIRA) measurements were found to be dominated by a1 modes.

From the usual SEIRA propensity rule that only vibrations giving dipole changes normal to the

surface are infrared active, it was concluded that each molecule is on average standing up with

the in-plane symmetry axis normal to the metal surface. The absence of an S-H stretching mode

in any of the surface spectroscopies confirms that an Ag-S chemical bond is formed. The 514

nm SERS spectrum of PMA on silver island films exhibited strong b2 ring-mode activity which

was assigned to molecular resonance. Within the charge-transfer model of Lombardi, et. al.,11

this invokes transfer of a metal electron from the ground state (mixed with metal orbitals) to a

1B2 affinity level of the molecule about 2.5 eV above the Fermi energy via Albrecht C-term

5

Raman scattering.11,25 In contrast, a1 mode enhancement was ascribed strictly to electromagnetic

effects (A-term scattering).

In more recent off-resonant experiments, Jackson, et al.,5,6 have measured SERS response

of PMA-coated nanoshells under carefully controlled conditions with 782 nm (1.6 eV)

excitation. While several weak modes are observed in some of the nanoshell experiments, there

are always three strong modes seen around 390, 1077 and 1590 cm-1, as well as ususally weaker

modes at 1003 and 1180 cm-1. In C2v geometry, these are expected to be of a1 symmetry since the

charge-transfer mechanism investigated by Osawa, et al., should not be activated at this longer

wavelength. This was examined by measuring the SERS intensity in each of these modes,

normalized to the number of nanoshells and molecules, for different core/shell ratios. For single-

shell surface plasmons in and out of resonance with the laser frequency, the relative intensity

variations for different nanoshells were found to be in good agreement with those calculated

from Mie theory using local field enhancement factors at incident and scattered frequencies.26

Enhancement factors (compared to free PMA Raman scattering) of 1012 in solution5 and up to

1010 for nanoshell films6 were obtained.

III. AgPMA and Ag3PMA electronic structure

In order to be more definite about the electronic states and vibrational modes involved in

the recent experiments, a series of ab initio calculations were undertaken for PMA covalently

linked to Ag instead of H. Both HF and DFT calculations were run for AgPMA and Ag 3PMA

using the Guassian program G03,27 with initial optimizations restricted to C2v geometry to assist

in vibrational assignments. For Ag3PMA, the additional Ag atoms were also kept in-plane and

optimized to a stationary geometry as the closest pair of an Ag3 isosceles triangle. Small

6

imaginary frequencies occur in C2v geometry even with 1 Ag atom since the latter prefers to bend

out of plane, though we are mostly concerned with the ring vibrations. The LANL2DZ basis

available in Gaussian is used,13,28 corresponding to columns SCF and DFT in Tables 1 and 2.

Extended calculations were also made using 6-31g* instead of the default D95 basis for non-

silver atoms (SCFx and DFTx).

Table 1 shows the calculated modes (frequency-ordered) falling in the primary ring-mode

range 1000-1800 cm–1. The SCF and SCFx columns are expected to overestimate the vibrational

frequencies by several percent. However, it can be seen in the case of the 18a mode

experimentally observed around 1000 cm-1 that no SCF/SCFx a1 mode occurs at less than 100

cm-1 higher. Both DFT and DFTx calculations correct this for both 1 and 3 Ag atoms, placing

the vibrational mode in the range of 1015-1022 cm-1. Table 2 shows our matching of calculated

modes to the observations of Jackson and Halas and the observations/assignments of Osawa, et

al. Only the DFT and DFTx calculations are used for this table, and the calculated frequencies

are expressed as a range covering the entries in the relevant columns of Table 1.

Table 1. Calculated ring-mode vibrations of AgPMA and Ag3PMA in planar C2v geometry.

AgSNH2 (C2v) Ag3SNH2 (C2v)

SCF SCFx DFT DFTx SCF SCFx DFT DFTx

b1 953 b1 920 a2 988 b1 943 b1 952 b1 919 a2 987 a2 935

b1 1095 b1 1060 b1 1002 a1 1019 b1 1093 b1 1058 a1 1018 a1 1022

a1 1108 a2 1077 a1 1015 b2 1035 a1 1110 a2 1075 b2 1035 b2 1035

a2 1115 a1 1102 b2 1035 a1 1094 a2 1114 a1 1103 a1 1089 a1 1101

b2 1126 b2 1113 a1 1083 b2 1146 b2 1124 b2 1111 b2 1148 b2 1148

7

a1 1186 a1 1200 b2 1146 a1 1210 a1 1189 b2 1199 a1 1213 a1 1212

b2 1232 b2 1202 a1 1212 b2 1307 b2 1230 a1 1203 a1 1319 b2 1314

a1 1316 b2 1279 b2 1318 a1 1337 a1 1316 b2 1276 b2 1325 a1 1335

b2 1322 a1 1303 a1 1321 b2 1349 b2 1319 a1 1302 b2 1361 b2 1351

a1 1402 a1 1411 b2 1359 b2 1459 a1 1398 a1 1406 b2 1448 b2 1465

b2 1460 b2 1453 b2 1443 a1 1536 b2 1459 b2 1452 a1 1516 a1 1537

b2 1567 b2 1579 a1 1514 b2 1606 b2 1565 b2 1577 b2 1607 b2 1610

a1 1654 a1 1670 b2 1603 a1 1654 a1 1652 a1 1668 a1 1654 a1 1657

b2 1754 b2 1761 a1 1651 a1 1680 b2 1753 b2 1760 a1 1682 a1 1680

a1 1795 a1 1799 a1 1682 a1 1796 a1 1799

Table 2. Comparison of observed ring mode frequencies for PMA adsorbed on silver with

DFT/DFTx frequencies calculated for AgnPMA.

Osawa, et al.18 Jackson, et al.6 DFT

Assignment SERS SEIRA SERS Range

18a (a1) 1006 1008 1003 1015-1022

7a (a1) 1077 1080 1077 1083-1101

9b (b2) 1142 1146-1148

9a (a1) 1180 1180 1210-1213

14 (b2) 1306 1307-1319

7a' (a1) 1280 1321-1335

19b (b2) 1440 1443-1465

19a (a1) 1488 1514-1537

8

8b (b2) 1573 1603-1610

8a (a1) 1590 1590 1651-1657

– (a1) 1629 1680-1682

Table 3 shows the calculated modes falling in the 300-500 cm-1 range. There is a b2 mode

in the DFT and DFTx calculations which is principally an in-plane wagging of the NH2 moiety

accompanied by slight torsional counter-movement by the aromatic ring. A b2 assignment of the

390 cm–1 mode is unlikely, however, given the persistent strength of this mode in the

experiments made by Jackson, et al. (Other b2 modes were occasionally observed but generally

weaker.) The nearest higher candidate in Table 3 is the a1 mode with a DFT/DFTx range of 436-

455 cm-1, despite the unusually large calculation error. Accepting this mode, Figure 1 shows our

C2v–geometry assignments of the five primary vibrational features observed. It can be seen that

only this low-frequency mode has a significant S-Ag bond stretching component. Consequently,

it was expected to be the most sensitive to deviations of the Ag atom from the plane of the

benzene ring and such deformations have been confirmed to lower the frequency significantly.

Full optimization of AgPMA in fact produces a Cs-geometry nearly-tetrahedral C-S-Ag bond

angle with the frequency dropping to the range of 370-380 cm–1. The other modes are

dominantly ring modes and remain almost unchanged under the C-S-Ag bending. For PMA

adsorved on silver, it is thus plausible that there is at least some tilting (benzene face down) away

from the standing-up geometry.

Table 3. Calculated vibrations in the range 330-480 cm–1 for AgPMA and Ag3PMA in planar C2v

geometry.

9

AgSNH2 (C2v) Ag3SNH2 (C2v)

SCF SCFx DFT DFTx SCF SCFx DFT DFTx

b1 336 b2 324 a2 321 a2 337 b1 342 b1 332 a2 313 a2 330

b2 424 b2 423 b2 384 b2 385 b2 425 b2 424 b2 390 b2 391

a2 466 a2 459 a2 424 a2 419 a1 457 a1 455 a2 408 a1 440

a1 469 a1 466 a1 453 a1 455 b1 473 a2 459 a1 436

390cm 1003cm–1 1077cm 1180cm 1590cm–1 –1 –1–1

Figure 1. Motions of five a1 vibrational modes dominant in SERS spectroscopy at

782 nm.

Figure 2 shows the dominant orbitals involved in HOMO-to-LUMO excitation for both

planar C2v and bent Cs geometries. The transition is clearly accompanied in either geometry by

significant transfer of charge from molecule to metal. In order to understand the spectroscopic

importance of the excited states, a series of CIS calculations were run for AgPMA and Ag3PMA

with Ag-S-C bond angles fixed between 90º and 180º and all other coordinates optimized for the

ground state. The resulting transition energies are shown in Fig. 3, sorted according to A' and A"

symmetry in Cs geometry. The curves are decorated with dots whose radii are proportional to the

10

oscillator strengths of the corresponding transitions. From this it can be seen that the LUMO is

A' in for both molecules, is in the range of 2–3 eV above the ground state (as opposed to ~4 eV

for free PMA), and has a significant increase in oscillator strength with decreasing bond angle.

Studies appropriate for PMA fully coating Ag nanoparticles would require more Ag atoms and

more PMA molecules, beyond the scope of the present work. On the basis of the spectroscopic

evidence, Ag-S-C bond angles near 180º are still expected. Nevertheless, the metal salt

calculations do yield weak transitions at significantly lower energies than in the free molecule,

and the possibility exists that closer approach to resonance with the laser frequency overshadows

the weaker transition moments.

Figure 2. Dominant electronic orbitals transitions from HOMO to LUMO for

AgPMA with Ag-S-C bond angles of 180º and 140‘: (a) C2v HOMO, (b) C2v

LUMO, (c) Cs HOMO, (d) Cs LUMO.

11

Figure 3. Transition energy correlation diagrams versus changes in the Ag-S-C

bond angle for the first 10 excited states of AgPMA (left) and Ag3PMA (right).

Dot radius is proportional to oscillator strength.

IV. Density matrix evolution

A density matrix model of the molecular response is used to include the different

relaxation phenomena that can occur in the SERS process along the lines of the recent work by

Xu, et al.23 and Johansson, et al.,24 The equation of motion for the molecular density matrix is

€

∂ρ∂t

= 1i h

H,ρ[ ] + L1 ρ + +L2 ρ (1)

Here H is the effective Hamiltonian

€

H = H0 + H ' (2)

12

where H0 is the pure molecular contribution and H' governs the interaction with classical external

electromagnetic field. The quantities L1 and L2 are, respectively, effective population decay and

dephasing relaxation operators. The calculations here are carried out for an AgPMA molecule

with multiple Raman-active modes near a silver nanoshell whose surface plasmon resonance is

tunable by variation of the core and shell radii.

A. Molecular Hamiltonian

The operator

€

H0 is the electronic and vibrational Hamiltonian of an AgPMA molecule

with fixed orientation. Within the Born-Oppenheimer approximation, the field-free molecular

eigenfunctions are

€

Ψn v q,Q( ) = ψnel q;Q( ) ψ n v

vib Q( ), (3)

where n and v are collective quantum numbers for electronic and vibrational motion, and q and

Q their respective collective coordinates. The electronic wave functions depend parametrically

on the nuclear coordinate geometry and, for a given electronic state n, the vibrational wave

functions are eigenfunctions of the nuclear Hamiltonian

€

Hnvib . The vibrational modes are the five

strongest bands observed by Jackson, et al., at 390, 1003, 1077, 1180 and 1590 cm–1, while all

other vibrations are regarded as spectator modes and not considered further.

We take Q = {Q1, Q2, Q3, Q4, Q5} as the normal modes for the ground n = 0 state.

Multiple excited electronic states can easily be included, but only one (n = 1) is included in the

present treatment. Expressed in terms of Q, the excited state normal mode Hamiltonian will

generally have different linear and quadratic force constants. The Raman spectra are expected to

be most sensitive to the linear terms29 for a1 modes, and the general lack of vibrational overtones

and combination bands in SERS30 provides little information about other force constants.

13

Quadratic force constants are therefore taken to be the same as in the ground state. The

vibrational Hamiltonians are taken as

€

Hnvib =

i∑ hωi − 1

2∂2

∂Qi2 + 1

2Qi − dQni( )

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥+ Δn , (4)

where the displacements

€

dQni and adiabatic electronic energy separation

€

Δn are zero for n = 0.

In this Parallel Modes Approxmation,29 all vibrational Hamiltonians are separable into single-

mode displaced oscillators. This model is the simplest for which the excited state potential

slopes at the equilibrium geometry are the structural parameters influencing the SERS spectra.

The Hamiltonian H0 is taken as diagonal in the vibronic basis with matrix elements

€

Δn +i∑ hω vni

, where the

€

vni are the modal occupation numbers in electronic state n.

B. Interaction Hamiltonian

The interaction Hamiltonian in the Multipolar Gauge31 and the Electric Dipole

Approximation is

€

H ' = d ⋅Etot r, t( ) , (5)

where d is the dipole operator of the molecule, r = (r, , ) is the molecular position, and

€

E r, t( )

is the sum of the incident monochromatic laser field Einc(r, t) and corresponding field Esp(r, t)

scattered by nanoshell surface plasmon oscillations. For a cw incident field of angular frequency

€

ω l , we may write the total field as

€

E r, t( ) = 12

E r( )e−iω l t + 12

E* r( )eiω l t , (6)

the first term corresponding to molecular absorption and the second to emission. Further

approximations are made, as is usually the case. The matrix elements of H' are evaluated within

the Rotating Wave approximation32 and a transformation is made to a frame rotating with the

14

laser frequency.24 Within the Condon Approximation, the transition dipole moment components

€

μ10 (matrix elements of d between excited and ground electronic wave functions) are taken as

constants. Combined with the Parallel Modes Approximation above, the vibrational portions of

the transition dipole matrix reduce to products of five one-dimensional Franck-Condon factors.

C. Generalized Mie theory for metallic nanoshells

The problem of cw excitation of two-layer spheres was solved long ago by Aden and

Kerker22 as an extension of Mie theory for solid spheres.21 [Finite-difference time-domain

calculations may be used for a much wider variety of particle geometries and morphologies.33]

The incident and scattered fields can be expanded in vector spherical harmonics

€

Mσ l mz k r( ) and

€

Nσ l mz k r( ) based on spherical Bessel

€

jl (z = j) and Hankel

€

hl1( ) (z = h) functions and classified as

even ( = e) or odd ( = o). These are given in more detail, for example, by Sarkar and Halas,34

whose conventions we follow. Then the

€

e−iω t component of the incident field propagating along

the z axis and polarized along the x axis has the multipole expansion

€

Einc r( ) = E0 ˆ x ei k z = E0l=1

∞

∑ il+2 2l +1l l +1( )

Mol1j k r( ) − iNe l 1

j k r( )[ ] . (7)

. The surface-plasmon-scattered field outside the shell has the corresponding expansion

€

Esp r( ) = E0l=1

∞

∑ il+2 2l +1l l +1( )

al Mol1h k r( ) − ibl Ne l1

h k r( )[ ] , (8)

where the coefficients

€

al and

€

bl are obtained by total field continuity conditions at the inner and

outer shell surfaces.22

For shell radii much smaller than the incident wavelength, the series converges very

quickly and is dominated by dipole (l = 1) and sometimes quadrupole (l = 2) contributions. All

15

of the differences from the case of solid spheres reside in the coefficients

€

al and

€

bl , which

depend on the core/shell aspect ratio a/b, the incident field frequency, and the dielectric functions

of the core, shell and surrounding medium. These plasmons are broadened by radiative damping

on time scales of ca. 10 fs35-37 as well as nonradiative damping (e.g., coupling to electron-hole

continua and electron scattering). Prodan, et al.,38,39 have shown that the nanoshell surface

plasmons may usefully be regarded as hybridizations

€

ω l± of those for a sphere and a cavity, in

strong analogy to quantum mechanical hybridization of atomic orbitals to form molecular

orbitals. For the simple nanoshell in the dipole case, the lower-frequency

€

ω1− is the active mode

relevant to the SERS process.

Consider the case of a 39/57 silver nanoshell, i.e. a 39 nm radius silica core with

dielectric constant 2.04 surrounded by an 18 nm radius silver shell with complex dielectric

function.40 For propagation along +z and linear polarization along +x, an incident CW field at

782 nm (1.6 eV) is off-resonant with the plasmon, so only mild local field enhancements are to

be expected. In Fig. 4, the scattered electric field

€

ˆ r ,

€

ˆ θ ,

€

ˆ φ ,

€

ˆ x ,

€

ˆ y and

€

ˆ z components are mapped as

a function of angular position just outside the outer radius. It is seen that

€

Esp is primarily radial

in character and focused along the initial polarization axis, but is neither a spherical nor a plane

wave. The maximum enhancement factor for this particular pair of core-shell radii is rather

small at 782 nm, but is ~40 around 500 nm. Other nanoshells used by Jackson, et al., are closer

to resonance and have 782 nm enhancements that are higher (see below).

16

Figure 4. Magnitudes of spherical and Cartesian components of scattered field

€

Esp

just outside a 39/57 nm Ag nanoshell. The incident field is a unit-magnitude

plane wave field propagating in +z direction and polarized in +x direction with

wavelength 782 nm. The components are mapped as functions of angular

position around the nanoshell.

D. Relaxation Processes

The population decay components of the master equation reside in the superoperator L1,

which is taken to be a sum of Lindblad forms,41,42

€

L1 ρ = 12 j,k∑Γk j 2σ k j ρ σ j k −σ j k σ k j ρ − ρ σ j k σ k j( ) (9)

These allow maintenance of positivity of probabilities extracted from the dynamically evolving

density matrix. (There are, however, constraints on N-level quantum systems.43) The j and k

indices run over the entire set of included vibronic levels n and v. Each

€

j k matrix has a unit (j,

k) element with all the rest zero, and the constant

€

Γk j is the transition rate from level j to level k.

17

Two types of population decay included in this model are spontaneous emission transitions

between electronic states and vibrational relaxation within each electronic state.

For ordinary spontaneous emission from a state in the n' manifold to one in the n

manifold, the width is given by

€

Γ0 v 1v'0 =

ω1v' 0 v3

3π ε0 hc3 μ012

F0 v1v'2, (10)

where

€

ω1v' 0 v =Δ1h

+i

∑ωi v1i '−v0i( ) (11)

and the

€

F0 v'1v' are the Franck-Condon factors. Spontaneous emission rates are modified by

placing molecules in electromagnetic cavities,44-48 near metallic planes49-56, dielectric and metallic

nanobodies,57-64 and photonic crystals.65 One may calculate rate enhancements (i) using

quantized electromagnetic fields, with rates calculated via Fermi's Golden Rule and the local

density of field states, or (ii) using classical field calculations based on modification of the

asymptotic energy flux for a radiating classical dipole near a dielectric or conducting surface.

For atoms or molecules near dielectric slabs, both lossless 52,53 and absorbing,66 it has been shown

that equivalent results are obtained classically and quantum mechanically. For dielectric spheres,

the quantum and classical results are equivalent at least through first order in time-dependent

perturbation theory.52,53,61

We consider this modification of emission rates around the core-shell nanoparticles from

the perspective of purely classical fields, paralleling the treatments of Kerker, et al.,26 and Chew60

for solid spheres. In particular, we assume a point dipole p at position r near the nanoshell

oscillating at the transition frequency

€

ωn' v' n v and use vector spherical harmonics to calculate the

secondary scattered field in the presence of the nanoshell. The radiative decay rate for the

18

excited state is obtained by integrating the far-field Poynting vector over all angles. At the end of

the calculations, one has a radiative enhancement factor (relative to the dipole in the absence of

the nanoshell)

€

Xr = 32 p2

l=1

∞

∑ 2l +1( ) pr2 l l +1( ) k3 r( )

−2 jl k3 r( ) + bl hl(1) k3 r( )

2 ⎧ ⎨ ⎩

€

+pθ

2 + pφ2

2jl k3 r( ) + al hl

(1) k3 r( )2

+ η lj k3 r( ) + bl η l

h k3 r( )2 ⎡

⎣ ⎢ ⎤ ⎦ ⎥

⎫ ⎬ ⎪

⎭ ⎪, (12)

€

ηlj x( ) = 1

xddx

x jl x( ), η lh x( ) = 1

xddx

x hl1( ) x( ) . (13)

This is of exactly the same form as for the solid sphere case,60,61 except that the coefficients

€

al

and

€

bl are the Mie coefficients for the nanoshell22 evaluated at the emitted photon frequency.

For complex dielectric functions, additional losses (Joule heating) will occur. The combined

decay rate in this case can be expressed in terms of the dyadic Green function of the field

(density of electromagnetic states).51,60,62 The total radiative and non-radiative decay rate

enhancement is then given by

€

Xrnr = 1+ 32

pr2

p2 Rel=1

∞

∑ 2l +1( ) l l +1( ) k3 r( )−2 bl

s hl(1) k3 r( )[ ]

2

€

+34

pθ2 + pφ

2

p2 Rel=1

∞

∑ 2l +1( ) al hl(1) k3 r( )[ ]

2+ bl η l

h k3 r( )[ ]2 ⎧

⎨ ⎩

⎫ ⎬ ⎭. (14)

As for pure spheres,60

€

Xr = Xrnr for the lossless case of purely real dielectric functions. Figure 4

shows the absorption and emission enhancement factors for a 58/65 nm shell. The absorption-

step field intensity enhancement |M|2 = |E/E0|2 a distance 0.1 nm outside the shell along the +x

direction is almost exactly matched by the emission enhancement Xr of a purely radial dipole.

However, |M|2 is generally smaller for other positions around the shell and Xr is generally smaller

for other orientations of the transition dipole. These results follow the general behavior for

19

metallic spheres in the analysis of Kerker, et al.,26 though the specific wavelength dependence

differs for shells and spheres.

80

60

40

20

0

Enhancement Factors

1000900800700600500400300λ (μ)

|E/E0|2, =π / 2, φ = 0

|E/E0|2, =φ =π / 4

Xρ, π ρ Xρ, π

Figure 5. Enhancement factors for absorption and spontaneous emission for 58/65

nm Ag nanoshell. The absorption factor |E/E0|2 for the +x direction and the

emission factor Xr for a radial dipole are essentially identical for the large

nanoshell dipole peak. Change of molecular position reduces the absorption

factor and change of dipole orientation reduces the emission factor.

The factor

€

Xrnr including non-radiative decay is used in the equations of motion for the

molecular density matrix. Electron-hole pair creation contributions to molecular decay are also

extremely important. In the study by Johansson, et al.,24 these contributions are evaluated by

consideration of a classical point dipole symmetrically located between two silver surfaces.

They use d-parameter theory to treat the nonlocal-dielectric response, imposing a momentum

cutoff in order to avoid double-counting the small-momentum-transfer contributions already

captured by the Mie theory contributions. For smaller molecule-nanoparticle distances d, the

20

distance dependence of the dipole field causes the Mie power losses to scale as d–3, while the

electron-hole contributions scale as d–4 and become much more important as d decreases below 3

nm. The current problem actually has the molecules chemically attached to the nanoparticle

surface, so that the appropriate distance to be used, and therefore the proportion of Mie and

electron-hole loss rates, would be somewhat ambiguous. The Mie and electron-hole formalisms

are in any case not meant to apply to such short distances, which would really need an atomistic

quantum treatment with direct unscreened Coulomb forces. For the current model, in lieu of a

more complete procedure, a crude accounting of these effects is made by simply introducing a

multiplicative factor Y for the Mie loss,

€

Γ0 v1v' = Y Xrnr Γ0 v1v'0 . (15)

This factor is meant to implicitly represent at least part of the electron-hole creation effects. It is

to be emphasized that Y is treated strictly as an adjustable parameter providing for needed extra

relaxation in the density matrix equations and to be determined by consideration of calculated

spectra. A more sophisticated treatment remains for future work, but this will be adequate for us

to examine approximate relaxation rates implied by experimental spectra.

For vibrational relaxation in either ground or excited electronic states, molecular changes

of only a single vibrational quantum at a time are allowed, just as in the single-mode

treatment.23,24 In the absence of further information about the distinct vibrational mode decays,

the rates in a given electronic state n are taken to be governed by a single phenomenological rate

constant

€

Γnvib. For a decay from vibrational state vi of mode i to state vi–1, the appropriate

coefficient in

€

L1 ρ is then

€

vi Γnvib . In order to maintain thermal equilibrium in the limit of low

field strengths (important in the presence of low-frequency modes), detailed balance must be

ensured. This can be satisfied by requiring that the corresponding excitation Lindblad operator

21

has the coefficient

€

vi Γnvib exp −hωi /kB T( ) , where the latter factor is the ratio of upper to lower

state equilibrium populations.67

The superoperator L2 provides for dephasing of the transition dipoles between electronic

states. In accord with Xu, et al., and Johansson, et al., this is taken to be of the form

€

L2 ρ =n>0∑ Γ0n

ph

j∈0

k∈n

∑ σ j k ρ j k + σ k j ρ k j( ) (16)

Fluorescence at the molecular vibronic transition frequencies is broadened by this dephasing.

D. Correlation functions and spectral intensities

The dipole operator of a molecule at position r may be expanded in the molecular basis,

€

d r( ) =v'∑

v∑ 1v' d r( ) 0v 1v' 0v +

v'∑

v∑ 0v d r( ) 1v' 0 v 1v'

€

= d +( ) r( ) + d −( ) r( ) . (17)

Within a quantized field treatment, the spectra are obtained by Fourier transformation of the

two-time dipole-dipole correlation functions,32,68

€

d −( ) r, t( )d +( ) r, t + τ( ) , written here in dyadic

form. The quantities

€

d ±( ) r, t( ) represent the time-evolved operators in the Heisenberg

representation. Within the frame rotating with the laser, the time-dependent correlation functions

may be approximately evaluated24 by use of the Quantum Regression Theorem.69,70 This requires

solving for the steady-state solution of the rotating-frame density matrix equations, post-

multiplying the result by the components of

€

d +( ) r,0( ) , integrating that result to time , and then

pre-multiplying by the components of

€

d −( ) r,0( ) . The doubly differential cross-section is then

given by the Fourier transform of the correlation function according to the Wiener-Khintchine

Theorem24,71 [see Eq. (32) of Ref. 20, whose notation is slightly different than ours],

22

€

d2 σdΩ d hωs( )

=ωs

4 Xr

Iin 8π 3 c2 ε0 hRe dτ

0

∞∫ eiωs t d −( ) r,0( )d +( ) r,τ( ) . (18)

There are additional geometrical factors dealing with the direction and acceptance angle of

detection that are not explicitly included here since the primary goal is examination of the

spectral dependence of the dynamical factors in Eq. (18).

V. Calculations of emission spectra

We consider the case of an AgPMA molecule at the surface of the nanoshell along the +x

axis in the standing-up C2v orientation. Strictly speaking, the density matrix treatment regards an

ensemble of identical molecules in this neighborhood that scatter independently. To correspond

more closely to the experiments, we would need to average over all positions at the surface of the

shell.6 This would have the main effect of scaling down the magnitude of the cross-section, so

we proceed with the stipulation that our results correspond to upper bounds. The excited

AgPMA state is taken to be of A1 symmetry in C2v geometry and to fall 2.60 eV above the

ground state. The transition dipole moment is taken in the radial direction with a value of 0.5

a.u., roughly the maximum value suggested by the CIS calculations. The vibrational

displacements dQ1i in the excited state [cf. Eq. (4)] are taken to be 0.35, 0.15, 0.4, 0.2 and 0.35

a.u. in order of increasing mode frequency so as to approximately reproduce the observed

relative SERS intensities.6 The vibrational decay constants are

€

Γ0vib = Γ1

vib = 3 ×1012 s−1, for which

the widths of the vibrational features also approximately match the ~24 cm–1 widths in the

experiments. The electronic dephasing constant is

€

Γ0nph = 8 ×1012 s−1 .

The rapid growth of density matrix size with number of vibronic states makes it

important to choose the latter carefully. The vibrational states included are those with (i) zero

23

quanta in all modes, (ii) one quantum in a single mode, (iii) two quanta in a single mode and (iv)

one quantum in the low-frequency 390 cm–1 mode with one quantum in another mode. This

yields a total of 15 vibrational states in each electronic level and a density matrix of size 900

900. States (i) and (ii) will be most important for the SERS process. Overtones (iii) and

combination bands (iv) can also contribute in principle from both optical pumping and thermal

excitation of the vibrational population distribution. Since all modes are of symmetric character,

the lowest overtones will be the first to gain in intensity and inclusion of states (ii) allows us to

monitor whether or not overtone intensity needs to be treated more extensively by including

states with higher quantum numbers. The same considerations apply for combination bands, but

the large number of possible mixed-quanta states leads us to consider as "monitor states" only

those of lowest-order in which the 390 cm–1 mode (that with the largest thermal population) is

one of the partners.

Generally, it is found in the calculations that fluorescence around 2.6 eV is vastly more

intense than Raman features if the extra relaxation parameter Y ~ 1. There could in principle be

strong fluorescence outside the Raman spectral region, but such information is not available at

present. If there is actually little or none, it is necessary to adjust the model by strongly

increasing Y. Figure 6 illustrates the total calculated emission spectra for different values of Y

around 106. This very rapid depletion of excited state population strongly damps the

fluorescence while producing only minor changes in the Raman intensities, as discussed in detail

in Ref. 20. Vibronic fluorescence is completely unresolved in this situation. If stronger

fluorescence is observed at wavelengths away from the Raman spectral region, smaller values of

Y would be required and vibronic structure would then appear. If Y is much smaller than 106,

however, there will also be a strongly sloping baseline extending into the Raman spectral region.

24

0.0006

0.0005

0.0004

0.0003

0.0002

0.0001

0.0000

CS (10

–26

cm2

/meV)

3.02.82.62.42.22.01.81.61.4Emitted Photon Energy (eV)

Y 1 x 106

2 x 106

3 x 106

Figure 6. Nonresonant Raman (left) and fluorescence spectra for an electronically

excited state of AgPMA 2.6 eV above the ground state calculated using an 81/98

nanoshell and different values of Y. The double differential cross section (CS) is

defined in Eq. (18).

In Figs. 7-9 are shown the Stokes and anti-Stokes cross-sections calculated for a variety

of nanoshells with different core/shell radii investigated by Jackson and Halas.6 A value of Y =

106 is assumed, the lowest value used in Fig. 6. From Fig. 7 for 39-nm-core nanoshells, one sees

that the enhancement factor |M|2 peaks at higher energies, and is smaller and roughly

independent of shell radius in the Raman region. This translates into essentially identical SERS

spectra. In Fig. 8, the 58/65 nanoshell has a dipole surface plasmon mode that is resonant with

the laser anti-Stokes transitions, and both Stokes and anti-Stokes SERS cross-sections are

considerably higher. For different shell widths with a 58 nm core, there is a much greater

variation in |M|2 for different shell widths and therefore also in the SERS intensities. Figures 9

shows series of nanoshells with lower-energy plasmon modes.

25

8x10-5

6

4

2

1.551.501.451.40Emitted Photon Energy (eV)

(a) 39/57 39/59 39/61

2.0x10-5

1.5

1.0

0.5

1.801.751.701.65Emitted Photon Energy (eV)

(b) 39/57 39/59 39/61

403530252015

2.82.62.42.22.01.81.61.4Photon Energy (eV)

(c) 39/57 39/59 39/61

Figure 7. (a) Stokes cross-section, (b) anti-Stokes cross-section, and (c)

absorption enhancement factor |M|2 for a series of silver nanoshells with inner

radius 39 nm. Cross-section (CS) units are as in Fig. 6.

4x10-4

3

2

1

01.551.501.451.40

Emitted Photon Energy (eV)

(a) 58/65 58/66 58/69 58/72

1.5x10-4

1.0

0.5

1.801.751.701.65Emitted Photon Energy (eV)

(b) 58/65 58/66 58/69 58/72

80

60

40

20

2.82.62.42.22.01.81.61.4Photon Energy (eV)

(c) 58/65 58/66 58/69 58/72

Figure 8. Same as Fig. 7, with core radius 58 nm.

26

5x10-4

43210

1.551.501.451.40Emitted Photon Energy (eV)

(a) 81/91 81/94 81/96 81/98

1.2x10-4

0.8

0.4

1.801.751.701.65Emitted Photon Energy (eV)

(b) 81/91 81/94 81/96 81/98

40

30

20

10

2.82.62.42.22.01.81.61.4Photon Energy (eV)

(c) 81/91 81/94 81/96 81/98

Figure 9. Same as Fig. 7 with core radius 81 nm.

Comparison of the intensity scales shows that, as expected, the highest Raman intensities

are obtained for surface plasmon modes peaking in the general region of the laser frequency.

The Raman intensities follow quite well the traditional composite enhancement factors

€

| M ωl( ) |2 | M ωs( ) |2 (see, e.g., Kerker, et al.26). The first factor indicates that second order

perturbation theory is still an appropriate description of the numerical density matrix results at

the experimental field strength used (E0 = 64.8 kV/m). The second factor is essentially

equivalent to

€

Xr ωs( ) as shown in Fig. 5, and the latter factor appears explicitly in Eq. (18). The

multimode cross-sections agree reasonably well in shape with the measured spectra of Jackson

and Halas6 for single-nanoshell surface plasmons nearly resonant with the laser excitation

energy. The individual mode displacements can be fine-tuned to improve this agreement, but the

more important issue is that the SERS signals are still relatively small in magnitude, the most

direct comparison being to the single nanosphere calculations carried out by Johansson, et al.,

27

and shown in Fig. 10 of their paper. One reason is our use of a transition dipole moment smaller

by a factor of 4, which enters raised to the fourth power. Another is our use of even closer

molecule-particle distances than theirs. Yet another, and more trenchant, reason is that we have

purposely taken the electronic energy interval off-resonant with respect to the incident photon

energy.

The importance of resonance enhancement of localized electronic levels of the metal-

molecule complex in SERS has been examined before. This underlies the AgnCO ab initio

calculations of Nakai and Nakatsuji12, where an intensity enhancement factor of up to 107 was

obtained at resonance even in the absence of extended surface plasmon modes. The effects of

resonance can be examined in the current calculations by varying the energy interval of the

ground and excited electronic states. Figure 10 shows the combined Raman scattering and

fluorescence for transition energies spaced by 0.1 eV around resonance. It is seen that, for

transition energies to either side of the Rayleigh line, individual vibronic transitions (even non-

fundamental transitions) can be resonantly enhanced relative to others. On the other hand,

resonance with the Rayleigh line itself produces a general enhancement of a range of Stokes and

anti-Stokes lines with the fundamental transitions clearly dominant. Intensities at peak are over

four orders of magnitude more intense than for the off-resonant case of Fig. 7. Taking use of

smaller transition dipole moments into account, the resonant cross-sections are comparable to the

single sphere resonant results.24 The overall Raman spectrum is considerably weaker when the

resonance is in the anti-Stokes region.

28

8

6

4

2

01.81.71.61.51.41.3

Emitted Photon Energy (eV)

(d)

( x 10 )

8

6

4

2

01.81.71.61.51.41.3

Emitted Photon Energy (eV)

(c)

8

6

4

2

01.81.71.61.51.41.3

Emitted Photon Energy (eV)

(b)

8

6

4

2

01.81.71.61.51.41.3

Emitted Photon Energy (eV)

(a)

Figure 10. Cross-section (CS)

€

d2σ /dΩd hωl( ) in Stokes and anti-Stokes spectral

regions for different electronic transition energies near resonance with the 782 nm

laser energy using an 81/98 Ag nanoshell and Y = 106. Arrows indicate transition

energies: (a) 1.4 eV, (b) 1.5 eV, (c) 1.6 eV, (d) 1.7 eV.

While the Raman and fluorescence spectra are overlapped, the latter are still clearly

identifiable as features considerably broader than the vibrational peaks and appearing at different

energies according to the position of the excited electronic state. This is in agreement with the

findings of Johansson, et al., which suggest that the nonuniform continuous backgrounds

29

frequently seen in SERS and correlated with the Raman spectra4,7,8,72 arise from fluorescence. In

the case of PMA bonded to Ag, this is interpreted as coming from the molecule-metal complex.

One readily accessible experimental quantity is the ratio R of integrated anti-Stokes and

Stokes SERS signals (after background subtraction). This was used by Kneipp, et al., 3 who

observed different laser power dependences for anti-Stokes (quadratic) and Stokes (linear) SERS

features of dyes on colloidal silver. Combined with values of R considerably above the

unenhanced thermal ratio, they argued that vibrational population pumping within the ground

electronic state may occur as a result of SERS excitation. Haslett, et al.,73 conducted subsequent

experiments which showed only linear dependences of both Stokes and anti-Stokes signals over

four orders of power magnitude, but nevertheless similarly obtained ratios R exceeded expected

values by factors of anywhere from 2 to over 70. They argued that this must be due to resonance

effects between the laser and the adsorbate-metal complex. In a recent examination of this issue,

Maher, et al.,74 have obtained SERS R6G spectra at both 514 and 633 nm excitation. The values

of R appear slightly power-dependent for 633 nm, but have an extrapolated low-power value

higher than the unenhanced thermal value. The value of R for 514 nm is constant with respect to

laser power, but with a value several times lower than the unenhanced thermal value. These

results further argue that the low-power values of R are widely scattered by resonance

phenomena. Related results have been obtained by Jackson75 for our specific case of PMA on

silver nanoshells and are in accord with this interpretation.

The resonance behavior of R can be examined in the same vein as Fig. 10, i.e., by varying

the electronic interval of the computational model while leaving the laser frequency and power

fixed. All such calculated spectra are linear in laser power. We focus on the 1077 cm --1 ring

mode using the same series of 81/98 nanoshells as in Fig. 9. The lineshapes of the individual

30

calculated vibrational features are fairly Lorentzian in shape and the 1077 cm–1 peak is slightly

merged with surrounding peaks. A multiple-peak-fitting algorithm was therefore used based on

the Levenberg-Marquardt least-squares method. Local fits using three Lorentzian peaks and

quadratic-polynomial baselines were found to give very good fits in both Stokes and anti-Stokes

regions in all cases. The results are shown in Fig. 11.

2.0

1.6

1.2

0.8

0.4

0.01.901.801.701.60

Electronic transition energy (eV)

81/98 81/96 81/94 81/91

(a)

0.06

0.05

0.04

0.03

0.02

0.01

0.002.42.01.61.2

Electronic transition energy (eV)

81/98 81/96 81/94 81/91

(b)

Figure 11. Ratio of the 1077 cm–1 mode anti-Stokes to Stokes integrated

intensities for a range of electronic transition energies around the 1.59 eV photon

energy. Ratios are shown for a variety of nanoshells with different Ag shell

thicknesses used in the experiments of Jackson, et al.6 (a) Horizontal close-up; (b)

Vertical close-up.

The ratio R for the 1077 cm–1 mode is large over a relatively narrow window in Fig. 11a

for which the electronic separation is resonant with the anti-Stokes photon energy. The

unenhanced ratio at room temperature is given primarily by the Boltzmann factor 0.0055, while

31

inclusion of the

€

ωs4 dependence74 raises this to 0.011. The 81/98 nanoshell yields a maximum

ratio ~200 times larger, with significant systematic differences for the other nanoshells. This is

in agreement with the conclusions of Haslett, et al.,73 from their kinetic modeling calculations.

Figure 11b focuses on the off-resonant behavior over a much larger range for which the

Stokes signal (not shown) decreases over three orders of magnitude from peak. For lower

electronic transition energies, the ratio is lower than the thermal value owing to the preferential

enhancment of the Stokes spectral region. For higher transition energies, the preferential

enhancement of the anti-Stokes spectra yields a ratio higher than thermal. Put in terms of

varying laser frequency with a fixed electronic separation, R is enhanced when the laser is tuned

to the blue and de-enhanced when it is tuned to the red. This is precisely the behavior seen by

Maher, et al.74 The interesting trend is that these deviations persist quite some range away from

exact resonance and, unlike the resonant behavior, are relatively insensitive to shell thickness.

Thus, for a range of electronic transition intervals, values of R that are factors of a few lower or

higher than the unenhanced value would be commonly expected, while values many times higher

would only result from a narrow window of resonance around the anti-Stokes transition.

VI. Conclusions

The multimode spectral calculations implemented here for SERS with PMA on Ag

nanoshells have followed the pattern of electromagnetic enhancement factors of Mie theory,

though near-resonance of the laser and electronic transition frequencies was required in order to

dramatically increase Raman intensities. Very near resonance, incompletely quenched

fluorescence from AgPMA overlaps the Raman spectra with broader features. The suggestion

that frequent observations of continuum background features underlying groups of vibrational

32

SERS bands and correlated with their intensity may be modeled as near-resonance fluorescence

leads to the question as to how near-resonance can be such a common occurence. From the

standpoint of metal-cluster/molecule calculations, it is clear that addition of metal atoms will

provide more low-lying electronic levels with probability density in both metal and molecule.

This will increase the chances of particular excitation frequencies being near-resonant with at

least one level. From the standpoint of these metal atoms and molecule being embedded within a

metal nanodevice, such levels will be broadened and may or may not survive as distinct features

in the local electronic density of states. Nevertheless, theoretical investigations of this nature

would be of interest. On the experimental side, despite inevitable averaging away from the

single-molecule SERS limit, it would also be of interest to have the ability to densely sample the

excitation frequencies. Excitation profiles in ordinary Raman scattering provide much

information complementary to emission spectra,76,77 and it is to be expected that the same will

hold true for SERS excitation profiles.

As this paper was being completed, the very recent work by Maher, et al.,78 was brought

to our attention. A detailed experimental investigation was made identifying "hidden

resonances" in a variety of nanoparticle/analyte combinations using two or more excitation

frequencies. The anti-Stokes/Stokes ratios were found to be lower than expected for 514 nm

excitation and higher than expected for 633 nm for completely different types of nanoparticle

geometries and for three different visible dyes. The hidden resonances were concluded to arise

from local metal/analyte interactions, and provide some support for our inference from the model

calculations for PMA on Ag nanoshells that the relevant electronic interval of the metal-

molecule complex must not be as off-resonant as initially assumed. The question does arise

about how the three principal analytes used by Maher, et al., could all possess such similar

33

resonances. Extrapolating from our calculations of the anti-Stokes/Stokes ratios for AgPMA, the

persistence of elevated or diminished ratios rather far from exact resonance (Fig. 11b) would

mean that each different analyte could indeed have a rather different resonance peak (or peaks)

within the 514–633 nm interval. They reach the same conclusion as do we that application of

continuously tunable laser excitation would be a valuable addition to SERS spectroscopy.

Acknowledgments

The authors acknowledge many valuable conversations with Joseph Jackson, Naomi

Halas, Peter Nordlander, Mikael Käll, Peter Johansson and Patrick Vaccaro, and thank the latter

for making available his nonlinear least squares routine. This work was supported by NSF grants

CHE-0111008, CHE-0518476 and EEC-0304097 and by DoD-OSD grant W911NF-04-01-0203.

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