Download Density Matrix Calculation of Surface Enhanced

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 pmercaptoaniline 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.
a
Current 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 pmercaptoaniline (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
1
B2 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 singleshell 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 Ag3PMA
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 nonsilver 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.
AgSNH2 (C2v)
Ag3SNH2 (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
Assignment SERS
Jackson, et al.6
DFT
SEIRA
SERS
Range
18a (a1)
1006
1008
1003
1015-1022
7a (a1)
1077
1080
1077
1083-1101
9b (b2)
1142
9a (a1)
14 (b2)
1180
19a (a1)
1180
1306
7a' (a1)
19b (b2)
1146-1148
1210-1213
1307-1319
1280
1321-1335
1440
1443-1465
1488
1514-1537
8
8b (b2)
1573
8a (a1)
1590
– (a1)
1603-1610
1590
1629
1651-1657
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 nearlytetrahedral 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
AgSNH2 (C2v)
Ag3SNH2 (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
390 cm–1
1003 cm –1
1077 cm –1
1180 cm –1
1590 cm–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

1

H,   L1   L2 
t
i
(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 H 0 is the electronic and vibrational Hamiltonian of an AgPMA molecule
with fixed orientation. Within the Born-Oppenheimer approximation, the field-free molecular
 are
eigenfunctions
n v q,Q  nel q;Q nvibv 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 H nvib . The vibrational modes are the five
strongest bands observed by Jackson, et al., at 390, 1003, 1077, 1180 and 1590 cm–1, while all
 and not considered further.
other vibrations are regarded as spectator modes
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
vib
Hn


i
 1  2 1
2 

Q

dQ
i
n i   n ,
2


 2 Qi 2

i 


(4)
where the displacements dQni and adiabatic electronic energy separation  n 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
 n    vni , where the vni are the modal occupation numbers in electronic state n.
i

B. Interaction Hamiltonian

The interaction Hamiltonian in the Multipolar Gauge31 and the Electric Dipole
Approximation is
H'  d  E tot 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
 l , we may write the total field as

Er,t  
1
i  t 1 *
i t
Ere l  E re l ,
2
2
the first term corresponding to molecular absorption and the second to emission.
(6)
Further
 approximations are made, as is usually the case. The matrix elements of H' are evaluated within
14
the Rotating Wave approximation32 and a transformation is made to a frame rotating with the
laser frequency.24 Within the Condon Approximation, the transition dipole moment components
10 (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 Mz l m k r  and
1
z
N l m k r  based on spherical Bessel j l (z = j) and Hankel hl  (z = h) functions and classified as

even ( = e) or odd ( = o). These are given in more detail, for example, by Sarkar and Halas,34


i  t
whose conventions we follow. Then the e
component of the incident field propagating along

the z axis and polarized along the x axis has the multipole expansion
Einc r   E 0 xˆ e
ikz


2l  1
j
j
 E 0  i l 2
Mol
1k r   iNel 1 k r  .
l l  1
l1


(7)
. The surface-plasmon-scattered field outside the shell has the corresponding expansion


Esp r   E 0  i l2
l1


2l 1
al Mhol 1k r  ibl Nhel 1k r  ,
l l 1
(8)
where the coefficients al and bl are obtained by total field continuity conditions at the inner and
 outer shell surfaces.22


15
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
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

radiative damping
of the core, shell and surrounding medium. These plasmons are broadened
by
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  l of those for a sphere and a cavity, in
strong analogy to quantum mechanical hybridization of atomic orbitals to form molecular
 case, the lower-frequency  is the active mode
orbitals. For the simple nanoshell in the dipole
1
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
ˆ , 
ˆ , xˆ , yˆ and zˆ components are mapped
be expected. In Fig. 4, the scattered electric field rˆ , 
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  

1
  2 k j  j k   j k  k j    j k  k j
2 j,k 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 k 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
0
0 v 1 v'

13v' 0 v
2
3  0 c
3
2
01 F0 v 1 v' ,
(10)
where

1 v' 0 v 
1

  i v1i 'v 0i

(11)
i
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 timedependent 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  n' v' n v and use vector spherical harmonics to

18
calculate the secondary scattered field in the presence of the nanoshell. The radiative decay rate
for the 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

3

2p
2

2
pr2 l l  1k3 r
 2l  1

j l k3 r bl hl(1) k3 r
2
l1
2
2 
p2  p2 

(1)
j
h

j
k
r

a
h
k
r


k
r

b

k
r
 l  3  l l  3 
l 3 
l l  3  ,
2





lj x  
1 d
x j x,
x dx l
lh x  
1 d
1
x hl x .
x dx
(12)
(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

X rnr
 1


2
2 

3 p  p
(1)
h

Re
2l

1
a
h
k
r

b

k
r

 .







3
l l
3
 l l

4 p2
l1
2


2
2
3 pr
2 s (1)
Re
2l

1
l
l

1
k
r
b
h
k
r









3
l
l
3
2 p2
l1
2




(14)
As for pure spheres,60 X r  X rnr 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.
19
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
metallic spheres in the analysis of Kerker, et al.,26 though the specific wavelength dependence
differs for shells and spheres.
|E/E0|2,  = / 2,  = 0
2
|E/E0| ,  = = / 4
Xr, p r
Xr, p
Enhancement Factors
80
60
40
20
0
300
400
500
600
700
 (nm)
800
900
1000
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 X rnr 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
important. In the study by Johansson, et al.,24 these contributions are evaluated by
extremely
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
20
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
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,
0 v1v'

0
Y X rnr 0 v1v' .
(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
vib
constant n . For a decay from vibrational state vi of mode i to state vi–1, the appropriate
coefficient in L1  is then vi n . In order to maintain thermal equilibrium in the limit of low
vib

21


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
has the coefficient vi nvib exp  i /kB T , where the latter factor is the ratio of upper to lower
state equilibrium populations.67
 superoperator L2 provides for dephasing of the transition dipoles between electronic
The
states. In accord with Xu, et al., and Johansson, et al., this is taken to be of the form
L2  

0phn
k n
  j k  j k   k j k j 
(16)
j 0
n0
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'
1v' dr  0 v 1v' 0 v    0 v dr  1v' 0 v 1v'
v
v'
v


 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    , 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

22

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
 20, whose notation is slightly different than ours],
Theorem24,71 [see Eq. (32) of Ref.
d2 

dd  s 
 s4 X r
Re
3 2
Iin 8  c 0

 0 d ei t
s


d r,0d r,   .
(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
vib
vib
12 1
relative SERS intensities.6 The vibrational decay constants are 0  1  3 10 s , for

23
which the widths of the vibrational features also approximately match the ~24 cm –1 widths in the
ph
12 1
experiments. The electronic dephasing constant is 0n
 8 10 s .
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
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
24
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.
0.0006
CS (10
–26
cm2/meV)
0.0005
Y
6
1 x 10
6
2 x 10
6
3 x 10
0.0004
0.0003
0.0002
0.0001
0.0000
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Emitted Photon Energy (eV)
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
25
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.
8x10
-5
6
CS
(a)
39/57
39/59
39/61
4
2
1.40
1.45
1.50
Emitted Photon Energy (eV)
-5
2.0x10
39/57
39/59
39/61
CS
1.5
1.55
(b)
1.0
0.5
|M|
2
1.65
1.70
1.75
Emitted Photon Energy (eV)
1.80
(c)
39/57
39/59
39/61
40
35
30
25
20
15
1.4
1.6
1.8 2.0 2.2 2.4
Photon Energy (eV)
2.6
2.8
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.
26
4x10
3
CS
(a)
58/65
58/66
58/69
58/72
-4
2
1
0
1.40
1.45
1.50
Emitted Photon Energy (eV)
58/65
58/66
58/69
58/72
-4
CS
1.5x10
1.0
1.55
(b)
0.5
1.65
1.70
1.75
Emitted Photon Energy (eV)
80
(c)
58/65
58/66
58/69
58/72
2
60
|M|
1.80
40
20
1.4
1.6
1.8 2.0 2.2 2.4
Photon Energy (eV)
2.6
2.8
Figure 8. Same as Fig. 7, with core radius 58 nm.
5x10
-4
4
3
2
1
0
CS
(a)
81/91
81/94
81/96
81/98
1.40
1.45
1.50
Emitted Photon Energy (eV)
81/91
81/94
81/96
81/98
-4
CS
1.2x10
0.8
1.55
(b)
0.4
1.65
1.70
1.75
Emitted Photon Energy (eV)
40
(c)
81/91
81/94
81/96
81/98
2
30
|M|
1.80
20
10
1.4
1.6
1.8 2.0 2.2 2.4
Photon Energy (eV)
Figure 9. Same as Fig. 7 with core radius 81 nm.
27
2.6
2.8
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
2
2
| Ml  | | M s  | (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 X r  s  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.,
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 metalmolecule 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
28
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 nonfundamental 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
8
(a)
6
4
2
0
1.3
1.4
1.5
1.6
1.7
Emitted Photon Energy (eV)
8
1.8
(b)
6
4
2
0
1.3
1.4
1.5
1.6
1.7
Emitted Photon Energy (eV)
8
1.8
(c)
6
4
2
0
1.3
–26
CS (10
1.4
1.5
1.6
1.7
Emitted Photon Energy (eV)
8
6
2
cm /meV-sr)
CS (10
–26
2
cm /meV-sr)
CS (10
–26
2
cm /meV-sr)
CS (10
–26
2
cm /meV-sr)
resonance is in the anti-Stokes region.
1.8
(d)
( x 10 )
4
2
0
1.3
1.4
1.5
1.6
1.7
Emitted Photon Energy (eV)
29
1.8
Figure 10. Cross-section (CS) d 2 /dd  l  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
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
30
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
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
Anti-Stokes/Stokes ratio
regions in all cases. The results are shown in Fig. 11.
2.0
1.6
81/98
81/96
81/94
81/91
(a)
1.2
0.8
0.4
0.0
1.60
1.70
1.80
Electronic transition energy (eV)
Anti-Stokes/Stokes ratio
0.06
0.05
81/98
81/96
81/94
81/91
1.90
(b)
0.04
0.03
0.02
0.01
0.00
1.2
1.6
2.0
2.4
Electronic transition energy (eV)
31
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
4
inclusion of the  s 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
32
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
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.
33
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 metalmolecule 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
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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