In this work, a Raman bond model that partitions the Raman intensity to interatomic charge flow modulations or Raman bonds is extended from the static limit to frequency dependent cases. This model is based on damped response theory and, thus, enables a consistent treatment of off-resonance and resonance cases. Model systems consisting of pyridines and silver clusters are studied using time dependent density functional theory to understand the enhancement mechanisms of surface-enhanced Raman scattering (SERS). The Raman bonds in the molecule, the inter-fragment bond, and the cluster are mapped to the enhancement contributions of the molecular resonance mechanism, the charge transfer mechanism, and the electromagnetic mechanism. The mapping quantifies the interference among the coupled mechanisms and interprets the electromagnetic mechanism as charge flow modulations in the metal. The dependence of the enhancement on the incident frequency, the molecule–metal bonding, and the applied electric field is interpreted and quantified. The Raman bond framework offers an intuitive and quantitative interpretation of SERS mechanisms.
Surface-enhanced Raman scattering (SERS) relies on large signal enhancements due to plasmon excitations and has been shown to be able to detect single molecules with high specificity.1–7 SERS has also been extended to other analytic techniques such as tip-enhanced Raman scattering (TERS),8–11 which has offered single molecule images with subnanometer resolutions.12,13 The plasmon resonance amplifies the local fields at the surface of metal structures, which enhances Raman signals, and the mechanism is classified to be electromagnetic (EM).14–19 Other enhancement mechanisms, such as the molecule–metal bonding, the charge transfer resonance (CT), and the molecular resonance (RRS), are grouped to be chemical (CM).20–27 Enhancements ranging from 105 to 1010 can be reached within the gaps between nanoparticles.28–33 Classic electrodynamics models show that the EM enhancements increase as the gaps close,34–38 but the EM enhancements can decrease when the gap is in the subnanometer scale where quantum mechanical effects such as the spill-out of electrons and tunneling become important.39–45 To correctly interpret SERS enhancements, it is crucial to describe EM and CM consistently especially when both CM and EM are important.46–49 Although we often discuss the two mechanisms separately, they are in fact coupled and understanding the synergy between them is important.
To consistently describe EM and CM, it is necessary to include quantum mechanical effects using electronic structure methods.26,27,50–58 Most of the studies using electronic structure methods focus on understanding CM using molecules interacting with either metal clusters26,27,50–54 or periodic slabs.55–58 Quantum mechanical methods that incorporate both CM and EM have been used to describe SERS.50,53,54 The cluster models have provided key insights into the SERS enhancement mechanisms but are restricted to small clusters due to high computational cost. It also remains challenging to determine the relative contributions of the different enhancement mechanisms using electronic structure simulations. One approach is to quantify the contributions of CT, EM, and RRS by the ratios of the enhancements at the charge transfer resonance, the plasmon resonance, and the molecular resonance vs the enhancement at the static limit, respectively.50 While this approach is simple, it assumes that different types of resonances (molecular, charge transfer, or plasmonic) are well separated in energy, and therefore, the coupling among different mechanisms is weak. An analysis of the different enhancements71 has also been done by using an orbital-based partitioning of the Raman intensity into molecule and surface contributions. This work found the largest contributions from the molecule with no significant contributions from the surface. As is common in orbital-based partitioning schemes, the results depend on the origin and on the specific basis set employed, and thus, care must be taken in the analysis. More recently, a semi-empirical approach based on Intermediate Neglect of Differential Overlap / Singles Configuration Interaction (INDO/SCI) has been applied to decompose the enhancements into contributions from CM and EM.54 The CM contribution was quantified by neglecting the overlap of orbital integrals between Ag and other elements in the simulations. This approach was used to gain insights into the importance of charge transfer excitations under external bias, but it is hard to be generalized to other electronic structure methods.54
Another difficulty in using electronic structure methods to understand SERS is the lack of an intuitive interpretation of EM using fully quantum mechanical methods. EM enhancements are often approximated as |E|4, where E is the local field enhancement.22,59,60 However, it has been shown theoretically that atomic features on metal nanoparticles can result in highly confined fields where the magnitude of the local field can vary greatly over the space of a few nanometers.4,28,61–63 For these highly confined fields, the electric field gradient can contribute significantly to the enhancement and the traditional selection rules of Raman scattering break down.2,64–67 Therefore, the results obtained using electronic structure methods cannot directly be interpreted in terms of the classical |E|4 enhancement since the small clusters used in the simulations naturally contain such atomic features. The advantage of electronic structure methods is that CM and field gradient effects are fully accounted for in the simulations, but it becomes important to develop new ways of interpreting the results.
In this work, we present a Raman bond model (RBM) for analyzing the frequency dependent Raman spectra of molecules interacting with metal clusters. The model is an extension of a RBM recently proposed to analyze the CM contribution to SERS.68 The RBM partitions the Raman intensities into Raman bonds based on interatomic charge flow modulations obtained from a Hirshfeld analysis69 of the induced density. Here, this model is combined with damped response theory and a short-time approximation to consistently treat off-resonance and resonance Raman simulations.70 Using a model system consisting of a pyridine interacting with a Ag20 cluster, we show how the Raman bonds in the molecule, the inter-fragment bond, and the metal cluster can be mapped to the enhancement contributions of RRS, CT, and EM. This mapping enables the different enhancement contributions of SERS to be quantified. Furthermore, we show that EM in electronic structure simulations can be interpreted as charge flow modulations in the metal. As a further illustration of this model, we use it to interpret how the SERS enhancement mechanisms depend on external electric fields.
To understand the frequency dependent Raman spectra of molecules interacting with metal clusters, we will adopt a time-dependent density functional theory (TDDFT) method, which uses a short-time approximation to evaluate the Raman scattering cross section.70,72 This short-time approximation makes it possible to calculate both normal and resonance Raman intensities from the geometrical derivatives of the frequency-dependent (complex) polarizabilities.
RBM partitions the polarizability derivative vs the vibrational mode ∂αab/∂Qk as68
where −rb is the electronic dipole operator in direction b and Ri,b is the coordinate of atom i in direction b. δρi,a is the induced electron density for atom i caused by the external field in direction a, which is calculated by adopting Hirshfeld partitioning.69,qij,a is the charge flow between atoms i and j induced by the external field in direction a, which is calculated by adopting the LoProp method.73 The use of the LoProp method ensures that the results are independent of the origin. The first term on the right-hand side of Eq. (1) describes the vibrational modulation of the induced atomic electron densities, which corresponds to the atomic contributions to the Raman intensity. The second term on the right-hand side of Eq. (1) describes the vibrational modulation of the interatomic charge flows, which corresponds to the bond contributions to the Raman intensity. The bond contributions are dominant and denoted as the Raman bonds.
At the static limit, the imaginary parts of the polarizability derivatives are negligible and the atomic and bond contributions are real numbers. When the incident frequency is larger than zero, the atomic and bond contributions are complex, and thus, the analysis needs to account for both real and imaginary contributions. To facilitate this, we will use that ∂αab/∂Qk can be defined as a vector, Rtotal, on the complex plane. The Raman intensity is determined by the magnitude of Rtotal, which we will denote as ptotal. Similarly, the atomic and bond contributions can also be defined as vectors Ratom and Rbond, respectively. The interference among the individual Raman atoms and bonds can be characterized by the phases of the Raman vectors. The phases of Ratom and Rbond can vary from −180° to 180° continuously in frequency dependent cases but are discrete (−180° or 180°) at the static limit.
To simplify the analysis, the individual contributions Ratom and Rbond can be grouped based on their spatial distributions within a SERS model system. In a SERS model system consisting of a molecule and a metal cluster, we showed previously68 that it is convenient to group the contributions as Rmol, Rclu, and Rinter, which correspond to the molecular, the cluster, and the inter-fragment contributions to the Raman intensity, respectively. Since the individual contributions are additive, Rmol + Rinter + Rclu = Rtotal is fulfilled. It is important to point out that the Raman intensity is determined by the square of Rtotal, and thus, there is interference among the individual terms. However, the analysis can be further simplified by projecting the grouped contributions Rmol, Rinter, and Rclu to the total vector Rtotal as
where pmol, pinter, and pclu are the group projections that can be used to quantify the contributions of each of the terms. Again, the projection ensures that the contributions are additive, and thus, pmol + pinter + pclu = ptotal is fulfilled.
III. COMPUTATIONAL DETAILS
All calculations in this work were performed using a local version of the Amsterdam density functional (ADF) program package.74,75 The Becke–Perdew (BP86) XC-potential76,77 and triple-ζ polarized slater type (TZP) basis set with large frozen cores from the ADF basis set library were used. The scalar relativistic effects were accounted for by the zeroth-order regular approximation (ZORA).78 For the systems in this work, full geometry optimization and frequency calculations were performed. The vibrational frequencies and normal modes were calculated within the harmonic approximation. Polarizability calculations were performed using the AOResponse module70 with the Adiabatic Local Density Approximation (ALDA). The phenomenological damping parameter is Γ = 0.004 a.u. in this work. The polarizability derivatives were calculated by numerical differentiation with respect to the normal mode displacements. For any system in this work, the molecule–cluster axis was aligned with the x-axis and only the xx components in the polarizabilities were considered. All Raman bond figures, in this work, were plotted using PyMOL.79
IV. RESULTS AND DISCUSSION
To demonstrate how RBM can be used to understand the different enhancement mechanisms of SERS, we will consider a model system consisting of a pyridine molecule (Py) bound to the surface of a tetrahedral Ag20 cluster (S-complex). Previous work has shown that this is a simple model system for understanding the enhancement mechanisms of SERS using first-principles simulations.50 In the following, we will show how the Raman bond projections pmol, pinter, and pclu can be used to gain insights into the different enhancement mechanisms.
In Fig. 1(a), we plot ptotal of Py vs the vibrations at the static limit. For the S-complex, we plot the frequency dependent ptotal obtained at 0.00 eV, 2.35 eV, 3.45 eV, and 6.20 eV in Figs. 1(b)–1(e), respectively. These frequencies have been chosen as they correspond to different types of resonances in the S-complex. The different types of resonances of the S-complex are fairly well separated in energy, and the previous work50 assumed that the coupling among different mechanisms is weak. At 0.00 eV, the S-complex is far from any resonance and we can learn about the SERS mechanism at the static limit, which results from changes to the electronic structure of the molecule when adsorbed on the metal cluster.19,21,27,51 Previously, we showed how RBM can be used to understand this enhancement mechanism as charge flow modulations across the molecule–metal interface.68 In the S-complex, the lowest strong charge transfer excitation occurs at 2.35 eV and corresponds to the transition from the highest occupied orbital (HOMO) of Ag20 to the lowest unoccupied orbital (LUMO) of the Py. Therefore, calculations of the Raman scattering at this energy should probe the enhancement from the charge transfer excitation. However, it is important to note that there are several weaker transitions within the silver cluster around this energy, which could also contribute to the enhancement. The strong transitions in the silver cluster are located around 3.45 eV and correspond to a superposition of many single-particle transitions and, thus, plasmon-like in nature. It is, therefore, expected that the excitation of these strong transitions in the silver cluster will reflect the EM mechanism. Finally, a strong molecular transition in the Py is located at 6.20 eV and calculations of the Raman scattering at this energy should be able to probe the RRS mechanism.
As shown in Fig. 1, the Raman spectrum of the S-complex depends strongly on the incident frequency used in the simulations. At the static limit, we see that the mode at 1462.04 cm−1 is strongly enhanced when the Py is adsorbed on the cluster. In addition, the relative intensities of the two ring-breathing modes around 1000 cm−1 are reversed. Going on, resonance with the charge transfer excitation leads to more significant spectral changes. In particular, the two strongest modes are now the modes at 1200.11 cm−1 and 1576.60 cm−1 instead of the two ring-breathing modes that are normally characteristic of the Raman spectrum of pyridine. At the strong cluster excitation, the ring-breathing modes are enhanced, and the Raman spectrum of the S-complex is dominated by four modes at 606.76 cm−1, 984.97 cm−1, 1017.45 cm−1, and 1576.60 cm−1. Finally, at the molecular resonance at 6.20 eV, the Raman spectrum looks similar to the Raman spectrum at the charge transfer resonance, although there are differences in the spectra particularly around the ring-breathing modes. The fact that these two spectra are similar is expected since they both involve excitations into the LUMO of Py.
To understand the enhancements and the spectral changes at different frequencies, we will focus on the ν1 ring-breathing mode that is strong at all frequencies. The Raman enhancement for this mode is a factor of 4 at the static limit, around a factor of 60 at the charge transfer resonance, around a factor of 7000 on resonance with strong silver transitions, and about a factor of 2000 around the pyridine excitation. To explain the dependence of the enhancement on the incident frequency, we can use the RBM partitioning. The vibrational pattern of mode ν1 in the S-complex is shown in Fig. 2(a). Ratom and Rbond at the static limit, the charge transfer resonance, the cluster resonance, and the molecular resonance are plotted in Figs. 2(b)–2(e), respectively. The phases of Ratom and Rbond are color coded using the color scheme shown in Fig. 2(f). The magnitudes of Ratom and Rbond are represented by the volumes of spheres and cylinders, respectively. Although all Rbond and Ratom are calculated and considered in the quantitative analysis, to keep the Raman bond plots in Fig. 2 clear, only the largest 70 Rbond are plotted. All Ratom are plotted.
At the static limit, the small enhancement is explained by the weak Raman bonds in the cluster. The weak molecule–metal bonding in the S-complex does not effectively connect the charge flows across the molecule–metal interface and the charge flows in the cluster cannot respond effectively to the charge flow modulations in the molecule. At the charge transfer resonance, the Raman bonds outside of the molecule, especially the inter-fragment Raman bond, are enhanced. The charge flow in the inter-fragment bond is enhanced at the charge transfer resonance and better connects the charge flows across the molecule–metal interface. The charge flows outside of the molecule can respond more effectively to the charge flow modulations in the molecule. At the cluster resonance, the Raman bonds in the cluster are drastically enhanced. The charge flows in the cluster are enhanced drastically due to cluster resonance and respond much more to the charge flow modulations in the molecule. At the molecular resonance, the Raman bonds in the molecule are predominantly enhanced. The charge flows in the molecule are enhanced due to molecular resonance, which leads to more charge flow modulations. The enhancement is the largest at the cluster resonance, which indicates that the polarizability of the cluster serves as a large reservoir of charge flows and a large enhancement can be obtained if the vibration manages to modulate the charge flows in the cluster.
By applying the interpretation constructed above to other vibrations, RBM is able to explain SERS selection rules. In the S-complex at the static limit, the charge flows outside of the molecule do not respond effectively to the charge flow modulations in the molecule due to poor charge flow connectivity, which explains the minor change of the spectral signatures in Fig. 1(b) compared with Fig. 1(a). At specific resonance conditions, the vibrations that can effectively modulate the most enhanced charge flows will have large Raman intensities. Thus, the spectral signatures change significantly in Figs. 1(c)–1(e) compared with Fig. 1(a). Some vibrations such as mode ν8a (1576.60 cm−1 in the S-complex) modulate the charge flows more globally and, thus, gain extra intensities over other modes at resonance conditions.
The Raman bond patterns shown in Fig. 2 at different frequencies generally reflect the different enhancement mechanisms of SERS based on the types of resonances involved. Maybe, the most interesting case is on resonance with the silver transitions. Here, the Raman bonds in the cluster are the largest and largely in phase, reflecting the large Raman enhancement at this frequency. The enhancement at the cluster resonance is usually related to the contribution of EM, which is interpreted using classic electrodynamics as the amplified local field amplifying the polarizability derivative of the molecule. Therefore, it is not obvious how the large Raman bonds in the cluster are related to the enhanced local field typically associated with the EM mechanism. To understand this further, we will consider a simple model for the EM enhancement of SERS that consists of treating the molecule and the nanoparticle as two polarizable dipoles.21,59 The total polarizability for such a system parallel to the molecule–nanoparticle axis is given by
where αM and αNP are the isotropic polarizabilities of the molecule and the nanoparticle, respectively. R is the separation distance between the molecule and the nanoparticle. It should be noted that in this model, the chemical bonding between the molecule and the nanoparticle is not included. Thus, the polarizability of the inter-fragment bonds is not considered in Eq. (3) and the charge transfer between the two subsystems is neglected. The polarizability derivative vs the vibrational mode Qk is80
where the factor scaling ∂αM/∂Qk can be denoted as |E∥|2 that is the squared local field enhancement in the parallel direction if the image field contribution in the denominator is ignored.21 The Raman intensity is proportional to the squared total polarizability derivative, which leads to the familiar |E|4 enhancement factor.
Alternatively, the polarizability in Eq. (3) can be partitioned as
where VM and VNP are the volumes of the molecule and the nanoparticle, respectively. Here, we have neglected the denominator in Eq. (3), which accounts for the image field effect.21 The term 4αMαNP/R3 describes the interaction between the molecule and the nanoparticle, which is partitioned based on the relative volumes of the molecule and the nanoparticle. This partitioning based on the volumes of the fragments is consistent with Hirshfeld partitioning used in RBM.
Effective polarizabilities of the molecule and the nanoparticle can then be defined as
In this way, we can write the polarizability derivative as
where the first term on the right-hand side can be mapped to Rmol and the second term can be mapped to Rclu in RBM. Since the partitioning is based on the relative volumes of the two subsystems, the second term will dominate the response. Therefore, we can see that when there is no charge transfer, we can map the EM mechanism onto Rclu in RBM. At the cluster resonance, Rclu or is dominant, and thus, the amplified local field can be interpreted as the Raman bonds induced in the cluster.
Because EM also contributes to the enhancements at other incident frequencies, the Raman bonds in the cluster can be mapped to and quantify the contributions of EM at other incident frequencies even when the Raman bonds in the cluster are not dominant. However, we should note that the Raman bonds in the cluster can also be induced by the charge transfer between the molecule and the cluster. The Raman bonds in the cluster can only be mapped directly to the local field when the charge transfer is small. As the Raman bonds in the cluster are mapped to the contribution of EM, the Raman bonds in the molecule and the inter-fragment bond can also be mapped to the contributions of RRS and CT. The mapping is consistent with the Raman bond patterns shown in Fig. 2 that the most enhanced Raman bonds at the molecular resonance, the charge transfer resonance, and the cluster resonance are the Raman bonds in the molecule, the inter-fragment bond, and the cluster, respectively.
Although Figs. 2 showing the Raman bonds and atoms provide an intuitive illustration of the SERS mechanisms, the main advantage of RBM is the ability to quantify the individual contributions to the enhancements. To illustrate this, we will quantify the dependence of the enhancement on the molecule–metal bonding. A model system that consists of a pyridine binding on the vertex of a tetrahedral Ag20 cluster (V-complex) is compared with the S-complex. The N–Ag bond length is shorter in the V-complex than in the S-complex, and thus, the charge flows in the V-complex are better connected across the molecule–metal interface. The percentages of pmol, pinter, and pclu of mode ν1 in the S-complex and the V-complex are shown in Figs. 3(a) and 3(b), respectively.
For the S-complex, this analysis shows that CT contributes 60% of the enhancement at the charge transfer resonance, EM contributes 100% of the enhancement at the cluster resonance, and RRS contributes 61% of the enhancement at the molecular resonance. This is consistent with the Raman bond pictures discussed above but also highlights the interference among the different enhancement mechanisms. In the V-complex, the charge transfer resonance shifts to 1.60 eV due to better charge flow connectivity. The RRS contribution decreases from 61% to 51% at the molecular resonance or from 93% to 77% at the static limit. At the charge transfer resonance, the CT contribution decreases from 60% to 26%. The lower RRS or CT contribution is due to stronger mixing of the molecular states with the cluster states, which results in a stronger contribution of the Raman bonds in the cluster. Likewise, at the cluster resonance, the EM contribution decreases from 100% to 78% due to this increased coupling. The reason is that the better charge flow connectivity in the V-complex enables the charge flows across the system to respond more effectively to the local charge flow modulations induced by the molecular vibration. This, in turn, reflects that in the V-complex, the Raman bonds are enhanced across the whole system, whereas the Raman bonds are enhanced more locally in the S-complex.
The enhancements at other incident frequencies can also be interpreted as the interference among RRS, CT, and EM, rather than explaining based on the resonance at discrete frequencies. pmol, pinter, and pclu of mode ν1 in the S-complex and V-complex vs the incident frequency from 0 eV to 6.50 eV are plotted in Figs. 3(c) and 3(d), respectively. The continuous change of the enhancement vs the incident frequency is interpreted by the evolution of the Raman bond pattern, which is simplified as the profile of the three components pmol, pinter, and pclu. The profile of the three components in the S-complex or V-complex forms a platform in the low frequency range until the charge transfer resonance is excited. The peaks corresponding to the charge transfer resonance (CT peaks) are around 2.35 eV for the S-complex and 1.60 eV for the V-complex. The largest enhancement is obtained when the cluster resonance is excited. The peaks corresponding to the cluster resonance (EM peaks) are around 3.40 eV for the S-complex and 3.50 eV for the V-complex. The profile fluctuates in the high frequency range and a large enhancement is obtained when the molecular resonance is excited. The peaks corresponding to the molecular resonance (RRS peaks) are around 6.20 eV for the S-complex and 6.25 eV for the V-complex.
An interesting point is that although the overall enhancements at the cluster resonance are similar for the two complexes, the contribution of pclu is smaller in the V-complex as compared to the S-complex. This indicates that the EM contribution decreases when the charge flow connectivity across the molecule–metal interface is improved. The drop in the EM contribution as the charge-flow connectivity increases is consistent with the reduction of the local field in nanoparticle dimers with very small gaps.40,44,45 The overall enhancement stays similar because pmol and pinter are larger in the V-complex than the S-complex, which indicates that the CM contribution becomes more significant due to better charge flow connectivity and compensates the reduction of the EM contribution. Therefore, a reduction in the EM contribution does not necessarily mean a reduction in the Raman intensity, which could explain the increased Raman intensities seen in single molecule TERS with very small tip-molecule separations.12,13
As an additional application of RBM, we will quantify the dependence of the enhancement on external electric fields. Historically, external bias has been used to examine the role of charge transfer resonance in SERS.81–83 The SERS relative intensities being a function of electric bias have often been interpreted as the electric bias tuning the charge transfer resonance frequency.52–54,58 The recent TERS studies have renewed the interest in understanding how the SERS enhancement mechanisms depend on the external bias.52,54 In Fig. 4(a), we plot ptotal of mode ν1 in the S-complex as a function of the incident frequency under different electric fields. The electric fields are applied in the x direction, which is along the molecule–metal axis. The positive direction is from the cluster to the molecule. The effect of electric fields on the geometry or the vibrational pattern is not considered. At the ground state, when no electric field is applied, the charge (0.056 e) transfers from the pyridine to the cluster. The negative fields allow less charge to transfer, while the positive fields allow more charge to transfer from the pyridine to the cluster. We find that the CT peak is red shifted by the negative fields and blue shifted by the positive fields. The Raman intensity at the CT peak is enhanced by the negative fields and reduced by the positive fields. The individual contributions to the Raman intensities at the CT peaks under different electric fields are plotted in Fig. 4(b). At the charge transfer resonance, compared with the Raman intensity under no electric field, the larger enhancements under the negative fields can be explained by the increases of pclu and pmol. Meanwhile, the smaller enhancements under the positive fields are explained by the reduced pclu and pinter. By perturbing the charge transfer by a small amount, the negative fields increase, while the positive fields decrease the charge flow connectivity. Thus, under the negative fields, the charge flows across the system are easier to modulate, while under the positive fields, the charge flows outside of the molecule are harder to modulate.
We find that the EM peak is slightly blue shifted by the negative fields and slightly red shifted by the positive fields, but the Raman intensity at the EM peak is decreased under both types of fields. In Fig. 4(c), we plot the individual contributions to the Raman intensities at the EM peaks under different electric fields. At the cluster resonance, the smaller enhancements under the negative or positive fields can be explained by the reduced pclu. The inter-fragment Raman bond is enhanced by the positive fields, while it is reduced by the negative fields, which indicates that the positive fields increase, while the negative fields decrease the charge flow connectivity across the molecule–metal interface. Thus, the decreased EM contribution under the positive fields can be interpreted as the reduction of the local field enhancement due to improved charge flow connectivity, which has also been shown in the comparison between the S-complex and the V-complex. Meanwhile, the decreased charge flow connectivity under the negative fields reduces the molecule–metal interaction and makes the charge flows in the cluster respond less effectively to the charge flow modulations in the molecule, which explains the decreased EM contribution under the negative fields. In other words, achieving the largest EM contribution requires an optimal charge flow connectivity. Increasing or decreasing the charge flow connectivity from the optimal value will reduce the EM contribution. It should be noted again that the effect of electric fields on the geometry or the vibrational pattern is not considered here and the conclusion of the optimal charge flow connectivity is valid for a fixed geometry and vibrational pattern.
In this work, we have presented a frequency dependent Raman bond model that partitions the Raman intensity to interatomic charge flow modulations or Raman bonds. The frequency dependent Raman scattering is obtained using damped response theory, which enables both off-resonance and resonance cases to be modeled. To understand the enhancement mechanisms of surface-enhanced Raman scattering (SERS), the Raman bond model was used to interpret the TDDFT simulations of a model system with a pyridine interacting with a small silver cluster. We show how the Raman bonds in the molecule, the inter-fragment bond, and the metal cluster are mapped to the enhancement contributions of RRS, CT, and EM, respectively. The mapping quantifies the interference among RRS, CT, and EM at any incident frequency and interprets EM in electronic structure simulations as charge flow modulations in the metal. We find that the EM enhancement is strongly affected by the charge flow connectivity between the molecule and the cluster. The consistent and quantitative interpretation potentially offers new insights into the SERS applications where both CM and EM are important such as quantifying the effect of tunneling on the single molecule images obtained by tip-enhanced Raman scattering.
See the supplementary material for details on the color-coding of the Raman bond phases and Raman bond patterns for the active modes of the S-complex, the excitations of S-complex near the charge transfer resonance frequency, and the comparison between the Raman scattering of the xx component and the orientational average Raman scattering.
The authors gratefully acknowledge financial support from the National Science Foundation, Grant No. CHE-1707657. Simulations in this work were conducted in part with the Advanced Cyber Infrastructure computational resources provided by the Institute for Cyber-Science at The Pennsylvania State University (https://ics.psu.edu/).
The data that support the findings of this study are available from the corresponding author upon reasonable request.