We present a first-principles model that partitions Raman intensities to atomic and bond contributions. This framework allows us to interpret the chemical mechanism in surface-enhanced Raman scattering (SERS) as interatom charge flow modulations, which we define as Raman bonds. Hirshfeld partitioning and charge density localization are applied to express polarizability derivatives as charge flow modulations. Model systems consisting of pyridines, thiols, and carbenes interacting with metal clusters are studied using time-dependent density functional theory. We demonstrate that the mode-specific enhancements can be explained as Raman bonds conjugated across the molecule-metal interface. We also illustrate that the changes in Raman intensities induced by electric fields or chemical substitutions can generally be interpreted as changes of charge flows. The model is shown to work consistently for different types of molecule-metal bonds. Furthermore, our work shows that increasing the Raman bond conjugation across the interface leads to stronger chemical enhancements. The Raman bond model developed in this work provides a quantitative and intuitive interpretation of the chemical mechanism in SERS.
Surface-enhanced Raman scattering (SERS) has been widely applied as a single-molecule technique in chemical and biological detection.1,2 Most of the enhancements can be attributed to the amplified local fields arising from the surface plasmon.3,4 Distinguished from other single-molecule techniques, SERS provides vibrational fingerprint information from spectral signatures. This advantage enables SERS to monitor chemical reactions5,6 or determine molecular adsorption conformations.7 The vibrational fingerprint information has been directly visualized in high resolution images using tip-enhanced Raman scattering (TERS),8 which combines SERS with scanning probe techniques.9–11 The spectral signatures of SERS are mainly determined by the chemical interactions between molecules and surfaces.12 Therefore, to extract the vibrational information, correct interpretations of the chemical mechanism in SERS are required.
The chemical mechanism is usually studied by modeling molecules interacting with metal clusters13,14 or periodic slabs15 using electronic structure methods. The interpretations are usually based on electronic structures. However, these interpretations are restrained to certain resonance conditions or types of molecule-metal bonds and offer semiquantitative analyses. Molecular resonance and charge transfer resonance have been proposed to contribute to the chemical enhancements by coupling to the surface plasmon.14,16 The chemical mechanism is interpreted as vibrational modulations on the excitation energy and the transition dipole of the resonance. The enhancement specificity of vibrational modes can be explained as different modulations on molecular orbitals due to atomic motions.15,17 However, it is difficult to quantify the orbital changes. The mode-specific enhancements have also been explained based on the symmetry of electronic and vibrational transitions,18 but this interpretation presumes molecular surface configurations and only works for weakly bonded systems. At off resonance conditions, a large number of transitions are expected to contribute to the chemical enhancements. The chemical enhancements can be approximately explained by only comparing the excitation energies of the low-lying charge transfer states with the molecular HOMO-LUMO gaps.19–22 However, a quantitative interpretation of the mode specific enhancements is still missing. To quantitatively describe the chemical mechanism in SERS, case-by-case studies using electronic structure methods are required because the chemical enhancements are system and mode dependent. Therefore, a general and intuitive framework is desired for interpreting simulation results.
In this work, a framework called Raman bond model (RBM) is presented to interpret the chemical mechanism in SERS. Raman bonds are defined as charge flow modulations between atoms. Contributions to Raman intensities are assigned to individual bonds and directly correlated with vibrational modes. We find in general that the mode-specific enhancements can be interpreted as Raman bonds conjugated across the molecule-metal interface and increasing this conjugation leads to stronger chemical enhancements.
An element of the polarizability tensor αab can be written as
where −rb is the electronic dipole operator in direction b and δρa is the induced electron density caused by the external field in direction a. The induced density δρa can be partitioned to atomic contributions δρi,a, which can be calculated as
where Wi is the Hirshfeld partitioning weight23 for atom i. αab can be rewritten as
where Ri,b is the nuclear coordinate for atom i in direction b. The first term on the right side describes the atomic contributions to the polarizability. The integral of the atomic induced density in the second term is the induced atomic charge qi,a,
The value of the second term in Eq. (3) depends on the choices of origin. The origin dependence can be removed by allocating atomic induced charges to charge flows between atoms,
where qij,a is the charge flow between atom i and j caused by the external field in direction a. To obtain the charge flows, we adopt the Loprop method24 where the objective function L is minimized using the method of Lagrange multipliers,
where λ is the Lagrange multiplier applying the constraint that the atomic charges are not altered due to the allocation. f(Rij) is a penalty function of the distance between atom i and j (Rij) limiting long-range charge flows. The penalty function proposed in the Loprop method is
where is the covalent radius for atom i25 and c is a coefficient tuning the penalty. To limit long-range charge flows, sufficient penalty is required. However, to reproduce the polarizability, excessive penalty needs to be avoided. In SERS studies, using one coefficient c is difficult to achieve this balance because the polarizabilities vary significantly from molecules to metal clusters. Therefore, in this work, the penalty function is modified,
c = 2 is applied to all the systems studied in this work. Charge flows are limited to short ranges, and the polarizabilities are reproduced. This specific choice for the penalty function is discussed in detail in the supplementary material. The partitioning of the polarizability αab is then expressed as
which is now origin independent. Because Raman intensities are proportional to squared polarizability derivatives with respect to vibrational modes Qk, the chemical mechanism in SERS can be interpreted by analyzing the polarizability derivative,
The first term in the above equation describes the changes in the atomic induced charge densities, which is denoted as Ratom. The second term is defined as Raman bonds (Rbond). It describes the modulations of charge flows between atoms. We find that this term is the dominant contributions to Raman intensities, which we will discuss in detail later in this article.
III. COMPUTATIONAL DETAILS
All calculations in this work were performed using a local version of the Amsterdam density functional (ADF) program package.26,27 The Becke-Perdew (BP86) XC-potential28,29 and an all-electron triple-ζ polarized (TZP) slater type basis set from the ADF basis set library were used. The scalar relativistic effects were accounted for by the zeroth-order regular approximation (ZORA).30 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 module31,32 at zero frequency with the Adiabatic Local Density Approximation (ALDA). Polarizability derivatives were calculated by numerical differentiation with respect to the normal mode displacements. For any system in this work, the molecule-cluster axis is aligned with the x-axis and only the xx components in the polarizabilities are considered in the analyses. All the figures in this work are plotted using PyMOL.33
IV. RESULTS AND DISCUSSION
The study of the chemical mechanism starts by interpreting the difference between enhancements of vibrational modes. The spectra of ∂αxx/∂Qk are plotted for a free pyridine (Py) and a pyridine on a Ag20 cluster (Py–Ag20) in Fig. 1(a). The enhancements of a ring breathing mode ν1 (976.07 cm−1 in Py and 996.17 cm−1 in Py–Ag20) and a symmetric ring deformation mode ν12 (1019.80 cm−1 in Py and 1025.30 cm−1 in Py–Ag20) are interpreted using the RBM.
In Py, modes ν1 and ν12 both belong to A1 symmetry in the C2v point group and have similar Raman intensities. Ratom and Rbond are plotted in Fig. 1(b) and represented as spheres and cylinders, respectively. The geometric volumes of the spheres and cylinders characterize the magnitudes of Ratom and Rbond. The details of the calculations are provided in the supplementary material. For both modes, the Raman bonds corresponding to the aromatic bonds are the main contributions to the Raman intensities. The reason is that the aromatic bonds provide major charge flows. The constructive Raman bonds are labeled red and the destructive Raman bonds are labeled blue, meaning that charge flows in different bonds are modulated with different phases. The distributions of constructive and destructive Raman bonds change from mode ν1 to ν12, indicating that the phases of bond deformation determine the phases of charge flow modulations. The interference between Raman bonds provides two general causes for weak Raman intensities: One is that the vibrations do not modulate charge flows effectively, expressed as weak Raman bonds, and the other is that the charge flow modulations cancel each other, expressed as symmetrically distributed constructive and destructive Raman bonds.
In Py–Ag20, mode ν1 is more enhanced than ν12 because mode ν1 has stronger Raman bonds and fewer destructive Raman bonds than ν12, which is shown in Fig. 1(c). The different enhancements result from different N–Ag Raman bonds. A more effective modulation on the charge flow in the N–Ag bond is exerted by mode ν1 than ν12 because of the larger N–Ag stretching at mode ν1. This is reflected as a stronger N–Ag Raman bond at mode ν1. Because the π backbonding between Py and Ag2019 connects the delocalized electrons in the fragments, the Raman bonds in the aromatic ring and Ag20 are conjugated via the N–Ag Raman bond. Here, we analogize the conjugation idea for chemical bonds to Raman bonds. The stronger N–Ag Raman bond at mode ν1 leads to a stronger Raman bond conjugation across the molecule-metal interface, which causes stronger Raman bonds and turns the destructive Raman bonds in the free pyridine to constructive. In contrast, the weaker N–Ag Raman bond at mode ν12 leads to a weaker Raman bond conjugation across the interface, which causes weaker Raman bonds and preserves the original Raman bond pattern in the free pyridine.
The RBM is also applied to interpret the influence of the interaction between Py and Ag20 on the Raman intensities. Electric fields are widely applied to tune the molecule-metal interaction, and their effect on SERS enhancements has been studied theoretically at the Hartree-Fock level of theory.34 Electric fields are applied to Py–Ag20, and the spectra of αxx derivatives for Py–Ag20 with different electric fields are shown in the supplementary material. Here, we focus on the changes in the Raman intensity at a C–H in plane bending mode. The frequency of the C–H in plane bending mode is 1200.90 cm−1 in Py–Ag20, 1200.44 cm−1 in Py–Ag20 with the negative field, and 1203.67 cm−1 in Py–Ag20 with the positive field. The changes in its Raman intensity are interpreted using the RBM in Fig. 2. The Raman intensity is enhanced by the negative field and reduced by the positive field, reflected as stronger Raman bonds in Fig. 2(a) and weaker Raman bonds in Fig. 2(c). Because a common mode is compared, the Raman bond difference can be attributed to the charge flow changes caused by the applied fields. The negative field increases the interfragment charge flow, which increases the Raman bond conjugation across the molecule-metal interface and leads to stronger Raman bonds. In contrast, the positive field decreases the charge flows and the Raman bond conjugation which leads to weaker Raman bonds and destructive Raman bonds. The destructive Raman bonds recover the Raman bond pattern in the free pyridine, which means that the Raman bonds in the molecule are less affected by the Raman bonds in the cluster due to the decreased Raman bond conjugation.
To understand the effect of chemical substitutions on Raman intensities, chemical substitutions are applied to Py–Ag20 to tune the Raman intensity at the same C–H in plane bending mode. Py is substituted at the para site by a chlorine atom (PyCl–Ag20) or an amino group (PyNH2–Ag20). The frequency of the C–H in plane bending mode is 1200.87 cm−1 in PyCl–Ag20 and 1202.86 cm−1 in PyNH2–Ag20. Corresponding Ratom and Rbond are plotted in Figs. 3(a) and 3(c). The chlorine substituent enhances the Raman intensity and the Raman bonds become stronger. The amino substituent reduces the Raman intensity, reflected as weaker Raman bonds and destructive Raman bonds. The destructive Raman bonds recover the Raman bond pattern in the free pyridine. The different Raman bond patterns also result from the changes of Raman bond conjugation. The chlorine substituent increases the conjugation, leading to stronger Raman bonds, and the amino substituent decreases the conjugation, leading to weaker Raman bonds and destructive Raman bonds. Considering that the Raman bonds are changed similarly by chemical substitutions or electric fields, we propose that chemical substitutions and electric fields are equivalent in tuning Raman intensities by changing charge flows. To further demonstrate the equivalence, a negative field is applied to PyNH2–Ag20 at the same C–H in plane bending mode whose frequency is changed to 1204.67 cm−1. Ratom and Rbond are plotted in Fig. 3(d). Because of the applied field, Raman bonds in general become stronger and destructive Raman bonds become constructive. The Raman bond changes caused by the amino substituent are reversed by the applied field, which supports that chemical substitutions and electric fields are equivalent in tuning Raman intensities by changing charge flows.
In summary, the negative field or the chlorine substituent increased the interfragment charge flow, while the positive field or the amino substituent decreased the interfragment charge flow. However, considering electron transfer from Py to Ag20 when Py is adsorbed on Ag20, the negative field or the chlorine substituent was expected to pull electrons back to Py and decrease the interfragment charge flow. Similarly, the positive field or the amino substituent was expected to push electrons to Ag20 and increase the interfragment charge flow. To address this confusion, we consider the ground state charge transfer qGCT defined as the sum of atomic charges in the molecule at the ground state. qGCT is decreased by the negative field or the chlorine substituent, while increased by the positive field or the amino substituent, which is consistent with our expectation. The concept of “pushing or pulling electrons” refers to the electron behavior at the ground state. Although qGCT and charge flows are intrinsically correlated, the derivation from one to the other is not trivial and their relation varies from system to system.
To study Raman bond patterns for different types of molecule-metal bonds, a propanethiol molecule on a Ag19 cluster (A–Ag19) is studied. The structure is shown in Fig. 4(a). Ratom and Rbond at the S–Ag stretching mode (368.98 cm−1 in A–Ag19) are plotted in Fig. 4(b). The S–Ag Raman bond is dominant, and no significant Raman bonds are generated in the fragments, indicating that the S–Ag Raman bond is not conjugated with other Raman bonds in the fragments. The reason is that S–Ag is a σ bond19,35 and not conjugated with other bonds. To further demonstrate the lack of conjugation between the S–Ag Raman bond and other Raman bonds, a double bond is introduced to the C2–C3 (2E–Ag19) or C1–C2 (1E–Ag19) bond. Ratom and Rbond at their S–Ag stretching modes (330.57 cm−1 in 2E–Ag19 and 313.88 cm−1 in 1E–Ag19) are shown in Fig. 4(b). Because the S–Ag σ bond cannot conjugate with the introduced double bond, the Raman bond pattern is not changed significantly, regardless of the position of the double bond.
Considering that the double bond has the capability of conjugating with the Ag–Ag bonds in the cluster, it is expected that stretching the double bond will cause conjugated Raman bonds. A stronger Raman bond conjugation is thus expected in 1E–Ag19 than in 2E–Ag19 as the double bond is closer to the cluster. Ratom and Rbond in 2E–Ag19 and 1E–Ag19 at their double bond stretching modes (1635.03 cm−1 in 2E–Ag19 and 1585.39 cm−1 in 1E–Ag19) are shown in Fig. 4(c). The Raman bonds outside the thiol molecule in 1E–Ag19 are stronger at the double bond stretching mode than at the S–Ag stretching mode. This means that the vibrations establishing the Raman bond conjugation across the molecule-metal interface result in stronger Raman bonds from the cluster and have larger Raman intensities. The comparison of the Raman bonds between 2E–Ag19 and 1E–Ag19 at their double bond stretching modes shows that a larger Raman intensity can be obtained by increasing the Raman bond conjugation across the interface.
To further demonstrate the importance of the Raman bond conjugation to the chemical enhancements, the contributions to the Raman intensity from the molecule (Rmol), the interfragment bonds (Rinter), and the cluster (Rcluster) are compared between 2E–Ag19 and 1E–Ag19. The contributions are averaged over the vibrations in the range from 300 cm−1–2000 cm−1 because 2E–Ag19 and 1E–Ag19 have different vibrational modes. The ratio of the averaged contributions ⟨Rmol⟩:⟨Rinter⟩:⟨Rcluster⟩ is 0.31:0.32:0.37 in 2E–Ag19. Normalized with respect to ⟨Rmol⟩ + ⟨Rinter⟩ + ⟨Rcluster⟩ of 2E–Ag19, ⟨Rmol⟩:⟨Rinter⟩:⟨Rcluster⟩ in 1E–Ag19 is 0.29:0.78:0.96. This means that the averaged Raman intensity of 1E–Ag19 is 4.1 times as large as that of 2E–Ag19. The main contributions to the larger Raman intensity in 1E–Ag19 are the S–Ag Raman bond and the Raman bonds in the cluster. The reason is that in 1E–Ag19, any molecular vibration stretching the double bond will generate stronger Raman bonds outside the thiol molecule than 2E–Ag19 because of the increased Raman bond conjugation in 1E–Ag19. This means that increasing the Raman bond conjugation across the molecule-metal interface leads to stronger chemical enhancements.
N-heterocyclic carbenes on gold surfaces are another group of SERS systems with strong chemical molecule-metal bonds. They are promising to provide better surface functionalizability than thiols because of their resilience under harsh chemical conditions.36 To understand the Raman bond behavior in this bonding environment, a system consisting of a N-heterocyclic carbene on a Au20 cluster (Cb–Au20) is studied using the RBM. The structure of Cb–Au20 is shown in Fig. 5(a), and the spectra of Rmol, Rinter, and Rcluster are plotted in Fig. 5(b). The spectra show that Rmol are the dominant contributions to the significant Raman peaks, indicating that major Raman bonds are located in Cb. The reason is revealed by the structure of Cb–Au20 where the gold atom connected to Cb is pulled out from the surface. The isolation of Cb makes the Raman bonds in Cb not conjugated with the Raman bonds in Au20, which leads to small Rcluster and Rinter. Another spectral pattern is the interference between Rinter and Rmol or Rcluster, showing that the C–Au Raman bond is not conjugated with the Raman bonds in Cb or Au20. The reason is that C–Au is a σ bond,36 which is also responsible for the lack of the Raman bond conjugation. Time-dependent density functional theory (TDDFT) studies of carbenes on gold surfaces employing the Au20 cluster offered Raman spectra that agreed with experiments.37 Because TDDFT tends to overestimate charge transfers between fragments, limited charge flow modulations between fragments are expected to alleviate the possible overestimation. This idea is supported by the small Rinter component in Fig. 5(b). Considering the small size of the Au20 cluster, the agreement also indicates that the charge flow modulations in the cluster are localized near the carbene. To represent the general Raman bond distribution in the cluster, Ratom and Rbond at the mode (1260.93 cm−1) with the largest Rcluster are plotted in Fig. 5(c). The weak and locally distributed Raman bonds in Au20 suggest that Au20 is sufficiently large to capture most charge flow modulations in the surface.
To obtain an intuitive and quantitative framework for interpreting the chemical mechanism in SERS, a Raman bond model was proposed in this work. Raman bonds are calculated from polarizability derivatives via Hirshfeld partitioning and charge density localization. Although, Hirshfeld partitioning was chosen for its basis independence and chemically intuitive results,38,39 other atomic charge models based on charge densities are also compatible and the limitations of Hirshfeld partitioning can, in principle, be mitigated using the iterative Hirshfeld scheme.40 Although charge flows are not uniquely determined given a set of atomic charges, a stable charge flow pattern can be achieved if long-range charge flows are limited and polarizabilities are reproduced. The applicability of this framework is shown for different types of molecule-metal bonds. We find that mode-specific enhancements can be interpreted as Raman bonds conjugated across the molecule-surface interface. The Raman bond conjugation can be established when molecular vibrations involve bonds conjugated with the bonds in clusters. Stronger chemical enhancements can be achieved by increasing the Raman bond conjugation, which can be induced by electric fields, chemical substitutions, and tuning molecular structures. The framework can also be potentially applied to quantify the effect of tunneling on TERS spectra41 and interpret SERS or TERS observations of electrochemical reactions.42 Facilitated by experimental Raman spectra, this framework can help us to understand charge transport in molecular electronic devices.43
See the supplementary material for the equations used to plot Ratom and Rbond. The spectra of αxx derivatives with respect to vibrational modes for Py–Ag20 with different electric fields are shown. Tests on the penalty functions are discussed.
The authors gratefully acknowledge financial support from National Science Foundation (Grant No. CHE-1707657). Simulations in this work were conducted in part with Advanced Cyber infrastructure computational resources provided by the Institute for Cyber-Science at the Pennsylvania State University (https://ics.psu.edu/).