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A full quantum mechanical approach assessing the chemical and electromagnetic effect in TERS

Tip-enhanced Raman spectroscopy (TERS) is a valuable method for surface analysis with nanometer to angstrom-scale resolution, however, the accurate simulation of particular TERS signals remains a computational challenge. We present a unique approach to this challenge by combining the two main contributors to plasmon-enhanced Raman spectroscopy and to the high resolution in TERS in particular, the electromagnetic and the chemical effect, into one quantum mechanical simulation. The electromagnetic effect describes the sample's interaction with the strong, highly localized and inhomogeneous electric fields associated with the plasmonic tip, and is typically the thematic focus for most mechanistic studies. On the other hand, the chemical effect covers the different responses to the extremely close-range and highly position-sensitive chemical interaction between the apex tip atom(s) and the sample, and, as we could show in previous works, plays an often underestimated role. Starting from a (time-dependent) density functional theory description of the chemical model system, comprised of a tin(II) phthalocyanine (SnPc) sample molecule and a single silver atom as tip, we introduce the electromagnetic effect through a series of point charges that recreate the electric field in the vicinity of the plasmonic Ag nanoparticle. By scanning the tip over the molecule along a 3D grid, we can investigate the system's Raman response on each position for non-resonant and resonant illumination. Simulating both effects on their own already hints at the achievable signal enhancement and resolution, but the combination of both creates even stronger evidence that TERS is capable of resolving sub-molecular features.


Tip-enhanced Raman spectroscopy (TERS) is a valuable method for surface analysis
with nanometer to angstrom-scale resolution, however, the accurate simulation of particular TERS signals remains a computational challenge. We present a unique approach to this challenge by combining the two main contributors to plasmonenhanced Raman spectroscopy and to the high resolution in TERS in particular, the electromagnetic and the chemical effect, into one quantum mechanical simulation. The electromagnetic effect describes the sample's interaction with the strong, highly localized and inhomogeneous electric fields associated with the plasmonic tip, and is typically the thematic focus for most mechanistic studies. On the other hand, the chemical effect covers the different responses to the extremely close-range and highly position-sensitive chemical interaction between the apex tip atom(s) and the sample, and, as we could show in previous works, plays an often underestimated role. Starting from a (time-dependent) density functional theory description of the chemical model system, comprised of a tin(II) phthalocyanine sample molecule and a single silver atom as tip, we introduce the electromagnetic effect through a series of point charges that recreate the electric field in the vicinity of the plasmonic Ag nanoparticle. By scanning the tip over the molecule along a 3D grid, we can investigate the system's Raman response on each position for non-resonant and resonant illumination. Simulating both effects on their own already hints at the achievable signal enhancement and resolution, but the combination of both creates even stronger evidence that TERS is capable of resolving sub-molecular features.

I. INTRODUCTION
Since its first experimental demonstration in the year 2000, tip-enhanced Raman spectroscopy [1][2][3] has become an immensely useful tool for the study of surface-bound molecules. Under certain conditions, even sub-molecular features can be resolved [4][5][6][7][8][9][10][11] , which was controversially discussed as this is way beyond the limit of conventional light microscopy. Similar to surface enhanced Raman spectroscopy (SERS) [12][13][14][15] , the basis of TERS is the surface plasmon generated by irradiating a nanoparticle (or nano-structured tip) close to the analyte. 16,17 The exact nature of the underlying mechanisms that enable such a high resultion are still under investigation, but are broadly categorized into two contributions 14 . The first of these is the electromagnetic effect [18][19][20][21][22][23][24] , describing how the highly inhomogeneous electromagnetic fields from the plasmonic tip interact with the sample and enhance the Raman response, especially in a so-called pico-cavity between the plasmonic tip and a possible metal substrate. [25][26][27][28][29][30] The other one being the chemical effect 23,31-36 , describing how the molecule's electronic structure changes while in proximity to the plasmonic particle. In TERS, this is especially relevant, since the scanning tip can be moved very precisely to different areas of the sample molecule, exhibiting site-specific responses to the changeing chemical environment.
Several studies focus heavily on the plasmon in different nanoparticle configurations and therefore mainly try to understand the electromagnetic effect, as published by the groups of Aizpurua 4,15,26,[37][38][39]Jensen 28,[40][41][42][43][44] . In particular, the discrete interaction model/quantum mechanics (DIM/QM) method 40,50,51 operates on a similar quantum chemical level of theory, and includes an atomistic model for the nanoparticle, however, no explicit metal atoms are present in the quantum chemical simulations. Most approaches typically assume a kind of 'medium-range' regime, where the plasmonic particle is sufficiently close to the sample to meaningfully enhance the Raman signal, but distant enough to neglect the chemical interactions. Since the electromagnetic effect is already relatively well-described with classical electrodynamics 52 , we follow a strategy that builds a model system from the chemical point of view.
Our quantum mechanical approach starts from a high-level simulation of the sample molecule and incorporates the plasmonic tip into those simulations as accurately as possible. To properly account for the chemical effect, a small part of the nanoparticle is fully incorporated into the quantum chemical simulations, in the form of the apex atom. Ex-tending this previously established 53-56 model, the electric field in vicinity to the plasmonic nanoparticle is added to these simulations through a series of point charges. This tip model (apex atom and field-creating point charges) is then scanned over the immobilized sample molecule, tin(II) phthalocyanine (SnPc), and allows the study of short-range chemical interaction between tip and molecule, as well as the influence of the much larger plasmonic nanoparticle on the whole system. Additionally, since both parts of the tip model are separable, we can evaluate and quantify the system's behaviour if only one of the two effects is active and therefore study how they influence each other.
The results of this new, purely quantum mechanical approach are presented in the following contribution. In the first part, the energy of the incident radiation is assumed to be far from any electronic excitations of the sample molecule, i.e. non-resonant conditions. Here, the chemical effect (no external point charges) and the electromagnetic effect (no explicit tip atom) are investigated separately first, and then how the full model system behaves, with respect to the two contributions. The second part focuses on resonant conditions, and how the included electric fields influence the molecular properties that are relevant for the system's resonant Raman response under the IMDHO model. While the same quantum chemical system (SnPc-Ag) has been investigated before 54 , the addition of the external electric fields proves to be a valuable extension of the model and provides further insight into the mechanism governing TERS.

A. Finite Element Simulations
The RF module of Comsol Multiphysics® 5.0 57 was used to solve Maxwell's equations in a finite element setup in order to obtain the field distribution in the vicinity of the silver tip. Here, the underlying equation that governs the electric field is: with the permittivity rc and the relative permeability µ r set to 1 and the wave number of the free space k 0 depending on the speed of light c 0 : The following computational setup is based on the procedure introduced by Trautmann et al. 27 The Simulations were performed in a cubic system of size 700x700x700 nm 3 with periodic boundary conditions (Floquet periodicity) at the side walls (x-and y-direction) and top and bottom walls (z-direction) defined as periodic port boundaries. For modelling the field distribution near the tip, a spherical nanostructure (radius 100 Å) above a flat surface, acting as substrate (glass, n glass = 1.520), was created. A substructure was modelled as solid of revolution from a Gaussian profile resembling a single atom that protrudes from the otherwise smooth surface, creating an atomic scale plasmonic substructure. The Gaussian profile was defined as: with χ = 2.11 Å determining the width of the Gaussian profile and ζ = 8.5 Å resembling the protrusion for a silver atom with an atomic radius of 1.65 Å. The gap between the substrate and the lowest point of the nanoparticle was set to z g = 10 Å. Material settings for the silver nanoparticle were taken from Ref. 58 .
The incident field, with an electric field strength of 5.142 × 10 8 V m −1 (10 −3 a.u.), 59 entered the system from the bottom port with an angle of θ = 60°, where the incident power was obtained as: with c 0 , 0 , n air = 1 being the speed of light, vacuum permittivity, and air refractive index, respectively. The minimal tetrahedral mesh size was set to 0.9 Å in a 100 nm sphere around the nanoparticle and 5 nm otherwise to reduce the computational cost.
The electric field distribution obtained by the finite element calculations was then mimicked by ten point charges -five above and five below the glass substrate at z = ±2, ±3, ±4, ±5, and ± 6 nm, respectively, see Fig. 1a)) (white dots). Two sets of these ten point charges were obtained to model the oscillating electric field at opposing field orienta-

B. Quantum Chemical Simulations
The simulation of the full TERS experiment follows our previously published [53][54][55] (TD)DFT-based protocol, with the inclusion of the above mentioned point charges to complement the tip model. All quantum chemical calculations were carried out with To elucidate the interplay of the chemical and the electromagnetic effect, in the SnPc-Ag±E system, two sub-systems were investigated as well. One without the field-creating point charges, SnPc-Ag, where only the chemical interaction between the Ag atom and the sample molecule are observed, and one without the Ag atom, SnPc±E, that only shows the molecule's response to the electric fields. The SnPc-Ag system, and with it the chemical effect, was discussed in-depth previously in Fiederling et al. 54 .
In line with previous studies, the detection of the Raman signal is assumed to be mostly in z-direction and therefore the intensity of each mode is dependent on the zz-component of the respective derivative of the (transition) polarizability tensor For the non-resonant case, the polarizability derivatives are calculated directly by the quantum chemistry package. In the resonant case, transition polarizability derivatives are required and are calculated using the independent-mode displaced harmonic oscillator model (IMDHOM) 66 Here, ∆ e,l is the dimensionless displacement, E e,g the vertical excitation energy from the ground-state g to excited-state e and Γ a damping factor describing homogeneous broadening (chosen as 3000 cm −1 =0.372 eV), while E L represents the energy of the irradiating laser. The function Φ e , neglecting Franck-Condon factors, is given by:

III. RESULTS
In this section, we present the simulated TERS response of the full SnPc

A. Non-Resonant
Applying the non-homogeneous electric field from the COMSOL calculations to the SnPc molecule without adding the Ag atom at the tip's apex barely affects the molecules TERS response, as Fig. 2 a) and e) shows. The integrated intensity slightly increases for the negative field direction, while the positive direction slightly decreases the intensity. In both cases, the variation in intensity over the whole map is small, but exhibits one common  hydrogen atoms are roughly 2.2 Å closer to the field-creating point charges above the tip than the central Sn atom, resulting in a 3 % deviation in field strength in z-direction.
For the other sub-system with only the tip atom and no external field, the response is very different. The overall intensity increases by roughly one order of magnitude and there are several equally bright, sub-molecular features. Those features correspond to three distinct regions over the molecule: the center directly around the Sn atom, the middle 'ring' on top of the alternating C-N atoms, and the edge above the C-H bonds.
Combining both sub-systems to form the full system with the Ag atom at the tip apex and the electric field (Fig. 2 b) and f)), signal intensity generally rises and at least some of the sub-molecular structures from Fig. 2 d) are still present. In case of the positive field direction, this increase in intensity is roughly one order of magnitude with regard to the chemical sub-system and two orders of magnitude with regard to the electrostatic subsystem. All three features from the field-free chemical sub-system are still present in the map, but the 'edge' and 'middle' areas show less intensity than the bright 'center' structure.
This corresponds well with the Ag-free case (Fig. 2 a), where the intensity in the center of the map is slightly higher. For the negative field direction, only the 'edge' structure remains bright, while 'middle' and 'center' vanish into the background. The maximum intensity increases barely by a factor of 2. Similar to the positive case, this fits the result from the respective Ag-free sub-system, where the center of the map exhibits a slightly lower intensity.
In both cases, the resulting maps for the whole system reveal that both effects are interacting heavily with each other, boosting the overall degree of detail and intensity.
The different behaviour of the three features under opposite field directions can be explained by looking at the field-induced partial charge on the Ag atom. For better visualization, the charges shown are relative to the charge at the 'middle' position in the respective sub-system, as the charge at this position is the least influenced by changing electric fields.
Additionally, the charges are normalized to the respective position's charge with no external electric field, leading to Fig. 2g). Here, it is evident that the 'center' structure is far more  Looking at the TERS spectra (Fig. 3) for the three selected positions and possible external electric fields, it is evident that in case of the 'middle' position ( Fig. 3b), e), and h)), the field's influence is relatively weak. The peak with the highest intensity belongs to the same normal mode in all cases, and its intensity stays rougly the same, regardless of the external field. However, small variations in the spectra are visible, such as the second-highest peak at 850 cm −1 slightly losing intensity in SnPc+E and gaining it in SnPc-E.
For the 'center' and 'edge' positions, the changes are much more pronounced. In the 'center', without field, three peaks are prominent in the TERS spectra (Fig. 3d)). For the SnPc-Ag+E case, those three peaks stay relevant, though their relative intensities change.
Furthermore, the overall intensity of the spectrum increases by more than one order of magnitude. In contrast, for the reversed field direction, SnPc-Ag-E, most of the spectrum stays at roughly the same intensity as without field, with two of the three prominent peaks mostly vanishing.
At the 'edge' position, similar phenomena are observed. For the positive field direction, the overall intensity grows, especially for peaks that were already dominating the spectrum, like the one at 1540 cm −1 . The negative field direction also shows a slight increase in the overall intensity in this case, mostly preserving the relative intensities across the spectrum compared to the field-free case.

B. Resonant
If the energy of the incident light matches that of a dipole-allowed electronic transition, the molecule's response is no longer governed by the (non-resonant) Raman effect, but rather by the resonant one. In this case, the signal intensity is proportional to the square of the derivative of the transition polarizability, see eq. 5. Therefore, according to eq. 6, two major contributors to a strong resonant Raman response are the transition dipole moment (in z-direction) of the state in resonance and its excitation energy. In line with previous results 54, 55 , local excitations on the molecule (mostly of π − π * character here) are relatively insensitive to tip positioning and applied electric fields, both in terms of excitation energy and transition dipole moment, as Fig. 4 shows. Typically, these states are also polarized in the molecular plane, which makes them nearly invisible for detection in z-direction. The SnPc molecule is not wholly planar, therefore these π − π * transitions might contribute to signal strength (and therefore sub-molecular resolution) at the slightly upwards-angled edges of the molecule, but are otherwise only of minor importance for signal intensity and resolution.
In case of SnPc-Ag, charge-transfer states between molecule and tip exist (in the observed energy window) in both directions, i.e. as metal-to-ligand charge transfer (MLCT)   However, the inclusion of the silver atom at the tip's apex introduces strong differences over the whole Raman map, hinting at a sub-molecular resolution. The combined systems exhibit the same features from the one only including the silver tip, folded together with the signal increase from the field, along with the highlighting of different features (center / edge). We rationalize this through changing partial charges at the tip atom, reacting to the different chemical environments over different parts of the molecule.
Under resonant conditions, we identify charge-transfer states between tip and molecule as the major contributors to high lateral resolution and signal intensity. The excitation energy of these states is highly dependent on tip position as well as field direction and strength. However, the transition dipole moment, an important factor for the resonance Raman intensity, is only dependent on the tip position, and therefore the chemical environment. Incorporating electric fields only increases the overall magnitude of the transition dipole moment, not its pattern.
Both non-resonant and resonant cases hint at the underlying idea that the chemical interaction between molecule and tip atom shapes the resolution, while the strong electric fields mainly increase signal intensity and function as a means to highlight certain chemical features, e.g. through choice of excitation wavelength. Nevertheless, the chemical contribution and the field effect are strongly influencing each other and should not be considered seperately.
Several additional factors could not be included in this model so far, including any chemical interaction with the substrate, as well as dynamic electric fields from the plasmon, and any sort of 'backwards' interaction from molecule to plasmon.