## 3 PiFM Measurement for a Single-Molecule
### Microscopic Interaction between Molecule and Plasmon  
We discuss the PiFM (attractive force) spectra for the molecule, which are shown in Figure 2 as solid colored lines when the PiFM tip placed at each position. The dashed line represents the spectrum of the gold substrate, that is, in the absence of the molecule. The PiFM spectra show that PiFM has a high sensitivity for the wavelength of the incident laser, and different resonance states can be observed depending on the position of the tip, resulting in a high spatial resolution. This is due to the significant localization and enhancement of localized surface plasmons by the atomic-level spatial structure called picocavity at the tip apex [22]. In addition, a Fano-like structure with an asymmetric peak-dip structure appears near the molecular resonance energy in the PiFM spectrum. This can be caused by the interference effect between the broad plasmon resonance peak and the sharp molecular resonance peak, which has been explained in our previous work [18]. When light with energy lower than the resonance energy of the molecule is incident, the polarization of the plasmon and the polarization of the molecule are in phase. As these enhance each other, the photoinduced force is clearly observed in the direction of attraction. In contrast, when light with an energy higher than the resonance energy of the molecule is incident, the polarization of the molecule and the plasmon are in anti-phase, and they suppress each other. In this case, the polarization of the molecule was very small and only slightly weakened the plasmon. Therefore, the peak in the attraction direction was large, and the dip in the repulsion direction was small. We calculated the PiFM images and polarization structures at each peak of the PiFM spectrum shown in Figure 2. As shown in Figure 3a-e, completely different PiFM images were obtained for each resonance energy of the molecule. The calculated spatial polarization structures for the respective PiFM images are shown in Figure 3f-j. Figure 3f,g are optically allowed transitions that can be excited under the long-wavelength approximation,  
Figure 1: (**a**) Schematic PiFM model. The tip has atomic-scale structures, which form a picocavity on its apex. (**b**) Simplified model of a six-membered ring molecule on the substrate. The susceptibility of each element of the molecule was defined using the Lorentz model. The entire space was discretized into 1 Å\({}^{3}\) cubic cells represented by grid lines. (**c**) Absorption spectrum of a six-membered ring molecule in vacuum. The peaks correspond to the resonance states under long-wavelength approximation. (**d**) The spatial structures of the dipoles in the molecule at each resonance energy, as shown in (**c**). whereas Figure 3h-j are forbidden transitions, where the average of the polarization inside the molecule vanishes. The difference between these PiFM images can be understood by the interaction with the electric field vector of the plasmon spreading isotropically from the tip apex. The features of these images agree with the discussion of a previous study on molecular imaging by TERS [13]. In the allowed transition, shown in Figure 3a,b, when the tip is at the center of the molecule, two robe-like PiFM images and the nodes are observed. Because the electric field vectors of the localized plasmon spread radially around the tip apex, when the polarizations of plasmons and molecules are in the same line, the molecule is excited and the force signal is obtained, as shown in Figure 4a. In contrast, when the polarizations of plasmons and molecules are orthogonal, the molecule cannot be excited, and the node appears in the PiFM image, as shown in Figure 4b. For optically forbidden transitions, the molecule can be excited when the tip is at a position where the dipoles are opposed to each other, as shown in Figure 4c.