Molecule-like and lattice vibrations in metal clusters

We report distinct molecule-like and lattice (breathing) vibrational signatures of atomically precise, ligand-protected metal clusters using low-temperature Raman spectroscopy. Our measurements provide fingerprint Raman spectra of a series of noble metal clusters, namely, Au25(SR)18, Ag25(SR)18, Ag24Au1(SR)18, Ag29(S2R)12 and Ag44(SR)30 (–SR = alkyl/arylthiolate, –S2R = dithiolate). Distinct, well-defined, low-frequency Raman bands of these clusters result from the vibrations of their metal cores whereas the higher-frequency bands reflect the structure of the metal–ligand interface. We observe a distinct breathing vibrational mode for each of these clusters. Detailed analyses of the bands are presented in the light of DFT calculations. These vibrational signatures change systematically when the metal atoms and/or the ligands are changed. Most importantly, our results show that the physical, lattice dynamics model alone cannot completely describe the vibrational properties of ligand-protected metal clusters. We show that low-frequency Raman spectroscopy is a powerful tool to understand the vibrational dynamics of atomically precise, molecule-like particles of other materials such as molecular nanocarbons, quantum dots, and perovskites.

Fragments due to (Au-PET) losses

Normal mode analysis
The similarity/differences between the normal modes computed for the Au 25 , Ag 25 and Ag 24 Au clusters were assessed by calculating normal mode overlaps, i.e., scalar products between the normalised nuclear displacement vectors. In addition, the localisation of the mode motion on the inner-cores (i.e., Au 13, Ag 13 and Ag 12 Au 1 ), the (Au 2 S 3 ) 6 ligand binding motifs and the remaining C and H atoms was computed using the following formula: is the localisation of the motion associated with normal mode j on the molecular ( , ) fragment X, is the square of the nuclear displacement vector S associated with the ith atom 2 ( ) and the jth normal mode, is the number of atoms in fragment X and is the total number of atoms in the molecule.
Three different fragments have been considered when computing the localisation of the normal mode motion. Fragment one (F1) represents the Au 13 , Ag 13 and Ag 12 Au inner-cores, fragment two (F2) the (Au 2 S 3 ) 6 ligand binding motifs and fragment three (F3) the remaining C and H atoms. Further, there are two important points that need to be made about the localisation of the normal mode motion. First, the localisation of the normal mode motion on fragment F3 will be significantly larger than on the other two fragments combined. This happens because the C and H atoms are lighter and more numerous than the Au/Ag and S atoms. Second, the nuclear displacement vectors associated with the C and H atoms have little relevance for the interpretation of the experimental spectra. This is because the CH 3 group is not only significantly lighter than the DMBT group it has substituted, but it also has a very different geometry. Consequently, when assigning the character of the Raman bands, we only take into account the localisation of the modes on the fragments F1 and F2. For this reason, there are two normal mode overlap values listed in Tables I, II and II (i.e., O1 is obtain by considering the nuclear displacement vectors of all atoms in the nanoclusters, while the O2 is calculated without the nuclear displacement vectors of the C and H atoms.) Table I lists the results obtained when comparing the modes of the Au 25 and Ag 25 clusters. The analysis was performed for all Au 25 modes with a significant Raman intensity. Furthermore, only the Ag 25 modes that exhibits an overlap larger than 0.6 with the Au 25 modes have been listed in Table I. Tables II and III list the results obtained when comparing the Ag 25 and Ag 24 Au 1 modes with significant Raman intensities. To illustrate the meaning of the values taken by a given normal mode overlap, in Figure S12 we considered one Ag 25 mode from each of the six Ag 25 bands (i.e., the most intense mode) together with their corresponding Ag 24 Au 1 modes and superimposed their nuclear displacement vectors. An overlap of 0.99 (e.g. modes 43 in Figure S12), means that the two modes are 98% (0.98 = 0.99 2 ) similar. In turn, this implies that all nuclear displacement vectors have very similar directions and magnitudes. Similarly, an overlap of 0.79 (e.g. modes 77 in Figure  S12), means that the two modes are only 62% (0.62 = 0.79 2 ) similar. While the two modes can still be considered as similar, their nuclear displacement vectors exhibit clear difference both in direction and magnitude.

Supporting Information 12
Figure S11. Overlaps between the nuclear displacement vectors of the Ag 25 and Ag 24 Au 1 clusters. The O2 normal mode overlaps were computed using only the nuclear displacement vectors of the atoms in the Ag 25 S 18 and Ag 24 Au 1 S 18 fragments. Table S1. Comparison of the nuclear displacement vectors of the Au25 and Ag25 clusters. Only the modes with significant Raman intensities were considered. All atoms were considered when computing the O1 overlap. For =O2, the C and H atoms were neglected. Units: Freq (cm -1 ), Raman (Ang 4 /amu). Table S2. Comparison of the nuclear displacement vectors of the Ag 25 and Ag 24 Au clusters. Only the modes with significant Raman intensities were considered. All atoms were considered when computing the O1 overlap. For =O2, the C and H atoms were neglected. Units: Freq (cm -1 ), Raman (Ang 4 /amu).

Supporting Information 20
Coordinates of the optimized geometry of Au 25 (SCH 3 ) 18