# The Effect of Anchor Group on the Phonon Thermal Conductance of Single Molecule Junctions
## 2 Results and Discussion  
To study vibrational and thermal properties of junctions formed by OPE3 and different anchor groups, the geometry of OPE3 in gas phase and between gold electrodes was relaxed to the force tolerance of 5 meV/A using the SIESTA [31] implementation of density functional theory (DFT), with a double-\(\zeta\) polarized basis set (DZP) and the local density approximation (LDA) functional with Ceperley and Alder (CA) parameterization. A real-space grid was defined with an equivalent energy cut-off of 350 Ry. Following the method described in [12; 24], a set of xyz coordinates were generated by displacing each atom from the relaxed xyz geometry in the positive and negative x, y and z directions with \(\delta qt=0.01\) A. The forces \(F_{i}^{q}=\left(F_{i}^{x},F_{i}^{y},\,F_{i}^{z}\right)\) in three directions \(q_{i}=\left(x_{i},y_{i},z_{i}\right)\) on each atom were then calculated and used to construct the dynamical matrix \(D_{ij}=K_{ij}^{qq^{\prime}}/M_{ij}\) where the mass matrix \(M=\sqrt{M_{i}M_{j}}\) and \(K_{ij}^{qq^{\prime}}=\left[F_{i}^{q}\left(\delta q_{j}^{q}\right)-F_{j}^{q} \left(-\delta q_{j}^{q}\right)\right]/2\delta q_{i}^{q}\) for \(i\
eq j\) obtained from finite differences. To satisfy momentum conservation, the \(K\) for \(i=j\) (diagonal terms) was calculated from \(K_{ii}=-\sum_{i\
eq j}K_{ij}\). The phonon transmission \(T_{p}(\omega)\) can then be calculated from the relation \(T_{p}(\omega)=Trace\left(\Gamma_{L}^{p}(\omega)G_{p}^{R}(\omega)\Gamma_{R}^{ p}(\omega)G_{p}^{R\dagger}(\omega)\right)\) where \(\Gamma_{L,R}^{p}(\omega)=i\left(\sum_{L,R}^{p}(\omega)-\sum_{L,R}^{p}{}^{ \dagger}(\omega)\right)\) describes the level broadening due to the coupling to the left (L) and right (R) electrodes, \(\sum_{L,R}^{p}(\omega)\) is the retarded self-frequencies associated with this coupling and \(G_{p}^{R}=\left(\omega^{2}I-D-\sum_{L}^{p}-\sum_{R}^{p}\right)^{-1}\) is the retarded Green's function, where \(D\) and \(I\) are the dynamical and the unit matrices, respectively. Figure 1 shows the phonon transmission coefficient \(T_{p}\) for phonons with energy \(\hbar\omega\) traversing from one gold electrode to the other through OPE3 derivatives with different anchor groups. \(T_{p}\) was limited to phonons with energies \(\hbar\omega<19\) meV, which is the Debye frequency of Au electrodes [24]. The amplitude of \(T_{p}\) was generally higher for OPE3 with  
Figure 1: Phonon transmission coefficient versus phonons with energy \(\hbar\omega\) for OPE3 with different anchor groups: (**a**) pyridyl PY, (**b**) thiol S, (**c**) methyl sulphide SMe and (**d**) carbodithioate CS. the pyridyl PY (Figure 1a) and carbodithioate CS (Figure 1d) anchors. The amplitude of \(T_{p}\) was noticeably lower for OPE3 with the methyl sulphide SMe anchor (Figure 1c) compared to that with the thiol S anchor (Figure 1b). This is due to a combination of two effects. First, our calculations show that the binding energy between Au-S in SMe (0.47 eV) was weaker than that of Au-S in thiol (2.08 eV). This is because SMe makes a coordination bond to Au whereas the bond between Au-S in thiol is a stronger covalent bond. This means that the vibrational coupling between Au-S is stronger with thiol compared to the SMe anchor. Secondly, phonon interference [17] due to CH\({}_{3}\) side group in SMe led to the suppression of \(T_{p}\). The phonon waves transmitted through sulphur atoms interfered destructively with the reflected waves by CH\({}_{3}\) groups for given frequencies, leading to the suppression of \(T_{p}\). This is like a guitar string, where waves with certain frequencies are suppressed by pressing the string at different points.