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%		 \noindent\LARGE{\textbf{The transformation mechanisms among cuboctahedra, Ino's decahedra and icosahedra structures of magic-size gold nanoclusters$^\dag$}} \\%Article title goes here instead of the text "This is the title"
%		\vspace{0.3cm} \\
%		\noindent\large{Ehsan Rahmatizad Khajehpasha\textit{$^{a}$}, Mohammad Ismaeil Safa\textit{$^{a}$}, Nasrin Eyvazi\textit{$^{a}$},}\\ 
%		\noindent\large{Marco Krummenacher\textit{$^{a}$},  Stefan Goedecker$^{\ast}$\textit{$^{a}$}} \\%Author names go here instead of "Full name", etc.
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	\begin{Huge}
		Supporting Information:\\
		\vspace{20pt}
		The transformation mechanisms among cuboctahedra, Ino's decahedra and icosahedra structures of magic-size gold nanoclusters
	\end{Huge}
	\vspace{40pt}
	
	\begin{Large}
		Ehsan Rahmatizad Khajehpasha, Mohammad Ismaeil Safa, Nasrin Eyvazi\\
		Marco Krummenacher, Stefan Goedecker
	\end{Large}
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\subsection{Trainig Dataset}
The training dataset was generated iteratively over five MH iterations. After each iteration, local minima together with high-energy structures on the MLIP PES were selected. Their energies and forces were computed using DFT with the settings listed in \cref{Si-tab:vasp_incar_au_static}. The MLIP model was then fine-tuned and used in the subsequent MH iteration. This procedure was repeated until a reliable MLIP was obtained. The energy and size distributions of the final dataset are shown in \cref{Si-img:hist_ener,Si-img:hist_nat}, respectively.
\Cref{Si-img:large_corr} depicts the energy correlation for nanoclusters of magic sizes that were excluded from the training dataset. Although only nanoclusters containing up to 90 atoms were included during training, the MLIP successfully extrapolates the PES to larger nanoclusters, achieving a root mean square error of $\approx 5$~meV/atom and a correlation of 99.7\% with the reference energies.

\begin{table}[H]
\small
\caption{INCAR parameters used for a non-spin-polarized DFT calculation of Au}
\label{Si-tab:vasp_incar_au_static}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}ll}
\hline
\textbf{Tag} & \textbf{Value} \\
\hline
% SYSTEM & Au \\
% ISTART & 0 \\
% LCHARG & .FALSE. \\
% LWAVE & .FALSE. \\
PREC & Accurate \\
ISPIN & 1 \\
LREAL & Auto \\
ROPT & 1E-4 1E-4 1E-4 \\
% LPLANE & .TRUE. \\
% NCORE & 1 \\
% KPAR & 1 \\
ALGO & Fast \\
ISIF & 0 \\
ISYM & 0 \\
ISMEAR & 0 \\
SIGMA & 0.05 \\
% IBRION & 2 \\
POTIM & 0.04 \\
% EDIFFG & $-0.01$ \\
EDIFF & $1\times10^{-6}$ \\
GGA & PE \#PBE\\
NSW & 0 \\
ENCUT & 250 \\
ENAUG & 800 \\
\hline
\end{tabular*}
\end{table}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\columnwidth]{SI-Figures/count.pdf}
    \caption{Histogram for the size of structures in the training dataset.}
    \label{Si-img:hist_nat}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\columnwidth]{SI-Figures/eners.pdf}
    \caption{Histogram for the energy of structures in the training dataset}
    \label{Si-img:hist_ener}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\columnwidth]{SI-Figures/total_ener_per_atom.pdf}
    \caption{Correlation of the energies calculated by the MLIP for the structures excluded from the training dataset of the MLIP.}
    \label{Si-img:large_corr}
\end{figure}

\subsection{Saddle Point Search}
\Cref{Si-fig:BiasedMH} shows an example of Python code for biased MH, with biasing weights listed in \cref{Si-tab:BiasVals} for different transformations and sizes. The high-symmetry transformation pathways from biased MH were further optimized using the parameters in \cref{Si-tab:neb_parameters}.

To assess the accuracy of the MLIP in describing energies and forces along transformation pathways, VASP NEB calculations were performed for Au$_{55}$ and Au$_{147}$ using the settings in \cref{Si-tab:vasp_neb_incar}, providing DFT-level validation of the predicted pathways and energy barriers.

While NEB captured high-symmetry transformations, it failed to locate saddle points for asymmetric transformations; these were instead obtained using COMPASS with the parameters in \cref{Si-tab:compassParams}.

\begin{figure}[H]
\centering
\begin{tcolorbox}[
    colback=white,
    colframe=black,
    boxrule=0.6pt,
    sharp corners,
    width=\columnwidth,
]
%\footnotesize
\begin{verbatim}
from ase.io import read
from nequip.ase import NequIPCalculator
from minimahopping.minhop import Minimahopping
from biascalc.laplace_calculator 
                      import LaplaceCalculator

target_configuration = read(
    'TargetStr.extxyz', format='extxyz'
)
configuration = read(
    'InputStr.extxyz', format='extxyz'
)
calculator = NequIPCalculator.from_deployed_model(
    model_path="b1_l1_r55n3f48.pth",
    device="cuda",
    species_to_type_name={"Au": "Au"},
)
bias_calculator = (
    LaplaceCalculator(
        target_configuration,
        bias=float('WBIAS')
    ).add_calculator(calculator)
)

configuration.calc = bias_calculator
with Minimahopping(
    configuration,
    md_calculator=bias_calculator,
    T0=300,
    dt0=1e-3,
    mdmin=1,
    n_soft=20,
    soften_positions=5e-2,
    fingerprint_threshold=0.1
) as mh:
    mh(totalsteps=500)
\end{verbatim}
\end{tcolorbox}
\caption{Example Python code for performing biased MH with an MLIP and Laplace fingerprint biasing within the ASE framework.}
\label{Si-fig:BiasedMH}
\end{figure}

\begin{table}[H]
\small
\caption{Bias weight ($\omega$ in biased MH for the high-symmetry transformations}
\label{Si-tab:BiasVals}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}lllll}
\hline
Transformation & Au$_{55}$ & Au$_{147}$ & Au$_{309}$ & Au$_{561}$ \\\hline
O$_h$ $\rightarrow$ I$_h$ & 0.4 & 1.3 & 1.1 & 1.0\\
I-D$_{5h}$ $\rightarrow$ I$_h$ & 2.2 & 2.1 & 1.6 & 1.4\\
I$_h$ $\rightarrow$ O$_h$ & 6.2 & 3.9 & 2.7 & 1.8\\
I$_h$ $\rightarrow$ I-D$_{5h}$ & 6.2 & 3.9 & 2.8 & 2.0\\
\hline
\end{tabular*}
\end{table}

\begin{table}[H]
\small
\caption{Parameters used for the nudged elastic band calculation with MLIP for 21 images, optimized using the FIRE algorithm}
\label{Si-tab:neb_parameters}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}lll}
\hline
\textbf{Component} & \textbf{Parameter} & \textbf{Value} \\
\hline
NEB & \texttt{climb} & True \\
NEB & Spring constant $k$ & 3.0 \\
NEB & \texttt{allow\_shared\_calculator} & True \\
NEB & \texttt{remove\_rotation\_and\_translation} & True \\
FIRE & \texttt{dtmax} & 0.01 \\
FIRE & \texttt{maxstep} & 0.01 \\
FIRE & \texttt{fmax} & 0.005 \#eV/\AA \\
\hline
\end{tabular*}
\end{table}

\begin{table}[H]
\small
\caption{INCAR parameters used for the nudged elastic band calculation of Au. The higher spring constant value compared to MLIP nudged elastic band calculation (\cref{Si-tab:neb_parameters}) is due to a lower number of images}
\label{Si-tab:vasp_neb_incar}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}ll}
\hline
\textbf{Tag} & \textbf{Value} \\
\hline
% SYSTEM & Au \\
% ISTART & 0  \\
% LCHARG & .FALSE. \\
% LWAVE & .FALSE. \\
PREC & Accurate \\
ISPIN & 1  \\
LREAL & Auto \\
ROPT & 1E-4 1E-4 1E-4 \\
LPLANE & .TRUE. \\
% NCORE & 1 \\
% KPAR & 1 \\
ALGO & Fast \\
NELM & 300 \\
NELMIN & 5 \\
EDIFF & $1\times10^{-8}$ \#eV \\
ISMEAR & 0 \\
SIGMA & 0.05 eV \\
ISYM & 0 \\
GGA & PE \#(PBE) \\
ENCUT & 300 eV \\
ENAUG & 900 eV \\
IBRION & 1 \\
POTIM & 0.5 \\
EDIFFG & $-0.010$ \#eV/\AA \\
ISIF & 0 \\
NSW & 1000 \\
IMAGES & 7 \\
SPRING & $-5.0$ \\
LCLIMB & .TRUE. \\
\hline
\end{tabular*}
\end{table}

\begin{table}[H]
\small
\caption{Input parameters of COMPASS calculations}
\label{Si-tab:compassParams}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}ll}
\hline
\textbf{Parameter} & \textbf{Value} \\
\hline
minimize\_input & True \\
nsteps & 10000 \\
nloops & 200 \\
forcetol & 0.005 \\
forcetolperdistance & 0.005 \\
distancestepmax & 1.0 \\
distancestepmin & 0.001 \\
targetdistancesaddle & 0.1 \\
targetdistanceother & 0.2 \\
energydifferencethreshold & 0.005 \\
nsplitdouble & 2 \\
nsplitsingle & 1 \\
alpha0 & 0.001 \\
trustradius & 0.05 \\
trustedforcedifference & 0.5 \\
extracurv & False \\
saddle\_cosp & 0.7 \\
nhistx & 10 \\
restart\_level & 2 \\
restart\_interval & 1 \\
basename & clog\_ \\
minimize\_fmax & 0.001 \\
minimize\_nsteps & 4000 \\
minimize\_maxrise & 0.005 \\
parallel & False \\
incremental\_projection & True \\
interpolation\_idpp & False \\
scalar\_pressure & 0.0 \\
\hline
\end{tabular*}
\end{table}
\newpage
\subsection{Ionized NEB of Au$_{55}$}
\phantomsection
\label{Si-sec:IonizedNEB}
To assess the effect of ionization on high-symmetry transformation pathways, we performed NEB calculations for Au$_{55}$ using the FHI-aims code with an intermediate basis set. The NEB was relaxed to a force tolerance of ten meV/atom. The parameters used in these calculations are listed in \cref{Si-tab:fhi-aims_neb_params}, with the total charge adjusted according to the ionization state~\cite{blum2009ab,marek2014elpa}. 

\begin{table}[!htb]
\centering
\caption{FHI-aims parameters used for NEB calculations of ionized Au$_{55}$}
\label{Si-tab:fhi-aims_neb_params}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}ll}
\hline
\textbf{Parameter} & \textbf{Value} \\
\hline
Exchange-correlation (xc) & PBE \\
Total charge & Ionization state dependent \\
Spin treatment & none \\
Relativistic treatment & atomic\_zora scalar \\
Compute forces & true \\
SCF energy convergence & $1\times10^{-6}$ \\
SCF force convergence & $1\times10^{-4}$ \\
\hline
\end{tabular*}
\end{table}
% \begin{figure}[H]
%     \includegraphics[width=0.99\columnwidth]{SI-Figures/aims_RMSD.pdf}
%     \caption{RMSD of structures along high-symmetry transformation pathways for different charge states of Au$_{55}$, computed relative to the corresponding neutral Au$_{55}$ structures. The RMSD is evaluated for three configurations along each pathway: the initial structure (either O$_h$ or I-D$_{5h}$), the saddle-point structure, and the final I$_h$ structure. For pathways without a saddle point (i.e., zero barrier height), the fifth image along the pathway is used as a reference to maintain consistency in the RMSD evaluation and to avoid comparisons with non-equivalent configurations.}
%     \label{Si-fig:Aims-RMSD}
% \end{figure}
% \begin{table}[htb]
%     \centering
%     \caption{RMSD values (\AA) for each image relative to the corresponding structure in the neutral Au$_{55}$ trajectory.}
%     \label{tab:rmsd_results}
% \begin{tabular}{lccccccccc}
%     \toprule
%     Transition & Charge & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} \\
%     \midrule
%     \multirow{4}{*}{$CO \rightarrow I_h$} & -2 & 0.024 & 0.033 & 0.050 & 0.053 & 0.039 & 0.025 & 0.014 & 0.008 \\
%     & -1 & 0.008 & 0.009 & 0.011 & 0.015 & 0.013 & 0.009 & 0.006 & 0.004 \\
%     & +1 & 0.008 & 0.011 & 0.018 & 0.027 & 0.023 & 0.016 & 0.008 & 0.004 \\
%     & +2 & 0.016 & 0.016 & 0.015 & 0.025 & 0.019 & 0.013 & 0.009 & 0.008 \\
%     \midrule
%     \midrule
%     \multirow{4}{*}{$DH \rightarrow I_h$} & -2 & 0.056 & 0.058 & 0.054 & 0.057 & 0.053 & 0.051 & 0.049 & 0.008 \\
%     & -1 & 0.026 & 0.029 & 0.033 & 0.025 & 0.024 & 0.024 & 0.023 & 0.004 \\
%     & +1 & 0.030 & 0.030 & 0.029 & 0.028 & 0.030 & 0.026 & 0.023 & 0.004 \\
%     & +2 & 0.022 & 0.027 & 0.030 & 0.029 & 0.024 & 0.019 & 0.013 & 0.008 \\
%     \bottomrule
% \end{tabular}
% \end{table}
\begin{table}[!htb]
\caption{RMSD values (\AA) for each image relative to the corresponding structure in the neutral Au$_{55}$ high-symmetry transformation pathway.}
\label{Si-tab:RMSD}
\begin{tabular}{lccccc}
\toprule
 & Image \# & \textbf{Au$_{55}^{-2}$} & \textbf{Au$_{55}^{-1}$} & \textbf{Au$_{55}^{+1}$} & \textbf{Au$_{55}^{+2}$} \\
\midrule
\multirow{8}{*}{\rotatebox{90}{O$_h$ $\rightarrow$ I$_h$}} & 1 & 0.024 & 0.008 & 0.008 & 0.016 \\
 & 2 & 0.033 & 0.009 & 0.011 & 0.016 \\
 & 3 & 0.050 & 0.011 & 0.018 & 0.015 \\
 & 4 & 0.053 & 0.015 & 0.027 & 0.025 \\
 & 5 & 0.039 & 0.013 & 0.023 & 0.019 \\
 & 6 & 0.025 & 0.009 & 0.016 & 0.013 \\
 & 7 & 0.014 & 0.006 & 0.008 & 0.009 \\
 & 8 & 0.008 & 0.004 & 0.004 & 0.008 \\
\midrule
\multirow{8}{*}{\rotatebox{90}{I-D$_{5h}$ $\rightarrow$ I$_h$}} & 1 & 0.056 & 0.026 & 0.030 & 0.022 \\
 & 2 & 0.058 & 0.029 & 0.030 & 0.027 \\
 & 3 & 0.054 & 0.033 & 0.029 & 0.030 \\
 & 4 & 0.057 & 0.025 & 0.028 & 0.029 \\
 & 5 & 0.053 & 0.024 & 0.030 & 0.024 \\
 & 6 & 0.051 & 0.024 & 0.026 & 0.019 \\
 & 7 & 0.049 & 0.023 & 0.023 & 0.013 \\
 & 8 & 0.008 & 0.004 & 0.004 & 0.008 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[H]
\caption{Minimum and maximum atomic displacements in \r{A} during Au$_{55}$ high-symmetry transformations calculated with FHI-Aims for different ionization states, excluding the innermost atom. The values in parentheses are from %\cref{tab:atom_movement}
table 2, i.e. were obtained from MLIP calculations.}
\label{Si-tab:atom_movement}
\begin{tabular}{llcc}
    \toprule
    Transformation & Ionization State & Min (\AA) & Max (\AA) \\
    \midrule
    \multirow{5}{*}{O$_h$ $\rightarrow$ I$_h$} & -2 & 0.6 & 1.3 \\
    & -1 & 0.6 & 1.3 \\
    & 0 & 0.6 (0.4) & 1.3 (1.3) \\
    & +1 & 0.6 & 1.3 \\
    & +2 & 0.7 & 1.3 \\
    \midrule
    \multirow{5}{*}{I-D$_{5h}$ $\rightarrow$ I$_h$} & -2 & 0.1 & 1.6 \\
    & -1 & 0.1 & 1.6 \\
    & 0 & 0.1 (0.1) & 1.6 (1.5) \\
    & +1 & 0.1 & 1.6 \\
    & +2 & 0.1 & 1.6 \\
    \bottomrule
\end{tabular}
\end{table}
\begin{figure}[!htb]
    \includegraphics[width=0.99\columnwidth]{SI-Figures/Energy_Pathway_CO.pdf}
    \includegraphics[width=0.99\columnwidth]{SI-Figures/Energy_Pathway_DH.pdf}
    \caption{The energy of NEB pathways calculated using FHI-aims for different ionization states of Au$_{55}$ for the O$_h$ $\rightarrow$ I$_h$ (top) and I-D$_{5h}$ $\rightarrow$ I$_h$ (bottom) transformations. The values in the legend indicate the corresponding barrier heights for each ionization state.}
    \label{Si-fig:ener_co_dh_ics}
\end{figure}

\Cref{Si-tab:RMSD} summarizes the RMSD of each image along the high-symmetry transformation pathway for different ionization states relative to neutral Au$_{55}$. \Cref{Si-tab:atom_movement} presents the corresponding atomic displacement amplitudes during these transformations. Notably, the RMSD values are approximately an order of magnitude smaller than the minimum atomic displacements observed, demonstrating in this way that the transformation pathway is quite insensitive to the ionization state.
Further comparison with \cref{Si-fig:ener_co_dh_ics} shows that, while ionization does not significantly alter the geometries along the high-symmetry transformation pathways, it does affect the associated barrier heights.

\newpage
\subsection{Global Optimization}
\Cref{Si-tab:minimahopping_params} presents the parameters used in MH to explore the PES in the pursuit of identifying the GM. Newly identified GM structures were further relaxed using VASP with the parameters listed in \cref{Si-tab:vasp_incar_au_dipole}.
\begin{table}[htb]
\centering
\caption{Parameters used for the Minima Hopping global optimization run}
\label{Si-tab:minimahopping_params}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}ll}
\hline
\textbf{Parameter} & \textbf{Value} \\
\hline
$T_0$ & 600 \\
$\Delta t_0$ & $1\times10^{-2}$ \\
$E_{\mathrm{diff},0}$ & 0.1 \\
Fingerprint threshold & 0.005 \\
Energy threshold & 0.010 \\
MD minimum steps & 10 \\
Softening steps ($n_{\mathrm{soft}}$) & 100 \\
$f_{\max}$ & $1\times10^{-3}$ \\
MPI enabled & True \\
% Log level & DEBUG \\
% Verbose output & True \\
% Graph output & True \\
% Collect MD data & False \\
Total steps & 4000 \\
\hline
\end{tabular*}
\end{table}

\begin{table}[H]
\small
\caption{INCAR parameters used for a spin-polarized DFT calculation of gold with dipole corrections. To disable geometry optimization for single-point calculations, NSW was set to zero}
\label{Si-tab:vasp_incar_au_dipole}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}ll}
\hline
\textbf{Tag} & \textbf{Value} \\
\hline
% SYSTEM & Au  \\
% ISTART & 0 \\
% LCHARG & .FALSE.  \\
% LWAVE & .FALSE. \\
PREC & Accurate \\
ISPIN & 2 \\
LDIPOL & .TRUE. \\
IDIPOL & 4 \\
DIPOL & 0.5 0.5 0.5 \\
LREAL & Auto \\
ROPT & 1E-4 1E-4 1E-4 \\
LPLANE & .TRUE. \\
% NCORE & 1 \\
% KPAR & 1 \\
ALGO & Normal \\
NELM & 1000 \\
ISIF & 0 \\
ISYM & 0 \\
ISMEAR & 0 \\
SIGMA & 0.0005 \\
IBRION & 2 \\
POTIM & 0.1 \\
EDIFFG & $-0.001$\\
EDIFF & $1\times10^{-9}$ \\
GGA & PE \\
NSW & 1000 \\
ENCUT & 300 \#eV \\
\hline
\end{tabular*}
\end{table}

\onecolumn
\clearpage
\subsection{Energy Profiles Along Low-Frequency Mode}
\Cref{Si-img:M7AuLjFcDh} shows the energy profile of structures displaced along the lowest positive mode. The softness of these modes introduces significant numerical noise in vibrational analysis using the finite-difference method.
\begin{figure*}[!htbp]
  \centering
  % First subfigure (side view)
  \subfloat[]{%
    \includegraphics[width=0.45\textwidth]{SI-Figures/All_FC_7_LJ.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.45\textwidth]{SI-Figures/All_DH_7_LJ.pdf}
  }\hfill\\
  \subfloat[]{%
    \includegraphics[width=0.45\textwidth]{SI-Figures/All_FC_7_Au.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.45\textwidth]{SI-Figures/All_DH_7_Au.pdf}
  }\hfill
  \caption{The energy profile for the saddle points of  LJ-O$_h$ $\rightarrow$ LJ-I$_h$ (a), LJ-I-D$_{5h}$ $\rightarrow$ LJ-I$_h$ (b), Au-O$_h$ $\rightarrow$ Au-I$_h$ (c), Au-I-D$_{5h}$ $\rightarrow$ Au-I$_h$ (d) transformations displaced parallel to the softest positive mode. The negative sign of the \textit{displacement amplitude x} represents a displacement antiparallel to the softest positive mode.}
  \label{Si-img:M7AuLjFcDh}
\end{figure*}

\onecolumn
\clearpage
\subsection{Role of Soft Modes in and asymmetric Transformations}
\Cref{Si-img:CompassPathAngle} illustrates the angles between the vector connecting the initial structure to the asymmetric transformation saddle point and the individual vibrational mode eigenvectors, thereby quantifying the contribution of each mode in initiating the transformation to the low-energy asymmetric saddle points. While high-symmetry transformation pathways are largely aligned only with the softest vibrational mode, asymmetric transformations involve contributions from multiple vibrational modes along the pathway to the saddle point, where soft modes play a dominant role. Since high-symmetry pathways and their dominance by the softest mode have already been discussed in section 3.2 
%\cref{subsec:minEnerPathway}
, they are not revisited here.
\begin{figure*}[htb]
  \centering
  % First subfigure (side view)
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000009_1_055_FC.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000009_1_055_DH.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000009_1_055_IH.pdf}
  }\hfill \\
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000003_1_147_FC.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000003_1_147_DH.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000003_1_147_IH.pdf}
  }\hfill\\
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000005_1_309_FC.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000005_1_309_DH.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000005_1_309_IH.pdf}
  }\hfill\\
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000000_3_561_FC.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000000_3_561_DH.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.30\textwidth]{SI-Figures/min000000_3_561_IH.pdf}
  }\hfill
  \caption{The angles between the vector connecting the initial structure to the asymmetric transformation saddle point and each vibrational mode are shown. Columns correspond to the initial structures: O$_h$ (left), I-D$_{5h}$ (middle), and I$_h$ (right). Rows correspond to nanocluster sizes: Au$_{55}$ (first), Au$_{147}$ (second), Au$_{309}$ (third), and Au$_{561}$ (fourth). }
  \label{Si-img:CompassPathAngle}
\end{figure*}

\clearpage
\subsection{Minimum Energy Pathways and Vibrational Modes}
The minimum energy pathways for high-symmetry structural transformations of Au nanoclusters are shown in \cref{Si-img:transformationPath}. The top row corresponds to the I-D$_{5h} \rightarrow$ I$_h$ transformations of Au$_{147}$, Au$_{309}$, and Au$_{561}$, while the bottom row depicts the O$_h \rightarrow$ I$_h$ transformations for the same nanocluster sizes. Each transformation pathway is accompanied by the corresponding softest vibrational mode, and the atomic coordinates for each step are provided in the \textbf{pathway} directory of the supplementary material for detailed visualization.
\begin{figure*}[htbp]
  \centering
  % First subfigure (side view)
  \subfloat[]{%
    \includegraphics[width=0.3\textwidth]{SI-Figures/147_DH_ICS_Side.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.3\textwidth]{SI-Figures/309_DH_ICS_Side.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.3\textwidth]{SI-Figures/561_DH_ICS_Side.pdf}
  }\hfill\\
  \subfloat[]{%
    \includegraphics[width=0.3\textwidth]{SI-Figures/147_FCC_ICS_100.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.3\textwidth]{SI-Figures/309_FCC_ICS_100.pdf}
  }\hfill
  \subfloat[]{%
    \includegraphics[width=0.3\textwidth]{SI-Figures/561_FCC_ICS_100.pdf}
  }\hfill
  \caption{High-symmetry transformation pathways of Au nanoclusters. Top row: side views of I-D$_{5h} \rightarrow$ I$_h$ transformations for Au$_{147}$ (a), Au$_{309}$ (b), and Au$_{561}$ (c). Bottom row: (100) views of O$_h \rightarrow$ I$_h$ transformations for the same nanoclusters (d–f). Golden spheres represent initial structures, green spheres indicate the final I$_h$ configuration, red lines show the softest vibrational mode, blue lines trace the NEB pathway, and orange highlights the saddle-point configuration.}
  \label{Si-img:transformationPath}
\end{figure*}
\bibliography{SI}
\bibliographystyle{SI}
\end{document}
