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\DOI{10.5802/crchim.448}
\datereceived{2025-12-19}
\daterevised{2026-02-19}
\datererevised{2026-03-16}
\dateaccepted{2026-03-20}
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\section*{Declaration of interests}
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\COI{The authors do not work for, advise, own shares in, or receive
funds from any organization that could benefit from this article, and
have declared no affiliations other than their research organizations.}

\begin{document}

%\dateposted{2026-02-16}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{DFT investigations of 2D raft-type heterometallic clusters 
[MMoCp(or C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2})(CO)\textsubscript{3}]\textsubscript{\textit{n}} with triangular 
(\textit{n} = 3) or square  (\textit{n} = 4) Cu, Ag, or Au metal cores}

\alttitle{\'{E}tude DFT des clusters h\'{e}t\'{e}rom\'{e}talliques 2D
[MMoCp(ou C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2})(CO)\textsubscript{3}]\textsubscript{\textit{n}} \`{a} c\oe{}urs
m\'{e}talliques Cu, Ag et Au de g\'{e}om\'{e}trie  triangulaire 
\textit{n} = 3) ou carr\'{e}e  \textit{n} = 4)}

\author{\firstname{Abdelatif} \lastname{Messaoudi}\CDRorcid{0000-0002-7089-1086}\IsCorresp}
\address{Laboratory of Materials Chemistry and the Living: 
Activity \& Reactivity (LCMVAR), Department of Chemistry, Faculty of Matter Sciences, 
University of Batna 1, Batna, Algeria}
\email[A. Messaoudi]{amessaoudi@univ-batna.dz}

\author{\firstname{Pierre} \lastname{Braunstein}\CDRorcid{0000-0002-4377-604X}\IsCorresp}
\address{Institut de Chimie (UMR 7177 CNRS), Universit\'{e} de Strasbourg, 
4 rue Blaise Pascal, CS 90032, Strasbourg Cedex F-67081, France}
\email[P. Braunstein]{braunstein@unistra.fr}

\editornote{Part of the PhD thesis of AM, Universit\'{e} Louis Pasteur,
Strasbourg (France), 2006  (national thesis number 
\href{https://theses.fr/2006STR13203}{2006STR13203}).} 

\keywords{\kwd{Metal--metal bonds}
\kwd{Transition-metal clusters}
\kwd{Heterometallic clusters}
\kwd{Optimised geometries}
\kwd{Coinage metals}
\kwd{DFT calculations}
\kwd{d\textsuperscript{10}--d\textsuperscript{10} interactions}}

\altkeywords{\kwd{Liaisons m\'{e}tal--m\'{e}tal}
\kwd{Clusters de m\'{e}taux de transition}
\kwd{Clusters h\'{e}t\'{e}rom\'{e}talliques}
\kwd{G\'{e}om\'{e}tries optimis\'{e}es}
\kwd{M\'{e}taux de la mine du platine}
\kwd{Calculs DFT}
\kwd{Interactions d\textsuperscript{10}--d\textsuperscript{10}}}

\dedicatory{\raggedleft Dedicated to Prof. Rick D. Adams, who has been a leader in
the field of organometallic chemistry for over forty years,  in
particular in cluster chemistry,  where he has made numerous
outstanding contributions.  He has beautifully served as 
Editor-in-Chief for the Journal of Organometallic Chemistry  for
twenty-seven years.\break\vspace*{.8pc}}

\begin{abstract}
We present theoretical investigations of 2D raft-type heterometallic
clusters  [MMoCp(or C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2})(CO)\textsubscript{3}]\textsubscript{\textit{n}} (M = Cu,
Ag, Au) with a triangular  (\textit{n} = 3) or square  (\textit{n} = 4) copper, silver,
or gold core edge-bridged by three or four metalloligands MoCp(or
C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2})(CO)\textsubscript{3},  respectively (\tralicstex{Cp = η\textsuperscript{5}}{$\mathrm{Cp}=\upeta^{5}$}-C\textsubscript{5}H\textsubscript{5}; 
C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2} = \tralicstex{η\textsuperscript{5}}{$\upeta^{5}$}-C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2}). Various molecular symmetries,
\textit{C}\textsubscript{1},  \textit{C}\textsubscript{s},  \textit{C}\textsubscript{2},  \textit{D}\textsubscript{2}, and \textit{S}\textsubscript{4}, were considered,
and our calculations reveal an excellent agreement between the most
stable computed structures and those determined experimentally by X-ray
diffraction when the Cp ligand is used. In contrast, clusters
incorporating the C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2} ligand display alternative
geometries that are energetically more stable than those found
experimentally, emphasising the crucial role of the \tralicstex{𝜋}{$\uppi$}-bound
ligand on cluster stability. For M = Cu, we demonstrate that square
cores with elongated Cu--Cu distances can be stabilised, consistent
with previously described systems. Energy decomposition analysis (EDA)
at the BP86 level shows that the Cu, Ag, and Au clusters are stabilised
by a strong interplay of electrostatic and orbital interactions, with
markedly stronger binding in the tetranuclear systems due to
cooperative  metal--metal and metal--ligand effects. Frontier molecular
orbital analysis was used to investigate the electronic structure and
potential reactivity of these clusters.  The results reveal that metal
nature, NMe\textsubscript{2} substitution, and cluster nuclearity strongly affect
the HOMO--LUMO gaps and charge-transfer behaviour.
NMe\textsubscript{2}-substituted Cu and Au clusters with higher nuclearity display
reduced HOMO--LUMO gaps and greater frontier orbital delocalisation,
indicating an increased propensity for redox processes. Our theoretical
study satisfactorily reproduces the experimental structures of these 2D
raft-type heterometallic clusters and highlights the possibility of
uncovering new potentially accessible geometries of transition-metal
clusters.
\end{abstract}

\begin{altabstract}
Nous pr\'{e}sentons une \'{e}tude th\'{e}orique de clusters
h\'{e}t\'{e}rom\'{e}talliques 2D  [MMoCp(ou
C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2})(CO)\textsubscript{3}]\textsubscript{\textit{n}} (M = Cu, Ag, Au)
poss\'{e}dant un c\oe{}ur triangulaire  (\textit{n} = 3) ou carr\'{e}  (\textit{n} = 4)
de cuivre, d'argent ou d'or, reli\'{e} par trois ou quatre
m\'{e}talloligands MoCp(ou C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2})(CO)\textsubscript{3}, 
respectivement (\tralicstex{Cp = η\textsuperscript{5}}{$\mathrm{Cp}=\upeta^{5}$}-C\textsubscript{5}H\textsubscript{5};
C\textsubscript{5}H\textsubscript{4}NMe\tralicstex{\textsubscript{2} = η\textsubscript{5}}{$_{2}=\upeta^{5}$}-C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2}).  Diverses
sym\'{e}tries mol\'{e}culaires, \textit{C}\textsubscript{1},  \textit{C}\textsubscript{s},  \textit{C}\textsubscript{2}, 
\textit{D}\textsubscript{2} et \textit{S}\textsubscript{4}, ont \'{e}t\'{e} consid\'{e}r\'{e}es, et nos 
calculs r\'{e}v\`{e}lent un excellent accord entre les structures
calcul\'{e}es les plus stables et celles d\'{e}termin\'{e}es
exp\'{e}rimentalement par diffraction des rayons X lorsque le ligand Cp
est utilis\'{e}. En revanche, les clusters contenant le ligand
C\textsubscript{5}H\textsubscript{4}NMe\textsubscript{2} pr\'{e}sentent des g\'{e}om\'{e}tries
alternatives qui sont \'{e}nerg\'{e}tiquement plus stables que celles
observ\'{e}es exp\'{e}rimentalement, soulignant ainsi le r\^{o}le
crucial du ligand $\uppi$-li\'{e} dans la stabilit\'{e} des clusters.
Pour M = Cu, nous montrons que des c\oe{}urs carr\'{e}s
pr\'{e}sentant des distances Cu--Cu allong\'{e}es peuvent \^{e}tre
stabilis\'{e}s, en accord avec des syst\`{e}mes pr\'{e}c\'{e}demment
d\'{e}crits. L'analyse des orbitales mol\'{e}culaires fronti\`{e}res a
\'{e}t\'{e} utilis\'{e}e pour \'{e}tudier la structure \'{e}lectronique
et la r\'{e}activit\'{e} des clusters contenant Cu, Ag et Au. Les
r\'{e}sultats montrent que la nature du m\'{e}tal, la substitution par
NMe\textsubscript{2} et la nucl\'{e}arit\'{e} des clusters influencent fortement
les \'{e}carts HOMO--LUMO et les transferts de charge. Les clusters de
Cu et d'Au substitu\'{e}s par NMe\textsubscript{2} et de plus grande
nucl\'{e}arit\'{e} pr\'{e}sentent des \'{e}carts HOMO--LUMO r\'{e}duits
et une plus forte d\'{e}localisation des orbitales fronti\`{e}res, ce
qui indique une plus grande activit\'{e} r\'{e}dox. Notre \'{e}tude
th\'{e}orique reproduit de mani\`{e}re satisfaisante les structures
exp\'{e}rimentales de ces clusters h\'{e}t\'{e}rom\'{e}talliques 2D de
type  \og radeau \fg et met en \'{e}vidence
la possibilit\'{e} de d\'{e}couvrir de nouvelles g\'{e}om\'{e}tries de
clusters de m\'{e}taux de transition, potentiellement accessibles.
\end{altabstract}

%\input{CR-pagedemetas}

\maketitle

\twocolumngrid

\end{noXML}

\dedication{Dedicated to Prof. Rick D. Adams, who has been a leader in
the field of organometallic chemistry for over forty years,  in
particular in cluster chemistry,  where he has made numerous
outstanding contributions.  He has beautifully served as 
Editor-in-Chief for the Journal of Organometallic Chemistry  for
twenty-seven years.}

\section{Introduction}\label{sec1}
After the pioneering report of Nyholm and coworkers of a heterometallic
complex containing a direct metal--metal bond between a group 11 metal
in the +I oxidation state (namely Cu(I), Ag(I), or Au(I)) and another
transition metal in  1964~\cite{1}, a new research area developed in
organometallic chemistry dedicated to the study of heterometallic
bonds, and numerous relevant transition metal dinuclear and cluster
complexes have since been synthesised and characterised with 1D, 2D, or
3D  structures~\cite{2,3,4,5,6,7}. Notably, several metal clusters were
found to display short intermetallic contacts between group 11 centres
that challenged the conventional understanding of metal--metal bonding
and were interpreted as evidence for d$^{10}$--d$^{10}$ metallophilic
interactions, a phenomenon that continues to attract significant
attention due to its implications in bonding theory, structural
diversity, and photophysics, with potential applications in material 
sciences~\cite{8,9,10,11,12,13,14,15,16,19}. Longoni and collaborators
have described octanuclear, tetra-anionic clusters
[NMe$_{3}$CH$_{2}$Ph]$_{4}$[Ag$_{4}$\{Fe(CO)$_{4}$\}$_{4}$]~\cite{17} 
and [NEt$_{4}$]$_{4}$[Au$_{4}$\{Fe(CO)$_{4}$\}$_{4}$]~\cite{18}, in
which all Ag--Ag and Au--Au bonds are bridged by the formally dianionic
metalloligand $\upmu$-[Fe(CO)$_{4}$]  (Scheme~\ref{sch1}). The inner
core of the Ag$_{4}$Fe$_{4}$ cluster adopts a square geometry, with
Ag--Ag distances ranging from 3.036(1) to 3.334(1)~\AA{}. In contrast,
the corresponding Au$_{4}$Fe$_{4}$ clusters display either a square
(2.973(2), 2.831(2)~\AA{}) or a rectangular arrangement (2.932(2) and
3.400(2)~\AA{}) of gold atoms.

\begin{scheme*}
\includegraphics{sc01}
\vspace*{4pt}
\caption{\label{sch1}Square or rectangular M$_{4}$ core in 
Ag$_{4}$Fe$_{4}$ and Au$_{4}$Fe$_{4}$  clusters~\cite{6,7,18,19}.}
\vspace*{-10pt}
\end{scheme*}

Furthermore, Kl\"{u}fers et~al.\ reported a series of clusters in which
a central square of silver or copper atoms is stabilised by the
formally monoanionic bridging metalloligand
$\upmu$-[Co(CO)$_{4}$]~\cite{20,21,22}. These diverse structures
illustrate the ability of group 11 metals to assemble into highly
symmetric frameworks when supported by carbonyl-based metalloligands.
In contrast to these square arrangements, a triangular Cu$_{3}$ core,
with an average Cu--Cu separation of 2.602~\AA{}, was later
characterised in the hexanuclear, trianionic cluster
[NEt$_{4}$]$_{3}$[Cu$_{3}$\{Fe(CO)$_{4}$\}$_{3}$]~\cite{23}, in which
the copper atoms are interconnected by $\upmu$-[Fe(CO)$_{4}$] units 
(Scheme~\ref{sch2}). The 
\mbox{characterisation} of the triangular Cu$_{3}$
and square Ag$_{4}$/Cu$_{4}$ 
\mbox{systems} 
highlights the structural
versatility of group 11 
\mbox{closed-shell} clusters, underscoring the
delicate balance between electronic and steric effects that determines
their nuclearity and overall structure.

\begin{scheme}
\includegraphics{sc02}
\vspace*{6pt}
\caption{\label{sch2}Triangular structure of the  Cu$_{3}$Fe$_{3}$
cluster~\cite{23}.}
\vspace*{-10pt}
\end{scheme}

Recent work has reported an unprecedented case of polymerisation
isomerism in heterobimetallic carbonyl clusters
[\{MFe(CO)$_{4}$\}$_{n}$]$^{n-}$ (M ${=}$ Cu, Ag, Au; $n=3$, 4), where
triangular and square nuclearities coexist for the same elemental
composition. Depending on the synthetic protocol, Ag and Au systems
selectively form either trinuclear or tetranuclear clusters, whereas Cu
stabilises only the triangular form. This behaviour has been
rationalised by a balance between Fe--M bonding and metallophilic
interactions, as revealed by structural and AIM analyses~\cite{24}.

Our group reported a series of hexa- and octanuclear heterometallic
clusters with a central core displaying d$^{10}$--d$^{10}$
interactions, obtained by reaction of the carbonylmetallate
[Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$^{-}$ with
[Cu(NCMe)$_{4}$]PF$_{6}$,  AgBF$_{4}$, or (NBu$_{4}$)[AuBr$_{2}$], 
respectively. These clusters were shown by X-ray diffraction to possess
a triangular copper core in
[Cu$_{3}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{3}$]  \textbf{A1},
and a square silver or gold core in
[Ag$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$] \textbf{A2}
and [Au$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$] 
\textbf{A3}, respectively  (Scheme~\ref{sch3})~\cite{19,25,26,27}.

\begin{scheme*}
\includegraphics{sc03}
\vspace*{4pt}
\caption{\label{sch3}Clusters
[MMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_n$  with a triangular copper
$(n=3)$ or square silver or gold central core  $(n=4)$.} 
\vspace*{10pt}
\end{scheme*}

Using the anion [MoCp(CO)$_{3}$]$^{-}$ (Cp ${=}$
$\upeta_{5}$-C$_{5}$H$_{5}$) had previously yielded similar clusters,
with a triangular copper core in [Cu$_{3}$\{MoCp(CO)$_{3}$\}$_{3}$]
\textbf{A4}, and a square silver or gold core in
[Ag$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$] \textbf{A5}, and
[Au$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$] \textbf{A6}, respectively 
(Scheme~\ref{sch4})~\cite{25,26}.

\begin{scheme*}
\includegraphics{sc04}
\vspace*{4pt}
\caption{\label{sch4}Clusters  [MMoCp(CO)$_{3}$]$_{n}$ with a
triangular copper $(n=3)$ or square silver or gold central  core
$(n=4)$.}
\vspace*{-6pt}
\end{scheme*}

Combined experimental and theoretical investigations on clusters
\textbf{A4--A6} demonstrated that the structural preference arises from
the delicate balance between steric effects, ligand coordination modes,
and d$^{10}$--d$^{10}$ metallophilic interactions between the
closed-shell metal centres~\cite{25,26}. These findings provided
important insight into the role of metallophilic interactions in
governing the geometry and stability of such heterometallic carbonyl
clusters. In the present work, we extend this investigation to related
systems incorporating the substituted metalloligand
[Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$^{-}$ in order to evaluate the
influence of ligand functionalisation on the 
\mbox{structure} and electronic
properties of the resulting clusters. 

The reason for introducing the C$_{5}$H$_{4}$NMe$_{2}$ ligand instead
of Cp in the metal carbonyl fragment was to examine its potential
influence on the nature and/or structure of the resulting clusters.
Because a bridging bonding mode of the ligand C$_{5}$H$_{4}$NMe$_{2}$
has been demonstrated earlier in the heterodinuclear complex
[Pt\{Mo($\upmu$-C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}(NCPh)Cl]
(Mo--Pt) \textbf{B}~\cite{28} (Scheme~\ref{sch5}), it was conceivable
that the NMe$_{2}$ donor group could engage in bonding with an adjacent
group 11 metal centre in the clusters of type \textbf{A} investigated
here. Although this situation was not observed, the
C$_{5}$H$_{4}$NMe$_{2}$ ligand in clusters \textbf{A1--A3} was found to
influence the orientation of the metalloligand with respect to the
central  metal core and the overall molecular structure.

\begin{scheme}
\includegraphics{sc05}
\vspace*{5pt}
\caption{\label{sch5}Complex \textbf{B} with a bridging
C$_{5}$H$_{4}$NMe$_{2}$ ligand~\cite{28}.}
\vspace*{-12pt}
\end{scheme}

In clusters \textbf{A1--A6}, the bonding behaviour of the
metalloligands [MoCp(CO)$_{3}$]$^{-}$ and
[Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$^{-}$ with respect to the
d$^{10}$--d$^{10}$ metal--metal bond is that of an anionic 
$\upmu_{2}$-unit bridging the M(I)--M(I) centres, acting formally as
four-electron donors, which confers the usual 14-electron count to the
coinage metals. This bridging bonding mode was observed for the first
time in the centrosymmetric butterfly clusters [Pd$_{2}$(or
Pt$_{2}$)\{MoCp(CO)$_{3}$\}$_{2}$(PR$_{3}$)$_{2}$]~\cite{29,30,31}
(Scheme~\ref{sch6}).

\begin{scheme}
\includegraphics{sc06}
\vspace*{5pt}
\caption{\label{sch6}Centrosymmetric clusters of type \textbf{C} with
centrosymmetric  Mo$_{2}$Pd$_{2}$ or Mo$_{2}$Pt$_{2}$ core.}
\vspace*{-12pt}
\end{scheme}

Although the structures of these centrosymmetric metal clusters did not
follow the original Wade--Mingos  rules~\cite{32,33}, they could be
readily explained by considering that the d$^{9}$--d$^{9}$ M$'$--M$'$
bond was doubly bridged by the carbonylmetallates formally acting as
four-electron donor anions, like $\upmu_{2}$-halides. This analogy was
extended to the even rarer  $\upmu_{3}$-bonding mode, first encountered
in a Pd$_{3}$Mo cluster, where the carbonylmetallate could be viewed as
an anionic six-electron donor, like  a $\upmu_{3}$-halide~\cite{34}
This electron-counting approach was subsequently successfully extended
to isoelectronic or isolobal analogues of these 
carbonylmetallates~\cite{35}.

In each of the clusters \textbf{A1}--\textbf{A6}, the short M--M
distances provide clear evidence for the occurrence and stabilising
role of metallophilic interactions between the closed-shell metal
centres. To gain a deeper insight into the bonding in these systems, we
carried out a detailed theoretical investigation aimed at elucidating
the reasons for the stabilisation of specific structural motifs. Our
study was designed to address two key aspects: (i)~to rationalise the
preference for the observed geometries of the six clusters under
consideration, and (ii)~to evaluate the relative stability of
structures with different possible symmetries. DFT calculations have
allowed us not only to reproduce the structural features observed
experimentally but also to better understand the electronic factors
underpinning their stability and structural diversity.

\section{Computational details}\label{sec2}
All quantum chemical calculations were carried out using the TURBOMOLE
software  package~\cite{36}. The BP86 density functional
(Becke--Perdew86) was 
\mbox{selected} because of its proven reliability in
describing diverse bonding situations, including those involving
transition-metal  complexes~\cite{37,38,39}.  This choice further
enabled the use of the efficient RI--J approximation for the treatment
of Coulomb two-electron  integrals~\cite{40,41,42}. For geometry
optimisations, SV(P) basis sets were  employed~\cite{43} consisting of
a single basis function for core orbitals, a double-$\upzeta$
description for the valence shells, and one set of polarisation
functions for all atoms except hydrogen. To refine the electronic
energies, single-point calculations were performed using the larger
triple-$\upzeta$ valence plus polarisation (TZVP) basis 
sets~\cite{44}. 
\mbox{Stationary} points on the potential energy surface were
verified through 
vibrational frequency analyses at the BP86/SV(P)
level: minima were confirmed by the absence of imaginary frequencies,
whereas the presence of a single imaginary frequency identified
transition  states~\cite{45,46,47}. In addition to Born--Oppenheimer
potential energies  ($E$, corrected for zero-point vibrational
effects), Gibbs free energies  $(G)$ were also evaluated. Unless
otherwise specified, all reported $G$ values correspond to standard
conditions of  25~\textdegree C and 1~bar CO, consistent with the
experimental  setup~\cite{26}.  Energy decomposition analysis (EDA),
originally developed by  Morokuma~\cite{47}, was employed to
investigate the nature and origin of the interactions between the 
metal cores M$_{n}$ (M ${=}$ Cu, Ag, Au; $n=3$, 4) and the metalloligand
fragments [MoCp(or C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_n$ in the
triangular  $(n=3)$ and square  $(n=4)$ heterometallic clusters. The
EDA calculations were carried out using the Amsterdam modeling suite 
(AMS)~\cite{48}. Single-point EDA calculations were performed at the
BP86/TZP level on geometries optimised at the BP86/SVP level with
Turbomole. Within this formalism, the total interaction energy $\Delta
E_{\mathrm{int}}$ between the interacting fragments is decomposed into
physically meaningful contributions according to Equation~(\ref{eq1}): 
{\begin{equation}\label{eq1}
\Delta E_{\mathrm{int}}=
\Delta E_{\mathrm{Pauli}}+
\Delta E_{\mathrm{elstat}}+
\Delta E_{\mathrm{orb}}
\end{equation}}\unskip
where $\Delta E_{\mathrm{Pauli}}$ represents the Pauli repulsion
arising from the overlap of occupied orbitals, $\Delta
E_{\mathrm{elstat}}$ corresponds to the classical electrostatic
interaction between the unperturbed charge distributions of the
fragments, and $\Delta E_{\mathrm{orb}}$ accounts for the stabilising
orbital interactions, including polarisation and charge-transfer
effects that occur upon relaxation of the fragment 
densities~\cite{50}. All frontier-molecular-orbital (FMO) calculations
were performed using the BP86 functional in combination with the
def2-TZVP basis  set~\cite{51,52,53}, as implemented in the Gaussian 16
package~\cite{53}. 

\section{Results and discussion}\label{sec3}

\subsection{Optimised geometries of the metal cores}\label{sec3.1}
The main results from the computationally optimised geometries and the
structures determined by X-ray diffraction of clusters
\textbf{A1}--\textbf{A6} are presented and compared below. A detailed
compilation of structural parameters, including bond lengths and bond
angles, available in  Tables~S1--S6 in the Supporting Information,
provide insights into the reliability of the theoretical methods
employed.

\subsubsection{Clusters with a copper core}\label{sec3.1.1}
In the structure of cluster \textbf{A1}, optimised in $C_{1}$ symmetry,
the calculated Cu--Cu bond distances are 2.642, 2.580, and 2.531~\AA{},
shorter than the experimental values by only 0.013, 0.036, and
0.048~\AA{}, respectively, while the calculated Cu--Mo distances are
longer by about 0.03~\AA{}. A noticeable difference concerns the
bridging carbonyl groups, where the calculated Cu--C bond lengths are
shorter by up to 0.117~\AA{} compared to the experimental values. The
calculated bond angles match well the experimental data, with only
small differences of 3.94\textdegree, 4.12\textdegree, 3.80\textdegree,
and 3.01\textdegree\ for Mo(2)--C(1)--O(1), Mo(2)--C(11)--O(4),
Mo(3)--C(21)--O(7), and Mo(3)--C(23)--O(9), respectively.

The X-ray diffraction study revealed that cluster \textbf{A4} with a
triangular copper core crystallises in two different systems, triclinic
and orthorhombic. The main difference between them lies in the presence
of a triply bridging carbonyl group in the orthorhombic form. For both
systems, geometry optimisation yielded similar structures, with the Cp
and carbonyl ligands arranged in an overall  $C_{3}$-symmetric
orientation. For our study, we considered cluster \textbf{A4} in its
orthorhombic crystalline form. The optimised Cu--Cu distances are, on
average, shorter by 0.05~\AA{} compared to those determined by X-ray
diffraction. As in the case of cluster \textbf{A1}, slight deviations
from the experimental values are observed in the calculated bond angles
Mo(1)--C(3)--O(3), Mo(2)--C(7)--O(7), Mo(3)--C(4)--O(4), and
Mo(3)--C(6)--O(6) of  3.36\textdegree, 3.49\textdegree, 
3.95\textdegree, and 4.86\textdegree, respectively.

\subsubsection{Clusters with a silver or gold core}\label{sec3.1.2}
The geometries of clusters \textbf{A2} and \textbf{A3} with a silver
and gold square core, respectively, were optimised in $C_{2}$ symmetry.
Compared to the experimental values, the calculated metal--metal
distances are longer by about 0.04~\AA{} in the silver square and by
0.12~\AA{} in the gold square. Furthermore, the diagonal distances in
the gold cluster are clearly overestimated, with a calculated
Au(1)--Au(3) distance of 4.151~\AA{} against 3.875~\AA{} determined
experimentally. In contrast, the M--Mo distances are in good agreement,
with a maximum deviation of 0.04~\AA{} for silver and 0.10~\AA{} for
gold. For the bridging carbonyl ligands, more pronounced discrepancies
are observed between the calculated and experimental values. For
instance, in the case of silver, the Ag(2)--C(1) distance is longer by
0.16~\AA{} while the Ag(2)--C(2) distance is shorter by 0.05~\AA{}.
This effect is even more pronounced in the gold square, where the
Au(2)--C(2) distance decreases by 0.12~\AA{}, while another Au--C bond
length increases by 0.37~\AA{} (Table~S3). A significant structural
difference between the silver- and gold-containing clusters lies in the
Au--CO bonds being shorter than the corresponding Ag--CO bonds.
Nevertheless, the calculated bond angles are in good agreement with the
experimental values. The differences in bond lengths and angles
sometimes observed between calculated and experimental values involving
the coinage metals and CO ligands can be explained by the relative
weakness of these interactions and their soft energy profile compared
to the Mo--CO interactions; it is mostly the orientation of the
Mo(CO)$_{3}$ cone that defines the position of the CO ligands with
respect to the coinage  metals~\cite{54,56}.

After discussing the C$_{5}$H$_{4}$NMe$_{2}$ derivatives (\textbf{A2}
and \textbf{A3}), we now turn to their Cp analogues, \textbf{A5} and
\textbf{A6}, which display notable structural differences in the
arrangement of their metal cores.

The silver square in \textbf{A5} and the gold square in \textbf{A6}
exhibit significant differences in the arrangement of their atoms. For
the geometry optimisation of cluster \textbf{A5}, it was necessary to
impose a constraint, under $C_{2}$ symmetry, by arbitrarily fixing the
torsion angle Ag(1)--Ag(2)--Ag(1a)--Ag(2a) at 21\textdegree, since
unconstrained optimisation led to a planar square structure like that
of the gold cluster, thereby causing convergence issues. The calculated
Ag--Ag distances show a maximum elongation of about 0.07~\AA{}. A
significant difference is also observed for the diagonal distances,
with calculated Ag(1)--Ag(1a) of 4.00~\AA{} compared to 3.50~\AA{} in
the experimental structure, while calculated Ag(2)--Ag(2a) is shorter
by 0.35~\AA{}. Moreover, a pronounced discrepancy is found in the bond
angles, with calculated values for Ag(1)--Ag(2)--Ag(1a) and
Ag(2)--Ag(1)--Ag(2a) of 87.41\textdegree\ and 88.76\textdegree,
compared to the experimental values of 75.92\textdegree\ and 
100.23\textdegree, respectively.

In the structure of cluster \textbf{A6}, the four gold atoms are
coplanar, and the diagonal distances are nearly identical, with
Au(1)--Au(3) at 3.940~\AA{} and Au(2)--Au(4) at 3.904~\AA{} 
(Table~S6). In contrast, the 
\mbox{diagonal} distances are very different in
\textbf{A5} for Ag(1)--Ag(1a) and Ag(2)--Ag(2a) at 3.530 and
4.405~\AA{}, respectively. Like in \textbf{A5}, the four Cp ligands and
the carbonyl groups in \textbf{A6} adopt a similar orientation,
consistent with an approximate $S_{4}$ symmetry  (Table~S5). When the
geometry optimisation of cluster \textbf{A6} was started from $C_{1}$
symmetry, it converged to a geometry close to $S_{4}$ symmetry. The
calculated Au--Au bond lengths are overestimated by up to ca.\ 
0.15~\AA{} compared to the experimental data. Nevertheless, the overall
set of calculated distances is in satisfactory agreement with the
experimental data, thereby validating the DFT method and basis set
employed. 

\subsection{Relative stabilities of the various models as a function of
their molecular symmetry}\label{sec3.2}
To better understand the relative stabilities of the different cluster
architectures, we carried out geometry optimisations using DFT at the
BP86 level of theory combined with the SV(P) basis set and explored,
for each system, several possible symmetries, namely $C_{2}$, $D_{2}$, 
$S_{4}$,  $C_{1}$, and $C_{\mathrm{s}}$, to identify the most stable
configuration.

The optimised geometries of the clusters containing the
C$_{5}$H$_{4}$NMe$_{2}$ ligands are presented in  Figure~\ref{fig1},
while the corresponding structures with unsubstituted Cp ligands are
displayed in  Figure~\ref{fig2}. This distinction allows a direct
comparison of the effect of the NMe$_{2}$ substituent on the cluster
stability and 
\mbox{geometry.}

\begin{figure*}
\includegraphics{fig01}
\vspace*{2.5pt}
\caption{\label{fig1}Optimised DFT/BP86 structures of the
[M$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$] and 
[M$_{3}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{3}$]
clusters (M  ${=}$ Cu, Ag, Au) under different symmetry constraints 
($C_1$,  $C_2$,  $D_2$,  $S_4$, and $C_{\mathrm{s}}$).}
\end{figure*}

\begin{figure*}
\includegraphics{fig02}
\caption{\label{fig2}Optimised DFT/BP86 structures of
[M$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$] and [M$_{3}$\{MoCp(CO)$_{3}$\}$_{3}$]
clusters (M ${=}$ Cu, Ag, Au) under different symmetry constraints
($C_{1}$, $C_{2}$, $D_ {2}$, $S_{4}$, and $C_{\mathrm{s}}$).}
\end{figure*}

For the eight-metal-atom clusters, the initial \mbox{geometries} were
constructed on the basis of the experimental data for the related
silver and gold square-type clusters. Specifically, we started from
$C_{2}$ symmetry for the clusters containing the
C$_{5}$H$_{4}$NMe$_{2}$ ligands, whereas we started from $S_{4}$
symmetric geo\-metries for those containing the Cp ligands. To complement
these models, we also examined hypothetical structures of $D_{2}$
symmetry (see structures \textbf{A8}--\textbf{A10} in  Figure~\ref{fig1} and
\textbf{A25}--\textbf{A27} in  Figure~\ref{fig2}), which provided useful
comparisons.

\subsubsection{Cu-containing clusters}\label{sec3.2.1}

\paragraph{With the C$_{5}$H$_{4}$NMe$_{2}$ ligands}\label{sec3.2.1.1}
For the hexanuclear clusters, the optimisations were guided by
experimental observations. The geometries of the
C$_{5}$H$_{4}$NMe$_{2}$-containing clusters were optimised starting
from $C_{1}$ symmetry, while for the Cp analogues (see below), we
adopted a slightly distorted $C_{1}$ geometry close to $C_{\mathrm{s}}$
symmetry. Furthermore, a hypothetical $C_{\mathrm{s}}$-like model was
considered, where the C$_{5}$H$_{4}$NMe$_{2}$ ligands are positioned on
opposite sides of the metal plane (structures
\textbf{A19}--\textbf{A21},  Figure~\ref{fig1}). This configuration was
introduced to probe the effect of the orientation of the aromatic
ligand on the overall cluster stability and electronic distribution.

Overall, this systematic optimisation approach, covering multiple
symmetries and both Cp and C$_{5}$H$_{4}$NMe$_{2}$ ligand types,
provided a coherent framework for evaluating how the metal (Cu, Ag, Au)
and the $\uppi$-ligand substitution (Cp or C$_{5}$H$_{4}$NMe$_{2}$)
affect the structural preferences and relative stabilities of these
polymetallic clusters.

The relative energies and the calculated metal--metal bond distances
for the different clusters optimised under the various symmetry
constraints are compiled in  Tables~\ref{tab1}--\ref{tab4}, which
allows a direct comparison between the alternative structural models,
highlighting the effect of symmetry on both the energetic stability and
the geometric features of the 
\mbox{clusters.}

\begin{table*}
\caption{\label{tab1}Main geometrical parameters (distances in~\AA{})
and relative energies (in Hartree and $\Delta E$ in kJ/mol) for the
models [Cu$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$]
and [Cu$_{3}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{3}$],
optimised under different symmetry constraints ($C_{1}$, $C_{2}$,
$D_{2}$, $S_{4}$, and $C_{\mathrm{s}}$)}
\begin{tabular}{ccccc}
\thead
Clusters & Symmetry & d$^{10}$--d$^{10}$ Distances (\AA{}) &
$E$ (Hartree) & $\Delta E$~(kJ/mol) \\ 
\endthead
\multicolumn{5}{c}{[Cu$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$]} \\ 
\textbf{A7} & $C_{2}$ & 2.55; 2.65 & ${-}$9508.231 & 0 \\ 
\textbf{A10} & $D_{2}$ & 3.32 & ${-}$9508.249 & ${-}$47.40 \\ 
\textbf{A13} & $S_{4}$ & 3.34 & ${-}$9508.252 & ${-}$53.37 
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Cu$_{3}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{3}$]} \\ 
\textbf{A1} & $C_1$ & 2.58, 2.60, 2.64 & ${-}$7131.182 & 0 \\ 
\textbf{A18} & $C_{\mathrm{s}}$ & 2.62 & ${-}$7131.178 & ${+}$10.18 \\ 
\textbf{A21} & $C_1$ & 2.60 & ${-}$7131.187  & ${-}$12.72
\botline
\end{tabular}
\end{table*}

\begin{table*}
\caption{\label{tab2}Main geometrical parameters (distances in~\AA{})
and relative energies (in Hartree and $\Delta E$ in kJ/mol) for the
models  [Cu$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$] and
[Cu$_{3}$\{MoCp(CO)$_{3}$\}$_{3}$], optimised
under different symmetry constraints  ($C_{1}$, $C_{2}$,  $D_{2}$,
$S_{4}$, and  $C_{\mathrm{s}}$)}
\begin{tabular}{ccccc}
\thead
Clusters & Symmetry & d$^{10}$--d$^{10}$ Distances (\AA{}) &
$E$ (Hartree) & $\Delta E$~(kJ/mol) \\ 
\endthead
\multicolumn{5}{c}{[Cu$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$]} \\ 
\textbf{A24} & $C_{2}$ & 2.58, 2.68 & ${-}$8972.149 & 0 \\ 
\textbf{A27} & $D_{2}$ & 2.52 & ${-}$8972.140 & ${+}$23.52 \\ 
\textbf{A29} & $S_{4}$ & 2.75 & ${-}$8972.159 & ${-}$25.45 \\ 
\textbf{A31} & $C_{2}$ & 2.70, 2.74 & ${-}$8972.154 & ${-}$13.74 
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Cu$_{3}$\{MoCp(CO)$_{3}$\}$_{3}$]} \\ 
\textbf{A34} & $C_{1}$ & 2.54, 2.55, 2.64 & ${-}$6729.124 & 0 \\ 
\textbf{A4} & $C_{\mathrm{s}}$ & 2.60 & ${-}$6729.127 & ${-}$9.55
\botline 
\end{tabular}
\vspace*{4pt}
\end{table*}

\begin{table*}
\caption{\label{tab3}Main geometrical parameters (distances in~\AA{})
and relative energies (in Hartree and $\Delta E$ in kJ/mol) for the
models [M$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$]
and [M$_{3}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{3}$]
(M ${=}$ Ag or Au), optimised under different symmetry constraints 
($C_{1}$,  $C_{2}$,  $D_{2}$,  $S_{4}$, and $C_{\mathrm{s}}$)}
\begin{tabular}{ccccc} 
\thead
Clusters & Symmetry & d$^{10}$--d$^{10}$ Distances (\AA{}) &
$E$ (Hartree) & $\Delta E$~(kJ/mol) \\ 
\endthead
\multicolumn{5}{c}{[Ag$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$]} \\ 
\textbf{A2} & $C_{2}$ & 2.84, 2.94 & ${-}$3533.290 & 0 \\ 
\textbf{A8} & $D_{2}$ & 2.91 & ${-}$3533.286 & ${+}$10.44 \\ 
\textbf{A11} & $S_{4}$ & 2.96 & ${-}$3533.294 & ${-}$10.92 
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Ag$_{3}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{3}$]} \\ 
\textbf{A14} & $C_{1}$ & 2.89, 2.95, 2.99 & ${-}$2649.958 & 0 \\ 
\textbf{A16} & $C_{\mathrm{s}}$ & 3.01 & ${-}$2649.958 & ${+}$1.81 \\ 
\textbf{A19} & $C_{1}$ & 2.98 & ${-}$2649.962 & ${-}$9.26 
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Au$_{4}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{4}$]} \\ 
\textbf{A3} & $C_{2}$ & 2.88, 2.93 & ${-}$3488.520 & 0 \\ 
\textbf{A9} & $D_{2}$ & 2.95 & ${-}$3488.513 & ${+}$17.25 \\ 
\textbf{A12} & $S_{4}$ & 2.90 & ${-}$3488.522 & ${-}$7.063 
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Au$_{3}$\{Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$\}$_{3}$]} \\ 
\textbf{A15} & $C_{1}$ & 2.96, 3.12, 3.18 & ${-}$2616.377 & 0 \\ 
\textbf{A17} & $C_{\mathrm{s}}$ & 3.17 & ${-}$2616.378 & ${-}$2.25 \\ 
\textbf{A20} & $C_{1}$ & 3.12, 3.13, 3.14 & ${-}$2616.379 & ${-}$6.20
\botline 
\end{tabular}
\vspace*{4pt}
\end{table*}

\begin{table*}
\caption{\label{tab4}Main geometrical parameters (distances in~\AA{})
and relative energies (in Hartree and $\Delta E$ in kJ/mol) for the
models [M$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$] and
[M$_{3}$\{MoCp(CO)$_{3}$\}$_{3}$]  (M ${=}$ Ag or
Au), optimised under different symmetry constraints ($C_{1}$,  $C_{2}$,
$D_{2}$,  $S_{4}$, and~$C_{\mathrm{s}}$)}
\begin{tabular}{ccccc} 
\thead
Clusters & Symmetry & d$^{10}$--d$^{10}$ Distances (\AA{}) & 
$E$ (Hartree) & $\Delta E$~(kJ/mol) \\ 
\endthead
\multicolumn{5}{c}{[Ag$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$]} \\ 
\textbf{A22} & $C_{2}$ & 2.88, 2.95 & ${-}$2997.209 & 0 \\ 
\textbf{A25} & $D_{2}$ & 2.85 & ${-}$2997.209 & ${+}$0.92 \\ 
\textbf{A28} & $S_{4}$ & 2.93 & ${-}$2997.216 & ${-}$18.14 \\ 
\textbf{A5} & $C_{2}$ & 2.88, 2.92 & ${-}$2997.212 & ${-}$8.72
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Ag$_{3}$\{MoCp(CO)$_{3}$\}$_{3}$]} \\ 
\textbf{A32} & $C_{1}$ & 2.88, 2.93, 3.02 & ${-}$2247.901 & 0 \\ 
\textbf{A35} & $C_{\mathrm{s}}$ & 2.99 & ${-}$2247.903 & ${-}$4.38 
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Au$_{4}$\{MoCp(CO)$_{3}$\}$_{4}$]} \\ 
\textbf{A23} & $C_{2}$ & 2.91, 2.92 & ${-}$2952.437 & 0 \\ 
\textbf{A26} & $D_{2}$ & 2.90 & ${-}$2952.434 & ${+}$8.18 \\ 
\textbf{A6} & $S_{4}$ & 2.91 & ${-}$2952.442 & ${-}$14.16 \\ 
\textbf{A30} & $C_{2}$ & 2.89, 2.91 & ${-}$2952.439 & ${-}$7.12 
\vspace*{6pt}\\ 
\multicolumn{5}{c}{[Au$_{3}$\{MoCp(CO)$_{3}$\}$_{3}$]} \\ 
\textbf{A33} & $C_{1}$ & 2.94, 3.07, 3.22 & ${-}$2214.317 & 0 \\ 
\textbf{A36} & $C_{\mathrm{s}}$ & 3.14 & ${-}$2214.318 & ${-}$1.56 
\botline 
\end{tabular}
%\vspace*{2pt}
\end{table*}

The optimisation of the three models with a copper square core led to
distinct geometries. Cluster \textbf{A7}, with $C_{2}$ symmetry,
converges to the same structure as that found for the silver square,
displaying short Cu--Cu distances of 2.55 and 2.65~\AA{}. In contrast,
for clusters \textbf{A10} and \textbf{A13}, with $D_{2}$ and $S_{4}$
symmetry, longer Cu--Cu distances of 3.32~\AA{} and 3.34~\AA{} were
\mbox{obtained,} respectively. Taking the energy of isomer \textbf{A7} as a
reference, isomers \textbf{A10} and \textbf{A13} were found to be
energetically more stable by 47.40 and 53.37~kJ/mol, 
\mbox{respectively.}

For the clusters with a triangular core \textbf{A1}, \textbf{A18}, and
\textbf{A21}, geometry optimisation led to the same structural motif,
with Cu--Cu distances in the  2.58--2.62~\AA{} range. This results in a
stabilisation of isomer \textbf{A21} by 12.72~kJ/mol and a
destabilisation of isomer \textbf{A18} by 10.18~kJ/mol, relative to the
experimentally observed structure of cluster \textbf{A1} (the energy of
isomer \textbf{A1} being taken as the reference).

\paragraph{With the Cp ligands}\label{sec3.2.1.2}
In the case of the square copper clusters containing the Cp ligands,
the Cu--Cu bond lengths are noticeably shorter than those obtained for
the systems containing the C$_{5}$H$_{4}$NMe$_{2}$ ligand. For
instance, cluster \textbf{A27} with $D_{2}$ symmetry exhibits Cu--Cu
distances of approximately 2.52~\AA{}, which are even shorter than the
experimental values determined by X-ray diffraction. By contrast,
cluster \textbf{A29} with $S_{4}$ symmetry displays d$^{10}$--d$^{10}$
separations very similar to those observed in cluster \textbf{A31} with
$C_{2}$ symmetry. Cluster \textbf{A29} was found to be more stable by
25.45~kJ/mol relative to \textbf{A24} taken as a reference and by
nearly 50~kJ/mol compared to cluster \textbf{A27}. These results
clearly indicate that the $S_{4}$-symmetric square arrangement
represents the energetically most favourable configuration among the
clusters containing a square copper core.

For the triangular copper clusters, the optimised geometries reveal
that the Cu--Cu bond lengths vary slightly depending on the considered
symmetry. In the case of cluster \textbf{A4} with $C_{\mathrm{s}}$
symmetry, the calculated Cu--Cu distances are uniform, around
2.60~\AA{}. For cluster \textbf{A34} with $C_{1}$ symmetry, the Cu--Cu
separations are more dispersed, ranging from 2.54 to 2.64~\AA{}. From
an energetic standpoint, cluster \textbf{A4} is more 
\mbox{stable} than
cluster \textbf{A34} by 9.5~kJ/mol, suggesting that the
$C_{\mathrm{s}}$-symmetric triangular arrangement is energetically
preferred over the $C_{1}$ variant. This stabilisation highlights the
role of symmetry in determining the most favourable structural
configuration for copper clusters containing a triangular\break core.

The energies calculated for clusters \textbf{A2} (M ${=}$ Ag) and
\textbf{A3} (M ${=}$ Au), optimised from the experimental data, are used
as reference values for the square systems, while the energies of
clusters \textbf{A14} (M ${=}$ Ag) and \textbf{A15} (M ${=}$ Au) are
arbitrarily chosen as reference values for the triangular systems.

\subsubsection{Ag- and Au-containing clusters}\label{sec3.2.2}

\paragraph{With the C$_{5}$H$_{4}$NMe$_{2}$ Ligands}\label{sec3.2.2.1}
In contrast to what was observed for the square copper clusters, the
geometry optimisation of the silver and gold square clusters yielded
calculated distances close to the experimental values for all three
symmetries, $C_{2}$, $D_{2}$, and $S_{4}$.

Cluster \textbf{A11} with $S_{4}$ symmetry, featuring an Ag--Ag
distance of 2.96~\AA{}, is more stable by 10.92~kJ/mol than isomer
\textbf{A2}, whereas cluster \textbf{A8} with $D_{2}$ symmetry is
destabilised by 10.44~kJ/mol relative to \textbf{A2}. For the
triangular clusters \textbf{A14}, \textbf{A16}, and \textbf{A19}, the
calculated Ag--Ag distances range from 2.89 to 3.00~\AA{} for
\textbf{A14} and are around 3.00~\AA{} for both \textbf{A16} and
\textbf{A19}. Cluster \textbf{A19} is more stable than \textbf{A14} by
9.26~kJ/mol.

For the triangular and square gold clusters, results similar to those
found for the silver clusters were obtained. The Au--Au distances in
the triangular clusters are significantly longer than those in the
square clusters, as exemplified by cluster \textbf{A15}, where the
distances range from 2.96 to 3.18~\AA{}. Among the square systems, the
$S_{4}$-symmetric cluster \textbf{A12} is the most favourable, while in
the triangular systems, the $C_{\mathrm{s}}$-like symmetric cluster
\textbf{A20} is preferred.

\paragraph{With the Cp ligands}\label{sec3.2.2.2}
We obtained similar results for the square silver clusters containing
Cp or C$_{5}$H$_{4}$NMe$_{2}$ ligands. Cluster \textbf{A28} with
$S_{4}$ symmetry is preferred by 18.14~kJ/mol over cluster \textbf{A22}
with $C_{2}$ symmetry, and by 10.58~kJ/mol over cluster \textbf{A5},
also with $C_{2}$ symmetry.

For the triangular silver clusters, \textbf{A32} with $C_{1}$ symmetry
exhibits different Ag--Ag distances of 2.88, 2.93, and 3.02~\AA{},
while cluster \textbf{A35} with $C_{\mathrm{s}}$ symmetry shows Ag--Ag
separations of about 2.99~\AA{}. The latter is more stable by
4.38~kJ/mol compared to cluster \textbf{A32}. Similar observations can
be made for the gold-containing clusters, where cluster \textbf{A6}
with $S_{4}$ symmetry is the most favourable among the square systems.
In contrast, the triangular clusters \textbf{A33} and \textbf{A36} are
nearly isoenergetic; however, their optimisation leads to longer Au--Au
distances, reaching up to 3.22~\AA{}.

Overall, the stability of the clusters studied in this work is strongly
influenced by both the nature of the coinage metal and the adopted
symmetry, which is itself influenced by the NMe$_{2}$-substituent at
the organic $\uppi$-ligand. In the case of copper, short M--M contacts
are favoured in square geometries, while triangular motifs show
moderate stabilisation depending on symmetry. In the case of silver and
gold, the optimised structures generally reproduce the experimental
distances more closely, with $S_{4}$ symmetry emerging as the most
stable for square clusters. For triangular clusters, the relative
stabilisations are smaller, but optimisation often leads to elongated
M--M distances, especially for gold. These trends highlight the key
role of symmetry in dictating both geometry and relative stability
across the series.

\subsection{Optimised geometries starting from\newline 
elongated metal--metal distances}\label{sec3.3}
Geometry optimisations were initiated from deliberately elongated
metal--metal distances to avoid any predefined interaction. This
approach allows an unbiased evaluation of whether a metal--metal bond
is intrinsically favoured, with any contraction arising solely from the
system's electronic effects.  After examining the stability of the
different models (\textbf{A1}--\textbf{A36}), we determined that the
square copper clusters \textbf{A10} ($D_{2}$ symmetry), \textbf{A13}
($S_{4}$ symmetry), and \textbf{A29} ($S_{4}$ symmetry) are the most
stable. The optimised structures of \textbf{A10} and \textbf{A13}
display large Cu--Cu separations of 3.32 and 3.34~\AA{}, respectively,
and the molybdenum and copper atoms are coplanar. In these systems,
each Cu--Mo--Cu unit contains one triply bridging carbonyl and two
doubly bridging carbonyls (see  Figure~\ref{fig3}). In contrast,
structure \textbf{A29} exhibits a shorter Cu--Cu distance of
2.75~\AA{}, with the molybdenum atoms displaced out of the plane
defined by the copper square, and the carbonyl ligands adopting only
doubly bridging coordination modes. Furthermore, the orientation of the
C$_{5}$H$_{4}$NMe$_{2}$ ligands in \textbf{A10} and \textbf{A13}
differs markedly from that of the Cp ligands in \textbf{A29}
(Figure~\ref{fig3}).

\begin{figure*}
\includegraphics{fig03}
\caption{\label{fig3}Optimised DFT/BP86 structures of \textbf{A10}, 
\textbf{A13}, and \textbf{A29}.}
\end{figure*}

We sought to optimise structures derived from models \textbf{A10} and
\textbf{A13} with $D_{2}$ and $S_{4}$ symmetries, which exhibit large
metal--metal distances for the three coinage metals, and Cp or
C$_{5}$H$_{4}$NMe$_{2}$ ligands. In the case of the copper square, we
successfully obtained two optimised structures, \textbf{A41} and
\textbf{A42}  
\mbox{(Figure~\ref{fig4})}, showing large Cu--Cu distances for
the $D_{2}$ and $S_{4}$ symmetries with the Cp ligand. These results
are comparable to those for structures \textbf{A10} and \textbf{A13}
presented in  Figure~\ref{fig3}. In contrast, for silver and gold, the
optimisation of the $S_{4}$ symmetric structures with both types of
ligands yielded geometries with short M--M distances of about
2.75~\AA{} for silver and 2.90~\AA{} for gold. However, optimisation of
the $D_{2}$ symmetric structures with both Cp or
C$_{5}$H$_{4}$NMe$_{2}$ ligands led to geometries with large
metal--metal separations, 3.38~\AA{} and 3.44~\AA{} for the silver
squares \textbf{A37} and \textbf{A39}, and 3.54~\AA{} and 3.56~\AA{}
for the gold squares \textbf{A38} and \textbf{A40} 
(Figure~\ref{fig4}).

\begin{figure*}
\includegraphics{fig04}
\caption{\label{fig4}Clusters with silver, gold, and copper squares
\textbf{A37}--\textbf{A41} with $D_{2}$ symmetry and \textbf{A42} with
${S}_{4}$ symmetry.}
\end{figure*}

\begin{figure*}
\includegraphics{fig05}
\vspace*{-4pt}
\caption{\label{fig5}Relative energies  ($\Delta E$ in kJ/mol) for the
fragment Cu[Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$].}
\vspace*{-6pt}
\end{figure*}

Optimisation attempts starting from elongated metal--metal separations
show that only square copper clusters can retain such large Cu--Cu
distances, while the silver or gold square 
clusters \mbox{predominantly} relax
to shorter M--M bonds in $S_{4}$ symmetry. However, $D_{2}$ symmetric
Ag and Au squares can retain elongated M--M distances, highlighting a
clear dependence of the final geometry on both the metal type and
cluster symmetry.

\subsection{Energies of the individual fragments
MMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$ and MMoCp(CO)$_{3}$ (M ${=}$ Cu,
Ag, Au)}\label{sec3.4}
To complement the previous results and clarify further the structural
preferences and relative stabilities of triangular and square clusters,
we conducted a comparative analysis of the relative energies of the
fragments MMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$ and MMoCp(CO)$_{3}$  
(M ${=}$ Cu, Ag, Au), which constitute building blocks in the final
clusters. For the square-type clusters, we normalised the total
electronic energy by dividing it by four, considering the four
constitutive fragments, while for the triangular-type clusters, the
total energy was divided by three. This normalisation procedure, which
allows a direct comparison between clusters with different
nuclearities, follows the approach previously introduced in our earlier
study~\cite{25}. As reference points, we used \textbf{A2}, \textbf{A3},
and \textbf{A7} for the MMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$ systems,
and \textbf{A22}--\textbf{A24} for the MMoCp(CO)$_{3}$ systems. The
relative energies thus obtained provide a reliable basis for evaluating
which geometries and symmetries are energetically more favourable. The
resulting energy differences were plotted as energy diagrams 
\mbox{(Figures~\ref{fig5}--\ref{fig8})},
clearly illustrating stability trends
across the coinage metals. These diagrams not only emphasise the
energetic preferences of each system but also enable visualisation of
the effect of ligand substitution (Cp or C$_{5}$H$_{4}$NMe$_{2}$) on
cluster stability. Overall, this comparative study offers a consistent
picture of how the interplay between the nature of the metal centre,
the $\uppi$-ligand, and the symmetry influences fragment stabilisation.
The analysis of these relative energies is key to understanding the
structural preferences of the copper-, silver-, and gold-containing
clusters and will underpin discussions of their electronic properties
in the following  sections.

\begin{figure*}
\includegraphics{fig06}
\caption{\label{fig6}Relative energies  ($\Delta E$ in kJ/mol) for the
fragments M[Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$] (M ${=}$ Ag and Au).}
\end{figure*}

\begin{figure*}
\includegraphics{fig07}
\caption{\label{fig7}Relative energies  ($\Delta E$ in kJ/mol) for the
fragment Cu[MoCp(CO)$_{3}$].}
\end{figure*}

\begin{figure*}
\includegraphics{fig08}
\caption{\label{fig8}Relative energies  ($\Delta E$ in kJ/mol) for the
fragments  M[MoCp(CO)$_{3}$] (M ${=}$ Ag and Au).}
\vspace*{-4pt}
\end{figure*}

\subsubsection{Fragment
Cu[Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]}\label{sec3.4.1}
The analysis shows that the square cluster \textbf{A13} with $S_{4}$
symmetry and the triangular cluster \textbf{A21} are energetically more
favourable compared to the 
\mbox{experimentally} observed structure
\textbf{A1}. This energetic preference highlights an intrinsic tendency
of copper to prefer the square rather than the triangular arrangement.
However, this stabilisation is only achieved when the Cu--Cu
separations are significantly elongated, with optimised distances
around 3.30~\AA{}. Such elongated Cu--Cu distances reduce the repulsive
interactions within the metallic core and allow a better electronic
distribution between the copper and molybdenum centres, thereby
favouring the square geometry. 

\subsubsection{Fragments M[Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]  (M
${=}$ Ag or Au)}\label{sec3.4.2}
In contrast to the case of copper, the most stable silver and gold
clusters with the C$_{5}$H$_{4}$NMe$_{2}$ ligands are the square
clusters \textbf{A11} and \textbf{A12}, respectively, both adopting
$S_{4}$ symmetry with short M--M distances.

The triangular structures \textbf{A14}--\textbf{A17}, \textbf{A19}, and
\textbf{A20}, as well as the square structures \textbf{A37} and
\textbf{A38} with elongated M--M separations, are less stable than
\textbf{A2} and \textbf{A3}. The preference for shorter M--M distances
in the case of silver and gold can be attributed to electronic and
steric factors: shorter metal--metal distances maximise metal--metal
interactions and allow a more favourable overlap with the CO ligands.
Overall, these results highlight that, unlike with copper, silver and
gold square arrangements with $S_{4}$ symmetry and short M--M
separations are energetically preferred, emphasising the role of the
intrinsic electronic properties of the coinage metal in determining the
cluster stability.

\subsubsection{Fragment Cu[MoCp(CO)$_3$]}\label{sec3.4.3}
In this case, the theoretical results are in excellent agreement with
the experimental structures determined by X-ray diffraction, confirming
that the triangular cluster \textbf{A4} indeed corresponds to the most
stable arrangement for the copper-containing clusters. We also
successfully obtained an optimised $S_{4}$-symmetric structure
\textbf{A29}, featuring short Cu--Cu distances of approximately
2.75~\AA{}. In this structure, the carbonyl ligands occupy positions
very similar to those observed experimentally in \textbf{A4},
demonstrating that the stereoelectronic environment is properly
captured by the DFT calculations. The \mbox{fragment} corresponding to
structure \textbf{A29} exhibits a stability comparable to that of the
fragment from structure \textbf{A4}, and it is also more stable than
the square structure \textbf{A42}, which has elongated metal--metal
distances. These observations highlight that triangular arrangements
are energetically favoured for copper when metal--metal distances are
moderate, while square {geometries} with long Cu--Cu separations can also
be stabilised under specific conditions. Overall, the results provide a
coherent picture of the interplay between geometry, metal--metal
distances, and ligand orientation in determining the relative stability
of the copper-containing clusters  (Figure~\ref{fig7}).

Overall, the calculations confirm that the triangular copper cluster
\textbf{A4} is the most stable, with a $S_{4}$-symmetric structure like
\textbf{A29} where shorter Cu--Cu distances provide additional
stabilisation, while square clusters with elongated metal--metal
separations are less favoured.

\subsubsection{Fragments  M[MoCp(CO)$_{3}$]  (M
${=}$ Ag or Au)}\label{sec3.4.4}
In the case of silver-containing clusters, our calculations reveal a
clear discrepancy with the experimental observations. Indeed, the
fragment corresponding to the experimentally determined structure
\textbf{A5} does not correspond to the most stable structure, since the
fragment of the square structure \textbf{A28} with $S_{4}$ symmetry is
more stabilised.

This suggests that, although the triangular geometry is accessible
experimentally for silver, the square arrangement with short Ag--Ag
distances is intrinsically more favourable from an energetic point of
view. In contrast, for the gold cluster \textbf{A6}, the theoretical
calculations are in perfect agreement with the experimental data
obtained by X-ray diffraction. This concordance strongly supports the
reliability of our computational approach and confirms that, unlike for
silver, the triangular geometry experimentally observed for gold indeed
corresponds to the most stable form.

\subsection{Bonding analysis}\label{sec3.5} 
Tables~S6--S11 summarise the energy decomposition analysis (EDA)
results for the interactions between the metal cores and the
metalloligand fragments [MoCp(CO)$_{3}$]$_{n}$, calculated at the BP86
level. Here, $\Delta E_{\mathrm{int}}$ denotes the total interaction
energy, while $\Delta E_{\mathrm{elstat}}$ and $\Delta
E_{\mathrm{orb}}$ correspond to the electrostatic and orbital
(covalent) contributions, respectively. Energy decomposition analysis
at the BP86 level was performed to elucidate the nature of the
interactions between the Cu$_{n}$, Ag$_{n}$, and
Au$_{n}$ metal cores and the metalloligand fragments
[MoCp(CO)$_{3}$]$_n$ and [Mo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_n$
for all investigated clusters. In every case, the strongly negative
$\Delta E_{\mathrm{int}}$ values indicate that cluster formation is
governed by highly stabilising metal--ligand interactions.

The BP86 EDA results show that all Cu-based complexes are strongly
stabilised by their interaction with the metalloligand fragments. The
Cu$_{3}$  systems (\textbf{A1}, \textbf{A18}, and \textbf{A21})
exhibit interaction energies of around  ${-}4.93\times10^{3}$~kJ/mol,
whereas the  Cu$_{4}$ clusters (\textbf{A7}, \textbf{A10}, and
\textbf{A13}) exhibit much larger stabilisations of  ${-}$7.28 to
${-}7.60\times10^{3}$~kJ/mol,  highlighting the stronger binding in the
square architectures. This difference mainly arises from the
electrostatic term, which increases markedly when going from Cu$_{3}$
{(${-}$4.46 to ${-}$4.56~MJ/mol) to Cu$_{4}$  (${-}$6.38 to
${-}$6.95~MJ/mol).  The orbital contribution also becomes significantly
more stabilising in the Cu$_{4}$ systems, indicating improved Cu--Mo
covalency and more efficient metal--ligand orbital mixing. Although
Pauli repulsion is larger in the more compact Cu$_{4}$ cores, it is
clearly outweighed by the enhanced electrostatic and orbital
attractions, resulting in superior overall stability of the
tetranuclear complexes.

All Ag-containing systems display highly favourable interaction
energies. The Ag$_{3}$ complexes (\textbf{A14},  \textbf{A16},
\textbf{A19})  show $\Delta E_{\mathrm{int}}$ values close to
${-}$4.45~MJ/mol, while the  Ag$_{4}$  species (\textbf{A2},
\textbf{A8},  \textbf{A11}, and \textbf{A37}) are much more stabilised,
with values approaching ${-}$7.0~MJ/mol. This enhanced binding in the 
Ag$_{4}$ series is primarily driven by the electrostatic term, which
becomes substantially more negative upon increasing the nuclearity of
the Ag core. In parallel, the orbital interaction also strengthens,
reflecting improved Ag--Mo covalent interactions and electronic
delocalisation within the square Ag$_{4}$ rings. Despite the rise in
Pauli repulsion, especially in \textbf{A37}, the dominant attractive
contributions lead to a pronounced stabilisation of the tetranuclear\break
assemblies.

For all gold complexes, the interaction between the Au cores and the
metalloligand is highly 
\mbox{stabilising.} The Au$_{3}$ species
(\textbf{A15}, \textbf{A17},  and \textbf{A20}) display interaction
energies around ${-}$5.0~MJ/mol, whereas the  Au$_{4}$ derivatives
(\textbf{A3}, \textbf{A9}, \textbf{A12}, and \textbf{A38}) reach
significantly larger values between ${-}$7.4 and ${-}$7.7~MJ/mol.
Electrostatic attraction is the leading stabilising factor in both
series, but it is strongly reinforced in the Au$_{4}$ clusters. The
orbital term also increases markedly with nuclearity, confirming a
substantial covalent contribution associated with Au--Mo bonding and
metal--metal cooperation. Although Pauli repulsion rises in the
Au$_{4}$ species, the net effect remains highly favourable, yielding
the most stable complexes in this family.

The Cu$_{3}$ complexes \textbf{A4} and \textbf{A34} exhibit interaction
energies of about  ${-}$4.87~MJ/mol, whereas the  Cu$_{4}$ derivatives
display a much broader but generally more stabilising range extending
down to  ${-}$7.51~MJ/mol. In all cases, electrostatic interactions
dominate the attractive forces, supported by sizable orbital
contributions that point to significant Cu--Mo covalent bonding. The
larger Pauli repulsion observed in some  Cu$_{4}$  systems is
counterbalanced by the stronger electrostatic and orbital
stabilisation, leading overall to enhanced binding in the tetranuclear
clusters. Although the EDA results indicate that the tetranuclear
systems generally exhibit stronger interaction energies with the
metalloligand fragments, this does not exclude the existence of
trinuclear  Cu$_{3}$ rafts. The EDA analysis mainly describes the
metal--ligand interactions and does not fully account for other factors
such as geometric constraints and intrinsic Cu--Cu bonding preferences.
These effects can stabilise the triangular Cu$_{3}$ arrangement,
allowing it to exist as a viable and locally stable structural motif
despite the greater stabilisation predicted for the tetranuclear
systems.

The Ag$_{3}$ complexes \textbf{A32} and \textbf{A35} show interaction
energies around  ${-}$4.40~MJ/mol, while the Ag$_{4}$  clusters
(\textbf{A5}, \textbf{A22}, \textbf{A25}, \textbf{A28}, and
\textbf{A39}) are substantially more stabilised, with} $\Delta
E_{\mathrm{int}}$ values approaching ${-}$6.9~MJ/mol.  As in the other
series, electrostatic attraction is the principal stabilising factor
and increases significantly with the size of the silver core. This is
accompanied by a marked rise in orbital stabilisation, evidencing
enhanced Ag--Mo covalent interactions. Even though Pauli repulsion
becomes larger in the Ag$_{4}$  clusters, the combined electrostatic
and orbital effects clearly favor the tetranuclear species.

The Au$_{4}$ complexes (\textbf{A6}, \textbf{A23}, \textbf{A26}, 
\textbf{A30},  and \textbf{A40}) exhibit much more negative interaction
energies (${-}$7.3 to ${-}$7.6~MJ/mol) than the Au$_{3}$ species
\textbf{A33} and \textbf{A36}, confirming the strong stabilising effect
of increasing gold nuclearity. This enhanced binding is the result of
both a stronger electrostatic attraction and a reinforced orbital
interaction, indicating more effective Mo${\rightarrow}$Au donation and
Au${\rightarrow}$Mo back-donation in the square  Au$_{4}$  cores. Among
them, \textbf{A23} and \textbf{A26} stand out by their particularly
large orbital contributions, reflecting pronounced covalent character
and efficient metal--metal electronic communication. Although Pauli
repulsion is higher in the  Au$_{4}$  systems, it is more than
compensated by the dominant attractive interactions.

The EDA results also rationalise the higher stability of the
tetranuclear Ag$_{4}$ and  Au$_{4}$ clusters compared with their
trinuclear analogues. The square M$_{4}$ cores provide a more
favourable electronic environment that enables a more efficient charge
redistribution between the metal core and the metalloligand fragments.
Consequently, both the electrostatic $\Delta E_{\mathrm{elstat}}$ and
orbital $\Delta E_{\mathrm{orb}}$ contributions become more
stabilising when moving from  M$_{3}$ to M$_{4}$ species. Although
Pauli repulsion increases in the more compact M$_{4}$ structures, it is
largely compensated by the stronger attractive interactions.

A clear and systematic trend is observed with increasing nuclearity of
the metal core. For all three metals, the tetranuclear M$_{4}$ systems
are significantly more stabilised than their trinuclear M$_{3}$ 
analogues. For Cu and Ag, the $\Delta E_{\mathrm{int}}$ values increase
from approximately ${-}$4.4 to ${-}$5.0~MJ/mol in the  M$_{3}$ species
to about ${-}$6.6 to ${-}$7.6~MJ/mol in the M$_{4}$  clusters, while
for Au, the stabilisation increases from around ${-}$5.0~MJ/mol in
Au$_{3}$  to nearly ${-}$7.7~MJ/mol in Au$_{4}$. This pronounced
enhancement reflects the cooperative effect of adding a fourth metal
centre, which enables a more efficient distribution of charge and
stronger metal--ligand and metal--metal interactions within the square
cores.

In all complexes, the electrostatic term $\Delta E_{\mathrm{elstat}}$
represents the dominant attractive contribution, accounting for the
largest fraction of the stabilisation. This highlights the strong
Coulombic attraction between the positively charged coinage-metal cores
and the anionic Mo-based metalloligand fragments. Importantly, $\Delta
E_{\mathrm{elstat}}$ becomes markedly more negative when moving from 
M$_{3}$  to  M$_{4}$  systems, 
\mbox{indicating} that the square arrangements
provide a more favourable electrostatic environment for binding the
four metalloligands around the metal core. The orbital interaction term
$\Delta E_{\mathrm{orb}}$ is also very large and systematically
increases with nuclearity, revealing a substantial covalent component
in the metal--ligand bonding. The much stronger $\Delta
E_{\mathrm{orb}}$ values in the M$_{4}$ complexes indicate enhanced
Mo${\rightarrow}$M $\upsigma$-donation and M${\rightarrow}$Mo
$\uppi$-back-donation, as well as more effective metal--metal
electronic communication within the cyclic  M$_{4}$ frameworks. This
effect is particularly pronounced for the gold clusters, consistent
with the well-known high polarisability and relativistic stabilisation
of Au orbitals, which promote strong covalent and aurophilic
interactions. Pauli repulsion $\Delta E_{\mathrm{Pauli}}$ increases
with the size and compactness of the metal core, especially in
tetranuclear clusters, as a consequence of greater overlap between
filled orbitals in crowded square geometries. Nevertheless, this
destabilising contribution is systematically outweighed by the much
larger electrostatic and orbital stabilisations. The steric term
$\Delta E_{\mathrm{ster}}$, which combines Pauli repulsion and
electrostatic interactions, remains overall favourable, further
supporting the structural viability of these clusters.

Taken together, the EDA results demonstrate that the superior stability
of the M$_{4}$  assemblies relative to the  M$_{3}$ analogues arises
from the cooperative interplay of strong electrostatic attraction,
enhanced covalent metal--ligand bonding, and efficient metal--metal
interactions within the square cores. This synergy explains the marked
preference for tetranuclear Cu, Ag, and Au architectures in these
Mo-based metalloligand systems and underlines the key role of
nuclearity in tuning the bonding and stability of coinage-metal
clusters.

\subsection{Frontier molecular orbital analysis}\label{sec3.6} 
The frontier molecular orbitals (FMOs), namely the highest occupied
molecular orbital (HOMO) and the lowest unoccupied molecular orbital
(LUMO), are fundamental to the electronic structure, chemical
stability, and reactivity of molecular  systems~\cite{51,52,57,58}. The
HOMO reflects a molecule's electron-donating capability, whereas the
LUMO indicates its electron-accepting potential. The energy separation
between these two orbitals, commonly referred to as the HOMO--LUMO gap
$(\Delta E)$, serves as an important descriptor of molecular stability
and chemical reactivity. To visualise the electronic structure of
clusters \textbf{A1}--\textbf{A42}, the HOMO and LUMO energy levels for the
hydride complexes were plotted. The corresponding FMO distributions are
presented in  Figure~\ref{fig9} (see also Supplementary
Figures~S1--S6).

\begin{figure*}
\includegraphics{fig09}
\caption{\label{fig9}Molecular orbital energy (eV) of
[CuMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_n$  ($n=3$ for \textbf{A1}, 
\textbf{A18}, \textbf{A21}, and $n=4$ for \textbf{A7}, \textbf{A10},
\textbf{A13}).}
\end{figure*}

The FMOs provide essential insight into the electronic stability and
reactivity of the Cu, Ag, and Au clusters investigated in this series
(\textbf{A1}--\textbf{A42}).
The calculated energy levels and HOMO--LUMO gaps clearly
demonstrate that the nature of the metal, the presence of the
electron-donating NMe$_{2}$ ligand, and the cluster nuclearity jointly
govern the overall electronic behaviour. For all complexes, the HOMO
corresponds to the highest doubly occupied molecular orbital and is
predominantly metal-centred, with major contributions from the d
orbitals of the coinage metal (Cu, Ag, or Au) and Mo, as well as
varying participation from the Cp and C$_{5}$H$_{4}$NMe$_{2}$ ligands.
The LUMO is systematically an unoccupied orbital dominated by Mo d and
CO $\uppi^*$ contributions. Accordingly, the lowest-energy electronic
excitation in all clusters corresponds to a metal/ligand-to-carbonyl
charge-transfer (MLCT) process. The energy separation between these two
orbitals, $\Delta E = E_{\mathrm{LUMO}}- E_{\mathrm{HOMO}}$, directly
reflects the ease of electron promotion, redox activity, and electronic
softness of the clusters. A dominant trend across all metal families is
the influence of the NMe$_{2}$ substituent. In the substituted series
[MMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_{n}$ (M ${=}$ Cu, Ag, Au), the
HOMO is significantly destabilised  relative to the unsubstituted 
[MMoCp(CO)$_{3}$]$_{n}$ analogues. This effect originates from the
strong  $\upsigma$-donor character of the NMe$_{2}$ group, which
injects electron density into the metal--Mo--Cp framework and increases
the population of metal--ligand bonding orbitals. Consequently, the
HOMO energies are shifted upward and the HOMO--LUMO gaps are reduced.
This behaviour is clearly reflected in the orbital plots  (Figures~S1,
S3, and~S5), where the HOMO of the NMe$_{2}$-substituted clusters shows
significant ligand participation in addition to metal d contributions.
The resulting increase in electron density at the metal centres
enhances back-donation to the carbonyl ligands and stabilises low-lying
antibonding orbitals, bringing the LUMO closer in energy to the HOMO.
As a result, the NMe$_{2}$ substituted clusters are more polarisable,
electronically softer, and 
\mbox{chemically} more reactive than their
unsubstituted counterparts. Among the three coinage metals, Cu-based
clusters exhibit the highest HOMO energies and the smallest HOMO--LUMO
gaps, making them the most electronically active systems. In the
substituted [CuMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_{n}$ series, the
HOMO lies in the range ${-}$4.49 to ${-}$4.84~eV, reflecting strong
Cu--Mo and Cu--NMe$_{2}$ interactions. The HOMO is fully occupied and
delocalised over the Cu--Mo core with substantial ligand contribution,
indicating efficient electron donation from NMe$_{2}$ into
metal-centred bonding orbitals. The corresponding LUMO is largely
Mo--CO $\uppi^*$ in character and remains relatively low in energy
(${-}$2.49 to ${-}$2.69~eV), resulting in small HOMO--LUMO gaps
(1.80--2.30~eV). In particular,  \textbf{A18}, which exhibits the
smallest gap, features a strongly delocalised HOMO and a low-lying
LUMO, enabling facile MLCT transitions and high redox activity. In the
unsubstituted [CuMoCp(CO)$_{3}$]$_{n}$ clusters, removal of the
NMe$_{2}$ donor stabilises the HOMO  (${-}$4.83 to ${-}$5.29~eV) and
leads to a concomitant increase in the energy gap. The HOMO becomes
more localised on the metal--CO framework, while the LUMO retains its
Mo--CO $\uppi^*$ character. This larger separation between occupied and
virtual orbitals explains why these systems are more electronically
robust but less reactive than their NMe$_{2}$-substituted counterparts.
Replacing Cu with Ag leads to a pronounced increase in electronic
stability. In [AgMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_{n}$,  the
HOMO energies are lower than in the Cu analogues (${-}$4.53 to
${-}$4.86~eV), while the LUMO levels remain relatively high (${-}$2.19
to  ${-}$2.43~eV), yielding HOMO--LUMO gaps of 2.14--2.64~eV.  The HOMO
in these systems is dominated by Ag--Mo d orbitals with limited ligand
participation, reflecting weaker ${\upsigma}$ donation from NMe$_{2}$
compared with the Cu system. In the unsubstituted
[AgMoCp(CO)$_{3}$]$_{n}$ series, the HOMO is further stabilised 
(${-}$5.02 to ${-}$5.24~eV), whereas the LUMO remains Mo--CO $\uppi^*$
in character, resulting in the largest HOMO--LUMO gaps of the entire
dataset (up to 2.73~eV for  \textbf{A35}). The diffuse nature of the Ag
4d orbitals reduces metal--ligand overlap and promotes electron
localisation, which is clearly visible in the FMO plots. Consequently,
Ag-based clusters are calculated to be the most electronically stable
and least reactive systems in this study. Au clusters display an
intermediate electronic character 
\mbox{between} Cu and Ag systems. In the
NMe$_{2}$-substituted [AuMo(C$_{5}$H$_{4}$NMe$_{2}$)(CO)$_{3}$]$_{n}$
series, the HOMO spans a broad energy range ({${-}$}4.25 to {${-}$}5.20
eV), reflecting sensitivity to nuclearity and coordination environment.
Relativistic stabilisation of the Au 5d orbitals enhances Au--Mo and
Au--ligand covalency, leading to efficient electronic delocalisation.
The LUMO remains dominated by Mo--CO $\uppi^*$ character, but for some
complexes, notably  \textbf{A9}, it lies unusually close to the HOMO,
giving rise to a small gap (1.76 eV) and pronounced electronic
softness. In the unsubstituted [AuMoCp(CO)$_{3}$]$_{n}$ family, two
distinct regimes emerge. The $n=4$ clusters display relatively small
gaps (1.95--1.99~eV), consistent with strong delocalisation over the
extended Au--Mo framework, whereas the $n=3$ species exhibit much
larger gaps (${>}$2.45~eV), indicative of more localised electronic
structures and higher stability. Across all metal families, increasing
the nuclearity from  $n=3$ to $n=4$ systematically reduces the
HOMO--LUMO gap. The larger metal framework allows greater
delocalisation of the HOMO over multiple metal centres and stabilises
low-lying charge-transfer states. This effect is particularly
pronounced for Au clusters but is also evident for Cu and Ag systems.
Taken together, the FMO analysis shows that the HOMO is always a fully
occupied metal--ligand bonding orbital, while the LUMO is a Mo--CO
antibonding orbital. The HOMO${\rightarrow}$LUMO transition therefore
corresponds to a MLCT process, which is most facile in Cu- and Au-based
NMe$_{2}$-substituted clusters and in  $n=4$ species. These systems are
thus predicted to be the most electronically soft, redox-active, and
catalytically promising, whereas Ag-based and unsubstituted clusters,
characterised by larger HOMO--LUMO gaps and more localised FMOs, are
expected to be the most electronically stable and resistant to
excitation.

\section{Conclusion}\label{sec4}
Taking advantage of the full characterisation of a unique series of 2D
raft-type heterometallic clusters, we could perform a systematic
investigation of the energies and relative stabilities of Cu(I), Ag(I),
and Au(I)-containing clusters with triangular and square core
geometries under different symmetries ($C_{1}$, $C_{\mathrm{s}}$,
$C_{2}$, $D_{2}$, and $S_{4}$), which allowed a clear comparison with
the experimentally determined X-ray structures. In the case of clusters
bearing the Cp ligand, an 
\mbox{excellent} agreement was observed between
theory and experiment, since the most stable computed structures
corresponded closely to those reported crystallographically. With
C$_{5}$H$_{4}$NMe$_{2}$ as $\uppi$-ligand, our calculations indicated
that alternative arrangements more stable than those experimentally
observed are possible, thereby highlighting the significant effect of
the $\uppi$-ligand substitution on the overall stability and geometry
of the clusters, although no direct interaction was observed between
the NMe$_{2}$ group and the coinage metals. It is not too surprising to
sometimes observe differences between calculated and experimental
values involving the carbonyl ligands and the coinage metals, since the
relevant interactions are much weaker than those involving the carbonyl
ligands and molybdenum.

Furthermore, previously reported square copper clusters such as
[Et$_{4}$N]$_{4}$[M$_{4}$Cu$_{4}$S$_{12}$O$_{4}$]  (M ${=}$ Mo,
W)~\cite{58} and
[($\upeta^{5}$-C$_{5}$Me$_{5}$)WS$_{3}$Cu]$_{4}$~\cite{59}  exhibit
elongated Cu--Cu distances, which are consistent with our theoretical
findings since we found that square copper clusters with long Cu--Cu
separations can indeed be stabilised. There is a subtle balance between
stabilisation through electronic factors and destabilisation due to
steric effects, while packing effects may also play a role. 

The energy decomposition analysis provides a clear and quantitative
understanding of the bonding mechanism in these coinage-metal clusters.
In all systems, electrostatic attraction between the positively charged
metal cores and the Mo-based metalloligands represents the dominant
stabilising force, while large orbital interaction terms reveal a
substantial covalent component associated with Mo${\rightarrow}$M
donation and M${\rightarrow}$Mo back-donation. The markedly higher
stabilisation observed for the tetranuclear M$_{4}$ (M ${=}$ Cu, Ag, Au)
complexes compared with their M$_{3}$ analogues demonstrates the
cooperative nature of metal--metal and metal--ligand interactions
within the square cores. These findings rationalise the enhanced
stability of the M$_{4}$ architectures and highlight nuclearity as a
key factor governing the electronic structure and bonding of
metalloligand-stabilised coinage-metal clusters. The electronic
properties of these clusters are controlled by a balance between metal
type, ligand donation, and nuclearity. Cu- and Au-based
C$_{5}$H$_{4}$NMe$_{2}$-substituted clusters, especially with $n=4$,
show enhanced delocalisation and superior charge-transfer capability.
In contrast, Ag-based and unsubstituted systems are more electronically
stable and better suited for applications requiring robust molecular
frameworks.

This work demonstrates that DFT calculations not only satisfactorily
reproduce experimental findings but can also predict new, potentially
accessible geometries, thereby offering valuable guidance for future
experimental investigations.

\section*{Acknowledgements}
The authors are grateful to Dr.~Peter Deglmann (University of
Heidelberg, Germany; now~BASF SE, Ludwigshafen, Germany),~for providing
computer resources and continuous encouragement.

\printCOI

\section*{Supplementary materials}
Supporting information for this article is available on the journal's
website under \printDOI\ or from the author.

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