Abstract
In this work ground and excited electronic states of Heisenberg cluster models, in the form of con-
figuration interaction many-body wave functions, are characterized within the spin-adapted Graph-
ical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric
group approach. Finite-size cluster models of well-defined point-group symmetry and of general
local-spin Slocal ≥ 21 are presented, including J1–J2 triangular and tetrahedral clusters, which are
often used to describe magnetic interactions in biological and bio-mimetic polynuclear transition
metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show
that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted
basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of
the model investigated. The block-diagonal structure is bound to the commutation relations between
cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the
cluster investigated. The many-body basis transformation, in the form of the orbital/site reorder-
ing, exposes such commutation relations. These commutation relations represent a rigorous and
formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression
of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block-
diagonal structure of the Hamiltonian matrix it is possible to selectively optimize electronic excited
states without the overhead of calculating the lower-energy states by simply relying on the initial
ansatz for the targeted wave function. Additionally, more compact many-body wave functions are
obtained. In extreme cases, electronic states are precisely described by a single configuration state
function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are
crucial in the electronic structure theory framework, for they offer a conceptual route towards wave
functions of reduced multi-reference character, that can be optimized more easily by approximated
eigensolvers and are of more facile physical interpretation. They open the way to study larger ab
initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping
the computational costs at their lowest.