of the wave function of the system one can thus be able to determine all the information about the
system of interest. For computational efficiency, however, density functional theory (DFT) serves as an
alternative to pursue such goal. In DFT one the primary objectives is the calculation of the electronic
density of the system, as opposed to the full wave function of all the electrons. Although it is common
to separate both, wave-function theory (WFT) and DFT, as separate fields, it can be argued that both
are intrinsically connected, especially from the algorithmic point of view.
DFT methods have been formulated on the basis of physical understanding of model systems and
small molecules. A notable example is the electron gas, which in many ways has led to functional
components that to date still remain an important part of a very large number of density functional
approximations (DFAs). These functionals are available for different energy “pieces” such as the
kinetic, exchange, correlation, and van der Waals energies. The kinetic energy is known to be the most
challenging energy to be expressed explicitly as a density-functional. For this reason, Kohn-Sham
(KS) DFT [1] is the most common theory within DFT that is utilized for practical calculations and
to derive concepts.[2, 3] As KS-DFT uses single-electron orbitals to determine a kinetic energy. As is
well known, even though KS-DFT practical calculations perform well for determining properties such
as molecular geometries, and optoelectronic properties of a very large number of compound types, it is
difficult for transition-metal systems [4], bond-breaking [5], and charge-transfer excitations [6], among
others, where erroneous charge delocalization [7–10] is a main manifestation of these adverse effects.
Extended DFAs that are free of incorrect charge delocalization should eliminate the main cause
for such adverse effect, the self-interaction error [11–14]. Additionally, improved methodologies must
also come with relatively low computational costs. Motivated by these considerations, and fueled by
advances in machine learning and the premise of new generation of computing technologies (classical
and quantum), theoretical methods are being advanced by the scientific community, with the goal of
extending the applicability of DFT methods [15–17]. These extensions include the development of force
fields, which are creating opportunities for detailed studies of systems at the mesoscopic scale [18]. For
example, artificial neural network (ANN) algorithms have been used to generate density functional
approximations [19, 20], and have been able to eliminate charge delocalization errors. On the other
hand, ANNs also have led to both transferable and specific force fields. This also includes ANNs being
used extensively in materials discovery and properties prediction [21–23]. Machine-learned interactomic
potentials, which are tailored for a particular system of interest demonstrate quite appealing theoretical
prospects for modeling mesoscale phenomena [24–30].
From a foundational perspective, the elimination of charge delocalization still remains a long sought
goal, where theoretical tools are still the subject of continued developments. This problem not only
manifests in DFT development, but also in WFT research. For example, it is known that there are
dynamically correlated post-Hartree-Fock methods that can also cause issues with size-consistency,
whereas the well-known exponential ansatz of WFT, in conjunction with spin-symmetry breaking,
offers a theoretically sound route to restore size-consistency (which implies size-extensivity as well).
We showed previously that this exponential operator, which in turn is determined by what is known
2