  In this work, we explore the quantum chemical foundations of descriptors for
molecular similarity. Such descriptors are key for traversing chemical compound
space with machine learning. Our focus is on the Coulomb matrix and on the
smooth overlap of atomic positions (SOAP). We adopt a basic framework that
allows us to connect both descriptors to electronic structure theory. This
framework enables us then to define two new descriptors that are more closely
related to electronic structure theory, which we call Coulomb lists and smooth
overlap of electron densities (SOED). By investigating their usefulness as
molecular similarity descriptors, we gain new insights in how and why Coulomb
matrix and SOAP work. Moreover, Coulomb lists avoid the somewhat mysterious
diagonalization step of the Coulomb matrix and might provide a direct means to
extract subsystem information that can be compared across Born-Oppenheimer
surfaces of varying dimension. For the electron density we derive the necessary
formalism to create the SOED measure in close analogy to SOAP. Since this
formalism is more involved than that of SOAP, we review the essential theory,
but also introduce a set of approximations that eventually allow us to work
with SOED in terms of the same implementation available for the evaluation of
SOAP. We focus our analysis on elementary reaction steps, where transition
state structures are more similar to either reactant or product structures than
the latter two are with respect to one another. The prediction of electronic
energies of transition state structures can, however, be more difficult than
that of stable intermediates due to multi-configurational effects. The question
arises to what extent molecular similarity descriptors rooted in electronic
structure theory can resolve these intricate effects.
