  The phenomenon of negative thermal expansion (NTE) deals with the increase of
the lattice parameters and the volume of the unit cell when the material is
thermally cooled. The NTE is typically associated with thermal phonons and
anomalous spin-lattice coupling at low temperatures. However, the underlying
mechanisms in the presence of strong electron correlations in multi-orbital
systems are not yet fully established. Here, we investigate the role of Coulomb
interaction in the presence of lattice distortions in setting out the NTE
effect, by focusing on the physical case of layered Ca$_2$RuO$_4$ with $d^4$
configuration at each Ru ion site. We employ the Slater-Koster parametrization
to describe the electron-lattice coupling through the dependence of the $d-p$
hybridization on the Ru-O-Ru bond angle. The evaluation of the minimum of the
free energy at finite temperature by fully solving the multi-orbital many-body
problem on finite size cluster allows us to identify the regime for which the
system is prone to exhibit NTE effects. The analysis shows that the nature of
the spin-orbital correlations is relevant to drive the reduction of the bond
angle by cooling, and in turn the tendency toward a NTE. This is confirmed by
the fact that a changeover of the electronic and orbital configuration from
$d^4$ to $d^3$ by transition metal substitution is shown to favor the
occurrence of NTE in Ca$_2$RuO$_4$. This finding is in agreement with the
experimental observations of a NTE effect which is significantly dependent on
the transition metal substitution in the Ca$_2$RuO$_4$ compound.
