  In this study, we examine the thermodynamics and spin dynamics of spin-1/2
and spin-3/2 heptamers. Through an exact diagonalization of the isotropic
Heisenberg Hamiltonian, we find the closed-form, analytical representations for
thermodynamic properties, spin excitations, and neutron scattering structure
factors. Furthermore, we investigate the {cluster-like excitations of quantum
spin heptamer} in the three-dimensional pyrochlore lattice material
MgCr$_2$O$_4$. Using a spin mapping of the spin-1/2 heptamer excitations, the
calculated structure factors of the spin-3/2 heptamer are be determined, which
provides clarification for the spin excitations in MgCr$_2$O$_4$. Overall, this
study demonstrates the ability to use the spin mapping of structure factors for
small spin systems to analyze more complex structures.
