Cluster orbitals for the mirror nuclei \boldmath{$^7\mathrm{Li}$} and \boldmath{$^7\mathrm{Be}$}*

Certain light nuclei are dominated by alpha particle clustering. In these nuclei, the alpha clusters form a molecular like structure and any additional nucleon(s) are orbiting in the potential created by the alpha clusters. The mirror nuclei $^7\mathrm{Li}$ and $^7\mathrm{Be}$ can be viewed as a $^8\mathrm{Be}$ core plus a proton or neutron hole, respectively. We derive the single-particle orbitals for $^7\mathrm{Li}$ and $^7\mathrm{Be}$ by solving the single-particle Schr\"{o}dinger equation for a proton (neutron) hole in the potential created by the $^8\mathrm{Be}$ core. Specifically, we derive the energies of the ground and excited states, the radii, and electromagnetic transition probabilities. We also compare the calculated observables against \textit{ab initio} no-core configuration interaction calculations using realistic interactions.