Accurate long-range coefficients for two excited like isotope He atoms: He($2 ^1P$)--He($2 ^1P$), He($2 ^1P$)--He($2 ^3P$), and He($2 ^3P$)--He($2 ^3P$)

A general formalism is used to express the long-range potential energies in inverse powers of the separation distance between two like atomic or molecular systems with $P$ symmetries. The long-range molecular interaction coefficients are calculated for the molecular symmetries $Delta$, $Pi$, and $Sigma$, arising from the following interactions: He($2 ^1P$)--He($2 ^1P$), He($2 ^1P$)--He($2 ^3P$), and He($2 ^3P$)--He($2 ^3P$). The electric quadrupole-quadrupole term, $C_{5}$, the van der Waals (dispersion) term $C_{6}$, and higher-order terms, $C_{8}$, and $C_{10}$, are calculated textit{ab initio} using accurate variational wave functions in Hylleraas coordinates with finite nuclear mass effects. A comparison is made with previously published results where available.