Effect of exciton hopping upon the mass of an exciton

A plausible formula derived in a previous paper for the mass $M_n^{\mathrm{*}}$ of an exciton in an nth bound state of the electron-hole binding potential is extended so as to include the effect of an exciton-hopping (or Heller-Marcus) mechanism upon $M_n^{\mathrm{*}}$. If $m_e^{\mathrm{*}}$ and $m_h^{\mathrm{*}}$ are the electron and hole masses, we find that $M_n^{\mathrm{*}}$= m ${}_e{}^{\mathrm{*}}{}_{}{}^{}$+$m_h^{\mathrm{*}}$ / 1- $K_n$ / W + $m_e^{\mathrm{*}}$+$m_h^{\mathrm{*}}$ / $M_F^{\mathrm{*}}$ $H_n$ / $H_F$, where $K_n$ and $H_n$ are, respectively, the kinetic and exciton-hopping energies in the nth bound state; W is one-half the sum of the electron and hole bandwidths, and $H_F$ is the value taken by $H_n$ for a Frenkel exciton of finite mass $M_F^{\mathrm{*}}$. For Wannier excitons, $K_n$=$H_n$≃0, so that $M_n^{\mathrm{*}}$≃$m_e^{\mathrm{*}}$+$m_h^{\mathrm{*}}$; while for Frenkel excitons, $K_n$≃W and $H_n$≃$H_F$, so that the mass of the Frenkel exciton $M_F^{\mathrm{*}}$ is finite as a consequence of the Heller-Marcus mechanism.