Rigid top propagator

For a rigid three dimensional asymmetric top the kernel is given by () (Eq. 15) ):$\rho _{\mathrm {rot} }^{t,t+1}(\beta /P)=\sum _{J=0}^{\infty }\sum _{M=-J}^{J}\sum _{{\hat {K}}=-J}^{J}\left({\frac {2J+1}{8\pi ^{2}}}\right)A_{{\hat {K}}M}^{(JM)}\exp \left(-{\frac {\beta }{P}}E_{\hat {K}}^{(JM)}\right)\sum _{K=-J}^{J}A_{{\hat {K}}K}^{(JM)}d_{MK}^{J}({\tilde {\theta }}^{t+1})\cos(M{\tilde {\phi }}^{t+1}+K{\tilde {\chi }}^{t+1})$

The contribution to the rotational energy of the interactions between beads $t$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t+1} is given by (Eq. 16):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{rot}^{t,t+1}= \frac{1}{ \rho_{\mathrm{rot}}^{t,t+1}} \sum_{JM\hat{K}} \left( \frac{2J+1}{8\pi^2} \right) A_{\hat{K}M}^{(JM)} E_{\hat{K}}^{(JM)} \exp \left( -\frac{\beta}{P} E_{\hat{K}}^{(JM)}\right) \sum_K A_{\hat{K}K}^{(JM)} d_{MK}^J (\tilde{\theta}^{t+1}) \cos( M\tilde{\phi}^{t+1}+K\tilde{\chi}^{t+1})}