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 $t+1$ is given by (Eq. 16):

$e_{rot}^{t,t+1}={\frac {1}{\rho _{\mathrm {rot} }^{t,t+1}}}\sum _{JM{\hat {K}}}\left({\frac {2J+1}{8\pi ^{2}}}\right)A_{{\hat {K}}M}^{(JM)}E_{\hat {K}}^{(JM)}\exp \left(-{\frac {\beta }{P}}E_{\hat {K}}^{(JM)}\right)\sum _{K}A_{{\hat {K}}K}^{(JM)}d_{MK}^{J}({\tilde {\theta }}^{t+1})\cos(M{\tilde {\phi }}^{t+1}+K{\tilde {\chi }}^{t+1})$