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):

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