Rigid top propagator: Difference between revisions

Revision as of 15:17, 25 February 2011

For a rigid three dimensional asymmetric top the kernel is given by (^[1]^[2]) (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})

References

[1] M. H. Müser and B. J. Berne "Path-Integral Monte Carlo Scheme for Rigid Tops: Application to the Quantum Rotator Phase Transition in Solid Methane", Physical Review Letters 77 pp. 2638-2641 (1996)

[2] Eva G. Noya, Carlos Vega, and Carl McBride "A quantum propagator for path-integral simulations of rigid molecules", Journal of Chemical Physics 134 054117 (2011)

[1]

[2]

@@ Line 1: / Line 1: @@
 {{stub-general}}
-For a rigid three dimensional asymmetric top the kernel is given by (<ref>[http://dx.doi.org/10.1103/PhysRevLett.77.2638  M. H. Müser and B. J. Berne "Path-Integral Monte Carlo Scheme for Rigid Tops: Application to the Quantum Rotator Phase Transition in Solid Methane", Physical Review Letters '''77''' pp. 2638-2641 (1996)]</ref><ref>[http://dx.doi.org/10.1063/1.3544214 Eva G. Noya, Carlos Vega, and Carl McBride "A quantum propagator for path-integral simulations of rigid molecules", Journal of Chemical Physics '''134''' 054117 (2011)]</ref>
+For a rigid three dimensional asymmetric top the kernel is given by (<ref>[http://dx.doi.org/10.1103/PhysRevLett.77.2638  M. H. Müser and B. J. Berne "Path-Integral Monte Carlo Scheme for Rigid Tops: Application to the Quantum Rotator Phase Transition in Solid Methane", Physical Review Letters '''77''' pp. 2638-2641 (1996)]</ref><ref>[http://dx.doi.org/10.1063/1.3544214 Eva G. Noya, Carlos Vega, and Carl McBride "A quantum propagator for path-integral simulations of rigid molecules", Journal of Chemical Physics '''134''' 054117 (2011)]</ref>) (Eq. 15)
+):<math>
+\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})
+</math>
+The contribution to the rotational energy of the interactions between beads <math>t</math> and <math>t+1</math> is given by (Eq. 16):
+:<math>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})</math>
 ==References==
 <references/>
 [[category: Quantum mechanics]]

Rigid top propagator: Difference between revisions

Revision as of 15:17, 25 February 2011

References

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