The centroid molecular dynamics approximation

The CMD method is equivalent to the following compact approximation for the time dependent QDO 10 

Equation (52) shows that in CMD the approximate centroid force is determined for the instantaneous centroid position vc (t) by the same functional form as for the zero time centroid force. Equation defines the zero time centroid force to be the negative gradient of the centroid potential of mean force, i.e, 

the CMD method is isomorphic to classical time evolution of the phase space centroids on the quantum centroid potential of mean force, Vcmd. It should be noted that in the harmonic, classical, and free particle limits, the CMD representation for the QDO [Eq. (50)] is also exact. Furthermore, it should also be noted that the approximation in Eq. (50) does not rely on any kind of mean field approximation. 

A second important property of CMD is that it will produce the exact equilibrium average of a dynamical variable A if the system is ergodic. That is, the 

This property may not be possessed by many other approximate methods based on, e.g., mean field or semiclassical approaches. Also, in low dimensional systems, the above property is not true for CMD, so to apply CMD to such systems is not consistent with spirit of the method (though perhaps still useful for testing purposes). 

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