The Feynman-Hibbs Variational Method

AAU)f(x) is an effective potential representing the quantum deviations from the classical interaction potential. This effective potential is variational since the only approximation made so far is from the inequality (11.25). 

Now if we assume that the deviations are small, and expand AUa(x — y) about the point x to second order, we get 

The QFH potential approximately captures two key quantum effects. When an atom is near a potential minimum, the curvature is positive and thus so is the QFH correction this models the zero-point effect. On the other hand, near potential maxima the curvature is negative, and the QFH potential models tunneling. 

The Feynman-Hibbs and QFH potentials have been used extensively in simulations examining quantum effects in atomic and molecular fluids [12,15,25]. We note here that the centroid molecular dynamics method [54, 55] is related and is intermediate between a full path integral simulation and the Feynman-Hibbs approximation  

We can view obtaining the QFH correction to the excess chemical potential in two ways. If we simply insert (11.27) back into (11.22), this suggests that we first compute the classical excess chemical potential and then insert the classical solute into the system and evaluate 

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