490 The Feynman path integral
Pvir({r(1),...,r(P)}) =dNkT
V−
1
V
∑N
i=11
P
r(ic)·∑P
k=1∂
∂r(ic)U(r( 1 k),...,r(Nk),V)−
1
P
∑P
k=1∂
∂V
U(r( 1 k),...,r(Nk),V), (12.6.42)which includes the possibility that the potential depends explicitly on the volume.
One final word is still needed to address the question of when path-integral simula-
tions are needed. Broadly speaking, they should be applied whenever nuclear quantum
effects are expected to be important, for example, when light nuclei such as hydro-
gen are present. Proton transfer reactions will often exhibit nontrivial quantum effects
such as tunneling and zero-point motion. In malonaldehyde (C 3 H 4 O 2 ), a small, cyclic
organic molecule with an internal O−H···O hydrogen bond, the hydrogen bond can
reverse its polarity and become O···H−O via a proton transfer reaction (see Fig. 12.14,
top). A free energy profile for this reaction can be computed usingthe blue moon en-
semble approach of Section 8.7 with a reaction coordinateδ=dO 1 H−dO 2 H, where
dO 1 HanddO 2 Hare the distances between the two oxygens and the transferringproton.
The free energy profile in this reaction coordinate exhibits a typicaldouble-well shaped
as illustrated in Fig. 8.7. Interestingly, even at 300 K, there is a pronounced quantum
effect on this free energy. The quantum free energy profiles can be computed using the
centroid of the reaction coordinate denotedδcin the figure (Vothet al., 1989; Voth,
1993). Fig. 8.7 predicts that quantum free energy barrier to the reaction decreases by
approximately 2 kcal·mol−^1 (Tuckerman and Marx, 2001) from 3.6 kcal/mol to 1.6
kcal/mol, as shown in the bottom part of Fig. 12.14. Since enzymatic reaction barriers
are in this energetic neighborhood, this simple example illustrates theimportant role
that nuclear quantum effects, particularly quantum tunneling, canplay in real biolog-
ical proton transfer reaction. Another interesting point is that ifonly the transferring
H is treated quantum mechanically, the reduction in the free energyis underestimated
by roughly 0.4 kcal/mol, which shows that secondary nuclear quantum effects of the
molecular skeleton are also important. These free energy profiles are generated using
ab initiomolecular dynamics (Car and Parrinello, 1985; Marx and Hutter, 2009) and
ab initiopath-integral techniques, in which a dynamical or path-integral simulation
is driven by forces generated from electronic structure calculations performed “on the
fly” as the simulation is carried out (Marx and Parrinello, 1994; Marx and Parrinello,
1996; Tuckermanet al., 1996).
Liquid water also has been shown to have nonnegligible quantum effects even at
room temperature (Chenet al., 2003; Fanourgakiset al., 2006; Paesaniet al., 2007;
Paesani and Voth, 2009; Morrone and Car, 2008). At low temperature, heavier nuclei
such as^3 He and^4 He also exhibit significant quantum effects. However, care is needed
when deciding whether to apply path integrals to a given problem. Onemust con-
sider both the physical nature of the problem and the source of the potential-energy
model used in each application before deciding to embark on a path-integral inves-
tigation. Consider, for example, an empirical potential-energy functionU(r 1 ,...,rN)
whose parameters are obtained by careful fits to experimental data. Since experiments
are inherently quantum mechanical, the potential modelUcontains nuclear quantum