Hybrid quantum mechanics/molecular mechanics (QM/MM) simulations have become a popular tool for

G
QM / MM

A(QM / MM) ^{A B}


B(QM / MM)

G
MM QM / MM
A


_GMM
MM QM / MM

G
B

A(MM)
^{A B} B(MM)

Fig. 6. Thermodynamic cycle for computing the free energy difference between states A and

B at the QM/MM level (DG ^QM/MM ). In the first step, the free energy difference between A and
A ! B _MM
B is determined at the MM level (DG_{A ! B} ), either by thermodynamic integration or free

A B

A ! B
energy perturbation. In the second step, the free energy required to transform the MM ensemble of A and B into the QM/MM ensemble (DG ^{MM ! QM/MM} and DG ^{MM ! QM/MM} ) are computed by free energy perturbation. The QM/MM free energy of converting A into B is calculated by adding up the free energy differences in going around the cycle from A(QM/ MM) to B(QM/MM). This procedure avoids computing the DG ^QM/MM directly.
MM degrees of freedom are assumed to be uncoupled. Whether such assumption is valid, depends on the process at hand. Another issue concerns finding a suitable reaction path along which the umbrella sampling will be carried out.

2. 
   Free Energy Perturbation
   

An alternative approach for extracting the free energy associated with the conversion between two states from QM/MM simulations is to use a combination of thermodynamic integration (32) and free energy perturbation (33). In thermodynamic integration (TI), the Hamiltonian is interpolated between the two states with a coupling parameter l:
H ðq; p; lÞ¼ ð1 — lÞH_Aðq; pÞþ lH_Bðq; pÞ; (10)

¼ ¼
where q and p are the positions and momenta of all atoms in the system. To obtain the free energy difference between state A, when l 0, and state B, when l 1, the system is sampled at fixed values of l between 0 and 1, followed by integration over the ensemble averages of h∂H / ∂li_l at these l values with respect to l:

DG ¼

dl: (11)
Z _{1 @H}

⁰

^@l

l

An advantage of the TI approach is that the pathway connecting the two states does neither have to be physically meaningful nor possible. For example, the free energy cost of changing or even disappearing atoms, can be computed efficiently this way. Such non-physical transformations are usually only possible at the MM level. To get the free energy change at the QM/MM level, an additional step is required (34).
Because the free energy is a state function, its magnitude does not depend on the pathway taken. Therefore, one can always construct a so-called thermodynamic cycle, as shown in Fig. 6. For the free energy of a transformation at the QM/MM level, the quantity of interest is the

free energy associated with the top process DG^QM=MM . Since the