SGE simulations in $\lambda$-space

In SGE simulations conducted in a generic $\lambda$-space at constant temperature, the dimensionless Hamiltonian is given by Eq. 6.3. In the ORAC program we use a Hamiltonian aimed to sample (i) the distance between two target atoms, (ii) the angle formed by three established atoms and (iii) the torsion formed by four established atoms or (iv) combinations of these coordinates. There are several ways to model such a Hamiltonian. Our choice is to use harmonic potential functions correlated to the given collective coordinates:

$$h_n(x,p,p_t) = \beta [H(x,p,p_t) + k (r - \lambda_n)^2],$$ (6.16)

where, as usual, $H(x,p,p_t)$ is the extended Hamiltonian. In Eq. 6.16, $r$ is the instantaneous collective coordinate (bond, bending, torsion) and $k$ is a constant. As in ST simulations, transitions from $n$ to $m$-ensemble occur at fixed configuration. However, in this case there is no need of rescaling momenta because they drop out of the detailed balance condition naturally. The resulting acceptance ratio is

$${\rm acc}[n \rightarrow m] = \min(1, {\rm e}^{\beta k [ (r - \lambda_n)^2 - (r - \lambda_m)^2 ] + f_m - f_n}).$$ (6.17)

In this kind of simulations, the free energy as a function of $\lambda$ corresponds to the biased PMF[51,52] along the coordinate associated with $\lambda$. Biasing arises from the harmonic potential added to the original Hamiltonian (see Eq. 6.16). However, reweighting schemes are available to recover the unbiased PMF along the real coordinate[53,54,125,126]. We will see later how $f_m$ and $f_n$ are determined.