into “hard” ($\mathrm h$) and “soft” ($\mathrm s$) components, $$E(\Gamma) = E_\mathrm h(\Gamma) + E_\mathrm s(\Gamma),$$ where $\Gamma$ is the configuration of the system. For example, let us assume that an all-atom force field, such as Amber [@cornell95], provides an accurate description of the system. (We recognize, of course, that present-day force fields are only approximate. However, our purpose here is to make the connection between an all-atom representation and a simpler rigid representation and, in this context, the details of the all-atom model are of secondary importance.) Then $E_\mathrm h$ might represent the bond stretching and bond bending terms, while $E_\mathrm s$ is given by the other terms (bond torsion and the non-bonded energies).

The constrained limit is now given by $E_\mathrm h \rightarrow \infty$. Before we consider this limit, it is useful to examine how the stiff system may be treated. Conventional Monte Carlo is inefficient because, in order to have an reasonably large acceptance rate, the step-size needs to be set to a small value (determined by $E_\mathrm h$) so that diffusion in the soft directions is very slow. However, we can apply MTS Monte Carlo in this case with $E_1 = E_\mathrm h$ and $E_2 =
