In order to realize the full benefits of imposing constraints we need to pass to the constrained limit ($E_\mathrm h \rightarrow \infty$). In this limit, the motion collapses onto a lower-dimensional sub-manifold in configuration space. Unfortunately, in contrast to the case of rigid molecules, we cannot appeal to symmetry to enable us to take this limit analytically. Instead, we use the adiabatic technique.

Adiabatically varying the stiffness
===================================

Let us rewrite the energy of the system, multiplying the $E_\mathrm
h(\Gamma)$ by $T/T^*$, where $T$ is the temperature of the system, and $T^*$ is a “constraint” temperature. The Boltzmann factor $\exp(-\beta E)$, will then have the form $$\exp(-\beta E) = \exp(-\beta E_\mathrm s - \beta^* E_\mathrm h)$$ where $\beta^* = 1/(kT^*)$.

In our application, where we are interested in the constrained limit $T^*\rightarrow 0$, a direct application of the MTS method leaves us with two bad choices. If we take $T^*$ to be sufficiently small that we can consider the constraints to be satisfied, we will have to chose the step size for the $E_\mathrm h$ moves to be so small that the change in configuration after $n$ $E_\mathrm h$ moves will be small. On the other hand, letting $T^*$ be 