Single Slice Moves

In classical MC, the particles are represented by points in $D$ dimensional space, which are moved in every MC step. Most simply, one can choose the displacement of particles according to a uniform distribution. In PIMC, particles are represented by path and the equivalent moves would shift the entire polymer to a new position without changing its internal structure. We call this displacement moves and use them for the protons because their paths stay very localized. They are more efficient than single or multi-slice moves discussed in the following.

In a single slice move, one selects a particle and a time slice $i$ and samples a new configuration ${\bf r}'_i$ while keeping ${\bf r}_{i-1}$ and ${\bf r}_{i+1}$ fixed. From now on, the subscript denotes the time slice. The optimal choice for the sampling distribution of ${\bf r}_i$ is given by the heat bath rule, which will be described in section 2.5.4. It states that the new coordinate should by chosen according to its equilibrium distribution,

$$T({\bf r}_i \to {\bf r}_i') \equiv T({\bf r}_i') = \frac{\... ...r}_{i+1};\tau)} {\rho({\bf r}_{i-1},{\bf r}'_{i+1};2\tau)} \;.$$ (71)

Unlike lattice MC methods, the normalization is difficult to compute, which is why one uses the distribution of non-interacting particles, which is a Gaussian centered around the midpoint ${\bf r}_{\rm m}=({\bf r}_{i-1}+{\bf r}_{i+1})/2$,

$$T({\bf r}_i) = (2 \pi \lambda \tau)^{-D/2} \exp \left \{ -... ...({\bf r}_i-{\bf r}_{\rm m})^2}{2\lambda \tau} \right \} \quad.$$ (72)

We call this implementation free particle sampling. For non-interacting particles, this leads to an acceptance ratio of 100% but interactions reduce this ratio. For very dense systems, like liquid $^4$He, which interacts approximately via a hard-sphere potential, it can become close to zero. For all hydrogen applications, the free particle sampling worked very well.