Backgroud

Here, we discuss the generalization of the chain structure beyond the simple straight chains. For this purpose, we introduce particle indices $(\alpha_{i}, \beta_{i})$ for connection (bond) $i$. Because the connection is completely characterized by the link between the two particles, the list of particle-pairs $(\alpha_{i}, \beta_{i})$ is enough for arbitrary structure.

Consider a system with $N$ particles. The positions of particles are given by $\{\bm{x}_{\mu}\}$ for $\mu = 0, \cdots, N-1$, where $N$ is the total number of particles in the system. Let $N_{b}$ be the number of bonds (connections) in the system. Each bond is characterized by the pair of particles $(\alpha_{i}, \beta_{i})$, where the bond index $i$ runs from 0 to $N_{b}-1$. We also define the ``group'' which is a set containing particles connecting each other. Particles without connection belong to different groups. We denote the number of groups in the system by $N_{g}$.

Contrast to the simple straight chain configuration, one extra care needs to be taken for general configurations. That is, for a single group of particles which are all connecting each other, sometimes the number of bonds (connections) becomes larger than the degrees of freedom of the group except for the center-of-mass, $N-1$. For straight chain configurations, the number of bonds are always $N-1$. This fact means that the extra bonds are redundant and the connector vectors for the redundant bonds can be derived from those for the independent bonds. See an example in Eq. (9.5) later.

The crucial point is forming the list of independent bonds. We denote the list by ${\tt bonds}[N-1]$. Detailed discussions for the construction of the list will be given in §9.1.3.

Let us define the connector vector $\bm{Q}_{i}$ for the bond $i$ as

$$\bm{Q}_{i} = \bm{x}_{\alpha_{i}} - \bm{x}_{\beta_{i}} .$$ (9.1)

Furthermore, we define the generalized (or canonical?) connector vector $\overline{\bm{Q}}_{i}$ where the index $i$ runs from 0 to $N-1$. For a system with single group with $N$ particles,

$$\overline{\bm{Q}}_{i} = \bm{x}_{\alpha_{{\tt bonds}[i]}} - \bm{x}_{\beta_{{\tt bonds}[i]}} ,$$ (9.2)

for $i = 0, \cdots, N-2$, and

$$\overline{\bm{Q}}_{N-1} = \bm{x}_{\text{COM}} := \frac{1}{N} \sum_{\mu=0}^{N-1} \bm{x}_{\mu} ,$$ (9.3)

where ${\tt bonds}[i]$ is the $i$th bond index of the independent bonds. Note that the number of elements of all connectors $\{\overline{\bm{Q}}_{i}\}$ is equal to that of all position vectors $\{\bm{x}_{\mu}\}$ and they have one-to-one correspondence. By introducing the transform matrix by $\overline{\mathsf{B}}$ whose dimension is $N\times N$, we can write

$$\overline{\bm{Q}}_{i} = \sum_{\mu=0}^{N-1} \overline{\mathsf{B}}_{i,\mu} \bm{x}_{\mu} .$$ (9.4)

The details about the transformation will be discussed later in §9.1.4.