molecule. Similarly, the atoms are divided into $F$ rigid fragments and we denote $F_a$ as the set of atoms making up the $a$th fragment. A typical pairwise energy term can then be written as $$E_g(\Gamma) = \sum_{0< i<j \le N} C_{g,ij} f_g(r_{ij}),$$ where $g$ denotes the type of energy term (electrostatic or Lennard-Jones), $i$ and $j$ are atom indices, $r_{ij}$ is the distance between atoms $i$ and $j$, $f_g$ is some function of distance, and $C_{g,ij}$ is a coefficient which depends on the atoms but not on their positions. Thus for electrostatic interactions, $C_{g,ij}$ depends on the partial charges on the two atoms (assumed to be constant in Amber) and on the bonding relation between the atoms. Physical energy functions satisfy $\lim_{r\rightarrow\infty}f_g(r) = 0$. When the fragments are separated sufficiently, we have $$E_g \rightarrow E_{g0} =
 \sum_{0< a \le F} \sum_{\stack{i<j}{i,j \in F_a}} C_{g,ij} f_g(r_{ij}),$$ which is independent of $\Gamma$. It is convenient to choose $E_{g0}$ as the “origin” for the $E_g$, i.e., we compute only $$E_{g1} = E_g - E_{g0} = \sum_{0< a<b \le F}
\sum_{\stack{i \in F_a}{j \in F_b}}
C_{g,ij} f_g(r_{ij}).$$ We note that only energy differences enter into the computation of observable quantities, and so we are 