Jacobi coordinates

In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. In words, the algorithm is described as follows:

For the N-body problem the result is:

${\displaystyle {\boldsymbol {r}}_{j}={\frac {1}{m_{0j}}}\sum _{k=1}^{j}m_{k}{\boldsymbol {x}}_{k}\ -\ {\boldsymbol {x}}_{j+1}\ ,\quad j\in \{1,2,\dots ,N-1\}}$

${\displaystyle {\boldsymbol {r}}_{N}={\frac {1}{m_{0N}}}\sum _{k=1}^{N}m_{k}{\boldsymbol {x}}_{k}\ ,}$

with

${\displaystyle m_{0j}=\sum _{k=1}^{j}\ m_{k}\ .}$

The vector ${\displaystyle {\boldsymbol {r}}_{N}}$is the center of mass of all the bodies and ${\displaystyle {\boldsymbol {r}}_{1}}$is the relative coordinate between the particles 1 and 2:

The result one is left with is thus a system of N-1 translationally invariant coordinates ${\displaystyle {\boldsymbol {r}}_{1},\dots ,{\boldsymbol {r}}_{N-1}}$and a center of mass coordinate ${\displaystyle {\boldsymbol {r}}_{N}}$, from iteratively reducing two-body systems within the many-body system.

This change of coordinates has associated Jacobian equal to ${\displaystyle 1}$.

If one is interested in evaluating a free energy operator in these coordinates, one obtains

${\displaystyle H_{0}=-\sum _{j=1}^{N}{\frac {\hbar ^{2}}{2m_{j}}}\,\nabla _{{\boldsymbol {x}}_{j}}^{2}=-{\frac {\hbar ^{2}}{2m_{0N}}}\,\nabla _{{\boldsymbol {r}}_{N}}^{2}\!-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N-1}\!\left({\frac {1}{m_{j+1}}}+{\frac {1}{m_{0j}}}\right)\nabla _{{\boldsymbol {r}}_{j}}^{2}}$

In the calculations can be useful the following identity

${\displaystyle \sum _{k=j+1}^{N}{\frac {m_{k}}{m_{0k}m_{0k-1}}}={\frac {1}{m_{0j}}}-{\frac {1}{m_{0N}}}}$.