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:

Jacobi coordinates for two-body problem; Jacobi coordinates are ${\boldsymbol {R}}={\frac {m_{1}}{M}}{\boldsymbol {x}}_{1}+{\frac {m_{2}}{M}}{\boldsymbol {x}}_{2}$ and ${\boldsymbol {r}}={\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{2}$ with $M=m_{1}+m_{2}$.

A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r₁, r₂, r₃ and the center of mass R. See Cornille.

Let m_j and m_k be the masses of two bodies that are replaced by a new body of virtual mass M = m_j + m_k. The position coordinates x_j and x_k are replaced by their relative position r_jk = x_j − x_k and by the vector to their center of mass R_jk = (m_j q_j + m_kq_k)/(m_j + m_k). The node in the binary tree corresponding to the virtual body has m_j as its right child and m_k as its left child. The order of children indicates the relative coordinate points from x_k to x_j. Repeat the above step for N − 1 bodies, that is, the N − 2 original bodies plus the new virtual body.

For the N-body problem the result is:

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

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

with

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

The vector ${\boldsymbol {r}}_{N}$ is the center of mass of all the bodies and ${\boldsymbol {r}}_{1}$ 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 ${\boldsymbol {r}}_{1},\dots ,{\boldsymbol {r}}_{N-1}$ and a center of mass coordinate ${\boldsymbol {r}}_{N}$, from iteratively reducing two-body systems within the many-body system.

This change of coordinates has associated Jacobian equal to $1$.

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

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

In the calculations can be useful the following identity

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