# Specialized to N=2, D=2

$Z=\int \mathrm{exp}\left(\left(-V\beta \right)\right)𝑑R$

specialize to N=2

$Z=\int \int \mathrm{exp}\left(\left(-V\beta \right)\right)𝑑\mathbf{{r}_{1}}𝑑\mathbf{{r}_{2}}$

replace potential with N=2

$Z=\int \int \mathrm{exp}\left(\left(-\beta V\left(\mathbf{{r}_{1}},\mathbf{{r}_{2}}\right)\right)\right)𝑑\mathbf{{r}_{1}}𝑑\mathbf{{r}_{2}}$

Switch variables

$Z=\int \int \mathrm{exp}\left(\left(-\beta V\left(\mathbf{{r}_{1}},\mathbf{{r}_{2}}\right)\right)\right)𝑑\mathbf{{r}_{\mathrm{12}}}𝑑\mathbf{{r}_{\mathrm{cm}}}$

Specialize to a potential that depends only on interparticle distance

$Z=\int \int \mathrm{exp}\left(\left(-\beta V\left(\mathbf{{r}_{\mathrm{12}}}\right)\right)\right)𝑑\mathbf{{r}_{\mathrm{12}}}𝑑\mathbf{{r}_{\mathrm{cm}}}$

Depend only on the magnitude of the distance

$Z=\int \int \mathrm{exp}\left(\left(-\beta V\left(\mathrm{Abs}\left(\mathbf{{r}_{\mathrm{12}}}\right)\right)\right)\right)𝑑\mathbf{{r}_{\mathrm{12}}}𝑑\mathbf{{r}_{\mathrm{cm}}}$

Integrate out r_cm (this step is still a hack)

$Z=\Omega \int \mathrm{exp}\left(\left(-\beta V\left(\mathrm{Abs}\left(\mathbf{{r}_{\mathrm{12}}}\right)\right)\right)\right)𝑑\mathbf{{r}_{\mathrm{12}}}$

Decompose into vector components

$Z=\Omega \int \int \mathrm{exp}\left(\left(-\beta V\left(\sqrt{{{r}_{\mathrm{12}x}}^{2}+{{r}_{\mathrm{12}y}}^{2}}\right)\right)\right)𝑑{r}_{\mathrm{12}x}𝑑{r}_{\mathrm{12}y}$

$Z=\Omega \underset{1.00000000000000e-5}{\overset{\phantom{\rule{1em}{0ex}}L}{\int }}\underset{1.00000000000000e-5}{\overset{\phantom{\rule{1em}{0ex}}L}{\int }}\mathrm{exp}\left(\left(-\beta V\left(\sqrt{{{r}_{\mathrm{12}x}}^{2}+{{r}_{\mathrm{12}y}}^{2}}\right)\right)\right)𝑑{r}_{\mathrm{12}x}𝑑{r}_{\mathrm{12}y}$