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\begin{document}

Daniel:

Let $x$ stand for $x-x'$ and $y$ stand for $y-y'$.  In place of
$$
	V(x,x',y,y'),
$$
write
$$
	V_2(x,y).
$$

The integral expression for the potential is, in terms of $x$ and $y$,
\begin{align}
\label{eqn:xy_int}
	V_2(x,y) &= \frac{a}{\sqrt{8\pi\beta}}
		e^{\frac{+\|x-y\|^2}{8\beta}}
		\int_0^1 \frac{1}{[s(1-s)]^{3/2}}
			e^{-\frac{\|x\|^2}{8\beta s}}
			e^{-\frac{\|y\|^2}{8\beta (1-s)}}
			ds.
\end{align}
If $\|x\|=\|y\|$ then we have the exact expression
\begin{align}
\label{eqn:xy_exact}
	V_2(x,y) &= \frac{2a}{\|x\|}
		e^{\frac{-\|x-y\|^2}{8\beta}}
		e^{\frac{-\|x\|^2}{2\beta}}
\end{align}

I want to write both of these in terms of $r_1$, $r_2$, and $\theta$.
First note that from the Law of Cosines we have
\begin{align*}
	\|x-y\|^2 &= \|x\|^2 + \|y\|^2 - 2\|x\|\,\|y\|\,\cos(\theta) \\
	&= r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta).
\end{align*}
When $\|x\|=\|y\|$, i.e. $r_1=r_2=r$, we have
\begin{align*}
	\|x-y\|^2 &= 2 r^2 (1 - \cos(\theta)) \\
	&= 4 r^2 \sin^2(\theta/2).
\end{align*}
Then the integral is
\begin{align}
\label{eqn:rrt_int}
	V_2(r_1, r_2, \theta) = \frac{a}{\sqrt{8\pi\beta}}
		e^{\frac{r_1^2+r_2^2-2r_1r_2\cos\theta}{8\beta}}
		\int_0^1 \frac{1}{[s(1-s)]^{3/2}}
			e^{-\frac{r_1^2}{8\beta s}}
			e^{-\frac{r_2^2}{8\beta (1-s)}}
			ds.
\end{align}
Likewise, the exact expression is
\begin{align}
\label{eqn:rrt_exact}
	V_2(r,r,\theta) &= \frac{2a}{r}
		e^{-\left(\,\frac{4r^2\sin^2(\theta/2) + 4r^2}{8\beta}\right)} \\
	&= \frac{2a}{r} e^{\frac{-r^2(1+\sin^2(\theta/2))}{2\beta}}.
\end{align}

Is it possible that equation \ref{eqn:xy_exact} should instead be
\begin{align}
\label{eqn:xy_exact_fix}
	V_2(x,y) &= \frac{2a}{\|x\|}
		e^{\frac{+\|x-y\|^2}{8\beta}}
		e^{\frac{-\|x\|^2}{2\beta}}?
\end{align}
Then equation \ref{eqn:rrt_exact} becomes
\begin{align}
\label{eqn:rrt_exact_fix}
	V_2(r,r,\theta) &= \frac{2a}{r}
	e^{\frac{-r^2(1-\sin^2(\theta/2))}{2\beta}}.
\end{align}

I do not know how you derived equation \ref{eqn:xy_exact} so I can't check it.
However, I do believe that either equation \ref{eqn:xy_int} or equation
\ref{eqn:xy_exact} is incorrect.  Using equations \ref{eqn:xy_int} and
\ref{eqn:xy_exact} I see the following:

\begin{verbatim}
 r  theta  integral     exact rel.error
--- -----  -------- --------- ---------
2.0   0.0 0.0135335 0.0135335  1.03e-15
2.0  15.0 0.0140026 0.0130802  7.05e-02
2.0  30.0 0.0154737 0.0118366  3.07e-01
2.0  45.0 0.0181390 0.0100974  7.96e-01
2.0  60.0 0.0223130 0.0082085  1.72e+00
2.0  75.0 0.0283989 0.0064494  3.40e+00
2.0  90.0 0.0367879 0.0049787  6.39e+00
2.0 105.0 0.0476551 0.0038434  1.14e+01
2.0 120.0 0.0606531 0.0030197  1.91e+01
2.0 135.0 0.0746102 0.0024548  2.94e+01
2.0 150.0 0.0874612 0.0020941  4.08e+01
2.0 165.0 0.0966500 0.0018950  5.00e+01
2.0 180.0 0.1000000 0.0018316  5.36e+01
\end{verbatim}

A priori, I do not know which one is incorrect.  However, a bridge from
$x$ to $-x$ should have a higher probability of intersecting a sphere
at the origin than a bridge from $x$ to $x$.  So, I believe it is
the exact expression which is incorrect.  Replacing
\ref{eqn:xy_exact} by equation \ref{eqn:xy_exact_fix}, I now see

\begin{verbatim}
  r theta  integral     exact rel.error
--- -----  -------- --------- ---------
2.0   0.0 0.0135335 0.0135335  1.03e-15
2.0  15.0 0.0140026 0.0140026  4.96e-16
2.0  30.0 0.0154737 0.0154737 -7.85e-16
2.0  45.0 0.0181390 0.0181390 -9.56e-16
2.0  60.0 0.0223130 0.0223130 -9.33e-16
2.0  75.0 0.0283989 0.0283989 -2.44e-16
2.0  90.0 0.0367879 0.0367879  3.02e-15
2.0 105.0 0.0476551 0.0476551  2.91e-16
2.0 120.0 0.0606531 0.0606531  0.00e+00
2.0 135.0 0.0746102 0.0746102  1.86e-16
2.0 150.0 0.0874612 0.0874612  6.35e-16
2.0 165.0 0.0966500 0.0966500  8.62e-16
2.0 180.0 0.1000000 0.1000000 -2.78e-16
\end{verbatim}

The integrals are computed using 1000 mesh points and na\"ive quadrature; we
have very good accuracy.

Here is a surface plot of $V_2(r,r,\theta)$ using equation
\ref{eqn:xy_exact_fix}:

\begin{center}
\includegraphics[scale=1.2]{../figures/vrrt.eps}
\end{center}

\end{document}
