Why does exchange integral 1s2p not vanish for Helium excited state?

I am studying first-order perturbation theory applied to the first excited state of Helium. I am facing the following problem:


I understand that the Coulomb integral for all 2p functions will not be $0$, but for me, all the exchange integrals involving the 2p functions will be $0$. For example, look at this two integrals for the 1s2pz state:


1s function:
$$1s=\sqrt{\frac{Z^3}{a\_0^3 ŽĆ}}e^{\frac{-Z}{a\_0}r}$$
2pz function:
$$2p\_z=\frac{1}4\sqrt{\frac{Z^5}{2a\_0^5 ŽĆ}}re^{\frac{-Z}{2a\_0}r}cos╬Ė$$


Coulomb integral:
$$J\_{p\_z}=H\_{(1s2pz;1s2pz)} =Ōł½Ōł½\frac{1s(1)2p\_z(2)e^21s(1)2p\_z (2)}{4ŽĆ╬Ą\_0r\_{12}}dr\_1 dr\_2$$
$$J\_{p\_z}=\frac{Z^8 e^2}{128a\_0^8 ŽĆ^3╬Ą\_0} Ōł½\_0^Ōł×e^{{\frac{-2Z}{a\_0}}r\_1} r\_1^2 dr\_1 Ōł½\_0^Ōł×e^{{\frac{-2Z}{a\_0}}r\_2} r\_2^4 dr\_2 Ōł½\_0^ŽĆ\frac{sen╬Ė\_1 d╬Ė\_1}{\sqrt{r\_1^2+r\_2^2-2r\_1 r\_2 cos╬Ė\_1}} Ōł½\_0^{2ŽĆ}dŽĢ\_1 Ōł½\_0^ŽĆcos^2 ╬Ė\_2 sen╬Ė\_2 d╬Ė\_2 Ōł½\_0^{2ŽĆ}dŽĢ\_2$$ 
Exchange integral:


$$K\_{p\_z}=H\_{(1s2pz;1s2pz)} =Ōł½Ōł½\frac{1s(1)2p\_z(2)e^22p\_z(1)1s(2)}{4ŽĆ╬Ą\_0r\_{12}}dr\_1 dr\_2$$
$$K\_{p\_z}=\frac{Z^8 e^2}{128a\_0^8 ŽĆ^3╬Ą\_0} Ōł½\_0^Ōł×e^{{\frac{-3Z}{2a\_0}}r\_1} r\_1^3 dr\_1 Ōł½\_0^Ōł×e^{{\frac{-3Z}{2a\_0}}r\_2} r\_2^3 dr\_2 Ōł½\_0^ŽĆ\frac{cos╬Ėsen╬Ė\_1 d╬Ė\_1}{\sqrt{r\_1^2+r\_2^2-2r\_1 r\_2 cos╬Ė\_1}} Ōł½\_0^{2ŽĆ}dŽĢ\_1 Ōł½\_0^ŽĆcos ╬Ė\_2 sen╬Ė\_2 d╬Ė\_2 Ōł½\_0^{2ŽĆ}dŽĢ\_2$$


In all cases, the electron 1 was placed along the z-axis for its $╬Ė$ angle to be equal to the angle between the two electrons.
So, the exchange integral (K) will be zero because the integral:
$$Ōł½\_0^ŽĆcos ╬Ė\_2 sen╬Ė\_2 d╬Ė\_2=-\frac{(cos^2ŽĆ-cos^20)}2=0$$
And the Coulomb Integral will not vanish because:
$$Ōł½\_0^ŽĆcos^2 ╬Ė\_2 sen╬Ė\_2 d╬Ė\_2=-\frac{(cos^3ŽĆ-cos^30)}3=\frac{2}3$$
But in all the materials I have read, both Coulomb and exchange integrals are not zero, but they do not show the calculations. I have searched a lot on the internet, and I still have not found where my mistake is.




**Introduction**


I start off this derivation more generally and analyize the configuration where $(n\_A,\ell\_A) \neq (n\_B,\ell\_B)$. By $E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}$. Later, I restrict my attention to the case where $(n\_A,\ell\_A) = (1,0)$.


Throughout this answer, the first two (of four) integrals in the equations tend to be the direct integrals, while the second two (of four) integrals tend to be the exchange integrals.


**Results**


Let's calculate the energy of where $(n\_A,\ell\_A) \neq (n\_B,\ell\_B)$. By $E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}$, I denote the energy of the configuration.  

\begin{align\*}
&E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\_{\mathbb{R}^{3\times 2}} 
\Psi^\*(\mathbf{r}\_1, \mathbf{r}\_2)
\, \frac{1}{ r\_{12}}
\,
\Psi (\mathbf{r}\_1, \mathbf{r}\_2)
\,
d^3\mathbf{r}\_1\,d^3\mathbf{r}\_2
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\_{\mathbb{R}^{3\times 2}} 
u^\*\_{n\_A\ell\_Am\_A}(\mathbf{r}\_1)
\,
u^\*\_{n\_B\ell\_Am\_B}(\mathbf{r}\_2) 
\,
u\_{n\_A\ell\_Am\_A}(\mathbf{r}\_1)
\,
u\_{n\_B\ell\_Am\_B}(\mathbf{r}\_2) 
\, \frac{1}{ r\_{12}}\,d^3\mathbf{r}\_1\,d^3\mathbf{r}\_2
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\_{\mathbb{R}^{3\times 2}} 
u^\*\_{n\_A\ell\_Am\_A}(\mathbf{r}\_1)
\,
u^\*\_{n\_B\ell\_Am\_B}(\mathbf{r}\_2)
\,
u\_{n\_B\ell\_Am\_B}(\mathbf{r}\_1)
\,
u\_{n\_A\ell\_Am\_A}(\mathbf{r}\_2) 
\, \frac{1}{ r\_{12}}\,d^3\mathbf{r}\_1\,d^3\mathbf{r}\_2
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\_{\mathbb{R}^{3\times 2}} 
u^\*\_{n\_A\ell\_Am\_A}(\mathbf{r}\_1)
\,
u\_{n\_A\ell\_Am\_A}(\mathbf{r}\_1)
\,
u^\*\_{n\_B\ell\_Bm\_B}(\mathbf{r}\_2) 
\,
u\_{n\_B\ell\_Bm\_B}(\mathbf{r}\_2) 
\, \frac{1}{ r\_{12}}\,d^3\mathbf{r}\_1\,d^3\mathbf{r}\_2
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\_{\mathbb{R}^{3\times 2}} 
u^\*\_{n\_A\ell\_Am\_A}(\mathbf{r}\_1)
\,
u\_{n\_B\ell\_Bm\_B}(\mathbf{r}\_1)
\,
u^\*\_{n\_B\ell\_Bm\_B}(\mathbf{r}\_2)
\,
u\_{n\_A\ell\_Am\_A}(\mathbf{r}\_2) 
\, \frac{1}{ r\_{12}}\,d^3\mathbf{r}\_1\,d^3\mathbf{r}\_2
\end{align\*}
Putting in the separable wave functions.
\begin{align\*}
&E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\_{\mathbb{R}^{3\times 2}} 
R^\*\_{n\_A\ell\_A}(\mathbf{r}\_1)
Y^\*\_{\ell\_Am\_A}(\mathbf{r}\_1)
\,
R\_{n\_A\ell\_A}(\mathbf{r}\_1)
Y\_{\ell\_Am\_A}(\mathbf{r}\_1)
\,
R^\*\_{n\_B\ell\_B}(\mathbf{r}\_2)
Y^\*\_{\ell\_Am\_B}(\mathbf{r}\_2)
\,
R\_{n\_B\ell\_B}(\mathbf{r}\_2) 
Y\_{\ell\_Bm\_B}(\mathbf{r}\_2) 
\, \frac{1}{ r\_{12}}\,d^3\mathbf{r}\_1\,d^3\mathbf{r}\_2
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\_{\mathbb{R}^{3\times 2}} 
R^\*\_{n\_A\ell\_A }(\mathbf{r}\_1)
Y^\*\_{ \ell\_Am\_A}(\mathbf{r}\_1)
\,
R\_{n\_B\ell\_B }(\mathbf{r}\_1)
Y\_{ \ell\_Bm\_B}(\mathbf{r}\_1)
\,
R^\*\_{n\_B\ell\_B }(\mathbf{r}\_2)
Y^\*\_{ \ell\_Bm\_B}(\mathbf{r}\_2)
\,
R\_{n\_A\ell\_A }(\mathbf{r}\_2) 
Y\_{ \ell\_Am\_A}(\mathbf{r}\_2) 
\, \frac{1}{ r\_{12}}\,d^3\mathbf{r}\_1\,d^3\mathbf{r}\_2
\end{align\*}
The computation of this integral is well served by the expansion of $1/r\_{12}$ in terms of spherical harmonics.
$$\frac{1}{ r\_{12}} = 
\begin{cases}
\frac{1}{r\_2}\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \left(\frac{r\_{1 } }{r\_{ 2} } \right)^{k } \sum\limits\_{q=-k}^{k} \,Y^{\*}\_{k,q}(\theta\_1, \phi\_1) \,Y \_{k,q}(\theta\_2, \phi\_2), 
&
\text{for}~r\_2>r\_1;~\text{and}
\\
\frac{1}{r\_1}\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \left(\frac{r\_{2 } }{r\_{ 1} } \right)^{k }\sum\limits\_{q=-k}^{k} Y^{\*}\_{k,q}(\theta\_2, \phi\_2) \,Y \_{k,q}(\theta\_1, \phi\_1),
&
\text{for}~r\_1>r\_2.
\end{cases}
$$
\begin{align\*}
&E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} 
\sum\limits\_{q=-k}^{k} \,
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A}(\mathbf{r}\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{k+2}\,r\_{ 2}^{1-k } 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y^\*\_{\ell\_Am\_A}(\theta\_1,\phi\_1)
Y\_{\ell\_Am\_A}(\theta\_1,\phi\_1)
Y^\*\_{\ell\_Am\_B}(\theta\_2,\phi\_2)
Y\_{\ell\_Bm\_B}(\theta\_2,\phi\_2) 
\, 
Y^{\*}\_{k,q}(\theta\_1, \phi\_1) 
\,
Y \_{k,q}(\theta\_2, \phi\_2), 
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2
\\
&+
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 \sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} 
 \sum\limits\_{q=-k}^{k} 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A}(r\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y^\*\_{\ell\_Am\_A}(\theta\_1,\phi\_1)
Y\_{\ell\_Am\_A}(\theta\_1,\phi\_1)
Y^\*\_{\ell\_Am\_B}(\theta\_2,\phi\_2)
Y\_{\ell\_Bm\_B}(\theta\_2,\phi\_2) 
\, 
Y^{\*}\_{k,q}(\theta\_2, \phi\_2) \,Y \_{k,q}(\theta\_1, \phi\_1)
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 \sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \sum\limits\_{q=-k}^{k} \,
 \int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{1}^{k+2}\,r\_{2}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y^\*\_{ \ell\_Am\_A}(\theta\_1,\phi\_1)
Y\_{ \ell\_Bm\_B}(\theta\_1,\phi\_1)
Y^\*\_{ \ell\_Bm\_B}(\theta\_2,\phi\_2)
Y\_{ \ell\_Am\_A}(\theta\_2,\phi\_2) 
\, 
Y^{\*}\_{k,q}(\theta\_1, \phi\_1) \,Y \_{k,q}(\theta\_2, \phi\_2), 
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \sum\limits\_{q=-k}^{k} \int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y^\*\_{ \ell\_Am\_A}(\theta\_1,\phi\_1)
Y\_{ \ell\_Bm\_B}(\theta\_1,\phi\_1)
Y^\*\_{ \ell\_Bm\_B}(\theta\_2,\phi\_2)
Y\_{ \ell\_Am\_A}(\theta\_2,\phi\_2) 
\,
Y^{\*}\_{k,q}(\theta\_2, \phi\_2) 
\,
Y \_{k,q}(\theta\_1, \phi\_1)
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2
\end{align\*}
Now, since
\begin{align}
& \int Y\_{l\_1 m\_1}(\theta, \phi) Y\_{l\_2 m\_2}(\theta, \phi) Y\_{l\_3 m\_3}(\theta, \phi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi \\
&\quad = 
\sqrt{\frac{(2l\_1 + 1)(2l\_2 + 1)(2l\_3 + 1)}{4\pi}}
\begin{pmatrix}
 l\_1 & l\_2 & l\_3 \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 l\_1 & l\_2 & l\_3\\
 m\_1 & m\_2 & m\_3
\end{pmatrix}
\end{align}
we rewrite as
\begin{align\*}
&E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} 
\sum\limits\_{q=-k}^{k} \,
(-1)^{m\_A+m\_B+q} 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A}(\mathbf{r}\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{k+2}\,r\_{ 2}^{1-k } 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y\_{\ell\_A,-m\_A}(\theta\_1,\phi\_1)
Y\_{k,-q}(\theta\_1, \phi\_1) 
Y\_{\ell\_A,m\_A}(\theta\_1,\phi\_1)
Y\_{\ell\_A,-m\_B}(\theta\_2,\phi\_2)
Y\_{k,q}(\theta\_2, \phi\_2), 
Y\_{\ell\_B,m\_B}(\theta\_2,\phi\_2) 
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2
\\
&+
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 \sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} 
 \sum\limits\_{q=-k}^{k} 
(-1)^{m\_A+m\_B+q} 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A}(r\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y\_{\ell\_A,-m\_A}(\theta\_1,\phi\_1)
Y \_{k,q}(\theta\_1, \phi\_1)
Y\_{\ell\_Am\_A}(\theta\_1,\phi\_1)
Y\_{\ell\_A,-m\_B}(\theta\_2,\phi\_2)
Y\_{k,-q}(\theta\_2, \phi\_2) 
Y\_{\ell\_Bm\_B}(\theta\_2,\phi\_2) 
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 \sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \sum\limits\_{q=-k}^{k} \,
 (-1)^{m\_A+m\_B+q}
 \int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{1}^{k+2}\,r\_{2}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y\_{ \ell\_A,-m\_A}(\theta\_1,\phi\_1)
Y\_{k,-q}(\theta\_1, \phi\_1)
Y\_{ \ell\_Bm\_B}(\theta\_1,\phi\_1)
Y\_{ \ell\_B,-m\_B}(\theta\_2,\phi\_2)
Y\_{k,q}(\theta\_2, \phi\_2), 
Y\_{ \ell\_Am\_A}(\theta\_2,\phi\_2) 
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2
\\
& \pm
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \sum\limits\_{q=-k}^{k} 
(-1)^{m\_A+m\_B+q} 
 \int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
\int\limits\_{0}^\pi 
\int\limits\_{0}^{2\pi} 
Y\_{ \ell\_A,-m\_A}(\theta\_1,\phi\_1)
Y\_{k,q}(\theta\_1, \phi\_1)
Y\_{ \ell\_B,m\_B}(\theta\_1,\phi\_1)
Y\_{ \ell\_B,-m\_B}(\theta\_2,\phi\_2)
Y\_{k,-q}(\theta\_2, \phi\_2) 
Y\_{ \ell\_A,m\_A}(\theta\_2,\phi\_2) 
\,
\sin\theta\_1\,d\theta\_1\,d\phi\_1 
\,
\sin\theta\_2\,d\theta\_2\,d\phi\_2.
\end{align\*}
Upon substitution,
\begin{align\*}
&E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} 
\sum\limits\_{q=-k}^{k} \,
(-1)^{m\_A+m\_B+q} 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A}(\mathbf{r}\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{k+2}\,r\_{ 2}^{1-k } 
\,
dr\_1\,dr\_2
\\
&\quad\times
 \frac{(2\ell\_A + 1) (2k + 1) (2\ell\_B + 1)}{4\pi} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 -m\_A & -q & m\_A
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 -m\_B & q & m\_B
\end{pmatrix}
\\
&+
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 \sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} 
 \sum\limits\_{q=-k}^{k} 
(-1)^{m\_A+m\_B+q} 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A}(r\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\frac{(2\ell\_A + 1)(2k + 1)(2\ell\_B + 1)}{4\pi} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 -m\_A & q & m\_A
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 -m\_B & -q & m\_B
\end{pmatrix}
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 \sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \sum\limits\_{q=-k}^{k} \,
 (-1)^{m\_A+m\_B+q}
 \int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{1}^{k+2}\,r\_{2}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\frac{(2\ell\_A + 1)(2k + 1)(2\ell\_B + 1)}{4\pi} 
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix}^2
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 -m\_A & -q & m\_B
\end{pmatrix}
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 -m\_B & q & m\_A
\end{pmatrix}
\\
& \pm
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\sum\limits\_{k=0}^\infty \frac{4\pi}{2k+1} \sum\limits\_{q=-k}^{k} 
(-1)^{m\_A+m\_B+q} 
 \int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\frac{(2\ell\_A + 1)(2k + 1)(2\ell\_B + 1)}{4\pi} 
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix}^2
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 -m\_A & q & m\_B
\end{pmatrix}
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 m\_A & -q & -m\_B
\end{pmatrix}
.
\end{align\*}
From \url{https://en.wikipedia.org/wiki/3-j\_symbol}, the Wigner $3-j$ symbol is zero unless (i) $m + q -m = 0$, and (ii) $|k-\ell| \leq \ell \leq k+\ell$. Consequently, the only values of $k$ and $q$ that result in a non-trivial definite intergal are $q=0$ and $k=0,1,\ldots,\ell$.
I now employ the fact\footnote{From \url{https://mathworld.wolfram.com/Wigner3j-Symbol.html}
} that for $J = 2\ell+k$
\begin{align\*}
&E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_A + 1) (2\ell\_B + 1) 
\sum\limits\_{k=0}^\infty 
(-1)^{m\_A+m\_B} 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A}(\mathbf{r}\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{k+2}\,r\_{ 2}^{1-k } 
\,
dr\_1\,dr\_2
\\
&\quad\times
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 -m\_A & 0 & m\_A
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 -m\_B & & m\_B
\end{pmatrix}
\\
&+
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_A + 1) (2\ell\_B + 1) 
 \sum\limits\_{k=0}^\infty 
(-1)^{m\_A+m\_B} 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A}(r\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 -m\_A & 0 & m\_A
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 -m\_B & 0 & m\_B
\end{pmatrix}
\\
& \pm 
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (-1)^{\ell\_A+k+q}
 \,
 (2\ell\_A + 1) (2\ell\_B + 1) 
 \sum\limits\_{k=0}^\infty \int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{1}^{k+2}\,r\_{2}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix}^2
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 -m\_A & m\_A-m\_B & m\_B
\end{pmatrix}^2
\\
& \pm
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
(-1)^{ \ell\_A + k + \ell\_B }
\,
 (2\ell\_A + 1) (2\ell\_B + 1) 
\sum\limits\_{k=0}^\infty 
 \int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\\
&\quad\times
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix}^2
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 -m\_A & m\_A-m\_B & m\_B
\end{pmatrix}^2
.
\end{align\*}
And
\begin{align\*}
&E\_{ n\_A\ell\_Am\_A,n\_B\ell\_Bm\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_A + 1) (2\ell\_B + 1) 
\sum\limits\_{k=0}^\infty 
(-1)^{m\_A+m\_B} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix} 
\begin{pmatrix}
 \ell\_A & k & \ell\_A \\
 -m\_A & 0 & m\_A
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 -m\_B & & m\_B
\end{pmatrix}
\\
&\quad \times
\begin{bmatrix}
+\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A}(\mathbf{r}\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{k+2}\,r\_{ 2}^{1-k } 
\,
dr\_1\,dr\_2
\\
+
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A}(r\_1)
\,
R\_{n\_A\ell\_A}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\end{bmatrix}
\\
& \pm
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_A + 1) (2\ell\_B + 1) 
\sum\limits\_{k=0}^\infty 
(-1)^{ \ell\_A + k + \ell\_B }
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix}^2
\begin{pmatrix}
 \ell\_A & k & \ell\_B \\
 -m\_A & m\_A-m\_B & m\_B
\end{pmatrix}^2
\\
&\quad\times
\begin{bmatrix}
+
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{1}^{k+2}\,r\_{2}^{1-k} 
\,
dr\_1\,dr\_2
\\
+ 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{n\_A\ell\_A }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
r\_{1}^{1-k} 
\,
dr\_1
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{n\_A\ell\_A }(r\_2) 
\,
r\_{2}^{k+2} 
\,dr\_2
\end{bmatrix}
.
\end{align\*}
Now for the case of interest here which includes one electron in the 1s state, I find that
\begin{align\*}
&E\_{ 1,0,0;n\_B,\ell\_B,m\_B}^{Z\_1=2,Z\_2=2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_B + 1) 
\sum\limits\_{k=0}^\infty 
(-1)^{ m\_B} 
\begin{pmatrix}
 0 & k & 0 \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix} 
\begin{pmatrix}
 0 & k & 0 \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & k & \ell\_B \\
 -m\_B & 0 & m\_B
\end{pmatrix}
\\
&\quad \times
\begin{bmatrix}
+\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{1,0}(\mathbf{r}\_1)
\,
R\_{1,0}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{k+2}\,r\_{ 2}^{1-k } 
\,
dr\_1\,dr\_2
\\
+
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{1,0}(r\_1)
\,
R\_{1,0}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{k+2}\,r\_{1}^{1-k} 
\,
dr\_1\,dr\_2
\end{bmatrix}
\\
& \pm
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_B + 1) 
\sum\limits\_{k=0}^\infty 
(-1)^{ k + \ell\_B }
\begin{pmatrix}
 0 & k & \ell\_B \\
 0 & 0 & 0
\end{pmatrix}^2
\begin{pmatrix}
 0 & k & \ell\_B \\
 0 & -m\_B & m\_B
\end{pmatrix}^2
\\
&\quad\times
\begin{bmatrix}
+
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{1,0}(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{1,0 }(r\_2) 
\,
r\_{1}^{k+2}\,r\_{2}^{1-k} 
\,
dr\_1\,dr\_2
\\
+ 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{1,0 }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
r\_{1}^{1-k} 
\,
dr\_1
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{1,0 }(r\_2) 
\,
r\_{2}^{k+2} 
\,dr\_2
\end{bmatrix}
.
\end{align\*}
According to the rules for the Wigner-$3j$ numbers, the only values of $k$ that result in non-trivial values here are $k=0$ for the direct integral, and $k=\ell\_B$ for the exchange integral. Thus,  

\begin{align\*}
&E\_{ 1,0,0;n\_B,\ell\_B,m\_B}^{Z\_1,Z\_2}
\\
&=
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_B + 1) 
(-1)^{ m\_B} 
\begin{pmatrix}
 0 & 0 & 0 \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & 0 & \ell\_B \\
 0 & 0 & 0
\end{pmatrix} 
\begin{pmatrix}
 0 & 0 & 0 \\
 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
 \ell\_B & 0 & \ell\_B \\
 -m\_B & 0 & m\_B
\end{pmatrix}
\\
&\quad \times
\begin{bmatrix}
+\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{1,0}(\mathbf{r}\_1)
\,
R\_{1,0}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{ 2}\,r\_{ 2}^{1 } 
\,
dr\_1\,dr\_2
\\
+
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{1,0}(r\_1)
\,
R\_{1,0}(r\_1)
\,
R^\*\_{n\_B\ell\_B}(r\_2)
\,
R\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{ 2}\,r\_{1}^{1 } 
\,
dr\_1\,dr\_2
\end{bmatrix}
\\
& \pm
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
 (2\ell\_B + 1) 
\begin{pmatrix}
 0 & \ell\_B & \ell\_B \\
 0 & 0 & 0
\end{pmatrix}^2
\begin{pmatrix}
 0 & \ell\_B & \ell\_B \\
 0 & -m\_B & m\_B
\end{pmatrix}^2
\\
&\quad\times
\begin{bmatrix}
+
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^\*\_{1,0}(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{1,0 }(r\_2) 
\,
r\_{1}^{\ell\_B+2}\,r\_{2}^{1-\ell\_B} 
\,
dr\_1\,dr\_2
\\
+ 
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^\*\_{1,0 }(r\_1)
\,
R\_{n\_B\ell\_B }(r\_1)
\,
r\_{1}^{1-\ell\_B} 
\,
dr\_1
\,
R^\*\_{n\_B\ell\_B }(r\_2)
\,
R\_{1,0 }(r\_2) 
\,
r\_{2}^{\ell\_B+2} 
\,dr\_2
\end{bmatrix}
.
\end{align\*}
Reducing further
\begin{align}
\label{eq:helium-energy-levels-l2-equals-0}
\boxed{
E\_{ 1,0,0;n\_B,\ell\_B,m\_B}^{Z\_1,Z\_2}
= 
\begin{matrix}
+\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^{Z\_1=2}\_{1,0}(\mathbf{r}\_1)
\,
R^{Z\_1=2}\_{1,0}(r\_1)
\,
R^{Z\_2=2}\_{n\_B\ell\_B}(r\_2)
\,
R^{Z\_2=2}\_{n\_B\ell\_B}(r\_2) 
\,
r\_{1 }^{ 2}\,r\_{ 2}^{1 } 
\,
dr\_1\,dr\_2
\\[2em]
+
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^{Z\_1}\_{1,0}(r\_1)
\,
R^{Z\_1}\_{1,0}(r\_1)
\,
R^{Z\_2}\_{n\_B\ell\_B}(r\_2)
\,
R^{Z\_2}\_{n\_B\ell\_B}(r\_2) 
\,
r\_{2}^{ 2}\,r\_{1}^{1 } 
\,
dr\_1\,dr\_2
\\[2em]
\pm
\frac{e^2}{ 4\,\pi\,\epsilon\_o }
\dfrac{\int\limits\_{0}^\infty 
\int\limits\_{r\_1=0}^{r\_2} 
R^{Z\_1}\_{1,0}(r\_1)
\,
R^{Z\_2}\_{n\_B\ell\_B }(r\_1)
\,
R^{Z\_2}\_{n\_B\ell\_B }(r\_2)
\,
R^{Z\_1}\_{1,0 }(r\_2) 
\,
r\_{1}^{\ell\_B+2}\,r\_{2}^{1-\ell\_B} 
\,
dr\_1\,dr\_2
}{ (2\ell\_B + 1) }
\\[2em]
\pm \frac{e^2}{ 4\,\pi\,\epsilon\_o }
\dfrac{\int\limits\_{0}^\infty 
\int\limits\_{r\_1=r\_2}^{\infty} 
R^{Z\_1}\_{1,0 }(r\_1)
\,
R^{Z\_2}\_{n\_B\ell\_B }(r\_1)
\,
r\_{1}^{1-\ell\_B} 
\,
dr\_1
\,
R^{Z\_2}\_{n\_B\ell\_B }(r\_2)
\,
R^{Z\_2}{1,0 }(r\_2) 
\,
r\_{2}^{\ell\_B+2} 
\,dr\_2
}{ (2\ell\_B + 1) }
\end{matrix}
.
}
\end{align}
In the boxed solution above, the first two (of four) integrals are the direct integrals, while the second two (of four) integrals are the exchange integrals.


**Discussion**


If you look at Appendix B of Atomic Physics by Mark Fox (Cambridge University Press, 2018), it appears that $Z\_1 = Z\_2 = 2$. However, in Atomic Physics by Christopher Foot, Foot argues that $Z\_1 = Z\_2 = 2$ is correct when calculating the 1s2s configuration. However, $Z\_1 = 2$ and $Z\_2 = 1$ is correct when, for example, calculating the 1s2p configuration. Further, Foot states (Oxford University Pres, page 55, 2005) that for the 1s2p configuration, upon plugging in the hydrogenic wave functions $Z\_1 = 2$ and $Z\_2 = 1$, that the two direct integrals yield
$$-\frac{e^2}{ 8\,\pi\,\epsilon\_o\,a\_o }\frac{{Z\_{2}}^5\,\left(108\,{Z\_{1}}^2+16\,Z\_{1}\,Z\_{2}+{Z\_{2}}^2\right)}{2\,{\left(2\,Z\_{1}+Z\_{2}\right)}^6} = -0.028~\text{eV}.$$ Using the boxed equation above, I obtain that same value. Foot also states (Oxford University Pres, page 56, 2005) that for the 1s2p configuration, upon plugging in the hydrogen wave functions ($Z\_1 = 2$, $Z\_2 = 1$), that the two exchange integrals yield $ +0.10$ eV (page 55, 2005 edition). However, using the boxed equation above, I obtain a different value; namely
$$\frac{e^2}{8\pi\epsilon\_o\,a\_o} \frac{2^{10}\,{Z\_{1}}^2\,{Z\_{2}}^2\,\left({Z\_{1}}^2+5\,Z\_{1}\,Z\_{2}+{Z\_{2}}^2\right)}{3^8\,{\left(Z\_{1}+Z\_{2}\right)}^5}'
 = 0.52~\text{eV}.$$
In both of my computations, I use the Bohr radius $a\_0 = 52.9~\text{pm}$. I do not understand why I get a separation 5 times greater than what I see in the book and on energy-level diagrams for helium.


 3