STO 3G basis expansion of p orbitals

I am expanding $2p\_{x},2p\_{y},2p\_{z}$ (slater type )orbitals in gaussian STO-3G basis as follows,


$$|2p\_{x}\rangle=C\_{1}x\exp(-\alpha\_{1}r^{2})+C\_{2}x\exp(-\alpha\_{2}r^{2})+C\_{3}x\exp(-\alpha\_{3}r^{2})$$


$$|2p\_{y}\rangle=C\_{1}y\exp(-\alpha\_{1}r^{2})+C\_{2}y\exp(-\alpha\_{2}r^{2})+C\_{3}y\exp(-\alpha\_{3}r^{2})$$


$$|2p\_{z}\rangle=C\_{1}z\exp(-\alpha\_{1}r^{2})+C\_{2}z\exp(-\alpha\_{2}r^{2})+C\_{3}z\exp(-\alpha\_{3}r^{2})$$


I thought the value of $C\_{1},C\_{2},C\_{3},\alpha\_{1},\alpha\_{2},\alpha\_{3}$ are the same for all the $p$ orbitals. I tried to evaluate the following two integral $$\langle2p\_{x}2p\_{y}||2p\_{y}2p\_{z}\rangle=\langle2p\_{x}2p\_{y}|2p\_{y}2p\_{z}\rangle-\langle2p\_{x}2p\_{y}|2p\_{z}2p\_{y}\rangle$$
where,
$$\langle2p\_{x}2p\_{y}|2p\_{y}2p\_{z}\rangle=\sum\_{i,j,k,l}C\_{p\_{x}i}C\_{p\_{y}l}C\_{p\_{y}j} C\_{p\_{x}m}\int d^{3}\vec{r\_{1}} \int d^{3}\vec{r\_{2}} x\_{1} e^{-\alpha\_{i}r\_{1}^{2}}y\_{1}e^{-\alpha\_{l}r\_{1}^{2}}\frac{1}{|\vec{r\_{1}}-\vec{r\_{2}}|}y\_{2}e^{-\alpha\_{j}r\_{2}^{2}}z\_{2}e^{-\alpha\_{m}r\_{2}^{2}}$$
The above integral vanishes completely. I think something went wrong in the basis I have chosen or does it really vanish ? I guess my STO-3G expansion is wrong. Can someone help me fixing this please ?


