 Gaussian interaction energy of a trapped condensate

For a normalized condensate orbital $\varphi ={\pi }^{-3/4}{a}_0^{-3/2}{e}^{-r^2/(2a_0^2)}$ and $\psi =\sqrt{N}\varphi$, the contact interaction functional is $E_I=(U_0/2)\int |\psi {|}^4{d}^{3}r$. The Gaussian integral gives $\int |\varphi {|}^4{d}^{3}r=[(2\pi {)}^{3/2}{a}_0^3{]}^{-1}$. With $U_0=4\pi {\hslash }^{2}a/m$, this becomes $E_I/(N\hslash \omega )=Na/(\sqrt{2\pi }{a}_0)$. An exact number-state pair count replaces $N^2$ by $N(N-1)$.