## Hamiltonian in atomic units

1. The problem statement, all variables and given/known data
So the question is I have to use some trial function of the form $$\sum c_if_i$$ to approximate the energy of hydrogen atom where $$f_i=e^{-ar}$$ for some number a (positive real number). Note that r is in atomic unit.

2. Relevant equations
Because r is in atomic unit, I think I should use the Hamiltonian in atomic unit, that is
$$H = -\frac{1}{2}\nabla^2 + \frac{1}{r}$$
or should I use the spherical Hamiltonian?

I try to compute $$H_{ij} = \int_0^\infty f_iHf_j$$ but there will be the term $$\int_0^\infty f_1\frac{1}{r}f_2dr$$ which cannot be integrated (not converged). So what's wrong with the way I approach the problem?

Thank you,

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 might your coefficients be r dependent since they are in the full solution of the hydrogen like atom
 $$\int_0^\infty f_1\frac{1}{r}f_2dr$$ will converge if you change 1/r to 1/(r+eps)