My basis set isn't orthonormal?

Question

I'm implementing a little QM calculation just for fun and to make sure I understand how it works (calculating the helium ground state energy). My problem is that my basis set doesn't seem to be orthonormal. I'm using the spherical harmonics for the angular part and the Slater type radial function. If I integrate just the angular part

$$\int_{\phi = 0}^{2\pi}\int_{\theta = 0}^\pi Y_l^{m*}(\theta, \phi) Y_{l'}^{m'}(\theta, \phi) \sin(\theta)\ d\theta \ d\phi = \delta_{ll'}\delta_{mm'}$$

then the result satisfies the requirement for orthonormality. But if I include the radial Slater component:

$$\int_{\phi = 0}^{2\pi} \int_{\theta = 0}^\pi \int_{r = 0}^{\infty} C r^{n-1} e^{-\alpha r} Y_l^{m*}(\theta, \phi) \cdot C r^{n'-1} e^{-\alpha r}Y_{l'}^{m'}(\theta, \phi) \cdot r^2 \sin(\theta)\ dr \ d\theta \ d\phi = \delta_{ll'} \delta_{mm'} \frac{(n + n')!}{\sqrt{(2n)!(2n')!}}$$

with

$$C = \left((2\alpha)^n\sqrt{\frac{2\alpha}{(2n)!}}\right)$$

I get some extra factor instead of what should be $\delta_{nn'}$. I've double-checked my references and can't figure out what I'm doing wrong.

Thanks!

Answer 1

Indeed, the Slater-type orbitals (radial wave functions) are not orthonormal – they are not even orthogonal to each other. The $\delta_{n,n'}$ Kronecker delta symbol doesn't appear in the inner product and it can't because the $r$-dependent integrand is positively definite and there is no room for cancellation.

Their not being orthogonal physically means that the orbitals for different $n$ are not mutually exclusive. For a given molecule, one is supposed to use one value of $n$ only.