Definition of atomic orbital

I compared results of HF calculation with that of another quantum chemistry cord. They yield different density matrix although their energy are same.This difference occurs to any basis set. I suspect that the definition of the atomic orbital(normalization factor etc.) could be different. Is there any description of the definition?

I got the density matrix represented with AO via this code.

The AO normalization in Psi4 does disagree with other codes because we omit a factor as described here. Other codes may also index their AOs differently.

Actually, I found that the AO ordering disagrees. In Psi4, the order is as same as that in basis sets; in another code(PySCF), the order is sorted to be s->p->d… I was also not careful that Psi4 is using spherical orbitals. Now I got the consistency with those conversions. Thank you for detailing.

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