How do we know the orbitals of multielectron atoms look like the orbitals of hydrogen?

Question

How do we know that the solutions of the time-independent Schrodinger equation for multielectron atoms are the same with the solutions for the hydrogen atom if we add a effective potential due to the shielding of the charge of the nucleus from inner shell electrons?

Answer 1

The electronic Hamiltonian for a hydrogen atom can be written as

$$\hat{H}_e = -\frac{Ze^2}{r_{Ne}} - \frac{\hbar^2}{2m_e} \nabla^2$$

The relative nuclear charge for an H-atom is $Z=1$ (the atomic number). When approximating the single electron Hamiltonian using an effective nuclear potential and otherwise ignoring electron-electron repulsion, which is missing in the above Hamiltonian, $Z$ is set to $Z_{eff}$, the difference of the nuclear charge and the screening constant.

The effect of an increase in $Z$ is to contract the size of an atom. The Hamiltonian could be written as

$$\hat{H}_e = \frac{e^2}{r^*_{Ne}} - \frac{\hbar^2}{2m^*_e} \nabla^{*2}$$

where $r^*_{Ne} = r_{Ne}/Z$ is an effective distance from the nucleus, $m^*_e = m_e Z^2$ is an effective electron mass and $\nabla^* = \sum_i \vec{e}_i \frac{\partial}{\partial x_i^*}$. The change in the dimensions of $r$ therefore does not alter the mathematical form of the eigenfunctions of the Hamiltonian above, it only rescales the mass and distance dimensions. The solutions will be the single electron hydrogenic wavefunctions but with the effective radius $r^*_{Ne} = r_{Ne}/Z$ in place of $r$ and the momentum of the electrons altered by the change in effective mass. An effective Bohr radius can be computed under these circumstances as (setting $\mu=Z^2 m_e$) $$a_\circ ^* = \frac{4 \pi \epsilon _\circ \hbar^2}{Z^2 m_e e^2} = \frac{a_\circ}{Z^2}$$ illustrating the contraction in the extent of the wavefunction due to the increase in $Z$.

Note that this is an oversimplification of the true situation, but the approximation suggests for instance that Slater-type orbitals might be useful as a first step to describing the properties of atoms. For molecules it is likewise inaccurate but can be helpful to grasp how behavior scales with nuclear charge.