# Energy levels in non-hydrogen-like atoms

The energy $E_n$ for a hydrogen like atom is given as

$$E_n = -hcR_\ce{H}\frac{Z^2}{n^2}$$

However, aside from on wikipedia where there is

$$E_n = -hcR\frac{Z_\text{eff}^2}{n^2},$$

I can't find anywhere that relates to the energy of an atom that has more than one electron. Is there an equation that gives the specific (or approximate) energy or is there an 'easy' way to calcuate a value for $Z_\text{eff}$?

Consider an atom to consist of an electron orbiting a positvely charged ion core (with one or more electrons associated to it). If you excite the electron to larger and larger values of the principal quantum number $n$ the classical electron orbit becomes larger and larger as well (in fact it scales as $n^2$). At sufficiently large distance, the postively charged ion core behaves as a point charge (like a single proton) and at long range you basically have the hydrogen problem with a different mass (assuming perfect shielding of the core electrons). In order to find the overall solution to this problem, one has to connect the long-range and short range behavior of the system. One way of doing this, is is by considering the system as a collision problem at negative energy (that is, a bound system). The effect of the nonhydrogenic ion core then results in a scattering phase shift between an incommming and outgoing electron. It can be shown that the solutions to this problem have a very similar appearance as the Bohr formula, namely:

$$E_{n,\ell}=E_\text{IE}-\frac{hcR_A}{(n-\delta_\ell)^2}$$ where $R_A=R_\infty(1-m_e/m_A)$ is the mass-corrected Rydberg constant, $E_\text{IE}$ is the ionization energy and $\delta_\ell$ is the so-called quantum defect for electrons with orbital angular momentum $\ell$. The quantum defect is related to the phase shift induced by the core and is different for $s, p, d, f$ etc. electrons. In principle, the quantum defect is a function of the binding energy of the electron to the ionic core, but in many cases this dependence may be neglected (in particular for large $n$).

What does this mean in practice? Let us for example look at the $n$s levels of the sodium atom. We find that $R_\text{Na}=109734.697205$ cm$^{-1}$ and $E_\text{IP}(\text{Na})=41449.451$ cm$^{-1}$. The second column in the table below gives the energy (in cm$^{-1}$ from the NIST website) and the third column gives the binding energy of the electron (also in cm$^{-1}$). The last column gives the so-called effective quantum number, that is $n^*=n-\delta_\ell$, with $\ell=0$ for $s$ orbitals

 3s         0.000   -41449.451  1.63
4s     25739.999   -15709.452  2.64
5s     33200.673    -8248.778  3.65
6s     36372.618    -5076.833  4.65
7s     38012.042    -3437.409  5.65
8s     38968.510    -2480.941  6.65
9s     39574.850    -1874.601  7.65
10s    39983.270    -1466.181  8.65
11s    40271.396    -1178.055  9.65


As you can see the quantum numbers take the form of the hydrogen solution if we start counting the principal quantum number form the 3$s$ levels and take $\delta_0\approx -0.65$.

To calculate the quantum defect is difficult as it requires ab initio calculations and then converting the energy levels to effective quantum numbers from which the defects can be extracted. However, experimentaly these numbers are very convenient, and one may use this quantum-defect theory for instance to determine the ionization energy of atoms and even simple molecules very accurately by extrapolating Rydberg series.