The ground state energy of hydrogen like species is proportional to :
\(\frac{-Rz^{2}}{n^{2}}\)
By Bohr theory the energy of an electron in level n of a one-electron species of nuclear charge z is \(E_{n} = -\frac{m_{e}e^{4}z^{2}}{8\varepsilon_{0}^{2}h^{2}n^{2}} = -\frac{Rz^{2}}{n^{2}}\), where R (in energy units, 13.6 eV) is the Rydberg constant.
The negative sign shows the electron is bound (its energy is below that of a free electron at rest, taken as zero), and the energy scales as \(z^{2}\) and as \(1/n^{2}\).
For the ground state n = 1, so \(E_{1} = -13.6z^{2}\) eV.
The expression without \(z^{2}\) applies only to hydrogen itself, the positive expression would mean an unbound electron, and a dependence on \(z/n\) does not follow from the theory.
Check: for H (z = 1) \(E_{1} = -13.6\) eV and for He+ (z = 2) \(E_{1} = -13.6 \times 4 = -54.4\) eV, matching their measured ionisation energies of 13.6 eV and 54.4 eV.
Hence, \(\frac{-Rz^{2}}{n^{2}}\).
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