(The energy of course depends on the zero-point used.) Note that if Z were able to be more than 137 (higher than any known element) then we would have a negative value inside the square root for the S 1/2 and P 1/2 orbitals, which means they would not exist. The Schrödinger solution corresponds to replacing the inner bracket in the second expression by 1. The accuracy of the energy difference between the lowest two hydrogen states calculated from the Schrödinger solution is about 9 ppm (90 μ eV too low, out of around 10 eV), whereas the accuracy of the Dirac equation for the same energy difference is about 3 ppm (too high). The Schrödinger solution always puts the states at slightly higher energies than the more accurate Dirac equation. The Dirac equation gives some levels of hydrogen quite accurately (for instance the 4P 1/2 state is given an energy only about 2 × 10 eV too high), others less so (for instance, the 2S 1/2 level is about 4 × 10 eV too low). The modifications of the energy due to using the Dirac equation rather than the Schrödinger solution is of the order of α, and for this reason α is called the fine-structure constant.