The first order energy correction in the | j , m , ℓ ℓ , 1 / 2 ⟩ ⟩ basis where the perturbation Hamiltonian is diagonal, is given by E s o = ℏ ℏ 2 e 2 4 m 2 c 2 j ( j + 1 ) − − ℓ ℓ ( ℓ ℓ + 1 ) − − 3 4 a 0 3 n 3 ℓ ℓ ( ℓ ℓ + 1 2 ) ( ℓ ℓ + 1 ) where a 0 is the Bohr radius. The total fine-structure energy shift is given by E f s = − − m c 2 α α 4 2 n 3 [ 1 / ( j + 1 / 2 ) − − 3 / 4 n ] for j = ℓ ℓ ± ± 1 2 .