Based on the tensor expression of the fine-structure Hamiltonian for many-electron atoms, and with the help of irreducible tensor theory, a general analytical theory of calculating the fine structure energy (including spin-orbit interaction, spin-other-orbit interaction and the spin-spin interaction) of many-electron atoms has been established explicitly. The fine-structure energies of the ground state of carbon-like atoms from Z=6 to 8 have been calculated. The results are in close agreement with the experimental data.