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The naive notion of the ground state of lithium atom's electron field is that there are 2 electrons with opposite spins in the 1s orbit, and 1 electron in the 2s state, i.e. $$\phi(\mathbf {r_1},\mathbf {r_2},\mathbf {r_3})=\left|\begin{matrix} \psi_{1s}(\mathbf {r_1})\alpha &\psi_{1s}(\mathbf {r_1})\beta &\psi_{2s}(\mathbf {r_1})\alpha \\ \psi_{1s}(\mathbf {r_2})\alpha &\psi_{1s}(\mathbf {r_2})\beta &\psi_{2s}(\mathbf {r_2})\alpha \\\psi_{1s}(\mathbf {r_3})\alpha &\psi_{1s}(\mathbf {r_3})\beta &\psi_{2s}(\mathbf {r_3})\alpha \end{matrix}\right|$$

When we take into account the spin-orbit coupling the 1s orbit splits. Why can't all 3 electrons occupy these new orbitals, lowering the ground state energy?

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The splitting breaks the degeneracy in the energies of the two 1s spin states, but it does not create more states. In other words, "splitting" means spreading of pre-existing states, not splitting old states into new states.

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