Atomic orbitals degenerate solutions

Exact solutions such as those given above have not yet been obtained for the usual many-electron molecules encountered by chemists. The approximate method which retains tile idea of orbitals for individual electrons is called molecular-orbital theory (M. O. theory). Its approach to the problem is similar to that used to describe atomic orbitals in the many-electron atom. Electrons are assumed to occupy the lowest energy orbitals with a maximum population of two electrons per orbital (to satisfy the Pauli exclusion principle). Furthermore, just as in the case of atoms, electron-electron repulsion is considered to cause degenerate (of equal energy) orbitals to be singly occupied before pairing occurs. 

For each value of n there is only one energy solution and for hydrogen-like species, all atomic orbitals with the same principal quantum number (e.g. Zs, Zp and Zd) are degenerate. 

As one can see from the quantum-number limits in Eqs. (1.11), there is a total of n2 degenerate (equal-energy) orbitals for each principal quantum number n and energy level. Thus, the orbitals are naturally grouped into shells a single orbital (Is) for n = 1, four (2s, 2px,2py, and 2pz) for n = 2, and so forth. Only the non-degenerate Is orbital is occupied in the ground-state H atom, whereas all other solutions are formally vacant. 

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