Orbital energies in a hydrogen-like species

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. 3s, 3p and 3d) are degenerate. It follows from eq. 1.16 that the orbital energy levels get closer together as the value of n increases. This result is a general one for all other atoms. 

Show that the energy of both the 2s and 2p orbitals for a hydrogen atom is —328kJmor.  

For a hydrogen atom, confirm that the energy of the 3s orbital is —1.51 eV. 

Besides providing information about the wavefunctions, solutions of the Schrodinger equation give orbital energies, E (energy levels), and equation 1.16 shows the dependence of E on the principal quantum number for hydrogen-like species. 

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Orbital energies in a hydrogen-like species Size of orbitals... 

The energy required to remove an electron from an orbital with principal quantum number n in a hydrogen-like species—the ionization energy—is the negative of that shown above, that is,... 

In Chapter 2, the development of atomic orbitals for many-electron atoms was built upon an understanding of the orbitals obtained from the exact solution for the one-electron hydrogen atom. Similarly, it is useful to examine the electronic structure for the simplest molecular species—the hydrogen molecule ion (Hj)—to begin our discussion of molecular orbitals. Like the H atom, the ion contains only one electron. The difference is the presence of two nuclei instead of one. The potential energy of the system is a sum of the Coulomb attraction of the electron to each of the two nuclei and Coulomb repulsion between the positively charged nuclei. 

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