Hydrogen atom degeneracy

There are three types of degeneracy of carbocations that have been observed carbon atom degeneracy, hydrogen atom degeneracy and combined... 

Hydrogen atoms chemisorbed on a metal surface may be bonded to just one metal atom or may be bonded to two atoms in a symmetrical bridge. In each case, there are three normal modes. Sketch what these are, and indicate any degeneracies (assume the metal atoms to be infinitely heavy). 

The hydrogen atom and one-electron ions are the simplest systems in the sense that, having only one electron, there are no inter-electron repulsions. However, this unique property leads to degeneracies, or near-degeneracies, which are absent in all other atoms and ions. The result is that the spectrum of the hydrogen atom, although very simple in its coarse structure (Figure 1.1) is more unusual in its fine structure than those of polyelectronic atoms. For this reason we shall defer a discussion of its spectrum to the next section. 

The hydrogenic atom energy expression has no 1-dependence the 2s and 2p orbitals have exactly the same energy, as do the 3s, 3p, and 3d orbitals. This degree of degeneracy is only present in one-electron atoms and is the result of an additional symmetry (i.e., an additional operator that commutes with the Hamiltonian) that is not present once the atom contains two or more electrons. This additional symmetry is discussed on p. 77 of Atkins. 

Degeneracy often arises in two- or three-dimensional problems, especially where there is a high degree of symmetry. We shall see in Chapters 4 and 5 that degeneracy is a feature of the solutions for the hydrogen atom, and is important in understanding atomic structure. 

In a hydrogen atom, the orbital energy is determined exclusively by the principal quantum number n—all the different values of / and mi are degenerate. In a multielectron atom, however, this degeneracy is partially broken the energy increases as / increases for the same value of n. 

As mentioned previously, energy E appears only in the radial equation [eq. (2.1.14)], which does not involve Hence we readily note that s orbitals are non-degenerate, p orbitals are triply degenerate, d orbitals have fivefold degeneracy, and f orbitals have sevenfold degeneracy. As it turns out, for the hydrogen atom only, E is also independent of l ... 

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