Why does bringing N 1-orbital atoms together yield N levels?

Question

A common example of this is that when bringing N hydrogen atoms together into a ring. Far apart, assume each electron exists in the 1s state. As we bring them together, instead of each electron staying at the original 1s level, or all of them changing by the same amount, the 1s level fans out into N.

For the case of 2 atoms, I can understand this as bonding or anti-bonding of the atoms. i.e., do the wavefunctions add between the protons, meaning each electron can share in the potential of both protons (bonding) or do the wavefunctions destructively interfere between the protons (anti-bonding).

With 3 atoms, I can't find 3 levels. Assuming Gaussian shaped wavefunctions, note that the sign of each wavefunction between any two atoms defines the wavefunction on the rest of the ring. Since the signs of the wavefunction are independent, there should be 2^3=8 possibilities since each wavefunction can be + or -. Yet, there are really only 2 energetically distinct arrangements that I see: all have the same sign (two cases) or 2 of 3 have the same sign (2*(3 choose 2)), to account for both sign cases). So I get 3 atoms yield 2 levels.

Can somebody shed light on what I've done incorrectly? Or is 3 too small to work correctly? Is there an argument about the shape of the orbitals I've neglected?

Thank you.

Answer 1

In your 3-orbital example, you are ignoring the fact that you can get degenerate, linearly independent states. Suppose that you have three orbitals $s_1, s_2,$ and $s_3$. Excluding normalization, you can form the following linearly independent combinations $s_1 + s_2 + s_3$, $s_1 - s_2 - s_3$, and $s_1 - s_2 + s_3$. The last two combinations are degenerate in energy if the orbitals are all identical and you bring them together in a $D_{3h}$ symmetry (i.e. perfect equilateral triangle). However, they are clearly distinct and will have different energy if you do not have this symmetry or interact with anything that breaks the symmetry.

In general, the answer to your question comes from basic linear algebra. Each orbital can be represented as a linearly independent vector; thus, when you bring together $N$ orbitals you can form only $N$ linearly idependent combinations with them, resulting in $N$ energy levels. Some of these levels may be degenerate because of symmetry.