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

by geologic
Tags: 1orbital, atoms, bringing, levels, yield
 P: 7 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.
 HW Helper Sci Advisor Thanks P: 9,127 Hint: Pauli exclusion principle.
P: 3,121
 Quote by geologic 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.
You are right in that you only get two distinct energy levels. However there are two independent wavefunctions giving the same energy in case of the upper level.
In the case of a ring with N atoms, you get (N+1)/2 distinct energy levels.

P: 7

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

 Quote by Simon Bridge Hint: Pauli exclusion principle.
Yeah, I suppose I'm unclear about what defines a state in this context. If I assume all of the electrons remain in the 1s state, while sort of forming a metal, then I have more electrons than available states. So, the creation of more states solves this problem, but isn't obviously the answer.

 Quote by DrDu You are right in that you only get two distinct energy levels. However there are two independent wavefunctions giving the same energy in case of the upper level. In the case of a ring with N atoms, you get (N+1)/2 distinct energy levels.
By the two independent wavefunctions in the upper (anti-bonding) level, do you mean the sign of them? Then this should yield 4 total levels since you could argue the same thing for the constructively interfering wavefunctions.

Also, the (N+1)/2 argument seems contrary to what I remember reading in Kittel: N atoms yield N levels. I assumed the levels were distinct.