Show that the helium ion is bound

[Oskar Paulsson]

Homework Statement

"Use the simple MO method to show that the He₂⁺ ion is bound"

Homework Equations

The hamiltonian for this system is;

H=-(ħ²/2m)∇² -2ke²[∑³_i=11/r_iB+∑³_i=11/r_iB-2/R_AB]

And as far as I know, the total wave function sould be;

ψ = φ₁φ₂φ₃ ... φ_n , n is the number of electrons and "a reasonable behaviour" for an electron would look like this;

φ= c_aφ_a + c_bφ_b

in the course literature they explain this as reasonable for one electron in a diatomic molecule because this one electron is "a little bit influenced by a and a little bit by b".

The a's and b's is for each atom/nuclei.

A bound state has the total wave function:

ψ = 1/√(2+2S)[φ_a+φ_b]

The Attempt at a Solution

So far the only 'attempts' I've made at solving this has been pretty much speculation - firstly I'm considering the Pauli principle. From the perspective of the Pauli principle we should have 1 electron in the 1s_a orbital and 2 in the 1s_b which is equivalent with 2 1s_a + 1 1s_b.

So the total wavefunction, I speculate to be some thing like;

ψ = c_a1s_a + c_b1s_b + 1/√(2)*(c_a1s_a + c_b1s_b)

The first term should be an electron which has a b-term that is almost 0 and the next one should be for an electron with an a-term that is almost 0. The third term is for an electron in "superposition" between 1s²_a and 1s²_b.

So - how to proceed? I am headed in the right direction? Does anyone have a source for some solid math on this whole thing because the course literature isn't very theoretical, it just shows some results and doesn't bother to properly explain anything.

 

[mfb]

The total wave function has to be antisymmetric, but as far as I remember, your approach, while not physically possible, leads to the same energy if you evaluate that. You can compare this energy to the unbound state.