Answer:
Linear combination of atomic orbitals (LCAO) is a simple method of quantum chemistry that yields a qualitative picture of the molecular orbitals (MOs) in a molecule. Let us consider H
+
2
again. The approximation embodied in the LCAO approach is based on the notion that when the two protons are very far apart, the electron in its ground state will be a 1s orbital of one of the protons. Of course, we do not know which one, so we end up with a Schrödinger cat-like state in which it has some probability to be on one or the other.
As with the HF method, we propose a guess of the true wave function for the electron
ψg(r)=CAψ
A
1s
(r)+CBψ
B
1s
(r)
where ψ
A
1s
(r)=ψ1s(r−RA) is a 1s hydrogen orbital centered on proton A and ψ
B
1s
(r)=ψ1s(r−RB) is a 1s hydrogen orbital centered on proton B. Recall ψ1s(r)=ψ100(r,ϕ,θ). The positions RA and RB are given simply by the vectors
RA=(0,0,R/2)RB=(0,0,−R/2)
The explicit forms of ψ
A
1s
(r) and ψ
B
1s
(r) are
ψ
A
1s
(r) =
1
(πa
3
0
)1/2
e−|r−RA|/a0 ψ
B
1s
(r) =
1
(πa
3
0
)1/2
e−|r−RB|/a0
Now, unlike the HF approach, in which we try to optimize the shape of the orbitals themselves, in the LCAO approach, the shape of the ψ1s orbital is already given. What we try to optimize here are the coefficients CA and CB that determine the amplitude for the electron to be found on proton A or proton B.
Explanation:
