# Determinantal wave function of Li using LCAO?

1. Dec 14, 2015

### magnesium12

1. The problem statement, all variables and given/known data

The wave function for lithium can be written as:

| 1sα(1) 1sβ(1) 2sα(1) | $\frac{1}{\sqrt(3)}$ = ψ(1,2,3)
| 1sα(2) 1sβ(2) 2sα(2) |
| 1sα(1) 1sβ(2) 2sα(1) |

How can each row be a linear combination of atomic orbitals that makes a new orbital in which the electron actually exists if these new functions are made up of the old functions which all have different spin states?

Like in the first row, |1sα(1) 1sβ(1) 2sα(1)| represents a wave function made up of 3 other wave functions. But two of these wave functions are functions of α (positive spin) and one has β (negative spin) in it. So how can we combine all three orbitals to make a new orbital? You can only have +1/2 or -1/2 spin, nothing in between. So what would the spin state of the new orbital be?

Could someone clarify this for me? I'm really sorry if I didn't word the question clearly.

2. Relevant equations

3. The attempt at a solution

2. Dec 14, 2015

### Staff: Mentor

That last row should be 1sα(3) 1sβ(3) 2sα(3)

What you get from the Slater determinant is a 3-electron wave function. Using that particular set of atomic orbitals, you get a 3-electron orbital in which 2 electrons are spin-up and one spin-down.

The total spin of 3 electrons can be 1/2 or 3/2. In the case at hand, the total spin is S = 1/2, and MS = +1/2.