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Homework Help: Determinantal wave function of Li using LCAO?

  1. Dec 14, 2015 #1
    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. jcsd
  3. Dec 14, 2015 #2


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    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.
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