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Constructing wave function for Beryllium (Z=4)

  1. Feb 1, 2012 #1
    1. The problem statement, all variables and given/known data

    Using an independent particle approximation to construct the wave functions of the ground and the first singlet and triplet excited states of the Beryllium atom in terms of the hydrogen-like atomic orbitals, which satisfy the Pauli Exclusion Principle.

    2. Relevant equations

    n/a

    3. The attempt at a solution

    Ok so it's been a while since I've done any molecular physics so I'm getting a bit stumped at the basics and need a bit of pointing in the right direction.

    So the ground state will be

    1s2 2s2 right?
    With spins like this:

    2s [itex]\uparrow[/itex] [itex]\downarrow[/itex]
    1s [itex]\uparrow[/itex] [itex]\downarrow[/itex]

    1s2 2s 2p would be the first excited state?

    2p [itex]\uparrow[/itex] or [itex]\downarrow[/itex]
    2s [itex]\uparrow[/itex] or [itex]\downarrow[/itex]
    1s [itex]\uparrow[/itex] [itex]\downarrow[/itex]

    Is this the triplet state? ie you can have 3 different orientations of the spins, both up, both down, or a mix as there are 2 unpaired electrons?

    Would the singlet state be 1s2 2p2? As the electrons are paired?

    Ok so from there I have to construct the wave functions, is this a case of summing the products of spatial and spin functions or is it simpler than that?

    I get that for a two electron system the spin functions are:

    [itex]\frac{1}{\sqrt{2}}[/itex][[itex]\alpha[/itex](1)[itex]\beta[/itex](2)-[itex]\alpha[/itex](1)[itex]\beta[/itex](2)] - singlet

    [itex]\alpha[/itex](1)[itex]\alpha[/itex](2) - triplet

    [itex]\frac{1}{\sqrt{2}}[/itex][[itex]\alpha[/itex](1)[itex]\beta[/itex](2)+[itex]\beta[/itex](1)[itex]\alpha[/itex](2)] -triplet

    [itex]\beta[/itex](1)[itex]\beta[/itex](2) - triplet

    How do you extend this to a 4 electron system, and will all spin functions be relevant?

    Obviously you'd have

    [itex]\alpha[/itex](1)[itex]\alpha[/itex](2)[itex]\alpha[/itex](3)[itex]\alpha[/itex](4)
    [itex]\beta[/itex](1)[itex]\beta[/itex](2)[itex]\beta[/itex](3)[itex]\beta[/itex](4)

    Im guessing the rest would be combinations of α & β, are there any specifics I should watch out for?

    Then spatial wavefunctions would just be of the form:

    u[itex]_{nlm}[/itex](r) = R[itex]_{nl}[/itex](r)Y[itex]_{lm}[/itex]([itex]\vartheta[/itex],[itex]\varphi[/itex])

    And I just pair the relevant spin functions with their spatial functions to get the wavefunctions I want.

    Any help would be appreciated.
     
  2. jcsd
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