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Homework 1

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    Write out the complete time-independent Hamiltonians for (each term is explicitly given):

    1. the helium atom
    2. the \(H_2^+\) ion
    3. the \(H_2\) molecule

    You may want to refresh yourself on the basics of Chem 110A material:


    What is the Born-Oppenheimer approximation and what do we use it? When would it fail?


    Confirm that the two sp-orbitals are normalized and orthonormal.


    Show that the sp2 hybrid orbital

    \[ | sp^2 \rangle = \dfrac{|s \rangle + \sqrt{2}|p \rangle }{\sqrt{3}}\]

    is normalized if the \(|s \rangle\) and \(|p_z \rangle\) atomic orbitals are also normalized.


    What is the average energy of a H atom \(|sp^3 \rangle\) hybrid orbital is the energy of the \(|s \rangle\) orbital is \(E_s\) and the energy of the \(|p \rangle\) orbital is \(E_p\)? Hint: this will require solving the variational energy:

    \[Energy = \dfrac{\langle sp^2 | \hat{H} | sp^2 \rangle }{ \langle sp^2 | sp^2 \rangle }\]


    When one s and two p atomic orbitals are used to generate hybrid orbitals, how many hybrid orbitals will be generated?

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