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Homework 9

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    Are the following two-electron wavefunctions symmetric, asymmetric or neither to electron permutation (note: spin and orbitals components are separated)

    1. \( [1s(1)2s(2) + 2s(1)1s(2)][\alpha(1) \beta(2) + \beta(1)\alpha (2)] \)
    2. \( [1s(1)2s(2) + 2s(1)1s(2)][\alpha(1) \beta(2) - \beta(1)\alpha (2)] \)
    3. \( [1s(1)2s(2) - 2s(1)1s(2)][\alpha(1) \beta(2) + \beta(1)\alpha (2)] \)
    4. \( [1s(1)2s(2) - 2s(1)1s(2)][\alpha(1) \beta(2) - \beta(1)\alpha (2)] \)
    5. \( [1s(1)2s(2) - 2s(1)1s(2)][\alpha(1) \alpha (2)] \)
    6. \(1s(1)2s(2)\)


    Construct the Slater determinant corresponding to the configuration for ground-state configuration of a Be atom.


    Find the spectroscopic terms originating from the following configurations (two electrons in two non-equivalent orbitals in the same atom):

    • 1s12s1
    • 1s13s1
    • 2p13p1
    • 2s13d1


    Give the electronic configurations and term symbols of the first excited electronic states of the atoms up to Z=13.


    Find the spin-orbit energy levels of the hydrogen atom with an electron in 2p and 3d orbitals.



    In valence bond theory, H2 is described by four wavefunctions as given below. Write these wavefunctions as Slater Determinants or linear combinations of Slater Determinants.

    \[\begin{align} \psi_1 &= \dfrac{1}{\sqrt{2}}[\phi_1(1)\phi_2(2) + \phi_1(2)\phi_2(1)][\alpha(1)\beta(2)-\beta(1)\alpha(2)] \\[5pt] \psi_2 &= \dfrac{1}{\sqrt{2}}[\phi_1(1)\phi_2(2) - \phi_1(2)\phi_2(1)][\alpha(1)\alpha(2)] \\[5pt] \psi_3 &= \dfrac{1}{\sqrt{2}}[\phi_1(1)\phi_2(2) - \phi_1(2)\phi_2(1)][\beta(1)\beta(2)] \\[5pt] \psi_4 &= \dfrac{1}{\sqrt{2}}[\phi_1(1)\phi_2(2) - \phi_1(2)\phi_2(1)][\alpha(1)\beta(2)+\beta(1)\alpha(2)] \end{align}\]


    Use normalization to identify the constants (\(c_1,c_2)\) for the LCAO approximation for the two molecular orbitals below \[\psi^+ = c_1\phi_A + c_2\phi_B\] and \[\psi^- = c_1\phi_A - c_2\phi_B\] with \(\phi\) is the 1s atomic orbital on the \(A\) and \(B\) nuclei, respectively.


    a. \(\psi^+\) and \(\psi^-\) are called the bonding and antibonding molecular orbitals. Which is higher in energy based on the nature of the electron density distribution?

    b. Which molecular orbital has a node and where?


    Q9 (you can do it in 1-dimension)

    Using your favorite plotting software (avoid Desmos and Excel), plot the following list. Don't forget to label your axis.

    • Plot the normalized wavefunction for the bonding and antibonding orbitals for the H2+ as a function of internuclear distance, R (Re=106 pm).
    • Plot the overlap integral, S as a function of R/a0.
    • Plot the Coulomb integral, J as a function of R/a0.
    • Plot the Exchange integral, K as a function of R/a0.
    • Plot the energies of the H2+ molecule as a function of internuclear distance, R.

    What is the most stable separation between these two atoms? (Look at Lectures and Readings for the expressions for the integrals as a function of internuclear distance.)


    Calculate the bond orders and spin multiplicity of the following molecules:

    • Be2
    • Se8
    • O2
    • N2
    • H2+
    • \( (\sigma_{1s})^2 (\sigma^*_{1s})^2(\sigma_{2s})^2 (\sigma^*_{2s})^2(\pi_{2p})^2 \)
    • \( (\sigma_{1s})^2 (\sigma^*_{1s})^2(\sigma_{2s})^1 (\sigma^*_{2s})^1 \)
    • \( (\sigma_{1s})^2 (\sigma^*_{1s})^2(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2p_z})^2 (\pi_{2p})^4(\pi^*_{2p})^2 \)

    Which are paramagnetic?


    Draw an electronic energy level arrow diagram (i.e., the electronic configuration with levels and spins either up or down per electron) for this secular determinant:

    \[\Psi(1,2,3,4,5) =\begin{vmatrix} 1s \alpha(1) & 1s \beta(1) & 2s \alpha(1) & 2s \beta(1) & 2p_x \beta(1) \\ 1s \alpha(2) & 1s \beta(2) & 2s \alpha(2) & 2s \beta(2) & 2p_x \beta(2) \\ 1s \alpha(3) & 1s \beta(3) & 2s \alpha(3) & 2s \beta(3) & 2p_x \beta(3) \\ 1s \alpha(4) & 1s \beta(4) & 2s \alpha(4) & 2s \beta(4) & 2p_x \beta(4) \\ 1s \alpha(5) & 1s \beta(5) & 2s \alpha(5) & 2s \beta(5) & 2p_x \beta(5) \end{vmatrix} \]