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Homework 10

  • Page ID
    92328
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    Name: ______________________________

    Section: _____________________________

    Student ID#:__________________________

    Q10.1 (Ignore if from Koski's Class)

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

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

    Q10.2 (Ignore if from Koski's Class)

    Give the electronic configurations and term symbols of the first excited electronic states of the atoms up to Z=10.

    Q10.3 (Ignore if from Koski's Class)

    Find the spectroscopic term originating from the ground states configuration of the sodium atom and the boron atom.

    Q10.4 (Ignore if from Koski's Class, delayed for Larsen class)

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

    Q10.5: Ab initio Calculation

    Consider the ionization of the beryllium atom.

    1. Write the "chemical reaction" associated used to calculate this property.
    2. Identify the (ground-state) term symbol for each atom and ion species in your "chemical reaction".
    3. Use ChemCompute to calculate the ionization energy using the above information at the HP level with the 6-311G** basis set.
    4. From the results of just the neutral calculation, identify the first ionization from Koopman's Theorem.
    5. How accurate is Koopman's theorem at this level of calculation for determining the ionization energy of the beryllium atom (relative difference of two calculations)? What is responsible for the accuracy of Koopman's theorem?

    Q10.6

    For the wavefunction

    \[ \psi = \begin{vmatrix} \psi_A(1) & \psi_A(2) & \psi_A(3) \\ \psi_B(1) & \psi_B(2) & \psi_B(3) \\ \psi_C(1) & \psi_C(2) & \psi_C(3) \end{vmatrix}.\]

    show that

    1. the interchange of two columns changes the sign of the wavefunction,
    2. the interchange of two rows changes the sign of the wavefunction, and
    3. the three electrons cannot have the same spin orbital.

    Q10.7

    Diatomic hydrogen in valence bond theory is described by four wavefunctions as given below. Write these wavefunctions as Slater Determinants or linear combinations of Slater Determinants.

    \[\psi_1 = \frac{1}{2}[\phi_1(1)\phi_2(2) + \phi_1(2)\phi_2(1)][\alpha(1)\beta(2)-\beta(1)\alpha(2)]\]

    \[\psi_2 = \frac{1}{\sqrt{2}}[\phi_1(1)\phi_2(2) - \phi_1(2)\phi_2(1)][\alpha(1)\alpha(2)] \]

    \[\psi_3 = \frac{1}{\sqrt{2}}[\phi_1(1)\phi_2(2) - \phi_1(2)\phi_2(1)][\beta(1)\beta(2)]\]

    \[\psi_4 = \frac{1}{2}[\phi_1(1)\phi_2(2) - \phi_1(2)\phi_2(1)][\alpha(1)\beta(2)+\beta(1)\alpha(2)] \]

    Q10.8 (delayed for Larsen class)

    Write the Hamiltonian for the \(H_2\) and \(H_2^+\) molecules and explain each term. What terms differ between these two systems?

    Q10.9 (delayed for Larsen class)

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

    Q10.10 (delayed for Larsen class)

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


    Homework 10 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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