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Chemistry LibreTexts

Homework 9

  • Page ID
    109927
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    Name: ______________________________

    Section: _____________________________

    Student ID#:__________________________

    Q9.1

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

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

    Q9.2

    Give the electronic configurations and term symbols of the first excited electronic states of the atoms up to Z=15.

    Q9.3

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

    Q9.4

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

    Q9.5: 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?

    Q9.6

    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.

    Q9.7

    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.

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

    Q9.8

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

    Q9.9

    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.

    Q9.10

    \(\psi^+\) 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?

    Q9.11 (you can do it in 1 dimensional)

    Using your favorite plotting software, plot the following:

    • 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 Lecture 26 for the expressions for the integrals as a function of internuclear distance)

    Q9.12 (not required - optional)

    Derive the overlap integral between an \(H_{2s}\) orbital and a \(H_{2s}\) orbital on hydrogen nuclei separated by a distance \(R\). Plot this function and find the separation for which the overlap is a maximum.

    Q9.13

    Contrast the bond orders for \(H_2^+\) and \(H_2\). The bond dissociation energy of \(H_2\) is 436 kJ/mol, which is less than twice that of \(H_2^+\); Why?

    Q9.14 (Optional)

    If a electron-sensitive probe of volume 2.00 pm3 is inserted into an H2+ molecule-ion in its ground state, what is the probability of finding the electron:

    1. at nucleus A
    2. at nucleus B
    3. halfway between A and B
    4. at a point 40 pm along the bond from A and 15 pm perpendicularly off the bond

    Do (a)-(d) again after the electron has been excited into the antibonding LCAO-MO.

    Consider R = 106 pm.

    Q9.15

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

    • B2
    • S8
    • O2
    • N2
    • 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?

    Q9.16

    Which of the three species is the is the least stable due to bond order: O2, O2+, O22-. Hint it may be helpful to draw molecular orbitals for each species although it is not required.

    Q9.17

    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:

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


    Homework 9 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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