Homework 9

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q1

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. \(1s(1)2s(2)\)

 

Q2

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

Q3

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

• 1s¹2s¹
• 1s¹3s¹
• 2p¹3p¹
• 2s¹3d¹

Q4

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

Q5

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

 

Q6

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.

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

Q7

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.

Q8

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, 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 (R_e=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.)

Q10

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

• Be₂
• 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?

Q11

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:

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