Homework 11A

 

 

Q10

If a electron-sensitive probe of volume 1.00 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.

Q11.1 (Ignore if from Koski's Class)

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

Q11.2 (Ignore if from Koski's 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.

Q11.3 (Ignore if from Koski's 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?

Q11.4 (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 (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 Lecture 26 for the expressions for the integrals as a function of internuclear distance)

Q11.5 (not required to do)

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.

 

Q11.6

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?

Q11.7

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.

Q11.8

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

• B₂
• O₂
• N₂
• N₂⁺
• H₂⁺
• \( (\sigma_{1s})^2 (\sigma^*_{1s})^2(\sigma_{2s})^2 (\sigma^*_{2s})^2(\pi_{2p})^2 \)
• \( (\sigma_{1s})^2 (\sigma^*_{1s})^2(\sigma_{2s})^1 (\sigma^*_{2s})^1 \)
• \( (\sigma_{1s})^2 (\sigma^*_{1s})^2(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2p_z})^2 (\pi_{2p})^4(\pi^*_{2p})^2 \) (this was changed since HW was assigned)

Which are paramagnetic?

Q11.9

Which of the three species is the is the least stable due to bond order: O₂, O₂⁺, O₂^2-. Hint it may be helpful to draw molecular orbitals for each species although it is not required.

 

Q11.10

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:

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

Q10 (From HW 9)

If a electron-sensitive probe of volume 1.00 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.