Quantum Chemistry

Information on the course

TBA

Exam questions

13th January 2025

Question 1 (6 points)

Given: a spectrum with 2 peaks (and their given energies in cm^(-1)(16978.0cm^-1 , 16960.9cm^-1) for Sodium (Z=11) with the caption: This is the spectrum for the first excitation of the sodium atom.

1. Obtain all atomic terms and levels of both the ground state and the first excited state of the sodium atom.
2. Use the selection rules to determine which transitions are allowed.
3. Use the spectra to estimate the value of the spin-orbit coupling constant ζ for the 3p-orbital of sodium.

Question 2 (8 points)

1. Show that the 2px and the 2pz functions of hydrogenlike atoms are orthogonal. (Given: real wave function of the 2px and 2pz hydrogenlike orbitals)
2. What would a trial function of the form "exp(-kr)" look like? (Given: Hamiltonian of H atom, Laplacian in spherical coordinates, use variational perturbation theory to obtain k, it should probably be like 1s of hydrogen))

Question 3 (6 points)

1. Given graph of radial wavefunction of hydrogen like atom with 3 nodes. Give the corresponding orbital.
2. Question about Slater determinants. Given Ф1 = /abcd>, Ф2 = /cdrs>. Evaluate <Ф1/F/Ф2> with F = 1-electron operator.

3th September 2024

1 (10 points)

• Obtain all the possible atomic terms and levels of a d^9 electronic configuration.
• How do the atomic levels of the ground state atomic term of the d^9 configuration split under the influence of SOCs? Order energetically the atomic levels and calculate the energy splittings. Leave the results in terms of the one electron SOC constante.

2 (4 points)

• For the ground state of the hydrogenlike atom, show <r> = 3a/2Z.

3 (6 points)

• Total wave function 3dz² van hydrogen atom given. Obtain the angular nodes.
• Graph of hydrogenlike atom radial wave function. Identify to which orbital it corresponds. X-as is based on R.

15th January 2024

Question 1 (6 points)

Given: a spectrum with 2 peaks (and their given energies in cm^(-1)(16978.0cm^-1 , 16960.9cm^-1) for Sodium (Z=11) with the caption: This is the spectrum for the first excitation of the sodium atom.

1. Obtain all atomic terms and levels of both the ground state and the first excited state of the sodium atom.
2. Use the selection rules to determine which transitions are allowed.
3. Use the spectra to estimate the value of the spin-orbit coupling constant ζ for the 3p-orbital of sodium.

Question 2 (8 points)

1. Show that the 2px and the 2pz functions of hydrogenlike atoms are orthogonal. (Given: real wave function of the 2px and 2pz hydrogenlike orbitals)
2. What would a trial function of the form "exp(-kr)" look like? (Given: Hamiltonian of H atom, Laplacian in spherical coordinates, use variational perturbation theory to obtain k, it should probably be like 1s of hydrogen))

Question 3 (6 points)

By using Perturbation Theory, obtain the second-order energy correction due to the interaction of the 1D2 and 3P2 states of the oxygen atom.

16th January 2023

Question 1 (6 points)

Given: a spectrum with 2 peaks (and their given energies in cm^(-1) for Sodium (Z=11) with the caption: This is the spectrum for the first excitation of the sodium atom.

1. Obtain all atomic terms and levels of both the ground state and the first excited state of the sodium atom.
2. Use the selection rules to determine which transitions are allowed.
3. Use the spectra to estimate the value of the spin-orbit coupling constant ζ for the 3p-orbital of sodium.

Question 2 (8 points)

1. Show that the 2px and the 2pz functions of hydrogenlike atoms are orthogonal. (Given: real wave function of the 2px and 2pz hydrogenlike orbitals)
2. What would a trial function of the form "exp(-kr)" look like?

Question 3 (6 points)

By using Perturbation Theory, obtain the second-order energy correction due to the interaction of the 1D2 and 3P2 states of the oxygen atom.

Monday 13th January 2020 (8:30 am)

The exam is open book, you can have your course notes, your exercises and the book "Quantum Chemistry" by Levine with you. You get 4 hours to make the exam. You write everything down and then you go to the professor for oral examination.

Question 1 (8 of 20 points)

1. Obtain all atomic terms and levels of a d9 configuration.
2. Rank them in order of energy and give their relative energies in terms of the one-electron spin-orbit coupling constants.

Question 2 (4 of 20 points)

1. Show that the 2px and the 2pz functions of hydrogenlike atoms are orthogonal. (Given: real wave function of the 2px and 2pz hydrogenlike orbitals)
2. For the ground state of a hydrogenlike atom, show that <R> = 3a/2Z. (Given: value of some difficult integral that you will need to calculate <R>)

Question 3 (4 of 20 points)

Given: all atomic states of the ground state configuration of a carbon atom and their energies, and all atomic states of the 1s22s2p3 configuration and their relative energies.

1. Comment on the order of the atomic states. (He expects that you talk about Hund’s rules, when are these valid, when not, talk about order of atomic states with different J values but from the same term…)
2. From the given energy values, estimate the value for ζ2p.
3. What spectral transitions are allowed between the ground state and the given excited state? Give the energies of these transitions.

Question 4 (4 of 20 points)

By using second-order perturbation theory, obtain the second-order energy correction due to interaction of the 1S0 and 3P0 states of a p2 configuration. (You should find the wave functions for both states in form of Slater determinants and calculate the matrix elements of ĤSO using the Condon-Slater rules)