CHE 110B
Final Exam – Due 9 December 2022
2022F
‖ Chemistry 110B final exam
This exam covers the entirety of CHE 110B with extra weight on NMR, gases, and statistical thermody- namics. It is due on Canvas by 11:59 pm on Friday, December 9; late submissions within an hour of the deadline will forfeit extra credit from homework score, and submissions afterwards will receive a 0. All times are Pacific. Here are the ground rules (see the FAQs on Canvas for a more complete list):
No collaboration The exam is for you to work on individually. Communicating with other people about the exam questions, whether or not they are in the class, is not allowed. Using any online forums or Q&A services is expressly prohibited, and posting exam content to the internet or viewing answers submitted in response to exam material uploaded online will result in an SJA referral.
Online resources You may use any resources that I have posted to Canvas for the exam, along with the course textbook and your own notes. I strongly discourage searching the internet for general information, as this could potentially run afoul of the “no collaboration” policy if you come across forum posts or similar that pertain to the exam material. However, Google etc. are not prohibited. You may also use online resources I have introduced to you, such as the Basis Set Exchange, the Otterbein Symmetry Gallery, online character tables, and the NIST Chemistry Webbook.
Questions If you need to ask a question about the exam, you may email me (my email address is at the bottom of the page) or send me a direct message on Discord. I may post clarifications on Discord as well, so please keep an eye out.
To avoid potentially being late due to network issues, do not wait until the last minute to upload your exam! I strongly recommend planning to finish the exam early enough that you have time to go somewhere with reliable internet should that become necessary. We will not have class during the exam time, but I will be available in my office if you have questions. Good luck!
1) This question refers to the paper “Spectroscopic detection of the stannylidene (H2 C −Sn and D2 C −Sn) molecule in the gas phase” (Smith et al. 2022, J. Chem. Phys. 157, 204306). A copy of the paper is available on Canvas. (50 pts)
1.1) Write the full Hamiltonian, the Born-Oppenheimer electronic Hamiltonian, and the single-electron Hamiltonian for H2 C −Sn. You may use summation notation for the first two, but you must write out all terms in the single-electron Hamiltonian explicitly. (10 pts)
1.2) Figure 1 shows three singlet electronic states of H2C−−Sn. Using the information available in the figure, prove that the symmetries of the X˜, A˜, and B˜ states listed in the paper are correctly assigned. (5 pts)
1.3) Figure 1 also shows three possible electronic transitions: the B˜ − X˜ and B˜ − A˜ transitions are allowed, while the A˜ − X˜ transition is forbidden. Explain what is meant by “allowed” and “forbidden,” and prove that each respective transition is correctly assigned as allowed or forbid-den. (5 pts)
1.4) Figure 2 shows simulated vibronic spectra for H2 C −Sn in absorption and emission. Draw an energy level diagram that illustrates the labeled transitions in the spectra. Ensure that your energy levels and transitions are appropriately labeled, and indicate the transitions with the same colors as used in the figure. (20 pts)
1.5) The paper discusses both H2 C −Sn and D2 C −Sn. At a temperature of 300 K, which of these two molecules has a greater rotational partition function (or are they the same), and why? (5 pts)
1.6) Determine the rotational contribution to the isochoric heat capacity for both H2C −Sn and D2C −Sn, assuming that each is an ideal gas:
Cv,rot = (∂T/∂⟨
∂T
Erot⟩)N,V , ⟨Erot〉(∂β/∂ ln
∂β
Qrot)N,V .
Which one is greater (or are they the same), and why? (5 pts)
2) NMR spectroscopy. (30 pts)
2.1) Deuterium is a spin-1 nucleus. Consider a system with 2 deuterium nuclei that are chemically inequivalent. The nuclear spin Hamiltonian (neglecting spin-spin coupling and quadupolar inter-actions) is:
Hˆ = −γB(1 − σ1)
ˆIz,1 − γB(1 − σ2)
ˆIz,2
Draw and label an energy level diagram showing the energies of the |m1, m2⟩ states and allowed transitions for this system in a magnetic field. Let σ1 > σ2, and use different colors to indicate transitions for each nucleus. (10 pts)
2.2) Calculate the first-order correction to the energies of the |m1, m2⟩ states due to dipolar spin-spin coupling using perturbation theory. The spin-spin interaction is represented by:
Vˆ = ℏ 2/hJ12 ˆI1 ·
ˆI2
Describe the splitting pattern that will be observed in the NMR spectrum (frequency and relative intensity). (5 pts)
2.3) When working with spin >1/2 nuclei, it is often convenient to use raising and lowering operators ˆI
+ and ˆI
−:
ˆI + = ˆIx + i
ˆIy,
ˆI
− = ˆIx − i
ˆIy
When applied to a nuclear spin wavefunction |I, mI ⟩, these operators raise (ˆI
+) or lower (ˆI
−) the value of mI by 1:
I
ˆ± |I, mI ⟩ = ℏ
p I(I + 1) − mI (mI ± 1)|I, mI ± 1⟩
Calculate the value of the ⟨1, −1| Vˆ |0, 0⟩ matrix element for the coupled deuterium atoms. Note that I = 1 for both deuterium nuclei, and the states are written as |m1, m2⟩ for simplicity. Hint: after expanding ˆI1·
ˆI2 into x, y, and z, components, rewrite in terms of ˆI
+
1
,
ˆI
−
1
,
ˆI
+
2
, and ˆI
−
2
. (10 pts)
2.4) When two nuclei are equivalent, the nuclear spin wavefunction must transform. as an irreducible representation of the S2 permutation group. Determine the normalized symmetry-adapted linear combinations of nuclear spin wavefunctions |m1, m2⟩ for two equivalent deuterium nuclei. (5 pts)
S2 E (12)
A 1 1
B 1 −1
3) Short answer (5 pts each)
3.1) In the paper “Virial equation of state as a new frontier for computational chemistry” (Schultz & Kofke 2022, J. Chem. Phys. 157, 190901), they write the virial equation of state as
kBT/p = ρ + B2ρ
2 + B3ρ
3 + · · · + Bnρ
n
where p is pressure, ρ is number density, and Bn is nth virial coefficient. Show that this is equivalent to the virial equation of state introduced in class, and say how B2 is related to B2v. A copy of the paper is available on Canvas, but you do not need the paper to answer this question.
3.2) Explain why NMR spectra are conventionally plotted against chemical shift, not frequency.
3.3) Molecule A has a lower B2v than molecule B at room temperature. In the van der Waals equation of state, which molecule do you expect to have a greater value of a, and why?
3.4) Some basis sets use an “effective core potential” in which some of the core electrons are treated implicitly to generate a potential that shields the outer electrons from the nuclear charge. For the I2 molecule in the def2-TZVP basis set, determine how many orbitals are calculated, and how many explicitly-treated electrons occupy those orbitals.