CH274 Electrons in molecules and solids
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ELECTRONS IN MOLECULES AND SOLIDS (CH274)
SECTION A
Answer EITHER question 1 OR question 2
1. Answer ALL parts
(a) Consider the 3 molecules, composed of 14 sp2 hybridised carbon atoms,
shown in Figure 1.
Figure 1
(i) Determine if the three molecules are alternant or non-alternant. Explain your rationale.
(ii) Why do all three molecules have exactly one non-bonded LUMO? (iii) Use an online Hückel calculator to determine the total energy and the
HOMO and LUMO energies of the three molecules. Report the total energies and rank the molecules from most stable to least stable. Provide a rationale of the stability ranking on the basis of the frontier orbital energies and the structures.
[25%]
(b) Consider the three molecules shown in Figure 1.
(i) Visualize the LUMO of each molecule by drawing shaded/unshaded circles on each carbon (depicting the pz orbital from a top view). The size of the circle is to indicate the magnitude of the wavefunction coefficient, the shading is to indicate the sign.
(ii) Explain why the three non-bonded LUMOs cannot have any non-zero
wavefunction coefficients at the central carbon atoms.
(iii) Using an online Hückel calculator, calculate the net charge populations to
identify which carbon atoms are the most likely site of a
nucleophilic/electrophilic attack. Which molecule is the most reactive? [25%]
(c) Consider a sample of a material with an electronic density of states (DOS) as depicted in Figure 2a. The sample is illuminated at room temperature by a beam of light with a uniform spectral density between 入1 = 400 nm and 入2 =
750 nm and zero elsewhere (see Figure 2b).
Figure 2
What would the maximum wavelength of the absorbed light be if the material was:
(i) a metal;
(ii) an intrinsic semiconductor?
(iii) Justify your answers.
[15%]
(d) A semiconductor material with the DOS given in Figure 2a has been p-doped. What would the value of the Fermi energy be, at T1 = 10 K and at T2 = 1000 K, assuming that these two temperatures lie in the extrinsic and intrinsic temperature regime, respectively? Justify your answers and give them with a precision of 0.5 eV.
[10%]
(e) The resistance of a sample made of an intrinsic semiconductor has been
measured at different temperatures, and the data were collected in Table 1.
Table 1
T / C |
10 |
30 |
50 |
100 |
120 |
150 |
R / |
9.12107 |
1.92107 |
4.91106 |
3.08105 |
1.24105 |
3.71104 |
Determine the band gap of the semiconductor.
[25%]
2. Answer ALL parts
Figure 3
(a) Figure 3 shows a two-dimensional infinite lattice of hydrogen atoms. The
system is described with a unit cell of length 2a in both directions that is twice as large as the primitive unit cell of length a. Each hydrogen atom contributes an S orbital and one electron to the system.
(i) Calculate the reciprocal lattice constant b and draw the reciprocal unit cell of this system. Identify the four high-symmetry points in the reciprocal unit cell:
Γ = (0,0),X = (b/2,0),X′ = (0, b/2),M = (b/2, b/2) (ii) Answer following questions:
1. How many atomic orbitals are present in the unit cell?
2. How many molecular orbitals can they form?
3. How many bands will be formed in the band diagram?
4. How many electronic states does the lattice have?
(iii) Visualize the four different molecular orbitals that can be formed within the
unit cell. Indicate sign changes of the wavefunction coefficient by shading of the s orbitals. Label the molecular orbitals as bonding, non-bonding, or antibonding (G,n, G ∗ ) and rank them by energy.
(iv) Visualize the following orbitals (with neighbouring unit cells in x and y)
1. G orbital at Γ, X, and M points
2. G∗ orbital at Γ, X, and M points
(v) Draw a schematic band diagram E(k) for the G and the G∗ bands along the following path: Γ → X → M that also indicates the energy ranking of all orbitals at different k points.
(vi) How will this band diagram change if the hydrogen atoms within the unit
cell form two H2 molecules in x direction?
[50%]
(b) At room temperature, the density of occupied states at the top of the valence
band of an intrinsic semiconductor is equal to 5.3 1017 cm−3 eV−1 . In order to increase this value, should the temperature be increased or decreased? Justify your answer.
[10%]
(c) Figure 4 shows the DOS of a material.
Figure 4
Give the energy intervals where the Fermi energy can be located if the material is:
(i) a metal;
(ii) an intrinsic semiconductor;
(iii) an n-doped semiconductor.
Justify your answers and give them with a precision of 0.1 eV.
[10%]
(d) Assume that the material in (c) is a semiconductor with an electron mobility µe = 1,200 cm2 V−1 s−1 , a hole mobility µh = 600 cm2 V−1 s−1 and an effective density of states NC = 6.81018 cm−3 . A cylindrical resistor is fabricated from this material with a diameter of 5 mm and a length of 20 mm. The resistor is connected to a battery of 3.0 V. Determine the current flowing through the resistor at 300 K if the semiconductor is:
(i) intrinsic;
(ii) n-doped with a donor density nd,1 = 5.0×1012 cm−3 (assume 300 K is in
the saturation regime);
(iii) n-doped with a donor density nd,1 = 3.0×1010 cm−3 (assume 300 K is in
the saturation regime).
[30%]
SECTION B
Answer EITHER question 3 OR question 4
3. Answer ALL parts
Part of the molecular orbital (MO) energy-level scheme (1) of pyrazine and its corresponding molecular structure (2), which has D2h symmetry (for character table,
see appendix), are shown below
1 2
(a) The first two lowest energy, spin allowed, transitions in pyrazine involve: (i)
promotion of an electron from the highest occupied molecular orbital (HOMO, b1u symmetry) to the lowest unoccupied molecular orbital (LUMO, b3u symmetry); and (ii) promotion of an electron from HOMO-1 (ag symmetry) to LUMO. Determine the term symbols for states (i) and (ii). Show all workings including appropriate energy level diagrams for the two excitations.
[30%]
(b) Evaluate the symmetries of the transition moment integrals for the spin allowed
transitions in (i) and (ii) (see point (a) above) and explain why one transition is orbitally allowed and one is orbitally forbidden.
[20%]
(c) What are the term symbols for the following diatomic molecules?
(i) C2+ : … (2Gg)2(2Gu)2(1u)3 (ii) O2+ : … (3Gg)2(1u)4(1g)1 (iii) BeF: … (4G)2(1)4(5G)1
[15%]
(d) The ground electronic state (X 1Σ ) of the HD molecule has a bond length of 74.14 pm. Two spectroscopically accessible electronically-excited states are the B 1Σu(+) state (129.0 pm), and the C 1 Πu state (103.3 pm), where the numbers in parentheses are the corresponding bond lengths.
Which of the two electronic transitions, B 1Σu(+) ← X 1Σ or C 1 Πu ← X 1Σ , would you expect to demonstrate a shorter v′′=0 vibronic progression (assume that the dissociation limit of the upper state is not reached in either case)? Explain your reasoning.
[15%]
(e) A molecule in its S1 electronic state decays via two different relaxation
pathways: fluorescence back down to the ground electronic state (with associated rate coefficient kF), and unimolecular dissociation (with rate coefficient kdiss). Data from an experiment monitoring the fluorescence intensity of the molecule as a function of time is plotted in figure 5 below.
Figure 5
(i) Use this plot to estimate the observed lifetime, T0, of the S1 state.
(ii) A second experiment reveals that the fluorescence quantum yield is 0 =
0.44. Using your answer to part (i), determine the values of kF and kdiss for this molecule.
[20%]
4. Answer ALL parts
The molecular structure of pyrazine (3), which has D2h symmetry, is shown below:
3
(a) Using the D2h character table provided (see appendix), determine the
symmetries of all the vibrational modes of pyrazine (3). Show all relevant working.
[40%]
(b) One of the vibrations determined in (a) is a C– H stretch with b1u symmetry.
Draw an appropriate vector displacement diagram for this vibration and provide reasoning for your choice, with reference to the appropriate characters of the b1u irreducible representation.
[10%]
(c) What are the term symbols for the following diatomic molecules?
(i) C2 : … (2g)2(2u)2(1u)4 (ii) O2– : … (3g)2(1u)4(1g)3 (iii) NO: … (5)2(1)4(2)1
[15%]
(d) The ground electronic state (X 1Σ ) of the HD molecule has a bond length of 74.14 pm. Two spectroscopically accessible electronically-excited states are the B 1Σu(+) state (129.0 pm), and the C 1 Πu state (103.3 pm), where the numbers in parentheses are the corresponding bond lengths.
Which of the two electronic transitions, B 1Σu(+) ← X 1Σ or the C 1 Πu ← X 1Σ , would you expect to demonstrate a longer v′′=0 vibronic progression (assume that the dissociation limit of the upper state is not reached in either case)? Explain your reasoning.
[15%]
(e) Spectroscopic constants for the ground-electronic state of the I2 molecule
and its spectroscopically-accessible 3 Πu excited-state are given in Table 2. Use these constants to determine the wavenumber value of the v ′ = 5 ← v ′′ = 1 vibronic transition.
Table 2
el/cm– 1
15769.01
0
[20%]
2022-09-12