Final Exam of Physical Chemistry 2004

(120 minutes)

*department*： *register number*：

*name*： *score*：

**No.** | **1** | **2** | **3** | **4** | **5** | **6** | **7** | **8** |

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**No.** | **9** | **10** | **11** | **12** | **13** | **14** | **15** | **16** |

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*This is a closed book exam. Use of a calculator and an English dictionary is permitted. Show all of your work and check your units carefully. ***Don’t give help to, or get help from, others**. Thanks for your cooperation.

*Some useful constants and results are:*

(atomic unit)

atomic mass: N-14

1. For a nitrogen moving at a speed of 333 , determine the wavenumber (in ) corresponding to the deBroglie wavelength. (4 points)

2. Write out the complete Hamiltonian operator for atomic Be in atomic unit. If a trial wavefunction for Be is to be constructed using various hydrogen-like atomic wave functions , , , etc., write the Slater determinant for the ground state of Be. (5 points)

3. A trial wavefunction is obtained by the linear combination of and

where and . Determine the normalization constant . (5 points)

4. Using the Bohr theory for the atomic hydrogen, calculate (in eV) for an electronic transition between and . (4 points)

5. Name the three principles/rules that are used when writing the ground state atomic electron configurations of polyelectronic atoms. (6 points)

6. In class we studied the quantum mechanical solution to H_{2} and H_{2}^{+} problems. For the H_{2}^{-} molecule ion

1) Write the complete hamiltonian operator for the internal energy of this system in atomic unit before and after B.O. approximation, respectively. (4 points)

2) Write the molecular orbital trial wavefunction using hydrogen atomic wavefunction. (2 points)

3) Predict the bond order and magnetic properties. (4 points)

4) Will H_{2}^{-} be more or will be less stable than H_{2}? Explain your conclusion by molecular orbital theory. (4 points)

5) Will the bond length in H_{2}^{-} be longer or will be shorter than H_{2}? Why? (3 points)

7. Consider the configuration of two non-equivalent -electrons.

a) What are the possible terms of this configuration? (6 points)

b) Sort these terms by increasing energy. (2 points)

c) What is the simplest ground-state atom X that has two equivalent -electrons? Describe the magnetic properties of the diatomic molecule X_{2}. (3 points)

8. Consider a hypothetical system consisting of energy states defined by the quantum numbers and .

(a) The system consists of the three energy states designated by the quantum number , 2, and 3.where the energy of each of these state is given by . Construct an energy diagram (to scale) showing these states. (2 points)

(b) In addition to the energy states, there are additional energy states designated by the quantum number , 1, 2, for each state and the energy of each of these states is given by . Complete the energy diagram (to scale) showing these states. (3 points)

(c) If the selection rule for transitions are and , draw vertical lines between the energy states in your diagram illustrating the permitted transitions. How many lines will be observed? (5 points)

9. Silicon has a face-centered cubic structure, just like diamond. Deslattes et al [ *Phys. Rev. Lett.*] found the following values for a single crystal of very pure silicon at 25°C: the length of the side of the unit cell is 543.1066pm, and the density is 2.328992g cm^{-3}. The atomic mass is 28.08541g mol^{-1}. What is Avogadro’s constant obtained from these values? (4 points)

10. Calculate the ratio of the radii of small and large spheres for which the small spheres will just fit into octahedral site in a close-packed structure of the large spheres.(4 points)

11. Please tell the relative band positions of the C-O stretching vibration for FeCO, FeCO^{+} and FeCO^{-}, and briefly discuss reasons for your judgment. (6 points)

12. 1) Work out the p molecular orbitals of cyclopropene cation using projection operators (6 points);

2) Show orbital energies and delocalization energy of the ground state cyclopropene cation with the HMO approximate hypothesis (6 points);

3) Dose the spectral possibility from the ground state to the first excited state equal to zero? (6 points)

13. Determine the point group of each of the following molecules.(8 points)

14. Prove the theorem that “The product of two reflections, intersecting at an angle f= 2p/2n, is a rotation by 2f about the axis defined by the line of intersection”.(8 points)

15. Write down the d^{5} and d^{7} with high and low spin configurations in tetrahedral ligand field; work out which configurations show Jahn-Teller distortion.(6 points)

16. Taking the four orbitals perpendicular to the molecular plane of trans-planar butadiene as bases.

1) Write out the characters of the representation; (4 points)

2) Determine whether it is an irreducible or reducible representations, if it is a reducible representation, reduce it to irreducible representations; (4 points).