Final Exam of Physical Chemistry June, 2003

(120 minutes)

*department*： *register number*：

*name*： *score*：

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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**. Some useful constants and results are:

(atomic unit)

Relative atomic mass: Fe-56 C-12 Na-23 Cl-35.5

1. Please draw the following molecular orbitals, (4 points)

1) 2) 3) 4)

2. X-ray diffraction analysis of crystalline C_{60} (*buckminsterfullerene or buckyball*) shows that the crystal structure at 300K is face-centered cubic (fcc) with one C_{60} molecule located at each lattice point and the unit cell edge length .

1) Calculate the density of the solid. (3 points)

2) If the solid is formed by *cubic closed-packing* (ccp) of “spherical” C_{60} molecules, calculate the “spherical” radius of C_{60}. (3 points)

3) If solid C_{60} is doped with K (potassium) atoms, a new solid is obtained, which has a superconducting transition temperature of 19.3K. The potassium atoms in the new solid occupy all of the octahedral and tetrahedral holes in the unit cell. What is the empirical formula of the new solid?(2 points)

3. The Balmer series for H atom transitions is given by:

1)What is the longest wavelength transition in nm? (3 points)

2)What is the period of the (minimum energy) ionizing radiation in units of fs (1fs=10^{-15}s)? (4 points)

3)What is the speed of an ejected electron following absorption of a 351nm photon? ( 3 points)

4)What is the de Brogile wavelength of the ejected electron in nm (nanometer) units? (2 points)

4. Please discuss the interpretation of and with brief explanations.(4 points)

5. A conjugated molecule containing 18 electrons has an effective delocalization length of 2000pm (1pm=10^{-12}m). What is the wavelength (in nm) corresponding to the lowest energy absorption as predicted by the Particle – in a one dimension box Model”.(5 points)

6. A hydrogen atom is in an eigenstate described by the wavefunction (in atomic units)

1) What is the value of the quantum numbers , and for this state?(6 points)

2) How many nodes are in the radial direction and angular direction?(2 points)

3) Calculate the energy and angular momentum for this state (in atomic units), respectively. (4 points)

7. For NaCl crystal:

1) If a chlorine ion is in (0,0,0), please give the fractional coordinate of all sodium ions in a face-centered cubic unit cell. (4 points)

2) How many NaCl units in per unit cell? (2 points)

3) If the edge length of the unit cell is 5.629A, and the radius of Cl^{-} is 133pm, what is the radius of Na^{+}. (2 points)

4) Please show the distance between (100) adjacent planes. (2 points)

8. Assume the spin wavefunction of the atoms is .

1) What is the average value of ? (3 points)

2) What is the probability that an atom will be found in the spin up state? (2 points)

9. A diatomic molecule of reduced mass is adsorbed onto a solid surface, with the molecular axis parallel to the (xy) plane of the surface. The molecule can rotate, but neither atom can change its position in the (z) direction perpendicular to the surface.

1) Write the Hamiltonian for the rotational motion of the molecule. (2 points)

2) Simplify the expression if the inter-nuclear distance of the molecule is fixed. (2 points)

10. Given the trial function for the Variational Method:

1) How many energy eigenvalues would one expect from the trial function if a secular equation (containing secular determinant) is constructed from this. (*There is no need to write the secular equation*). ( 2 points)

2) If these wavefunctions, , , are orthonormal, prove the following two overlap integrals:

i) ; ii) (4 points)

11. Write down the term symbols for the following electronic configurations:

1) (2 )^{1}; 2) (3 )^{10}; 3) (1 )^{3}; 4)( )^{1} (4 points)

12. For lithium (Li) atom,

1) Write down the Hamiltonian in atomic unit; (2 points)

2) List the term(s) that preclude(s) us from solving the Schrodinger equation exactly; (2 points)

3) If one removes the term(s) from part 2). Please briefly discuss (2-4 sentences & equations) how to solve the Schrodinger equation exactly. (4 points)

4) Show the complete wavefunction for Li in the ground state. (2 points)

13. Please show the packing fraction of hexagonal close packing.(4 points)

14.

Determine the point group of each of the following molecules.(6 points)

(5) The staggered configuration of ferrocene; (6) The eclipsed configuration of ferrocene

15. Taking the four p orbitals perpendicular to the molecular plane of cis-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)

**3) **The four p molecular orbitals are listed below, work out which irreducible representations they belong to.

** **

** **

** **(4 points)

4) Determine the spectral possibility from the ground state to the first excited state.(6 points)

16. Write down the d^{5} high and low spin configurations in (a) octahedral and (b) tetrahedral ligand field; work out which configurations show Jahn-Teller distortion.(8 points)