Download Problem Set II

CHEMISTRY 158a, FALL, 2008
PROBLEM SET II
1) The earth and the sun form a system of two bodies. How far is the center of the sun
from the center of mass of this system? Calculate a value for the angular momentum of
the earth-sun system. Obtain any necessary astronomical data from the CRC Handbook
of Chemistry and Physics.
2) The discussion of the two-body problem yielded the following expression for the
energy
H = [pr2/2μ + V(r)] + [P2/2μr2]
The second term in the expression provides the dependence of the energy on angular
momentum. It is coupled to the first through the denominator that depends on r, the
distance between the two bodies.
a) Expand P2/2μr2 in a Taylor series in r about r = re. Perform the expansion
through second order; that is, include terms up to (r - re)2. Don’t expand P2; it is a
constant of the motion.
b) You will have converted the Hamiltonian into the following expression
H = [pr2/2μ + V(r)] + [P2/2Ie] + H’
What are Ie and H’? To the extent that H’ is small and can be ignored, i.e. it is a
small perturbation, you have separated the problem into a one-dimensional
problem depending on r and one depending on angular momentum. This is the
celebrated rigid rotor approximation. H’ is the vibration-rotation interaction term
which couples vibrational and rotational motion.
3) The expression for the Rydberg constant on page 22 of M&S, R = mee4/8ε02ch3 =
1.097373147 x 107 m-1, is not quite correct. The hydrogen atom is a two-body problem
and the correct expression is R = μe4/8ε02ch3 where μ is the reduced mass for the
electron-nucleus system.
a) Using the value of R, calculate RH, RD, and RT, the values of the Rydberg
constant for the normal hydrogen, deuterium, and tritium atoms.
b) Normally, one replaces μ in electronic calculations with me. Is this
simplification justified?
c) An unusual short-lived atomic species is positronium where the nucleus is
replaced with a positron, the anti-particle of the electron. Calculate R for
positronium. In this case, can one replace μ with me.?
4) Consider the case in which j= 1. This is exemplified by the deuterium nucleus, a
particle with a spin of 1. This problem uses the NMR conventions where I, the nuclear
spin, is used as the symbol for the angular momentum quantum number j, I2|I m> =
I(I+1)|I m>, and Iz|I m> = m|I m>. Operators are shown in Italic bold face.
a) Three angular-momentum eigenstates with I = 1 are possible. What are they?
b) For each of the three eigenstates, calculate the angle that the angular
momentum vector makes with the z axis.
c) Apply the raising (I+) and lowering operator (I-) to each of the eigenstates.
d) Calculate values of the following matrix elements: <1 1|I+|1 1>, <1 0|I+|1 0>,
<1 1|J+|1 1>, <1 1|J+|1 1>
e) A nucleus with a spin greater than 1/2 has an asymmetric charge distribution
measured by Q. Q is called the quadrupole moment. In a molecule, the nuclear
charge asymmetry can interact with the electronic asymmetric charge distribution.
The asymmetry of the electronic charge distribution is described by 3 gradients:
Vzz, Vxx, and Vyy. The sum of these gradients is zero so there are only two
independent parameters: Vzz and the asymmetry  = (Vxx - Vyy)/Vzz. The
Hamiltonian describing the interaction between the nuclear and electronic charge
asymmetries is given by
H = {e2qQ/[4I(2I-1)]}[3Iz2 - I2 + (/2)(I+2 + I-2)]
i) An alternate expression for H replaces (I+2 + I-2)/2 with (Ix2 - Iy2). Show
that these two expressions are equivalent.
ii) Use first order perturbation theory to calculate the energy of the three
eigenstates.
iii) If one employs perturbation theory to second order, which eigenstates
contribute to |1 1>? To |1 0>? To |1 -1>?
5) This problem addresses the carbon-13 NMR spectrum of CDCl3.
a) The proton/carbon-13 coupling constant in CHCl3 is 150.0 Hz. J for coupling
between spins X and Y is directly proportional to gXgY. Calculate a value for
1
JCD, the deuterium/carbon-13 coupling constant in CDCl3.
b) The NMR frequency of spin X is directly proportional to gX. What are the
carbon-13 and deuterium NMR frequencies in Hz on a 400 MHz instrument?
c) Compare |C -D| with 1JCD. Will the use of first order perturbation theory
applied to the NMR spectrum of CDCl3 yield an accurate result?
d) Derive analytical expressions for the eigenenergies of the spin system CDCl3.
e) Derive expressions for the frequencies in the carbon-13 NMR spectrum of
CDCl3. How many peaks are in the spectrum?
6) The Morse potential, V(r) = De{1 - exp[-(r - re)]}2 where De is the bond dissociation
energy, is a fairly accurate representation of the vibrational potential energy for all r.
a) Expand the Morse potential in a Taylor series about r = re through (r - re)3.
b) Using the result from part (a), derive an expression for , the harmonic
oscillator vibration frequency in wave number.
c) Use first order perturbation theory and derive expressions for the energy
eigenvalues of the ground (v = 0) and first excited (v = 1) vibrational states.