Page 288 - Essentials of physical chemistry

250 Essentials of Physical Chemistry

4. The PIB model can be used to estimate the electronic transition wavelengths of linear
polyenes by using 1.4 Å per bond in the conjugated chain, although slightly longer values
of the total length L give better answers. The simple model with V ¼ 0 in the box gives
qualitatively useful interpretation of p ! p* transitions in organic compounds assuming no
electron–electron repulsion or exchange and simple spin pairing of two electrons per orbital.
2 2
n
h
and
5. A similar pi-electron model can be solved for the POR where E n ¼ 2
2ma
e inu
n
c ¼ p ffiffiffiffiffiffi , n ¼ 0, 1, 2, 3, .... In this case, the n ¼ 0 level is not zero and the levels
2p
above n ¼ 0 are doubly degenerate; this leads to a correspondence with the 4n þ 2 rule for
aromatic character in organic chemistry.
6. The POR model can be extended to other aromatic pi-electron ring systems by ignoring the
inner cross-ring bonds and using a ring defined by the radius ‘‘a’’ of a circle whose
circumference is obtained as 1.4 Å times the number of bonds in the outer circumference
of the pi-electron ring system. This provides a qualitative model for aromatic ring systems
assuming perfect spin pairing of two electrons per orbital and neglect of any repulsion or
exchange between electrons.
7. A number of theorems are introduced related to the properties of solutions of the Schrö-
dinger equation. It is especially noted that valid wave functions must be (a) finite,
(b) continuous, and (c) single-valued.

PROBLEMS
11.1 Estimate the wavelength of the HOMO ! LUMO p ! p* of all-trans octatetraene.
11.2 Calculate the value of the box length to bring the PIB HOMO ! LUMO p ! p*
wavelength of octatetraene into agreement with the experimental wavelength of 3040 Å.
2
11.3 Use integration by parts twice to derive the expression for the average value of <x >

ð L r ffiffiffi r ffiffiffi
2
2 2 px 2 2 px L 3
<x > ¼ sin (x ) sin dx ¼ 1 2 2 :
L L L L 3 2n p
0
11.4 Show that [P x , x] ¼ [x, P x ], hint apply the operators to a dummy function f(x).
11.5 Use the uncertainty relationship to estimate the uncertainty in the product (Dx)(DP x ) for the

h
n ¼ 1, 2, 3, 4, and 5 levels of the PIB problem. Use factors of .
2
11.6 Estimate the HOMO ! LUMO p ! p* wavelength for anthracene (C 14 H 10 ) using the Platt
Perimeter extension of the POR model.
11.7 Compare (calculate) the HOMO ! LUMO p ! p* wavelength for azulene (C 10 H 8 ) using
the Platt Perimeter extension of the POR model; compare your result for the example
calculation for naphthalene in the text. What does this say about the Perimeter model?
11.8 Evaluate the energy of the n ¼ 1and n ¼ 2 levels of an electron-in-a-box for L ¼ 10 Å in
ð
joules and show the wave functions are orthogonal by direct integration of c * c dt. Use
1 2
trigonometry relationships as needed. (sin(2u) ¼ 2 sin(u) cos(u) might help).
11.9 Evaluate the angular momentum of the n ¼ 0, n ¼ 1 and n ¼ 2 levels for a POR where the
radius is 1 Å, in h-bar units. Note the PIR wave functions are eigenfunctions of the angular
momentum as well as the energy!
11.10 Prove the wave functions of the n ¼ 1 and n ¼ 2 levels of the POR model are orthogonal by
direct integration of the product of the normalized functions.