Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

PROBLEMS AND EXAMPLES 285

Problem 17.1

Evaluate the following determinants:

∣

∣

∣

∣

∣

10

01

∣

∣

∣

∣

∣

,

∣

∣

∣

∣

∣

x 1

1 x

∣

∣

∣

∣

∣

,

∣

∣

∣

∣

∣

sinθ cosθ

−cosθ sinθ

∣

∣

∣

∣

∣

Problem 17.2

What is the energy increase relative to the ground state when one of the quantum

numbers, saynz, for the particle in a cubic box is raised from 1 to 2? What is the

degeneracy of the resulting wave function and probability distribution?

Problem 17.3

1,3-Pentadiene shows a strong absorption peak at 45,000 cm−^1 (Ege, 1994). What

is the wavelength of this radiation in (a) centimeters, (b) meters, (c) nanometers,

(d) picometers, and (e) angstroms? What is its frequency in hertz? What is its energy

in joules?

Problem 17.4

What is the actual energy increase for the single excitationnz= 1 →nz=2in

Problem 17.2 if the trapped particle is an electron and the dimension of the box is

approximately a bond length, 1.5A? Give your answer in joules. What wavelength of ̊

light will promote an electron from the ground state to one of the degenerate excited

states? Give your answer in nanometers.

Problem 17.5

The length of an ethene molecule is about 153 pm according to MM3. Using Huckel ̈

theory and the particle in a one-dimensional box as a model, in what region of

the electromagnetic spectrum (X-ray, UV, vis, IR, etc.) is the radiation necessary to

promote an electron from the highest occupied molecular orbital (HOMO) to the

lowest unoccupied molecular orbital (LUMO)?

Problem 17.6

Find the Slater-type orbital (STO) of nitrogen.

Problem 17.7

Write down the unnormalized Slater determinant for He in the ground state. The He

atom is a three-particle, two-electron system. Set the Slater determinant, including a