Physical Chemistry Third Edition

19.6 Atoms with More Than Two Electrons 821

b.Use the variation method to find the energy of a

hydrogen atom in an electric field in thezdirection

with magnitudeE. Use the variation trial function

φA(ψ 1 s+cψ 2 pz)

Minimize the energy with respect to c. Explain any

difference between your result and that of part a.

c.Find the polarizability of a hydrogen atom from your

answer to part b, using the formula for the energy of

an induced dipole

Edipole−

1

2

αE^2

19.45 a.Each element in the second row of the periodic table
has a higher ionization potential than the element to
its left, except for nitrogen, which has a higher
ionization potential than either carbon or oxygen, and
beryllium, which has a higher ionization potential
than either boron or lithium. Explain both the general
trend and these exceptions to the general trend.
b.The ionization potential is found generally to decrease
toward the bottom of a column of the periodic table.
Explain this fact in terms of effective nuclear charge
and electron shielding.
19.46Search the World Wide Web and write a brief report on
the current status of the controversy about the production
of elements 116 and 118.
19.47Identify each statement as either true or false. If a
statement is true only under special circumstances, label
it as false.
a.The orbital energy of a 4ssubshell is always lower
than that of a 3dsubshell in the same atom.
b.The ground state of every inert gas corresponds to a
filled valence shell.
c.The inert gases are the only elements with spherically
symmetric electron charge distributions.

d.An electron configuration that contains only filled

subshells can correspond to only one term symbol.

e.Orbital occupations that do not correspond to Hund’s

first rule cannot occur.

f.The self-consistent field theory can deliver the best

possible orbital wave function for a multielectron

atom.

g.A second-order perturbation result is always more

nearly correct than a first-order result.

h.An electron configuration with two unpaired electrons

cannot correspond to a doublet term symbol.

i.An antisymmetrized orbital wave function contains no

electron correlation.

j.An antisymmetrized orbital wave function contains no

dynamical correlation.

k.If a variation energy equals the correct ground-state

energy, the variation trial function must be equal to the

correct ground-state wave function.

19.48Gaussian orbitals are frequently used in computational

chemistry. Use the Gaussian function

φAe−br

2

as a variational function for the ground state of a

hydrogen atom. Minimize the energy and compare it with

the correct ground-state energy.

19.49Hoffman has published an article describing an Excel

worksheet that carries out the Hartree self-consistent

calculation on the ground state of helium.^36 Obtain and

use this spreadsheet to carry out the calculation and make

an accurate plot of the radial factor of the approximate

orbital obtained. Compare it with a plot of the orbital

obtained with the simple variational calculation in the

chapter.

(^36) G. G. Hoffman,J. Chem. Educ., 82 , 1418 (2005).