ground state of the anharmonic oscillator in terms of c, as-
suming that ˆHH° is the ideal harmonic oscillator Hamiltonian
operator. Use the integral table in Appendix 1 in this book.
12.20.Why would a perturbation Hˆcx^3 not work for an
energy correction to a ground-state harmonic oscillator? (Hint:
try evaluating the energy explicitly, then consider how to ar-
rive at the answer without evaluating the integral.)
12.21.Calculate a 3 for the real wavefunction in Example 12.10.
12.22.For a true polyene (that is, an organic molecule hav-
ing multiple conjugated carbon-carbon double bonds), there
may be a small potential energy change on the ends that can
be approximated by Vk(xa/4)^4 where kis some constant.
Apply this perturbation to the ground-state particle-in-a-box
and determine its energy. You will have to multiply the func-
tion out into a polynomial and evaluate each term individually.
12.23.The Stark effect is the change in energy of a system
due to the presence of an electric field (discovered by German
physicist Johannes Stark in 1913). Consider the hydrogen
atom. Its normally spherical 1sorbital distorts slightly when
exposed to an electric field. If the electric field is considered to
be in the zdirection, then the field acts to introduce, or mix,
some 2pzcharacter in with the 1sorbital. The atom is said to
be polarizable,and the extent to which it changes is consid-
ered a measure of the atom’s polarizability(which is desig-
nated by the letter , not to be confused with the spin func-
tion !). The perturbation Hamiltonian is defined as Hˆ
eErcos , where eis the charge on the electron, Eis the
strength of the electric field, and rand represent coordinates
of the electron. Evaluate the perturbation energy of the hy-
drogen atom. You will have to integrate all three spherical po-
lar coordinates in your evaluation of Hˆ. (A similar effect, the
Zeeman effect,exists for magnetic fields. It too can be treated
using perturbation theory.)
12.7 & 12.8 Variation Theory
12.24.Which of the following unnormalized functions can be
used in a variation theory treatment of a particle-in-a-box
having length a?
(a)cos (AxB), Aand Bare constants
(b)ear (c)ear^3
(d)x^2 (xa)^2 (e)(xa)^2
(f)a/(ax) (g)sin (Ax/a) cos (Ax/a)
12.25.Confirm equation 12.29 by substituting the defini-
tions in equation 12.28 into equation 12.27.
12.26.Show that a variation theory treatment of H using
ekras an unnormalized trial function yields the correct
minimum-energy solution for the hydrogen atom when the
specific expression for kis determined.
12.27.Explain why assuming an “effective nuclear charge,”
as used for our treatment of the helium atom in Example
12.11, is unnecessary for a treatment of the hydrogen atom.
12.28.Show that the two real wavefunctions determined in
Example 12.12 are orthonormal.
12.29.Consider a real system. Assume that a real wavefunc-
tion is a combination of two orthogonal functions such that
H 11 15, H 22 4, and H 12 H 21 2.5 (arbitrary
energy units). (a)Evaluate the approximate energies of the
real system and evaluate the coefficients of the expansion
aca,1 1 ca,2 2. (b)Compare your answer to the an-
swers in exercise 12.12 and comment.
12.30. (a)What does the secular determinant look like for a
system that is described in terms of four ideal wavefunctions?
(b)Comment on the complexity of a secular determinant as
the number of ideal wavefunctions increases. How many H
and Sintegrals need to be evaluated?
12.31.Prove the variation theorem. Assume that the lowest
possible energy of a system is E 1. Then, assume that any trial
wavefunction can be written as a sum of the true wave-
functions iof the system:
icii where HHˆiEii
Determine Eusing as the trial wavefunction and show that
EE 1 , equaling E 1 if is identically equal to 1 and greater
than E 1 if is not identically equal to 1.12.9 Comparing Variation and
Perturbation Theories
12.32.In introducing both the variation and the perturbation
theories, examples were given that had calculable answers,
leaving the impression that the systems under consideration
have ideal solutions. However, in all cases approximations were
made. Identify the point in each theory introduction where an
approximation is made that ultimately leads to an approxi-
mate, not exact, solution.12.10 & 12.11 Born-Oppenheimer
Approximation;
LCAO-MO Theory
12.33.State the Born-Oppenheimer approximation in words
and mathematically, and indicate how the mathematical form
is implied by the statement.
12.34.Consider the diatomic molecules H 2 and Cs 2. For
which is the Born-Oppenheimer approximation likely to intro-
duce less error, and why?
12.35.Spectroscopy deals with differences in energy between
levels. Derive an expression for E, the difference in energy,
between the two molecular orbitals of H 2 .
12.36.Repeat the determination of H 2 ,1and H 2 ,2as well
as E 1 and E 2 for R1.00, 1.15, 1.45, and 1.60 Å. Combine
these with the determinations from Example 12.14 and con-
struct a simple potential energy diagram for this system.
12.37.What is the bond order for the lowest excited state of
H 2 ? From this single result, propose a general statement
about unstablediatomic molecules and bond orders.
12.38.The helium atom was defined as two electrons and a
single nucleus, and the hydrogen molecule ion was defined as
a single electron and two nuclei. It seems that the only differ-
ence is an exchange in the identity of the particles in the sys-
tem; however, their quantum mechanical treatment is com-
pletely different, as are the results. Explain why.Exercises for Chapter 12 417