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\noindent {\bf Problem Set 1:}  Semiclassical quantization of
molecular vibrations
\smallskip

As an example combining several basic mathematical operations,  we
consider the problem of describing a diatomic molecule such as
$O_{2}$, which consists of two nuclei bound together by the electrons
that orbit about them.  Since the nuclei are much heavier than the
electrons we can assume that the latter move fast enough to readjust
instantaneously to the changing position of the nuclei
(Born-Oppenheimer approximation).  The problem is therefore reduced to
one in which the motion of the two nuclei is governed by a potential
$V$, depending only upon $r$, the distance between then:

\begin{displaymath}
\nonumber
V(r) = 4V_{O}\left [ \left (\frac{a}{r} \right )^{12} -
\left ( \frac{a}{r}\right )^{6} \right ] ,
\end{displaymath}

The vibrational states of relative motion, with energies $E_{n}$, are
described by the bound state solutions, $\psi_{n}(r)$, of a
one-dimensional Schroedinger equation, 

\begin{displaymath}
\nonumber
\left [ - \frac{\hbar^{2}}{2m} \frac{d^{2}}{dr^{2}} + V(r) \right ]
\psi_{n} = E_{n}\psi_{n} .
\end{displaymath}
Here, $m$ is the reduced mass of the two nuclei.

Our goal in this example is to find the energies $E_{n}$, given a
particular potential.  This can be done exactly by solving the
differential eigenvalue equation (1.17).  However, the great mass of
the nuclei implies that their motion is nearly classical, so that
approximate values of the vibrational energies $E_{n}$ can be obtained
by considering the classical motion of the nuclei $V$ and then
applying ``quantization rules'' to determine the energies.

Confined classical motion of the internuclear separation in the
potential $V(r)$ can occur for energies $-V_{0}<E<0$.  The distance
between the nuclei oscillates periodically (but not necessarily
harmonically) between inner and outer turning points, $r_{in}$ and
$r_{out}$.  During these oscillations, energy is exchanged between the
kinetic energy of relative motion and the potential energy such that
the total energy,

\begin{displaymath}
\nonumber
E = \frac{p^{2}}{2m} + V(r) ,
\end{displaymath}
is a constant ($p$ is the relative momentum of the nuclei).  We can
therefore think of the oscillations at any given energy as defining a
closed trajectory in phase space (coordinates $r$ and $p$) along which
Eq. (1.18) is satisfied, as shown in the lower portion of Figure 1.3.
An explicit equation for this trajectory can be obtained by solving
(1.18) for $p$:

\begin{displaymath}
\nonumber
p(r) = \pm [2m(E-V(r))]^{\frac{1}{2}} .
\end{displaymath}

The quantization rules state that, at the allowed energies $E_{n}$,
the action is a half-integral multiple of $2\pi$.  Thus, upon using
(1.19) and recalling that the oscillation passes through each value of
$r$ twice (once with positive $p$ and once with negative $p$), we have

\begin{displaymath}
\nonumber
S(E_{n} = 2 \left ( \frac{2m}{\hbar^{2}} \right )^{\frac{1}{2}}
\int^{r_{out}}_{r_{in}} \left [ E_{n} - V(r) \right ]^{\frac{1}{2}} dr
= (n + \frac{1}{2}) 2\pi
\end{displaymath}
where $n$ is a nonnegative integer.  At the limits of the integral,
the turning points $r_{in}$ and $r_{out}$, the integrand vanishes.

It is useful to define the dimensionless quantities

\begin{displaymath}
\nonumber
\epsilon = \frac{E}{V_{0}}, x = \frac{r}{a}, \gamma = \left (
\frac{2ma^{2}V_{O}}{\hbar} \right )^{\frac{1}{2}} ,
\end{displaymath}

The quantity $\gamma$ is a dimensionless measure of the quantum nature
of the problem.  In the classical limit ($\hbar$ small or $m$ large),
$\gamma$ becomes large.  By knowing the moment of inertia of the
molecule (from the energies of its rotational motion) and the
dissociation energy (energy required to separate the molecule into its
two constituent atoms), it is possible to determine from observation
the parameters $a$ and $V_{0}$ and hence the quantity $\gamma$.  For
the $H_{2}$ molecule, $\gamma = 21.7$, while for the HD molecule,
$\gamma = 24.8$ (only $m$, but not $V_{0}$, changes when one of the
protons is replaced by a deuteron), and for the much heavier $O_{2}$
molecule made of two  $^{16}O$ nuclei $\gamma = 150$.  These rather
large values indicate that a semiclassical approximation is a valid
description of the vibrational motion.

The goal of this problem set is to find all the (semiclassical) bound
iiiistates for a given value of $\gamma$.
\bigskip

{\bf Exercise 1.7}  One of the most important aspects of using a
computer as a tool to do physics is knowing when to have confidence
that the program is giving the correct answers.  In this regard, an
essential test is the detailed quantitative comparison of results with
what is known in analytically soluble situations.  Modify the code fo
use a parabolic potential for which the Bohr-Sommerfeld quantization
gives the exact eigenvalues of the Schroedinger equation:  a series of
equally-spaced energies, with the lowest being one-half of the level
spacing above the minimum of the potential.  For several values of
$\gamma$, compare the numerical results for the case with what you
obtain by solving analytically.
\bigskip

{\bf Exercise 1.8}  Another important test of a working code is to
compare its results with what is expected on the basis of physical
intuition.  Restore the code to use the Lennard-Jones potential and
run it for $\gamma = 50$.  Note that, as in the case of the purely
parabolic potential discussed in the previous exercise, the first
excited state is roughly three times as high above the bottom of the
well as is the ground state and that the spacings between the few
lowest states are roughly constant.  This is because the Lennard-Jones
potential is roughly parabolic about its minimum.  By calculating the
second derivative of $V$ at the minimum, find the ``spring constant''
and show that the frequency of small-amplitude motion is expected to
be

\begin{displaymath}
\nonumber
\frac{\hbar \omega}{V_{0}} =  \frac{6\times 2^{5/6}}{\gamma} \approx
\frac{10.691}{\gamma}
\end{displaymath}
Verify that this is consistent withthe numerical results and explore
this agreement for different values of $\gamma$.  Can you understand
why the higher energies are ore densely spaced than the lower ones by
comparing the Lennard-Jones potential with its parabolic
approximation?
\bigskip

{\bf Exercise 1.9}  Invariance of results under changes in the
numerical algorithms or their parameters can give additional
confidence in a calculation.  Change the tolerances for the turning
point and energy searches (line 120) or the number of Simpson's rule
points (line 130) and observe the effects on the results.  Note that
because of the way in which the expected number of bound states is
calculated (lines 1190-1200), this quantity can change if the energy
tolerance is varied.
\bigskip

{\bf Exercise 1.10}  Replace the searches for the inner and outer
turning points by the Newton-Raphson method or the secant method.
Replace the Simpson's rule quadrature for $s$ by a higher-order
formula (Eqs. (1.13a) or (1.13b)) and observe the improvement.
\bigskip

{\bf Exercise l.ll}  For the $H_{2}$ molecule, observations show that
the depth of the potential is $V_{0} = 4.747 eV$ and the location of
the potential minimum is $r_{min} = 0.74166 \AA$.  These two
quantities, together with Eq. (1.23), imply a vibrational frequency of

\begin{displaymath}
\nonumber
\hbar \omega = 0.492 V_{0} = 2.339 eV ,
\end{displaymath}
more than four times larger than the experimentally observed energy
difference between the ground and first vibrational state, 0.515 eV.
The Lennard-Jones shape is therefore not a very good description of
the potential of the $H_{2}$ molecule.  Another defect is that it
predicts 6 bound states, while 15 are known to exist.  A better
analytic form of the potential, with more parameters, is required to
reproduce simultaneously the depth and location of the minimum, the
frequency of small amplitude vibrations about it, and the total number
of bound states.  One such form is the Morse potential

\begin{displaymath}
\nonumber
V(r) = V_{0} [(1-e^{-\beta(r-r_{mm})})^{2} - 1] ,
\end{displaymath}
The Morse potential has a minimum at the expected location and th
parameter $\beta$ can be adjusted to fit the curvature of the minimum
to the observed excitation energy of the first vibrational state.
Find the value of $\beta$ appropriate for the $H_{2}$ molecule, modify
the program above to use the Morse potential, and calculate the
spectrum of vibrational states.







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