Exercise:
Anharmonic Vibrations of Molecular Hydrogen
The applet uses the Morse oscillator potential to model the
vibrations of the two hydrogen atoms (H) that make up the
H2 molecule. The equilibrium separation of the
atoms in the molecule is a = 0.074 nm,
U = 4.79 eV is the potential energy far from
equilibrium, and b = 19.3 nm−1 is a
decay constant derived from the stiffness of the effective
spring at equilibrium. These values, along with the reduced
mass of the H-atom pair, m = 469.19
MeV/c2, appear on the Math
tab of the applet. On the tab labeled Graphics:
[r] the Morse potential is plotted over the interval
[0, 0.3 nm]. The listing to the right of the graph includes
placeholders for four rotation-vibration states of the
molecule, all initialized as s-waves.
Radial waves in the
applet are indexed simply by the order in which they appear.
For rotation-vibration states, the node count is identical to
the vibrational quantum number.
This exercise has two parts. In Part I we will find the
energies of the three lowest-lying pure vibrational states of
the H2 molecule, as well as the maximum vibrational
energy it can have without coming apart. In Part II we explore
the rotational spectrum associated with the vibrational ground
state.
Instructions for use (Part I)
In the context of rotation-vibration states,
s-waves describe pure vibrational excitations.
Show the first-listed s-wave – designated
g0 in the applet – by right-clicking
its placeholder, clicking the visibility icon beside the
"Real" label in the Colors | Visibilities
field, then choosing the OK button. The displayed
wavefunction has a noticeable discontinuity; since quantum
wavefunctions must be everywhere continuous, the energy of
this state cannot be one of the vibrational energies of the
molecule.
Employ an automated search to find the energy of the lowest pure vibrational
state. Switch to the Math tab, where the energy of this state
is recorded as E0 = 0. Right click
anywhere in the value field for
E0, and select "Edit Parameter..."
from the popup menu. This brings up the Energy Editor, where
the energy of this state is displayed in the energy field, labeled E.
Next,
activate the automated search function by typing "0.2" in the auto-search
range field (labeled E±) and pressing the Enter key.
The editor responds by inspecting energies in the range E ± 0.2,
looking for a waveforms with fractional wave mismatch no larger than the
[default] value shown in the tolerance field, labeled δψ.
If the search succeeds, the new energy is displayed
in the energy field and the tolerance field is updated with
the actual wave discontinuity (see 1st Technical Note below);
a failed search is marked by the requested tolerance being
displayed in red, to indicate that the desired wave
discontinuity was not, in fact, realized.
In the event of a failed search,
you will have to adjust the search parameters – either shift the mid-range
energy E upward or widen the search range – and repeat. The lowest
vibrational state is nodeless; if you have found another,
continue searching at lower energies. Finish by selecting
OK to end the edit session with the current settings.
When an automated search finishes
(successfully or not), the editor reverts to manual mode; as a consequence,
after editing any search parameters, you must return
to the auto-search range field and press Enter
to initiate another search using the new values.
Even with an automated search, it is
good practice to follow-up a successful search result by resetting the match point
(click on ), as this will generate an
improved waveform and a more accurate value for the wave discontinuity at
the match point.
Repeat the above instructions to find the energies and
display the radial waveforms for the next two lowest-lying
pure vibrational states from their placeholders in the
list. These states have one and two nodes, respectively
(not counting the node at r = 0).
If an automated search returns
the value from a previous step, narrow the auto-search range entry
so as to exclude that value from the search.
Use the remaining placeholder to locate the highest-lying
vibrational state of the molecule. Reduce the auto-search
range to 0.1 eV to account for the fact that higher Morse
levels lie closer together in energy. The energy of this
highest-lying state cannot exceed 4.79 eV (why?).
Use the first-listed s-wave placeholder to find
the energy and display the waveform of the
rotation-vibration ground state, as outlined in Part I.
Convert the next s-wave placeholder (labeled
g1) to a p-wave.
To do this, switch to the Math tab and click in
the value field for the parameter l1,
the orbital quantum number for this state. Position the
cursor as appropriate and type in the new value
l1 = 1, then type Ctrl+Z to
register the changed value.
Use the automated energy search described in Part I to
find the p-wave with lowest energy (nodeless).
The same auto-search range E± 0.2 eV used to find
the lower lying vibrational levels in Part I continues to
be an acceptable choice here.
Repeat the last two instructions, this time converting the
next s-wave placeholder (labeled
g2) to a d-wave
(l2 = 2), and finding the
d-wave with lowest energy (nodeless).
With rotational excitation, the molecular energy should
increase by
ℏ2÷ICM for the
l = 0 → 1 transition and twice this value for
the l = 1 → 2 transition
(ICM is the moment of inertia for the
molecule about an axis passing through its center of mass
and perpendicular to the interatomic line). Are your
findings consistent with this expectation? If so, extract from your
numerical results
a value for ICM of the H2
molecule.
The Numerov method is used to construct stationary waves
for the given (trial) energy. Using boundary conditions
derived from the correct asymptotic form first at
r = 0, and then again at the interval endpoint,
the Schrödinger equation is integrated inward to a preset
match point. The right-side wave is then scaled to make its
slope agree with that of the left-side wave at the match
point. If the trial energy is allowed, the wavefunction
values also will agree at the match point; otherwise, a
discontinuity results. The [fractional] discontinuity is
recorded in the editor tolerance field, labeled
δψ. The technique is most accurate when the match point
coincides with an extremum of the wavefunction. Each press
of the reset button
re-positions the match point to an extremum of the current
waveform.
In automated search mode, a variant of Newton's method is
used to find the zeros of a function representing the
waveform discontinuity at the match point.
Automated mode is activated by entering a non-zero value into
the auto-search range field, labeled E±. This
action generates a target value
for the [fractional] waveform discontinuity (the default is 10−6)
that appears in the tolerance field, labeled δψ. The search
proceeds within the auto-search range centered on the
current value in the energy field. If successful, the new
energy replaces the current one and the tolerance field is
updated with the [fractional] wave mismatch at the new value. An
unsuccessful search results in the requested tolerance being
displayed in red to indicate that the desired wave discontinuity was not realized.
Exercise:
Anharmonic Vibrations of Molecular Hydrogen
Radial waves in the
applet are indexed simply by the order in which they appear.
For rotation-vibration states, the node count is identical to
the vibrational quantum number.
beside the
"Real" label in the Colors | Visibilities
field, then choosing the OK button. The displayed
wavefunction has a noticeable discontinuity; since quantum
wavefunctions must be everywhere continuous, the energy of
this state cannot be one of the vibrational energies of the
molecule.
Even with an automated search, it is
good practice to follow-up a successful search result by resetting the match point
(click on 
), as this will generate an
improved waveform and a more accurate value for the wave discontinuity at
the match point.
