The applet uses the divided square well potential to model
electron states in a diatomic molecule. U = 100 eV
is the height of the dividing barrier and 2w =
0.50 Å is its width. The barrier is centered at x =
b and 2a = 4.50 Å is the combined width of
wells plus barrier. The case b = 0 describes
potential wells of equal width (= 2.00 Å); this is the
homonuclear molecule, i.e., one formed from identical atoms.
Heteronuclear molecules can be modeled by taking b
nonzero. In this exercise we will find the bonding and
antibonding orbitals for the electron that derive from the
'atomic' ground states of two dissimilar atoms, modeled as
square wells of the same height but unequal widths, 1.75 Å
and 2.25 Å (requires b = 0.25 Å – see 1st
Technical Note below). All parameter values appear in the
Equation View of the applet, along with the
electron mass = 511 keV/c2. The list to
the right of the graph includes placeholders for two
stationary states of the electron in this environment.
Instructions for use
Show one of the stationary state waveforms by
right-clicking the first placeholder in the list, 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 allowed energies for the
electron.
Employ an automated search for the bonding orbital.
Right-click anywhere in the graph and select "Display in
Window" from the popup menu. This frees the graph to
'float' in full view while we make adjustments to the
energy. Reposition the graph as desired and proceed to
Equation View, where the energy of this state
is recorded as E0 = 0. Right click
anywhere in the equation field for
E0, and select "Edit parameter.."
from the popup menu. In the autosearch range field (labeled
E±) type "1.0" and press the "Enter" key. Accept
the default value in the tolerance field (labeled δψ) by
pressing "Enter"; this initiates a search for an allowed
energy in the specified range centered on the value
E entered in the energy field. If the search
succeeds, the allowed energy replaces the mid-range value
in the energy field and the tolerance field is updated with
the actual wave mismatch (see 2nd Technical Note below);
otherwise, increment the mid-range energy E by
1.0 eV (= the autosearch range) and repeat. The bonding
orbital has the least energy and 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. Finally, restore the graph to its rightful place
by clicking the close button in the upper right corner of
the graph window.
Repeat the above procedure for the second placeholder in
the list, using it to find the antibonding orbital. This
state has one node and is the next lowest in energy.
If incrementing the
mid-range energy by the autosearch range returns a
previous value, boost the mid-range value just a bit more
to initiate a search in the adjacent range.
Notice that the symmetry evident in the stationary states
for the homonuclear case is absent here; the bonding
orbital is concentrated in the wider well, the antibonding
orbital in the narrower well. Compared to the homonuclear
case, the energy 'splitting' also is far greater in the
present case. In fact, this is not a simple 'splitting' at
all, since each well in isolation has a different ground
state energy.
Any nonzero value for b leads to unequal well
widths (= a ± b – w); this is the
heteronuclear case. No generality is lost by taking
b > 0, which makes the left side well the
wider of the two.
In automated search mode, a variant of Newton's method is
used to find the zeros of a function representing the
[fractional] waveform discontinuity at the match point.
Automated mode is entered by typing a non-zero value into
the autosearch range field (labeled E±). This
activates the tolerance field (labeled δψ) where a value
for the target discontinuity must be entered. The search
proceeds within the autosearch range centered on the
current value in the energy field. If successful, the
allowed energy replaces the mid-range value and the
tolerance field is updated with the actual mismatch at the
new energy. An unsuccessful search results in the target
tolerance being displayed in red to emphasize that this
value is not the actual waveform discontinuity.
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 the left
endpoint, and then again at the right 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 parameter 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.
Exercise: A
Heteronuclear Diatomic Molecule
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 allowed energies for the
electron.
If incrementing the
mid-range energy by the autosearch range returns a
previous value, boost the mid-range value just a bit more
to initiate a search in the adjacent range.
re-positions the match point to an extremum of the current
waveform.
