more complex wavefunctions than either s or p orbitals. Each of the five
has a dependency on all three variables. However, they can be converted
to Cartesian equivalents as in the p-Orbital case.
As in the
p-Orbital case, the conversion of the angular portions of the
wavefunctions to Cartesian co-ordinates has reduced the unique angular
elements shapes to two. The following are the equations of the
wavefunctions for the 3d orbitals.
= (5/16π)1/2 (3cos2θ-1) = (5/16π)1/2 ●3z2-r2/r2
Y1 = (5/4π)1/2 cos Θ sin Θ cos Φ =
(5/4π)1/2 ● xz/r2
Y2 = (5/4π)1/2 cos Θ sin Θ sin Φ = (5/4π)1/2
Y3 = (5/4π)1/2 sin2 Θ cos Φ = (5/4π)1/2
Y4 = (5/16π)1/2 sin2Θ (cos2 Φ – sin 2 Φ) = (5/16π)1/2
● (x2 – y2)/r2
Using the spherical conversions, this leads to two unique angular
elements.The most common set is that of cos Θ sin Θ cos Φ.
The last four graphs are the angular elements positioned around
different axes.The second is from Y0, which
is the angular portion of 3cos2θ-1.
The first is given by the graph The net result is a
set of four lobes that resemble a clover:
The top left shows the angular behaviour between 0 and 90, where the
radius is positive. Between 90 and 180 (top right) , the radius is
negative. The radius is positive again between 180 and 270 (bottom
left), and has a negative value in the last quarter (bottom right).
The net result of the angular portions is this picture above. There are
two lobes of both phases. In each of the xy, yz, and xy orbitals, the
axes lie in the plane of the screen, such that the two axis meet at the
center of four lobed figure. The axis form planes of symmetry that fall
in between the lobes.. In the case of the x2
the x and y axis form two planes of symmetry that bisect the figure
along the lobes.
angular function turns out to have this strange shape. In 3D, has a
p-orbital type center, along with a “ring” of opposite phase that goes
around the middle.
Between 0 and approx. 57 degrees, the radius is positive (top left). As Θ varies between 57
and 125, the radius remains negative (top left). The next major lobs has
positive phase (bottom left), and the last completed lobe, between 152
and 214, has a negative radius.
As Θ goes back to 360
degrees, the radius is positive, and completes the last lobe. The net
result is a set of four lobes of non-equal size and shape. This orbital
is generally depicted such that the axis between 0 and 180 degrees forms
a vertical axis The figure is then rotated around this axis to sweep out
a figure that consist of a larger major lobe, along with a smaller ring
around its middle that is of opposite phase.
Introduction to Quantum Chemistry