d-Orbitals

d-Orbitals are 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.

Angular Elements

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.

Y0 = (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 ● yz/r2
Y3 = (5/4π)1/2 sin2 Θ cos Φ =  (5/4π)1/2 ● xy/r2
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 - y2 , the x and y axis form two planes of symmetry that bisect the figure along the lobes.

The second 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
Introduction to Quantum Chemistry