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.

Y_{0}
= (5/16π)^{1/2} (3cos^{2}θ-1) = (5/16π)^{1/2 }●3z^{2}-r^{2}/r^{2}

Y_{1 }= (5/4π)^{1/2} cos Θ sin Θ cos Φ =
(5/4π)^{1/2} ● xz/r^{2}

Y_{2 }= (5/4π)^{1/2} cos Θ sin Θ sin Φ = (5/4π)^{1/2}
● yz/r^{2}

Y_{3 }= (5/4π)^{1/2} sin2 Θ cos Φ = (5/4π)^{1/2}
● xy/r^{2}

Y_{4} = (5/16π)^{1/2} sin2Θ (cos2 Φ – sin 2 Φ) = (5/16π)^{1/2}
● (x^{2} – y^{2})/r^{2}

Y

Y

Y

Y

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 Y

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 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

Periodic Table

MO Theory

References