p-Orbitals

p-Orbitals are governed by their radial functions and two angular elements. Many of the principles of radial orbitals for the p-Orbitals are same as in the s-Orbitals. As such, their angular elements are discussed here. The three equations for the 2p orbitals are given by:

Y+ = (3/4π)1/2 sin Θ cos Φ
Y- = (3/4π)1/2 sin Θ sin Φ
Y0 = (3/4π)1/2 cos Θ

Each of the three orbitals has the different radial elements, since two of the functions depend on Φ and Θ. It will turn out that the three orbitals are each aligned along a different axis in the Cartesian plane. Using the Cartesian-Spherical conversion, we use the following conversion formulas:

x = sin Θ cos Φ
y = sin Θ sin Φ
z= cos Θ

we obtain:

Ypx(3/4π)1/2 sin Θ cos Φ = (3/4π)1/2 x/r
YpyY- = (3/4π)1/2 sin Θ sin Φ = (3/4π)1/2 y/r
Ypz = (Y0 = (3/4π)1/2 cos Θ = (3/4π)1/2 z/r

This means that we can graph one function with respect to an axis (x , y, or z), and then rotate 90° to the other two axes, and we have a full set of p orbitals.The angular element is given by cos Θ. We graph in polar co-ordinates on a polar grid:

From 0 to 90 degrees, the radius is positive, so the curve is drawn as blue (Top Left). As the curve goes between 90 and 180 degrees, the radius becomes negative, and is graphed in red. While this has no significant realistic interpretations, it does have

Since there is no dependency on Φ, the graph is sketched as a pair of tangential spheres. However, when we evaluate probabilities, we often are square integrating the function, so we have to take the square of function (in the case, the square of the radius).

Introduction
Introduction to Quantum Chemistry