Spherical Harmonics and Orbitals

Rigid Rotator

In order to approximate atomic systems, we can think of the atom as containing a very heavy nucleus (with mass m1) with a relatively light electron (m2) that rotates around the nucleus as a satellite. The distance of the nucleus and electron from the center of mass is given by r1 and r2 respectively. The actual derivation of the solution is quite lengthy, and involves the use of operators, and solving a second order differential equation. The result is simply stated here:

The solution to the Schrodinger Equation for the Rigid Rotator:

EL = (h2/8π2I) ● L(L+1)

where:
I is the moment of inertia for the system (calculated for m1, m2, r1, r2),
and L is a non-negative integer.

Notice that the energy is also discrete, and can only take on certain values (as governed by the L term). It turns out that the wave functions of the rigid rotator are spherical harmonics. A spherical harmonic is analogous to the sinusoidal wave from particle-on-a-line example. A spherical harmonic can be though of as a 3D-path that a particle can travel without “destroying” itself energetically. However, this 3D-path is NOT fixed, and can take on many different shapes, even for one energy level.

Orbitals In General

With the spherical harmonics, we can describe the 3D motion of a electron around a nucleus. As such, the Schrodinger wave equation is decomposed into two separate parts: the radial (r) and angular elements (Φ,Θ) which arise from spherical harmonics. While the radial parts are relatively easy to describe, however angular parts are quite subtle. The common decomposition for the wavefunction is written as:

Ψ(r, Φ, Θ)   = R(r) ● Y(Φ,Θ)

where: