Spherical Harmonics and Orbitals
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:
= (h2/8π2I) ● L(L+1)
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.
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:
Φ, Θ) = R(r) ● Y(Φ,Θ)
R(r) is the radial portion,
and Y(Φ,Θ) is the angular portion.
wavefunctions specify information about the behaviour of a particle, we
often square integrate over a limited range of r, Φ, or Θ in order to
get the probability that the particle lies in a particular volume, at
any one given time.
Introduction to Quantum Chemistry