Rigid Rotator

In order to approximate atomic systems, we can think of the atom as containing a very heavy nucleus (with mass m

The solution to the Schrodinger Equation for the Rigid Rotator:

E_{L}
= (h^{2}/8π^{2}I) ● L(L+1)

where:

I is the moment of inertia for the
system (calculated for m_{1}, m_{2}, r_{1}, r_{2}),

and L is a non-negative integer.

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:

R(r) is the radial portion,

and Y(Φ,Θ) is the angular portion.

and Y(Φ,Θ) is the angular portion.

While 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

Introduction to Quantum Chemistry

Periodic Table

MO Theory

References