Atomic Orbitals

Light has wave properties, which we characterize by frequency and wavelength, and particle properties as shown by the the photoelectric effect.

In 1923 a French physicist Louis de Broglie  postulated, by analogy with light, that matter must have both particle and wave properties.  De Broglie proposed that a particle with a mass m moving at a speed u will have a wavelength given by

= h / m · u

The mathematical description of atomic structure based on the wave properties of subatomic particles is called quantum mechanics.  A model of the hydrogen atom based on the wave nature of the electron was developed principally by the Austrian physicist Erwin Schrodinger.  Mathematical equations describing the nature of electron waves in atoms that give acceptable (acceptable here means physically reasonable) solutions to the Schrodinger equation are known as wave functions.  In order to obtain one of these wave functions, we must assign integral values known as quantum numbers to three quantities in the wave equation, similar to the integral value of n required in the Bohr equation for the hydrogen atom.

A given set of these three quantum numbers yields a wave function for an electron called an atomic orbital.  Although they are only mathematical expressions, atomic orbitals allow us to visualize a three-dimensional region in an atom in which there is a high probability of finding electrons.  A fourth quantum number refers to a magnetic property of electrons called spin.  A description of the quantum numbers follows

• Principal Quantum Number (n)  The first or principal quantum number designates the main or principal energy level.  This is the most important quantum number because values of the other two depend on the value assigned to n.  In the case of the hydrogen atom n is the only quantum number determining the energy ( given by the Bohr formula).  The value of n must be a positive integer n = 1, 2, 3,....   The size of an orbital depends on n.  Energy levels are sometimes called shells and designated by letters, K (n=1), L (n=2), M (n=3), N (n=4), ..... 

• Angular Momentum Quantum Number (l) (Azimuthal Quantum Number)  The angular momentum quantum number designates the particular energy sublevel or subshell within a principal energy level.  It can have positive integral values from zero to one less than the value of n.  That is l = 0, 1, 2, ...,(n-1).  This quantum number distinguishes orbitals of given n having different shapes.  The different sublevels are usually denoted by a letter as follows  

• Magnetic Quantum Number (m_l)  This quantum number designates a particular orbital within a given sublevel.  It distinguishes orbitals of given n and l ( of given energy and shape) but having a different orientation in space.  The allowed values for this quantum number are integers from -l to +l including zero.  Each orbital of a given sublevel has the same energy.  In a sublevel l there will be 2l + 1 orbitals.

The quantum numbers n, l and m_l arise from Schrodinger's treatment of the electron in a hydrogen atom as a matter wave; by analogy, this concept can be applied to electrons in other atoms as well.  Although these three quantum numbers are sufficient to fully characterize the orbitals in an atom, there is a need for a fourth quantum number to describe the electrons that occupy these orbitals.  

• Spin Quantum Number (m_s)  This quantum number refers to the two possible orientations of the spin axis of an electron.  The name, electron spin quantum number, implies that electrons have a spinning motion.  There is however no way to attach a precise physical reality to electron spin.  What we do know is that experiments have shown that electrons give atoms properties that can be explained by assuming that the electron is spinning.  The possible values of the electron spin quantum number are +1/2 and -1/2.