In the 1920s and 1930s, Erwin Schroedinger, Paul Dirac, Wolfgang Pauli, Max Planck, Albert Einstein, and others, developed a mathematical description of the behavior of electrons in atoms.

• The key to the success of this description was the realization that at the scale of atoms, energy is detectably quantized.

• That is, the energy of an electron in an atom or molecule must be expressed as a multiple of some fundamental quantity of energy:

  E = ( n )( basic energy unit ), where n = 0, 1, 2, ....

Height = ( n )( 1.5 ), where n = the number of the rung on which you are standing

On the second rung, the distance you are above the ground and your potential energy are exactly twice what they were on the first rung, on the third, three times, and so on.

In each of the expressions above, the symbol n, the counting variable, is called a quantum number.

A second consequence of the small scale of atoms is that specifying both the energy and the position of an electron simultaneously is impossible (the Heisenberg principle).

• The very act of locating an electron changes its energy or its position.

• Consequently, the mathematical theory described regions of space near the atomic nuclei in which an electron of a given energy has a high probability of being found.

• These regions are called orbitals.

• We describe them in shorthand by listing the quantum numbers that specify their energy and other fundamental properties. The Table below lists these numbers.

  Quantum Number	Possible Values	Describes
  n	1, 2, 3, ...	Total Energy
  l	0, 1, 2 ... n - 1	Kinetic vs Potential Energy = shape of orbital
  m	-l ... +l	Spatial Orientation of Orbital
  s	± 1/2	Orientation of Electron "Spin"

To avoid confusion between the numbers, we commonly replace the numerical value of the "l" quantum number by a letter:

l = 0 = s orbital

l = 1 = p orbital

l = 2 = d orbital

Pauli established the principle that no two electrons in an atom or molecule may have the same four quantum numbers; this would amount to being in exactly the same place at the same time, which is impossible for any physical object.

• The first three quantum numbers describe the energy of an electron occupying a given region of space

• Therefore, the Pauli principle means that each such regions of space, each orbital, can be occupied by no more than two electrons, one having spin + 1/2 and the other spin - 1/2.

With the restrictions on the values of the quantum numbers, we can easily see that the number of orbitals of each type is limited. For example, when n = 1, l and m must both be zero. Therefore, only one orbital in an atom can have the 1s designation, and it can hold a maximum of two electrons, each having one of the two possible spin values.

Electrons of opposite spin, occupying an orbital with the same values of n, l, and m, are said to be paired.

When n = 2, l can be either 0 or 1, and m can be -1, 0, and +1. Here are the resulting orbitals:

You should construct a similar Table to show that when n = 3 and l = 2, five orbitals can be constructed that receive the designation, 3d.

We need two final rules to be able to describe where electrons are found in atoms:

• Electrons in atoms occupy orbitals of the lowest possible energy

• When placing electrons in orbitals of the same energy, such as the 2p orbitals, place one electron in each before pairing any electrons (Hund's Rule)

The result of these rules is to construct atoms in which the electrons are placed as close to the nucleus, and as far from each other, as possible.

To write a description of where the electrons are, we start by using the atomic number of the neutral atom, which gives us the number of electrons as well. Here's sodium:

The lowest energy orbital available is the 1s.

• We put in two electrons, which will have n = 1, l = 0, and m = 0, but opposite values of s.

• No more orbitals are available for n = 1.

If n = 2, as shown above, we have a 2s orbital, which will hold two more electrons, and three 2p orbitals, which will hold two each.

• Since these are full, we "lump" them together as 2p⁶.

• Add up what we've done, and you'll find that 10 electrons have been placed in orbitals.

One electron remains, but no more orbitals are available having n = 2. So the last electron must go into a 3s orbital.

Nitrogen is element 7. We can fill the 1s and 2s orbitals just as we did for Na. This accounts for four electrons and leaves three to be placed. We have three 2p orbitals, so we place one electron in each, to keep them as far apart as possible.

When the species for which we are writing an electron congifuration is an ion, we must add or subtract electrons from the atomic number. For each negative charge, add one; for each positive charge, subtract one.

Cl^- therefore has 18 electrons.

• The first eleven fill orbitals exactly as in Na.

• The 12th pairs with the 11th to fill the 3s orbital.

• This leaves six electrons, which will just fill the three 3p orbitals, giving what is called a closed shell configuration

Atoms at the left and right of the Periodic Table gain or lose electrons in order to obtain this esepcially stable configuration.

As we will see, many of the rules described here apply also when we combine atomic orbitals to make molecular orbitals.

The characteristic shapes described by the l quantum number (s, p, etc) are shown below:

Elements in the first full row of the Table use only the s and p electrons for bonding.

If you have Java installed on your computer (it is standard on Apples), here is a link that will allow you to manipulate the pictures in thee dimensions.