Atomic Orbitals.

The wave pattern for electron in a hydrogen atom will depend on n, which is by the way called the principle quantum number. Since electron, as a standing wave, is spread out, no longer his motion around the nucleus can be described by orbits. You can not describe its position precisely at a given time. Instead, electron’s whereabouts can be characterized by a probability function to find it in a certain space called an orbital. The probability is proportional to the wave amplitude squared. It also often called the electron density. Orbitals with the same n form a shell.

Each orbital for the same principle quantum number n, as a standing wave, should have n -1 nodal surfaces, i.e. surfaces where amplitude of the standing wave changes sign and probability to find an electron equals zero.

1) n = 1. The lowest energy shell, n = 1, has an orbital with no nodal surfaces, called an s-orbital. This orbital is spherical, i.e. looks the same from any direction.

2) n = 2. Orbitals from the shell with n = 2, each have one nodal surface. Such a surface can be arranged as a layer inside the sphere, i.e. maintaining a spherical symmetry of the orbital. That orbital would be called 2s, where 2 states the principle number and s that it is spherical. Another way of making that nodal surface would be through unfolding it into a plane dividing a space into two halves. Such a plane provides an angular rather than a radial nodal surface we had for 2s. This plane can be oriented in three independent ways resulting in three orbitals (p_x, p_y and p_z) from the 2p subshell. You can think of p orbitals as pointing orbitals where p_x, p_y and p_z are pointing along different axes. Number of angular nodal surfaces is called the orbital angular momentum quantum number or the azimuthal quantum number and is labeled l.

l £ n -1 - since the overall number of nodal surfaces equals n-1

One can show that each subshell has 2l + 1 orbitals. Indeed, for l = 0 (s-orbital), it is 2´ 0 +1 = 1, for l = 1 (p-orbital), it is 2´ 1 + 1 = 3. And indeed, we have three of them, p_x, p_y and p_z. Each of the orbitals can be assigned to have individual quantum number, call the magnetic quantum number m_l , an integer which changes from - l to l, i.e. it can be -l, -l+1, …0, … l-1, l. As you can see by summing the number of the orbitals, it is equal 2l + 1. You should not think of m_l as an artificial mathematical trick because it is certainly not, it can be experimentally measured. Nor you should try to assign m_l = -1 to p_x, m_l = 0 to p_y etc. because it won’t be correct either. But some linear combinations of the orbitals p_x , p_y and p_z will have a definite m_l.

3) n = 3. Continues on. Now we have two nodal surfaces, which can be realized as:

• both radial nodes, thus making 3s subshell (1 orbital)
• one radial and one angular - 3p subshell (3 orbitals) or
• both angular, i.e. a new type of subshell with l = 2, called 3d (2 ´ 2 + 1 = 5 orbitals, m_l = -2,-1,0,1,2). See the sketch above.

4) The sequence continues and goes on to infinity. With every increase in n, a new subshell is introduced. After d (l = 2) goes f (l = 3), then g (l = 4), then h (l = 5), i (l = 6), j, k, and so on, following the alphabet. Each subshell would have 2l + 1 orbitals with the shapes looking more and more complicated because of l different angular nodal surfaces.