Chapter
2

Introducing Orbitals

The hydrogen atom wavefunctions

Read Engel and Reid

Sections 20.4, 20.5 and 20.6

In your studies in 1^st year chemistry or physics you will have already covered some of the hydrogen atom solutions, and you should readily identify the wavefunctions with atomic orbitals, and the quantum number l with the orbital labels s, p, d etc. (see p. 439 of the text). Here we will revise some of these general details, as well as clarify some of the ideas behind the more common graphical representations of atomic orbitals.

The radial functions display several interesting characteristics, which can be deduced from the r-dependence of the wavefunctions (see Table on pp.438-439):

1. s-orbitals (i.e. l = 0) have a finite, non-zero, value at the nucleus; all other orbitals have a node there;
2. all atomic orbitals can be characterised by their nodal character (the number of times they cross the axis and change sign); e.g. the 2s orbital has one radial node, 2p orbitals have none, 3s has two nodes, 3p orbitals have one, etc.

The term orbital is used here in place of the classical mechanical term orbit - it is something that is probabilistic, rather than deterministic. In an orbit, the electron always occupies a position on a curve in space, and its position can in principle be calculated. In an orbital, we can only talk in terms of the electron density- which we can think of as the probability of finding an electron somewhere- which we can obtain from the square of the wavefunction and which is non-zero everywhere in the universe. When an electron is described by one of the complete wavefunctions, we say that it occupies that orbital.

All the orbitals of a given value of n are said to form a single shell of the atom. It is common to refer to successive shells by the letters:

Thus all orbitals with n = 2 form the L shell. Orbitals with the same n but different l values form the subshells of a given shell. These subshells are generally referred to by the letters:

Hence we can get a 3p subshell, or a 4d subshell, etc. Since the energy levels depend only on n, all except s-orbitals are degenerate, and in general the degeneracy of each shell is n².