Electronic Structure of Atoms

Quantum Mechanics and Atomic Orbitals


Quantum Mechanics and Atomic Orbitals

1926 Erwin Schrödinger

Schrödinger's wave equation incorporates both wave- and particle-like behaviors for the electron.

Opened a new way of thinking about sub-atomic particles, leading the area of study known as wave mechanics, or quantum mechanics.

Schrödinger's equation results in a series of so called wave functions, represented by the letter y (psi). Although has no actual physical meaning, the value of y2 describes the probability distribution of an electron.

From Heisenberg's uncertainty principle, we cannot know both the location and velocity of an electron. Thus, Schrödinger's equation does not tell us the exact location of the electron, rather it describes the probability that an electron will be at a certain location in the atom.

Departure from the Bohr model of the atom

In the Bohr model, the electron is in a defined orbit, in the Schrödinger model we can speak only of probability distributions for a given energy level of the electron. For example, an electron in the ground state in a Hydrogen atom would have a probability distribution which looks something like this (a more intense color indicates a greater value for y2, a higher probability of finding the electron in this region, and consequently, greater electron density):

Orbitals and quantum numbers

Solving Schrödinger's equation for the hydrogen atom results in a series of wave functions (electron probability distributions) and associated energy levels. These wave functions are called orbitals and have a characteristic energy and shape (distribution).

The lowest energy orbital of the hydrogen atom has an energy of -2.18 x 10­18 J and the shape in the above figure. Note that in the Bohr model we had the same energy for the electron in the ground state, but that it was described as being in a defined orbit.

The Bohr model used a single quantum number (n) to describe an orbit, the Schrödinger model uses three quantum numbers: n, l and ml to describe an orbital.

The principle quantum number 'n'

The azimuthal (second) quantum number 'l' The magnetic (third) quantum number 'ml' For example, the electron orbitals with a principle quantum number of 3 (i.e. n=3) would have the following available values of 'l' and 'ml':
 
n
(principle quantum number)
l
(azimuthal)
(defines shape)
Subshell
Designation
ml
(magnetic)
(defines orientation)
Number of Orbitals in Subshell
3
0
3s
0
1
 
1
3p
-1,0,1
3
 
2
3d
-2,-1,0,1,2
5
  Thus: Restrictions on the possible values for the different quantum numbers (n, l and ml) gives rise to the following patterns for the different shells:  
Subshell
Number of orbitals
s
1
p
3
d
5
f
7
The number and relative energies of all hydrogen electron orbitals through n=3 are shown below:


1996 Michael Blaber