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'

• Has integral values of 1, 2, 3, etc.
• As n increases the electron density is further away from the nucleus
• As n increases the electron has a higher energy and is less tightly bound to the nucleus
The azimuthal (second) quantum number 'l'
• Has integral values from 0 to (n-1) for each value of n
• Instead of being listed as a numerical value, typically 'l' is referred to by a letter ('s'=0, 'p'=1, 'd'=2, 'f'=3)
• Defines the shape of the orbital
The magnetic (third) quantum number 'ml'
• Has integral values between 'l' and -'l', including 0
• Describes the orientation of the orbital in space
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

• A collection of orbitals with the same value of 'n' is called an electron shell
• A collection of orbitals with the same value of 'n' and 'l' belong to the same subshell
Thus:
• the third electron shell (i.e. 'n'=3) consists of the 3s, 3p and 3d subshells (each with a different shape)
• The 3s subshell contains 1 orbital, the 3p subshell contains 3 orbitals and the 3d subshell contains 5 orbitals. (within each subshell, the different orbitals have different orientations in space)
• Thus, the third electron shell is comprised of nine distinctly different orbitals, although each orbital has the same energy (that associated with the third electron shell) Note: remember, this is for hydrogen only.
Restrictions on the possible values for the different quantum numbers (n, l and ml) gives rise to the following patterns for the different shells:
• Each shell is divided into a number of subshells equal to the principle quantum number (e.g. the fourth shell is divided into four subshells: s, p, d,and f; whereas the first shell has a single subshell: s)
• Each subshell is divided into orbitals (increasing by odd numbers):

 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:
• At ordinary temperatures essentially all hydrogen atoms are in their ground states
• The electron may be promoted to an excited state by the absorbtion of a photon with appropriate quantum of energy

1996 Michael Blaber