Humboldt State University ® Department of Chemistry

Richard A. Paselk

Chem 110	General Chemistry	Fall 2003
Lecture Notes::Lec 19_15 October	© R. Paselk 2003
PREVIOUS		NEXT

Quantum Models of Covalent Bonding

With the successes and failures of classical bonding models in mind, let's explore how we might view covalent bonding from a more modern, quantum, point of view.

To do this we will need to look briefly again at atomic orbitals and ask what they can tell us about how atoms might share electrons.

But before we even do that, I want to look at some simple atoms and molecules calculated at the highest level of theory, and thus the best approximation we have of what real atoms and molecules look and behave like. In order to do these calculations, we are assuming our atoms or molecules are in a vacuum, and essentially alone in the Universe. For todays discussion the images etc are available in the initial section on ionic bonds in the Supplement.

A Quantum View of Bonding

We have looked at a quantum model for ionic bond formation, now I want to begin our discussion with a model for covalent bond formation using two well studied diatomic molecules: Cl₂ and H₂. The animations and images are available at the Bonding Supplement.

In viewing these models we should keep in mind that:

• When atomic orbital sets are filled, or half-filled they become completely symmetrical.
• We should expect orbitals in molecules to be different than those in atoms since the electrons are shared by two nuclei rather than distributed around a single nucleus.
• Orbitals are orbitals
  • Only two electrons can be accommodated in any orbital
  • No two electrons can have the same "address" (the same set of quantum numbers).
    • For a molecules the "address" becomes the molecule over which the electrons are shared rather than the atom.
  • We have conservation of orbitals - a molecule will have the same number of orbitals as the atoms which make up the molecule.
    • For our purposes we can also assume a conservation of orbital energy.

With these thoughts in mind, lets look further at bonding and bond formation.

For both Cl₂ and H₂ you will note that we have a cylindrical distribution of the electrons in the single bond around the axis between the nuclei. Obviously in both cases the shapes of the orbitals have changed.

In order to understand this change, let's go back and review the shapes and electron distribution of atomic orbitals. The animations and images from this discussion are available at the Atomic Orbital Supplement.

For our discussion of bonding we need to look at s, p, and d orbitals. Higher orbitals are not involved in any of the substances we are interested in in this course.

Electronic Energy Levels Review:

• We will designate the primary energy level, corresponding to the average radial distance of the electron from the nucleus as a shell, and give it the symbol n. The lowest possible energy level is then the ground state with n = 1.
• The value of n also gives the number of nodes in each of the orbitals in that shell, with each shell having one node at infinity, where:
  • A node is a region of zero probability of finding an electron.
  • Nodes can have two general geometries:
    • radial (or spherical, since they describe a spherical shell at a specific radial distance from the nucleus), with each atom having at least one radial (spherical) node at infinity;
    • angular (either planar, e.g. as in the planar p-node and diagonal d-nodes, or cone shaped, e.g. as in the cone-shaped nodes of the d_z² orbitals which results in the donut shaped orbitals).
• Shells with n > 1 may have subshells which are different geometrical patterns of electron distribution. Thus:
  • The lowest energy pattern is spherical and given the designation s.
  • The next lowest energy distribution is bi-lobed with a planar symmetry. It is given the designation p.
  • The third lowest energy distribution has diagonal planes of symmetry and is designated d.
  • The fourth lowest energy distribution is designated f. This is the highest subshell type occupied by ground state electrons in any atom, so we will not look any further (an infinite number of subshells exist in theory for excited states, but they are not important to our understanding).
• The average energies of the different subshells are the energy of the shell, thus when subshells are present the energy of the shell is split. For example, in the n=2 shell the 2s orbital becomes lower in energy than the shell, while the 2p orbitals become higher in energy.
• The regions of electron occupancy in subshells are called orbitals.
  • For each shell there is one s orbital.
  • For each shell with n = 2 or greater there are three p orbitals: p_x, p_y, and p_z.
  • For each shell with n = 3 or greater there are five d orbitals: d_xz, d_yz, d_xy, d_{x²- y²}, and d_z²

© R A Paselk

Last modified 16 October 2003