Atomic Interactions: Wavefunctions, Orthogonality, Orbitals, Hybridization, & Bonding


Home / Atomic Interactions: Wavefunctions, Orthogonality, Orbitals, Hybridization, & Bonding

The double-bond between carbons is drawn as if the two halves were identical, even though in reality, one is sigma, one is pi. Copyright image by Decoded Science, all rights reserved

Nobel Prize-winning theoretical chemist Linus Pauling wrote a most famous book in 1939, entitling it, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry.

There have been few facets of chemistry of greater importance.

Here, we will discuss basic concepts that we hope will lead our readers to a better understanding of what goes on between atoms. We will consider the bonding between carbon atoms in particular.


Quantum mechanics is a mathematical discipline that defines the nature and behavior of tiny particle systems, such as atoms and molecules.

At the atomic and subatomic level, particles and waves share similar properties, each isolated system containing one or more particles is described by a mathematically descriptive equation called a wavefunction.


In mathematics, numbers can be “operated on” to produce certain results. For instance, 3 and 4 can be operated on by multiplying them with each other, to produce the result, 12. Wavefunctions can be operated on in various ways to produce mathematical results that correspond to physical properties. If two wavefunctions are multiplied together yielding the value zero, the wavefunctions are said to be orthogonal.


An approximate definition for the word orthogonal is perpendicular – this applies to wavefunctions that are orthogonal – they do not interfere with each other. Thus atomic orbitals are orthogonal – they don’t interfere with each other.

Effectively, there is no overlap. When the electron orbitals of two atoms form a chemical bond, on the other hand, they overlap. That means they are not orthogonal.

Atomic Orbitals

Obviously, two atoms require electron participation if they are to form a chemical bond. For instance, in the formation of a diatomic (“two atoms”) molecule of hydrogen, the electron orbital of the one hydrogen atom must overlap the electron orbital of the other hydrogen atom, establishing a bond between them.

However, hydrogen atoms possess only one electron, and thus only one kind of orbital. They each have an “s” orbital and no other kind of orbital. Such is not the case for carbon. There are also, for instance, “p” orbitals. This leads to the important discussion of orbital hybridization.

Orbital Hybridization

Bonding is all about energy. Just as two partners may come together to form a business because to do so is more commercially viable than going into business for themselves, so two (or more) atoms bond together because it is energetically advantageous to do so.

A bond with greater overlap is a stronger bond. If the atoms involved in bond formation can sacrifice certain atomic orbitals to form hybridized orbitals that result in lower-energy molecular orbitals than would be otherwise achieved, so much the better.

Σ-Bonds (Sigma Bonds)

In the field of organic chemistry there are various kinds of bonds. The most common is the σ-bond (sigma-bond). A sigma bond lies directly between the two participant atoms (for instance, carbon) like a segment of string between two beads, C―C. Both regular atomic orbitals and hybrid orbitals are orthogonal. We will now consider hybrid orbitals for the element carbon.

Molecular Orbitals

Hybrid orbitals do not form unless a molecule is in the making, since atomic orbitals are orthogonal and have no tendency to interact by themselves. Hence they are also called molecular orbitals. There are three common hybrid or molecular orbitals for carbon, namely sp³, sp², and sp.

These all come from combining “s” and “p” atomic orbitals of a carbon atom. Here is the simple description of the orbitals of the carbon atom. Having 6 protons, the atom also has 6 electrons, distributed in the 1 shell and 2 shell as follows,

1s² 2s² 2p²

Interestingly, the p orbitals are actually 3 in number, each capable of holding the usual 2 electrons for a total of six. The three-p orbitals can be visualized as aligning with the x-axis, the y-axis, and the z-axis. Carbon leaves 4 spots unoccupied. Additional carbon, hydrogen, or other atoms can bond to it.

If that is the case, those atoms provide the missing 4 electrons. Although those electrons seemingly could occupy the other p-electron orbitals, but in reality, those 4 electrons along with carbon’s own 4 bonding electrons enter hybridized orbitals comprised of 1 s orbital and 3 p orbitals.

The 1s² electrons are buried within the atom, said to be a closed shell, and do not engage in the formation of chemical bonds. Rather, the 2s² and the 2p² electrons, plus four electrons from other atoms (which correlate to the other 4 p electron slots), enter the hybrid orbitals generated from 1s and 3 p orbitals. There are 4 of these sp³ orbitals. These 4 orbitals are arranged in tetrahedral fashion about the carbon atom.

On the other hand, if two atoms have a double bond between them, the hybridization is not sp³, but the trihedral sp2. Hybridization for a triple bond is dihedral, or sp. The reader may wish to visualize all of this. One excellent way is to do so is to watch this Khan Academy video presentation.

π-Bonds (Pi-Bonds)

All of the above applies in the case of single bonds between carbon atoms, or between carbon and some other atom. Sometimes carbon is bonded to itself by means of a so-called double bond. This bond does not merely consist of 2 σ-bonds, but of a σ-bond plus a π-bond (pi-bond). What is a π-bond?

Since two atoms with a double bond are sp2, each retains an unbonded p-orbital these bond above and below the plane of the carbon atoms. For sp-bonded triple bonds, four p orbitals are retained, yielding the two pi-bonds, up-and-down, left-and-right.

Keeping in mind the principle of orthogonality, a π-bond does not “interfere” with a σ-bond. Nor does it add to a σ-bond. It is orthogonal, and lies either “above and below” the σ-bond, or is to the “left and right” of the σ-bond. The triple bond has both varieties of π-bond – two π–bonds – one above and below the carbon atoms, and one to the left and the right of the carbon atoms.

Atomic Bond Properties

The end result for the organic (or, basically carbon) chemist is that bonding between carbon features one σ-bond, and in the case of multiple bonds, either one or two π-bonds.

All these bonds are hybrid molecular bonds, derived from s and p atomic orbitals.

We’ve discussed atomic bonding properties in detail here – do you have any questions?

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