Orbital Overlap

The extent to which two orbitals overlap is quantified by the overlap integral S = ∫ ψ_A · ψ_B dV, where ψ_A and ψ_B are the wave functions of orbitals on atoms A and B respectively, and the integral is taken over all space. A positive value of S (S > 0) means the wave functions reinforce each other — this is constructive interference, producing a bonding interaction with increased electron density between the nuclei. A negative value of S (S < 0) means destructive interference — this is an antibonding interaction where electron density is removed from between the nuclei. A value of S = 0 means the orbitals are orthogonal; they do not interact at all, producing a non-bonding situation. The magnitude of S also correlates with bond strength: larger positive overlap means a stronger, more stable bond.

A sigma bond forms when two orbitals overlap directly along the internuclear axis — the imaginary line connecting the two nuclei. This head-on overlap can occur between: (a) two s orbitals (as in H–H, where two 1s orbitals overlap); (b) an s orbital and a p orbital (as in H–F, where H's 1s overlaps with F's 2p_z); or (c) two p orbitals oriented end-to-end (as in F–F). Because the electron density in a σ bond is concentrated symmetrically along and around the internuclear axis, σ bonds allow free rotation — rotating one end of a single bond does not break the overlap. Sigma bonds are the first bond formed between any two atoms and are generally the strongest type of covalent bond.

A pi bond forms when two p orbitals (or d orbitals in transition metal complexes) overlap laterally — side by side — with their lobes above and below the internuclear axis. In a π bond, the electron density is concentrated in two regions: one above and one below the nodal plane that contains the internuclear axis. This means π bonds have a nodal plane through the bond axis, unlike σ bonds. Pi bonds are always the second (or third) bond in a double or triple bond; they never exist alone. For example, in ethylene (\ce{C2H4}), each carbon forms a σ bond with the other carbon using sp² hybrid orbitals, and the remaining unhybridized p orbitals on each carbon overlap sideways to form one π bond, giving the C=C double bond. Because π bond overlap is less effective than σ bond overlap (the lobes are farther apart), π bonds are weaker than σ bonds. Crucially, π bonds prevent free rotation — rotating around a C=C double bond would break the π overlap, which is why cis/trans isomers exist.

Zero overlap (S = 0) occurs when orbitals are orthogonal to one another — their positive and negative contributions to the integral exactly cancel. A classic example is the interaction of an s orbital on one atom with a p orbital oriented perpendicular to the internuclear axis on another atom. The s orbital sees equal amounts of the positive lobe and the negative lobe of the p orbital, so the net overlap is zero. Another example is two p orbitals on the same axis but pointing in opposite directions, or a p_x orbital trying to interact with a p_z orbital when they are at 90° to each other. Non-bonding interactions do not lower the energy of the system and therefore do not contribute to bond formation. Recognizing zero-overlap situations explains why only specific orbital combinations are chemically relevant.

In VBT, a single bond consists of one σ bond. A double bond consists of one σ bond plus one π bond (e.g., \ce{C=C} in ethylene or \ce{C=O} in formaldehyde). A triple bond consists of one σ bond plus two π bonds oriented perpendicular to each other (e.g., \ce{C#C} in acetylene or \ce{N#N} in dinitrogen). As bond order increases, bonds become shorter and stronger: C–C single bond ≈ 347 kJ/mol and 154 pm; C=C double bond ≈ 614 kJ/mol and 134 pm; C≡C triple bond ≈ 839 kJ/mol and 120 pm. The additional strength from going single → double → triple is not additive: each π bond contributes less than the σ bond because lateral overlap is inherently weaker than head-on overlap.

• S_{AB} — Overlap integral (dimensionless): A number between −1 and +1 quantifying how much two orbitals share the same region of space. S > 0 indicates bonding overlap; S < 0 indicates antibonding overlap; S = 0 indicates no interaction.
• \psi_A^ — Complex conjugate of orbital A's wave function (m^{-3/2}): The wave function of the orbital on atom A. For real wave functions (s and p orbitals), ψ = ψ.
• \psi_B — Wave function of orbital B (m^{-3/2}): The wave function of the orbital on atom B whose interaction with orbital A is being evaluated.
• dV — Volume element (m^3): An infinitesimally small volume element over which the product of the two wave functions is summed across all of three-dimensional space.

Problem. Acetylene (\ce{HC#CH}) contains a carbon–carbon triple bond. (a) How many σ bonds and π bonds are present in one molecule of acetylene? (b) Describe the orbital overlap responsible for the C≡C bond. (c) Explain why acetylene is a linear molecule.

Answer. Acetylene contains 3 sigma bonds (2 C–H and 1 C–C) and 2 pi bonds (both in the C≡C). The C–C σ bond arises from sp–sp head-on overlap; the two C–C π bonds arise from p_y–p_y and p_z–p_z sideways overlap. sp hybridization forces 180° bond angles, making the molecule linear.

• Hybridization (sp³, sp², sp) — Hybridization explains which specific orbitals are available for the head-on and sideways overlaps that create σ and π bonds in polyatomic molecules.
• Molecular Orbital Theory (MOT) — MOT is the complementary quantum model to VBT; it uses the same overlap integral concept but combines atomic orbitals into delocalized bonding and antibonding molecular orbitals across the whole molecule.
• Lewis Structures and Bond Order — Lewis structures identify which pairs of atoms share single, double, or triple bonds — the starting point for assigning the number of σ and π bonds using VBT.
• VSEPR Theory and Molecular Geometry — VSEPR predicts molecular shapes based on electron pairs, and the resulting geometries directly reflect the directionality imposed by σ bond overlap in VBT.
• Bond Energy and Bond Length — The relative strengths and lengths of single, double, and triple bonds are a direct consequence of the number and quality of σ and π overlaps between the bonded atoms.