SCIENCE · INVESTIGATE AND EXPLAIN

Audit sigma, pi and hybrid descriptions without literal orbital machinery

Relate localized overlap representations to bond multiplicity and rotation constraints.

Learning outcome: Compare all three carbon-pair orders, retain total sigma/pi counts, then rotate the supplied overlap basis. Explain why the single pi orientation differs from a complete two-dimensional pi subspace.

Starting recap

Review the electron and formal-charge budget, bond multiplicity, signed interaction energies and the distinction between a model and a measured property.

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Try

Ethene C2H4 has five sigma bonds including C–H and one pi bond; ethyne C2H2 has three sigma bonds and two pi components.

Open the experiment

Predict

Does a triple bond contain three sigma bonds, or one sigma plus two pi components?

Your prediction and reason

Experiment

Compare all three carbon-pair orders, retain total sigma/pi counts, then rotate the supplied overlap basis. Explain why the single pi orientation differs from a complete two-dimensional pi subspace.

Audit sigma/pi bookkeeping and orientation

Worked teaching model · synthetic values · Neutral acyclic C2H6/C2H4/C2H2 valence-bond representations. Supplied overlap-squared cos²θ illustrates one pi orientation, not a calculated torsional energy. A triple bond has two orthogonal pi components; rotating their basis does not remove the complete pi subspace.

Central carbon–carbon bond order: 2 order Relative orbital rotation: 30 degrees

Reset keeps your prediction, captures and practice history.

Axes rescale. Connected samples represent the supplied continuous parameter relationship; stems denote discrete pairs or steps, not a time trajectory.

Numerical plot coordinates

relative basis rotation (degrees)	supplied overlap-squared sum
0	1
15	0.9330127
30	0.75
45	0.5
60	0.25
75	0.0669873
90	0

Calculated evidence under the declared representation
Quantity	Value	Unit
Hydrogen count in neutral C2 skeleton	4	atoms
Total sigma bonds including C–H	5	bonds
Carbon–carbon pi bonds	1	bonds
One selected pi orientation overlap squared	0.75	relative
Complete pi-subspace overlap sum	0.75	relative

Ethene model: sp2 carbon, one C–C sigma plus one pi bond.

A single bond has one sigma component; adding bond order adds pi components in this localized representation. Restricted double-bond rotation follows loss of pi overlap, but cos²θ is an illustration, not a torsional energy calculation. Hybrid orbitals are mathematical combinations; they are not directly observed boxes prepared before bonding. Complete triple-bond pi space remains invariant under rotation of its two perpendicular basis directions.

Ethene model: sp2 carbon, one C–C sigma plus one pi bond.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Count overlap types separately

In a localized valence-bond description, sigma overlap lies along the internuclear direction. Pi overlap arises from sideways overlap with a nodal plane containing that direction. A single bond has one sigma component, a double bond one sigma and one pi, and a triple bond one sigma and two orthogonal pi components. These are representations of electronic states, not material rods stacked between atoms.

Build a complete molecular count

Neutral ethane has six C–H sigma bonds and one C–C sigma bond, seven total. Ethene has four C–H sigma plus one C–C sigma, five total, together with one pi. Ethyne has two C–H sigma plus one C–C sigma, three total, and two pi components. Counting only the central bond answers a different question from counting every sigma bond in the molecule.

Connect hybrid descriptions to geometry

The elementary localized descriptions associate tetrahedral carbon with sp3 combinations, trigonal-planar carbon with sp2 and linear carbon with sp. Combining atomic functions is a mathematical choice of basis that helps describe bonding. It is not a directly observed chronological process in which isolated atoms first build boxes and then mechanically assemble a molecule. More complete calculations can use other equivalent representations.

Investigate orientation cautiously

For a single selected pi orientation, the workbench supplies an overlap-squared factor cos²θ. At0° it is1, at60° it is1/4 and at90° it is0. Losing suitable pi overlap explains why unrestricted double-bond rotation is not equivalent to rotating a simple sigma bond. The supplied factor is not a calculated energy barrier: nuclear geometry and the full electronic state also change in a real twist.

Do not overextend the double-bond picture

A triple bond has two perpendicular pi components. Rotating their complete basis redistributes overlaps among those components without removing the whole two-dimensional pi subspace. The model therefore preserves its complete overlap sum. This distinguishes a change of representation from a physical loss of bonding. To discuss an actual rotational energy surface, specify the molecule, geometry and electronic model rather than treating a diagram angle as measured energy.

Sources and evidence boundary

Conceptual sources checked 30 October 2026: Hybrid orbital models, Multiple bonds. All prose, cases, code and diagrams are original. The supplied models are not measured structures, a performed laboratory investigation or certification of independent mastery. No chemical handling, heating, pressure vessel or cryogenic experiment is requested. Retain predictions, contrasting evidence and a changed-case explanation.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Why is an overlap factor not a measured torsional barrier, and hybridization not a preparatory event that was directly observed?

Your explanation and limitations

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Previous: Determine molecular polarity from vector evidence Next: Use orbital occupancy to compare bond order and magnetism

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