Separate electron geometry, molecular shape and polarity evidence

Count regions before naming a shape

A Lewis diagram gives connectivity and nonbonding electrons. Its page layout does not determine the angles between atoms. In the valence-shell electron-pair repulsion model, VSEPR, regions of electron density are organized spatially around a central atom. A bonded neighbor counts as one region whether the bond is single, double or triple. A lone pair also counts as one region.

In AX2E2, A is the central atom, X marks each bonded end and E each lone-pair region. Two bonds plus two lone pairs make four regions. The ideal electron geometry is tetrahedral, but only two endpoints are nuclei, so the molecular shape is bent. Selecting AX4 retains four regions but makes all four endpoints nuclei: both electron geometry and molecular shape are tetrahedral.

Capture those two cases and compare their region tables. The diagram is an oblique projection of explicitly listed three-dimensional unit directions. Its screen angles are distorted; do not measure a projected angle with a ruler and call it a molecular bond angle.

Use ideal directions as a model, not a measurement

Two ideal regions are opposite, three lie in a plane 120° apart, and four point toward tetrahedral corners. The tetrahedral angle calculated from the displayed directions is about 109.4712°. An AX2E2 drawing retains that ideal angle here. It does not claim that water has that measured angle or that real lone pairs leave all angles unchanged.

For two unit directions u and v, their scalar product is the sum of matching-coordinate products, uₓvₓ + uᵧvᵧ + u_zv_z. The ideal angle has cosine equal to that sum. This optional calculation explains the table independently of the projection. The essential task is to identify which directions are bonds and which are lone pairs, then use the table to compare them.

Five regions have distinct axial and equatorial positions. In a trigonal bipyramid, an axial direction has three 90° contacts and an equatorial direction has two. The usual qualitative repulsion argument favors equatorial lone pairs. Inspect AX4E, AX3E2 and AX2E3: one, two and three equatorial nonbonding regions leave seesaw, T-shaped and linear molecular arrangements. The contact counts expose the geometric reason without pretending to calculate repulsion energies or equal interaction strengths.

Six ideal regions form an octahedron. Removing one axial nuclear endpoint gives square pyramidal AX5E. Assigning opposite regions to lone pairs gives square planar AX4E2. All lone-pair regions remain in the electron geometry even though they are excluded from the molecular shape name.

Test a cancellation claim with a changed condition

The lab assigns each included bond an outward unit contribution by default. This is an original vector model in relative units, not a measured dipole. Symmetric equal contributions cancel in linear AX2, trigonal-planar AX3, tetrahedral AX4, trigonal-bipyramidal AX5, octahedral AX6 and square-planar AX4E2. AX2E3 also retains two opposite axial bonds and cancels under these assumptions.

Choose square planar AX4E2, then increase only the first bond contribution from 1 to 1.5. The former contributions cancel, leaving an additional 0.5 along the changed direction. The shape did not change, but its equal-weight cancellation condition did. This is a stipulated perturbation, not a prediction for a particular chemical substitution.

Now set every contribution to zero. The sum is zero for every shape, including bent and pyramidal arrangements. This does not claim that a real lone-pair-containing molecule must be nonpolar. The simplified sum omits explicit lone-pair and other electron-density contributions and does not calculate measured molecular dipoles. Geometry, bond polarity and full electronic distribution are separate evidence.

Keep the orbital vocabulary within its scope

In the restricted local model, two, three and four regions can be described using sp, sp² and sp³ labels. For five or six regions this activity names geometry without assigning d-orbital hybridization. The current AP Chemistry framework, topic 2.7, explicitly excludes d-orbital hybridization assessment while retaining the corresponding shape reasoning.

Return to formal-charge and resonance diagnostics when a supplied Lewis proposal needs checking first. Continue to sigma/pi and local orbital models to test why multiple bonds do not multiply the number of electron directions. These finite ideal cases are not an unrestricted geometry optimizer or a completed Unit 2 investigation.