VSEPR molecular geometry

Result

VSEPR molecular geometry

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Statement

The three-dimensional shape a molecule adopts is set, to good approximation, by the number of electron domains (bonding regions plus lone pairs, treating a multiple bond as one domain) around the central atom, which arrange themselves to maximise their mutual angular separation. The steric number \(SN=(\text{number of }\sigma\text{ bonds})+(\text{number of lone pairs on the central atom})\) determines an idealised electron-domain geometry; removing the lone-pair positions from that geometry (while they still occupy space and distort the remaining angles) gives the observed molecular geometry.

Why it matters

Molecular shape is not a cosmetic detail: it is what determines whether bond dipoles cancel (a symmetric shape like linear \(\text{CO}_2\) is nonpolar despite polar C=O bonds, while a bent shape like \(\text{H}_2\text{O}\) is strongly polar) — the direct link to this unit's electronegativity-polarity result. Shape also determines which orbitals must combine to reproduce the observed bond angles (orbital-hybridisation, the next result in this unit) and is the primary factor controlling reaction pathways in organic chemistry (steric hindrance, discussed throughout CU-105).

What makes VSEPR remarkable pedagogically is that it requires no wave-mechanics calculation at all: a five-minute electron count on a correct Lewis structure predicts molecular shape with an accuracy that would otherwise require solving the molecular Schrödinger equation.

Hypotheses

Electron domains around the central atom behave, for geometric purposes, like mutually repelling points confined to a sphere centred on that atom, and adopt the arrangement of maximum mutual separation.This is an electrostatic argument (like charges repel, and repulsion increases as domains are pushed closer together), not a literal claim that electron pairs are point charges; it is justified after the fact by how well the resulting geometries match observation, and independently by the fact that the predicted geometries for \(SN=2\)–\(6\) are exactly the known maximum-separation arrangements of 2–6 points on a sphere (linear, trigonal planar, tetrahedral, trigonal bipyramidal, octahedral) — the same family of arrangements that arises in the unrelated, purely electrostatic Thomson problem for small point counts. Lone pairs occupy more angular space than bonding pairs, and multiple bonds occupy more angular space than single bonds, but a multiple bond still counts as exactly one electron domain.A lone pair is held by only the central atom's nucleus and so spreads out more diffusely than a bonding pair, which is pulled toward two nuclei and consequently held in a narrower angular region; a double or triple bond has more electron density than a single bond but is not counted as two or three separate domains, since both/all of its electron pairs occupy the same region between the same two nuclei.

Proof

\text{Construct the correct Lewis structure (Result of lewis-structures-octet).}

VSEPR presupposes a correct electron-counting and bonding pattern; an incorrect Lewis structure (wrong skeleton, wrong number of lone pairs) propagates directly into an incorrect geometry prediction. A

SN = n_{\sigma} + n_{\text{lone pairs, central atom}}

Count \(\sigma\) bonds from the central atom to each directly bonded neighbour (exactly one per neighbour, regardless of bond order) and lone pairs residing on the central atom itself; this is the steric number, the count of electron domains around that one atom. A

\text{Domains adopt the maximum-separation arrangement for } SN \text{ points on a sphere (Hypotheses).}

For \(SN=2,3,4,5,6\) these maximum-separation arrangements are, respectively: linear (\(180^\circ\)), trigonal planar (\(120^\circ\)), tetrahedral (\(109.5^\circ\)), trigonal bipyramidal (\(90^\circ\)/\(120^\circ\)/\(180^\circ\), two geometrically distinct positions), and octahedral (\(90^\circ\)/\(180^\circ\)). This is the electron-domain geometry, describing where all \(SN\) domains point, whether occupied by a bonded atom or a lone pair. A

\text{Molecular geometry} = \text{electron-domain geometry, naming only atom positions (not lone pairs).}

Where all domains are bonding (no lone pairs on the central atom), molecular geometry and electron-domain geometry coincide. Where one or more domains are lone pairs, the molecular geometry name describes only where the atoms are, but the lone pairs remain physically present and still occupy their electron-domain positions, still exerting repulsion on the bonding domains. A

\text{Repulsion strength: lone-lone} > \text{lone-bond} > \text{bond-bond}\ \Rightarrow\ \text{observed angles compress below the idealised value as lone pairs increase.}

Since a lone pair occupies more angular space (Hypotheses), it pushes neighbouring bonding domains closer together than the idealised, all-bonding value; each additional lone pair on the same central atom compounds this compression. This is why, within a fixed \(SN\), bond angles decrease monotonically as the lone-pair count on the central atom increases. A

Result

SN	Electron-domain geometry	Idealised angle(s)	Lone pairs → molecular geometry
2	Linear	180°	0: linear
3	Trigonal planar	120°	0: trigonal planar · 1: bent
4	Tetrahedral	109.5°	0: tetrahedral · 1: trigonal pyramidal · 2: bent
5	Trigonal bipyramidal	90°/120°/180°	0: trig. bipyramidal · 1: seesaw · 2: T-shaped · 3: linear
6	Octahedral	90°/180°	0: octahedral · 1: square pyramidal · 2: square planar

Reading. A single integer, the steric number, indexes into a short, fixed table of geometries; the lone-pair count within a given \(SN\) then further refines which named molecular shape results.

Scope. Applies to main-group central atoms with \(SN\le6\); for \(SN=5\), lone pairs preferentially occupy equatorial (not axial) positions of the trigonal bipyramid, minimising the number of close (\(90^\circ\)) lone-pair–bond-pair contacts — the origin of the seesaw, T-shaped, and linear (not the alternative axial-lone-pair) shapes listed.

Corollaries & converses

Bond angles compress in a predictable, monotonic order within fixed \(SN=4\): \(\text{CH}_4\) (0 lone pairs, \(109.5^\circ\) exactly) \(>\) \(\text{NH}_3\) (1 lone pair, \(\approx107^\circ\) observed) \(>\) \(\text{H}_2\text{O}\) (2 lone pairs, \(\approx104.5^\circ\) observed) — each additional lone pair pushes the remaining bonding domains measurably closer together, exactly as Step 5 predicts, though VSEPR (a qualitative repulsion argument, not a quantitative force calculation) predicts only the direction and rough size of the compression, not these exact decimal values.
A symmetric electron-domain geometry with all domains identical (all bonding, to identical atoms, e.g. \(\text{CH}_4\), linear \(\text{CO}_2\)) always gives a nonpolar molecule regardless of individual bond polarity, since the bond-dipole vectors cancel by symmetry — the geometric precondition for the electronegativity-polarity result's molecular-polarity criterion.
Converse: an observed bond angle significantly different from any of the table's idealised values, for a molecule with an otherwise ordinary Lewis structure, is a signal to re-examine the Lewis structure itself (wrong steric number, miscounted lone pairs) before concluding VSEPR has failed outright.

Fails without

Drop the lone-pair vs. bond-pair asymmetric repulsion strength (Step 5), treat all domains identically: every \(SN=4\) species would be predicted to have exactly \(109.5^\circ\) angles regardless of lone-pair count, contradicting the well-established, systematic decrease seen from \(\text{CH}_4\) to \(\text{NH}_3\) to \(\text{H}_2\text{O}\) (Corollaries). The asymmetric-repulsion assumption is what turns VSEPR from "same \(SN\), same angle" into a model that correctly predicts systematic, ordered angle compression.
Miscount a multiple bond as more than one domain: e.g. treating \(\text{CO}_2\)'s two C=O double bonds as four separate domains would predict \(SN=4\) (tetrahedral, bent \(\approx109^\circ\)) instead of the correct \(SN=2\) (linear, \(180^\circ\)) — directly contradicted by \(\text{CO}_2\)'s well-established linear, nonpolar structure.

Common errors

Counting a double or triple bond as two or three electron domains instead of one (Fails without covers the consequence directly).
Confusing electron-domain geometry with molecular geometry — e.g. calling water's shape "tetrahedral" (the electron-domain geometry) rather than "bent" (the molecular geometry, naming only the three atoms' positions).
Forgetting to count lone pairs on the central atom in the steric number, which silently converts (for example) a correct \(SN=4\)/bent prediction for \(\text{H}_2\text{O}\) into an incorrect \(SN=2\)/linear one.
For \(SN=5\) species, placing a lone pair in an axial rather than equatorial position, which predicts the wrong molecular geometry (an axial lone pair in \(\text{SF}_4\), for instance, would predict a different shape than the correct, experimentally observed seesaw).

Discussion

The core repulsion idea traces to Nevil Sidgwick and Herbert Powell in 1940, but the systematic theory, its now-standard name, and the full \(SN=2\)–6 geometry table were developed by Ronald Gillespie and Ronald Nyholm in a influential 1957 review, building the qualitative repulsion argument into a genuinely predictive, easily-taught framework — VSEPR is sometimes called the Gillespie–Nyholm theory for this reason.

VSEPR and orbital hybridisation (this unit's next result) are complementary but logically distinct: VSEPR is a purely geometric, electron-counting argument that predicts shape without reference to specific atomic orbitals, while hybridisation asks which combination of atomic orbitals on the central atom would reproduce that already-known shape. Historically hybridisation was taught as if it explained VSEPR's geometries from first principles, but the modern view (supported by more detailed molecular-orbital calculations) is that hybridisation is best understood as a convenient bookkeeping description consistent with an already-determined shape, not an independent derivation of it — VSEPR's predictive shape argument stands on its own, using only electron-pair repulsion.

Common misconception: that VSEPR predicts exact bond angles to high precision. It predicts only the correct idealised angle set for \(SN\) and the correct direction of any lone-pair-driven compression — the exact observed angles (\(104.5^\circ\) for water, not some VSEPR-derived number) come from experiment (or, from first principles, from solving the full molecular Schrödinger equation), not from the qualitative repulsion argument itself.

Worked examples

\text{CH}_4:\ SN=4\ (\text{4 }\sigma\text{ bonds, 0 lone pairs}) \Rightarrow \text{tetrahedral}, \ 109.5^\circ \ (\text{exact, by symmetry})

With no lone pairs, electron-domain geometry and molecular geometry coincide exactly, and the tetrahedral angle is exactly \(109.5^\circ\) by the geometry of a regular tetrahedron (\(\arccos(-\tfrac13)\)), with no lone-pair distortion to compress it. A

\text{NH}_3:\ SN=4\ (\text{3 }\sigma\text{ bonds, 1 lone pair}) \Rightarrow \text{trigonal pyramidal (electron-domain: tetrahedral)}, \ \approx107^\circ\ (\text{observed})

Same electron-domain geometry (tetrahedral) as \(\text{CH}_4\), but one domain is now a lone pair; the molecular geometry name reflects only the three N–H bond positions (trigonal pyramidal), and the single lone pair compresses the H–N–H angle below the \(109.5^\circ\) idealised value, exactly as Step 5 predicts. A

\text{CH}_4:\ 109.5^\circ\ (\text{exact}); \qquad \text{NH}_3:\ \approx107^\circ\ (\text{observed, compressed by 1 lone pair})

Reading. Identical steric number, identical electron-domain geometry, but a different lone-pair count produces both a different molecular-geometry name and a measurably different bond angle — the two effects (naming, and angle compression) tracked in Steps 4 and 5 respectively.

Scope. The same steric-number-then-lone-pair-count procedure applies uniformly across \(SN=2\) through \(6\), as tabulated in the Result.

Problems

Beryllium chloride, \(\text{BeCl}_2\) (16 valence electrons, Be central with two Be–Cl single bonds and no lone pairs on Be, an example of the electron-deficiency pattern discussed for boron in lewis-structures-octet), has steric number \(SN=2\). State its electron-domain geometry, molecular geometry, and idealised bond angle.
Solution
\(SN=2\) (2 \(\sigma\) bonds, 0 lone pairs on Be) gives electron-domain geometry linear; with no lone pairs, molecular geometry is also linear, with an idealised \(\text{Cl}\)–\(\text{Be}\)–\(\text{Cl}\) bond angle of \(180^\circ\) — matching \(\text{BeCl}_2\)'s well-established linear structure.
Sulfur tetrafluoride, \(\text{SF}_4\) (34 valence electrons: S has 4 S–F single bonds and, after completing each F's octet with 3 lone pairs, exactly 1 remaining lone pair on S). Determine \(SN\), the electron-domain geometry, and the molecular geometry.
Solution
Electron count: \(34-2(4)-6(4)=34-8-24=2\) electrons \(=1\) lone pair remaining on S. \(SN=4\ (\sigma\text{ bonds})+1\ (\text{lone pair})=5\). Electron-domain geometry: trigonal bipyramidal. With the single lone pair occupying an equatorial position (minimising \(90^\circ\) lone-pair–bond-pair contacts, per the Result's scope note), the molecular geometry is seesaw.
Xenon tetrafluoride, \(\text{XeF}_4\) (36 valence electrons: Xe has 4 Xe–F single bonds and 2 lone pairs remaining after completing each F's octet). Determine \(SN\) and the molecular geometry, and explain why the two lone pairs are placed on opposite sides of the octahedron rather than adjacent to each other.
Solution
Electron count: \(36-2(4)-6(4)=36-8-24=4\) electrons \(=2\) lone pairs on Xe. \(SN=4+2=6\), electron-domain geometry octahedral. Placing the two lone pairs opposite each other (\(180^\circ\) apart, both along one axis) means every lone-pair–bond-pair angle is exactly \(90^\circ\) with no lone-pair–lone-pair \(90^\circ\) contact at all; placing them adjacent (\(90^\circ\) apart) would introduce one highly unfavourable lone-pair–lone-pair \(90^\circ\) repulsion, the strongest repulsion type per Step 5 — so the opposite arrangement is lower-energy and correct. The resulting molecular geometry (4 F atoms in a plane, both lone pairs perpendicular to it) is square planar, the well-known observed shape of \(\text{XeF}_4\).
Explain why \(\text{CO}_2\) is a nonpolar molecule despite each C=O bond being individually quite polar (oxygen being significantly more electronegative than carbon), using the VSEPR geometry established for \(\text{CO}_2\) in Fails without.
Solution
\(\text{CO}_2\) has \(SN=2\) (two C=O double bonds, each counted as one domain, no lone pairs on C), giving a linear molecular geometry with the two C=O bond dipoles pointing in exactly opposite directions (\(180^\circ\) apart). Two equal-magnitude vectors pointing in opposite directions sum to zero, so the individual bond dipoles cancel exactly by symmetry, and the molecule has zero net dipole moment despite each bond being individually polar — a direct illustration of the Corollaries' claim that a symmetric electron-domain geometry with identical bonded atoms always gives a nonpolar molecule.