Molecular geometry

Why do methane, Ammonia, and water have different shapes when each central chemical atom is surrounded by eight valence electrons? The answer separates the positions of atomic nuclei from the electron domains used in a VSEPR model.

Molecular geometry describes the three-dimensional arrangement of the atomic nuclei in a molecule, including bond lengths and bond angles. For a larger molecule, a shape name often describes just the neighbours of one central chemical atom. A Lewis structure records connections and electron bookkeeping; its flat layout does not fix molecular geometry.

In the VSEPR model, electron pair geometry includes both bonding electron domains and lone pairs, whereas molecular geometry counts only the chemical atoms. In methane, four bond pairs and no lone pairs give tetrahedral molecular geometry. In Ammonia, three bond pairs and one lone pair give trigonal pyramidal molecular geometry. In water, two bond pairs and two lone pairs give bent molecular geometry. All three use the same ideal tetrahedral electron pair geometry; this does not make their actual bond angles equal. See Flowers, Theopold, Langley and Robinson, Chemistry 2e, §7.6: electron-pair and molecular geometries.

A static drawing also needs a physical interpretation. The atomic nuclei undergo oscillations; an arrangement at a minimum of potential energy and an experimentally averaged arrangement need not have identical bond lengths or bond angles. This distinction matters when comparing measurements with quantum mechanics calculations; see the IUPAC Gold Book: experimental bond-length conventions. Gas electron diffraction gives a concrete route from a laboratory signal to molecular geometry rather than inferring shape from a drawing alone.

Interconversion of molecular arrangements that exchanges chemically distinguishable sites. In NMR spectroscopy, exchange fast compared with the relevant frequency difference can merge signals, so observed equivalence need not be a static molecular symmetry.

A connected molecular framework described by a polyhedral vertex graph. Terminal ligands are distinguished from atoms belonging to the skeletal framework.

A polyanionic main-group cluster whose electron count can support delocalized skeletal bonding. Localized lone pairs must be separated before applying a polyhedral count.

An open cluster framework formally obtained by removing two vertices from a closo parent. Its usual electron count is $n+3$ skeletal pairs; hydrogen positions and cage distortions require additional structural information.

An open cluster framework formally obtained by removing one vertex from a closo parent. In the usual electron-counting model, an $n$-vertex nido framework has $n+2$ skeletal pairs.

A closed deltahedral cluster described by $n+1$ skeletal electron pairs in the usual Wade model. borane examples include $[{\mathrm{B}}_{12}{\mathrm{H}}_{12}{]}^{2-}$.

A molecular framework in which two rings share exactly one atom.

A spirocyclic framework containing a dienone ring. An acid-promoted dienone-phenol rearrangement can convert a suitably substituted spiro framework into fused-ring connectivity.

A five-coordinate geometry with two axial sites and three equatorial sites. Ideal axial–equatorial and equatorial–equatorial angles are $90^{\circ }$ and $120^{\circ }$.

A molecular geometry in which three attached chemical atoms and their central atom lie in one plane. The ideal symmetric arrangement has three $120^{\circ }$ angles and D3h molecular point group symmetry, as in an isolated boron trifluoride molecule.

A ring fusion shares two neighboring atoms and their bond between rings. Cis or trans fusion specifies the relative faces of junction substituents; axial/equatorial labels additionally require a conformation.

Square-pyramidal molecular geometry has four basal donor positions in a square and one axial donor. Removing one axial ligand from an octahedral molecular geometry lowers orbitals with $z$ character. A common $d$-level order is $d_{x^2-y^2}>d_{z^2}>d_{xy}>d_{xz}=d_{yz}$, although the middle pair depends on axial strength and distances.

In square planar molecular geometry, four donor atoms surround a center in a plane. Two matching ligands can be cis or trans. Low-spin $d^8$ complexes of many late transition metals commonly have this geometry.

In octahedral molecular geometry, six donor directions point toward the vertices of an octahedron. Three mutually cis donors can occupy one face; three coplanar donor directions can occupy a meridian.

A bond angle is the angle between two chemical bonds sharing a central chemical atom. In H–O–H, the middle symbol identifies the vertex: the angle is at oxygen. Let $u$ and $v$ be the nonzero displacement vectors from that atomic nucleus to the two neighbours. The actual three-dimensional bond angle satisfiesusing the dot product and Euclidean norm. Rotating the whole molecule preserves this expression; projecting it onto a page generally does not.

There is another useful way to find a bond angle. If the two bond lengths are $a,b$ and the distance between the outer atomic nuclei is $d$, the law of cosines givesFor equal bond lengths $a=b$, this becomes $d=2b\sin (\theta /2)$. Thus a measured outer-atom separation contains information about a bond angle even though those outer chemical atoms are not directly joined by a chemical bond.

For a stipulated rigid, equal-bond length model using water's $104.4776^{\circ }$ equilibrium bond angle, $d/b\approx 1.5811$, whereas a straight H–O–H arrangement would give $d/b=2$. These are calculated geometric ratios, not new measured distances. Gas electron diffraction illustrates how pair-distance information can constrain molecular geometry. Substituting independently averaged distances into the rigid triangle formula does not automatically recover an equilibrium bond angle, because averaging and this nonlinear calculation need not commute.

In tetrahedral molecular geometry, four neighbouring chemical atoms surround a central chemical atom in the directions of the vertices of a tetrahedron. In the regular idealization, the central atomic nucleus lies at the centre of a regular tetrahedron, and every pair of chemical bonds makes the same bond angle. Methane, ${\mathrm{CH}}_4$, provides the standard example: its carbon has four bonding electron domains and no lone pair. See Chemistry 2e, §7.6: tetrahedral methane.

The exact ideal bond angle follows from four directions proportional to $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, and $(-1,-1,1)$. Each has squared Euclidean norm $3$, and each distinct pair has dot product $-1$. HenceThis is a statement about a regular tetrahedron, rather than a universal exact value for every molecule labelled tetrahedral. It also explains why a cross-shaped flat drawing with $90^{\circ }$ gaps is misleading: four equal directions can spread out more evenly in three dimensions.

The same vectors sum to zero. Consequently, four equal bond-dipole contributions along those directions cancel. The symmetric methane molecule is nonpolar, but a tetrahedral shape alone does not imply a nonpolar molecule: replacing one neighbour can make the electronic distribution and bond contributions unequal. Chemical polarity depends on both molecular geometry and electron density. See McMurry, Organic Chemistry, §2.2: molecular dipoles and methane.

In trigonal pyramidal molecular geometry, three neighbouring chemical atoms form a triangular base and the central chemical atom lies outside its plane. The pyramid has four atomic nuclei; its central chemical atom is the apex, not the centre of the base. Ammonia, ${\mathrm{NH}}_3$, has this arrangement. The ^[ref] lists an H–N–H bond angle of $106.67^{\circ }$, commonly rounded to $106.7^{\circ }$ or about $107^{\circ }$.

The Lewis structure of Ammonia allocates eight valence electrons as three bond pairs and one lone pair. Four electron domains therefore suggest tetrahedral electron pair geometry; removing the lone pair marker leaves three hydrogen chemical atoms below nitrogen. The VSEPR model treats the lone pair as more space-demanding and predicts compression relative to $109.4712^{\circ }$, but it does not calculate the precise $106.67^{\circ }$ value. See Chemistry 2e, §7.6: lone-pair effects.

Three equal bond-dipole contributions in this pyramid cancel sideways, but retain a component along its axis. By contrast, three equal contributions lying in one plane at $120^{\circ }$ would cancel completely. This shows how a small change in molecular geometry can permit a permanent electric dipole moment. Ammonia is a polar molecule; NIST lists a measured magnitude of about $1.476$ debye in the same ^[ref]. The vector argument predicts allowed direction and cancellation, not the full measured magnitude.

In bent molecular geometry, two neighbours and their central chemical atom form a noncollinear triangle, with an interior bond angle strictly between $0^{\circ }$ and $180^{\circ }$. Water, ${\mathrm{H}}_2\mathrm{O}$, is the standard example. The NIST CCCBDB experimental water entry, internal coordinates, citing Hoy and Bunker (1979) gives the equilibrium H–O–H bond angle as $104.4776^{\circ }$ with quoted uncertainty $0.0019^{\circ }$, usually rounded to $104.5^{\circ }$.

The oxygen in water has two bonding electron domains and two lone pairs. Its ideal electron pair geometry is tetrahedral, while the three atomic nuclei give bent molecular geometry. Three points always lie in a plane; this does not make the four-domain scaffold planar. Nor does the word bent by itself determine an electron domain count: the same atomic shape can arise from other domain arrangements.

In an illustrative model with two equal bond-dipole contributions of magnitude $p$ separated by bond angle $\theta$, their sum of vectors points along the bisector and has magnitude $2p\cos (\theta /2)$. At $180^{\circ }$ it vanishes; at $104.4776^{\circ }$ it is approximately $1.2247p$. This ratio is a geometric calculation, not a prediction of water's measured electric dipole moment. The entire electron density, including contributions represented by lone pairs, matters. Water is a polar molecule; chemical polarity helps explain its interactions with other polar molecules. See McMurry, Organic Chemistry, §2.2: water, lone-pair contributions, and polarity.

Gas electron diffraction infers molecular geometry from the diffraction of electrons by a gaseous sample. It makes a useful experimental test of shape: distances between both bonded and nonbonded atomic nuclei affect the signal, so a proposed straight arrangement and a bent arrangement can be distinguished through their pair separations.

A concrete modern apparatus is the Bielefeld instrument described by ^[ref]. A hot cathode supplies electrons; apertures and magnetic lenses shape the beam. A temperature-controlled nozzle injects sample vapour across it inside a vacuum chamber. Pumps and a cold trap reduce unwanted residual gas. A central beam stop catches the strong unscattered beam; off-axis scattered electrons expose imaging plates. A rotating sector can attenuate the intense small-angle region to accommodate detector range. The authors checked both the spatial scale of the plate scanner and its response to electron dose. These checks matter because an error in detector position or intensity calibration can masquerade as a structural difference.

What is measured is detector signal as a function of position, converted to scattering angle using the sample-to-detector distance. Randomly oriented molecules give radial information. After extracting the molecular intensity, compare it with predictions from a candidate arrangement and adjust the bond lengths and bond angles to improve the fit. The Wann Electron Diffraction Group's account of time-averaged GED, experiment and theory describes radial averaging and this refinement. The apparatus does not photograph one molecule with labelled atomic nuclei.

Why can distances survive random orientation? In a simplified rigid pair model with separation $r$, write $s$ for the magnitude of the difference between incident and scattered wave vectors. Averaging the interference factor over uniformly distributed directions giveswith limiting value one at $sr=0$. This follows by integrating the cosine; $t$ is the cosine of the angle between the pair and the scattering direction. The oscillations depend on $r$, although the orientation has disappeared. A real calculation weights many pairs by their scattering amplitudes and includes nuclear motion. The wave interference explanation and superposed pair-distance contributions are discussed in the same ^[ref].

Once a consistent structural model supplies two equal bond lengths $b$ and an outer-atom separation $d$, the law of cosines converts these to a bond angle. For example, the illustrative $d/b=1.5811$ corresponds to approximately $104.5^{\circ }$, while $d/b=2$ is straight. This geometric step explains what a shape measurement is trying to establish; it is not a new water measurement.

There are two essential limits. First, recovering labelled three-dimensional positions from a summed radial signal is an inverse problem: composition, candidate connectivity, and other structural information can be needed to distinguish competing fits. Karle's authored review Structure determination of gaseous and amorphous substances by diffraction methods (1977), abstract explains why too few independent observations make supplementary structural information indispensable. A good fit alone does not prove that the proposed arrangement is unique. Second, atomic nuclei are moving. Distances inferred by gas electron diffraction are motion-dependent averages; corrections are required before identifying them with a minimum-potential energy arrangement. For experimental distance conventions, see the IUPAC Gold Book: bond length. These limitations explain why model assumptions and the averaging convention must accompany a precise bond angle.