Chemical bonding II: molecular shape, polarity and bonding theories

A Lewis structure tells us which atoms are bonded and where the lone pairs are, but it is drawn flat on paper. Real molecules are three-dimensional, and their shapes control their properties: whether they are polar, how they pack in solids, how they fit into the active sites of enzymes, and how they smell and taste. This lesson predicts molecular shapes with VSEPR theory, uses shape to decide whether a molecule is polar, and then looks at two quantum-mechanical descriptions of bonding, valence bond theory and molecular orbital theory, that explain what Lewis structures cannot.

VSEPR theory

The valence-shell electron-pair repulsion (VSEPR) model, developed by Ronald Gillespie and Ronald Nyholm in 1957 (Gillespie spent most of his career at McMaster University in Hamilton, Ontario), rests on one idea: electron groups around a central atom repel each other and arrange themselves as far apart as possible. An electron group, or electron domain, is any region of electron density around the central atom: a lone pair, a single bond, or a multiple bond. A double or triple bond counts as one group, because its electrons all lie in the region between the same two atoms.

The number of electron groups determines the electron-group geometry: two groups are linear (180°), three are trigonal planar (120°), four are tetrahedral (109.5°), five are trigonal bipyramidal (90° and 120°) and six are octahedral (90°). The molecular geometry, the shape we name, describes only the positions of the atoms, not the lone pairs.

Lone pairs take up more room than bonding pairs, because they are held by only one nucleus and spread out closer to it. They therefore push bonding pairs together slightly. The bond angle shrinks from 109.5° in methane, to 107° in ammonia (one lone pair), to 104.5° in water (two lone pairs). The repulsions rank lone pair–lone pair > lone pair–bonding pair > bonding pair–bonding pair. Multiple bonds also take up a little more room than single bonds.

In the trigonal bipyramid, the five positions are not equivalent: three equatorial positions lie in a plane at 120°, and two axial positions lie above and below at 90° to the plane. Lone pairs always occupy equatorial positions, where they have fewer 90° neighbors. That is why SF₄ is seesaw-shaped and XeF₂ is linear, with its three lone pairs around the equator.

# A VSEPR lookup: (bonding groups, lone pairs) -> molecular geometry
GEOM = {
    (2, 0): "linear", (3, 0): "trigonal planar", (2, 1): "bent (from trigonal planar)",
    (4, 0): "tetrahedral", (3, 1): "trigonal pyramidal", (2, 2): "bent (from tetrahedral)",
    (5, 0): "trigonal bipyramidal", (4, 1): "seesaw", (3, 2): "T-shaped", (2, 3): "linear",
    (6, 0): "octahedral", (5, 1): "square pyramidal", (4, 2): "square planar",
}

def shape(central_valence, n_bonded, charge=0, multiple_bond_electrons=0):
    """Lone pairs on the central atom = (valence e - charge - electrons used in bonds to it) / 2.
    Each single bond to a terminal atom uses 1 electron from the central atom;
    multiple_bond_electrons counts the extra central-atom electrons in double/triple bonds."""
    used = n_bonded + multiple_bond_electrons
    lone = (central_valence - charge - used) // 2
    return lone, GEOM[(n_bonded, lone)]

molecules = [("CH4", 4, 4, 0, 0), ("NH3", 5, 3, 0, 0), ("H2O", 6, 2, 0, 0),
             ("SF4", 6, 4, 0, 0), ("ClF3", 7, 3, 0, 0), ("XeF4", 8, 4, 0, 0),
             ("SO2", 6, 2, 0, 2), ("CO2", 4, 2, 0, 2)]
for name, val, nb, q, extra in molecules:
    lone, g = shape(val, nb, q, extra)
    print(f"{name:5s} bonded atoms = {nb}, lone pairs on central atom = {lone}: {g}")

CH4   bonded atoms = 4, lone pairs on central atom = 0: tetrahedral
NH3   bonded atoms = 3, lone pairs on central atom = 1: trigonal pyramidal
H2O   bonded atoms = 2, lone pairs on central atom = 2: bent (from tetrahedral)
SF4   bonded atoms = 4, lone pairs on central atom = 1: seesaw
ClF3  bonded atoms = 3, lone pairs on central atom = 2: T-shaped
XeF4  bonded atoms = 4, lone pairs on central atom = 2: square planar
SO2   bonded atoms = 2, lone pairs on central atom = 1: bent (from trigonal planar)
CO2   bonded atoms = 2, lone pairs on central atom = 0: linear

Molecular polarity

A molecule is polar if it has a net dipole moment: a separation of positive and negative charge, measured in debyes (D). Whether a molecule is polar depends on two things: whether its bonds are polar (lesson 9), and whether its shape lets the bond dipoles cancel. Each bond dipole can be treated as a vector pointing toward the more electronegative atom; the molecular dipole is their vector sum.

Carbon dioxide has two very polar C=O bonds, but it is linear, so the two bond dipoles point in opposite directions and cancel exactly: CO₂ is nonpolar. Water also has two polar bonds, but its bent shape means they add to a net dipole of 1.85 D pointing toward the oxygen side. Similarly, symmetric molecules such as BF₃, CH₄, CCl₄, PCl₅ and SF₆ are nonpolar despite polar bonds, while NH₃, CHCl₃ and SO₂ are polar.

import numpy as np

def net_dipole(bond_dipole, directions):
    """Sum equal bond dipoles pointing along the given unit-vector directions."""
    total = sum(bond_dipole * np.array(d) / np.linalg.norm(d) for d in directions)
    return np.linalg.norm(total)

theta = np.radians(104.5)       # water bond angle
water = [(np.sin(theta / 2), np.cos(theta / 2), 0), (-np.sin(theta / 2), np.cos(theta / 2), 0)]
co2 = [(1, 0, 0), (-1, 0, 0)]
tetra = [(1, 1, 1), (1, -1, -1), (-1, 1, -1), (-1, -1, 1)]

print(f"H2O (bond dipole 1.51 D): net = {net_dipole(1.51, water):.2f} D")
print(f"CO2 (bond dipole 2.3 D):  net = {net_dipole(2.3, co2):.2f} D")
print(f"CCl4 (bond dipole 1.5 D): net = {net_dipole(1.5, tetra):.2f} D")

H2O (bond dipole 1.51 D): net = 1.85 D
CO2 (bond dipole 2.3 D):  net = 0.00 D
CCl4 (bond dipole 1.5 D): net = 0.00 D

The value of about 1.5 D for each O–H bond dipole in water was chosen to reproduce the measured molecular dipole; the calculation shows how geometry turns two bond dipoles into one molecular dipole, and how symmetry cancels them in CO₂ and CCl₄. Polarity governs solubility (“like dissolves like”), boiling points (lesson 12) and how molecules respond to electric fields. A microwave oven heats food because the oscillating field makes polar water molecules rotate, converting the energy into heat.

Valence bond theory

Lewis structures and VSEPR say nothing about where the shared electrons are in terms of orbitals. Valence bond (VB) theory, developed by Walter Heitler, Fritz London and Linus Pauling in the late 1920s and 1930s, says a covalent bond forms when an orbital on one atom overlaps an orbital on another and the two orbitals share a pair of electrons with opposite spins. The greater the overlap, the stronger the bond.

Overlap along the line between the nuclei gives a sigma (σ) bond, with electron density concentrated on the internuclear axis. Side-by-side overlap of parallel p orbitals gives a pi (π) bond, with electron density above and below the axis. A single bond is one σ bond; a double bond is one σ and one π; a triple bond is one σ and two π. Pi bonds prevent rotation about a double bond, which is why cis and trans isomers of compounds such as 2-butene are different, stable substances.

Hybridization

VB theory runs into a problem with methane. Carbon's ground-state valence configuration is 2s² 2p², with only two unpaired electrons, yet carbon forms four identical bonds pointing to the corners of a tetrahedron. Pauling's solution was hybridization: the 2s orbital and three 2p orbitals mix to form four equivalent sp³ hybrid orbitals, each pointing toward a tetrahedral corner. Each hybrid holds one electron and overlaps a hydrogen 1s orbital to form a C–H σ bond.

Different mixtures give different geometries. The number of hybrid orbitals equals the number of atomic orbitals mixed, and the hybridization of a central atom matches its VSEPR electron-group geometry.

In ethene, H₂C=CH₂, each carbon is sp² hybridized. Its three sp² hybrids form σ bonds to two hydrogens and the other carbon, in a plane at 120°. The remaining unhybridized p orbital on each carbon stands perpendicular to the plane, and these two p orbitals overlap side by side to form the π bond. In ethyne (acetylene), HC≡CH, each carbon is sp hybridized, giving a linear molecule with one σ and two π bonds between the carbons. For the expanded-octet molecules, the sp³d and sp³d² labels are still widely taught as a convenient description of shape, although modern calculations show that d orbitals contribute little.

Molecular orbital theory

VB theory keeps electrons localized between pairs of atoms, which works well for most molecules but fails in some striking cases. The simplest is oxygen. The Lewis structure of O₂, O=O with two lone pairs on each atom, has all its electrons paired, so O₂ should be diamagnetic. Yet liquid oxygen is visibly attracted to a magnet: it is paramagnetic, with two unpaired electrons. Molecular orbital (MO) theory explains this.

In MO theory, atomic orbitals on all the atoms combine to form molecular orbitals that belong to the molecule as a whole. Two atomic orbitals combine to give two molecular orbitals. Adding them in phase gives a bonding orbital, lower in energy than the atomic orbitals, with electron density between the nuclei. Combining them out of phase gives an antibonding orbital (marked with an asterisk), higher in energy, with a node between the nuclei. Electrons fill molecular orbitals following the same rules as atomic orbitals: lowest energy first, two electrons per orbital with opposite spins, and Hund's rule for orbitals of equal energy.

For H₂, the two electrons fill the σ1s bonding orbital: bond order 1. For He₂, four electrons fill both σ1s and σ*1s: bond order 0, so He₂ does not exist as a stable molecule, as observed. For the second-period diatomic molecules, the 2s and 2p orbitals combine to form σ2s, σ*2s, two π2p, σ2p, two π*2p and σ*2p orbitals. For B₂, C₂ and N₂, mixing between the 2s and 2p orbitals pushes σ2p above π2p; for O₂ and F₂, σ2p lies below π2p.

# Fill molecular orbitals for second-period homonuclear diatomics and report bond order and unpaired electrons
ORDER_BCN = [("s2s", 2, True), ("s*2s", 2, False), ("p2p", 4, True), ("s2p", 2, True), ("p*2p", 4, False), ("s*2p", 2, False)]
ORDER_OF = [("s2s", 2, True), ("s*2s", 2, False), ("s2p", 2, True), ("p2p", 4, True), ("p*2p", 4, False), ("s*2p", 2, False)]
VALENCE = {"Li2": 2, "Be2": 4, "B2": 6, "C2": 8, "N2": 10, "O2": 12, "F2": 14, "Ne2": 16}

def analyse(mol):
    order = ORDER_OF if mol in ("O2", "F2", "Ne2") else ORDER_BCN
    left, bonding, anti, unpaired = VALENCE[mol], 0, 0, 0
    for name, cap, is_bonding in order:
        n = min(cap, left)
        left -= n
        if is_bonding: bonding += n
        else: anti += n
        orbitals = cap // 2                        # 1 for sigma, 2 for the pi pair
        unpaired += n if n <= orbitals else 2 * orbitals - n
    return (bonding - anti) / 2, unpaired

for mol in VALENCE:
    bo, up = analyse(mol)
    mag = "paramagnetic" if up else "diamagnetic"
    print(f"{mol:4s} bond order = {bo:3.1f}, unpaired electrons = {up}, {mag}")

Li2  bond order = 1.0, unpaired electrons = 0, diamagnetic
Be2  bond order = 0.0, unpaired electrons = 0, diamagnetic
B2   bond order = 1.0, unpaired electrons = 2, paramagnetic
C2   bond order = 2.0, unpaired electrons = 0, diamagnetic
N2   bond order = 3.0, unpaired electrons = 0, diamagnetic
O2   bond order = 2.0, unpaired electrons = 2, paramagnetic
F2   bond order = 1.0, unpaired electrons = 0, diamagnetic
Ne2  bond order = 0.0, unpaired electrons = 0, diamagnetic

The results match experiment. O₂ has bond order 2, consistent with its double bond, and two unpaired electrons in the π*2p orbitals, explaining its paramagnetism. N₂ has bond order 3, matching its very strong triple bond. B₂ is paramagnetic, which is evidence for the reversal of the σ2p and π2p levels in the lighter diatomics. Be₂ and Ne₂, with bond order 0, are not stable molecules. MO theory also explains why removing an electron from N₂ weakens its bond (a bonding electron is lost), while removing one from O₂ strengthens it (an antibonding electron is lost).

Each theory has its place. Lewis structures and VSEPR are quick and predict shapes reliably. Valence bond theory with hybridization gives an intuitive picture of localized bonds that organic chemists use every day. Molecular orbital theory is more rigorous, handles delocalized electrons and magnetism naturally, and underlies the computational chemistry used throughout modern research; CHEM-241 and CHEM-271 develop it further.