Chemistry · Chemical bonding · Year 1

The molecule: orbitals and symmetry

Why does liquid oxygen hang between the poles of a magnet while liquid nitrogen runs off?

The situation

In nitrogen plants such as Navoiyazot, an air separation unit cools air to about −190 °C until it liquefies, then distils it into nitrogen and oxygen. In the plant laboratory, a technician pours both liquids between the poles of a powerful electromagnet: liquid nitrogen falls straight through, but pale blue liquid oxygen hangs between the poles. Both molecules are diatomic, and both gases are colourless — so where is the difference? The answer lies in the molecular orbitals that hold the electrons, and in the shape and symmetry of the molecule.

Topic

Molecular structure: orbitals, symmetry and polarity. Air separation unit at Navoiyazot: liquid oxygen hangs between the poles of a strong magnet, while liquid nitrogen runs off. Why? The answer is molecular…

Key ideas

MO diagram
Point groups
Dipole and refraction

Formulas

ψ = c_{A} φ_{A} \pm c_{B} φ_{B}

MO LCAO: a molecular orbital is a linear combination of atomic orbitals; “+” is bonding, “−” is antibonding (with a node).

n_{bond} = \frac{N_{bonding} - N_{antibonding}}{2}

Bond order: N₂ — 3, O₂ — 2, F₂ — 1, He₂ — 0. As the order increases, the bond becomes shorter and stronger.

E_{k} = α + 2 β \cos \frac{k π}{n + 1}

Hückel method: levels of a linear π-system of n carbon atoms ( $β < 0$ ); for a ring, $E_{k} = α + 2 β \cos (2 π k / n)$ .

R_{M} = \frac{n^{2} - 1}{n^{2} + 2} \cdot \frac{M}{ρ} = \frac{4 π}{3} N_{A} α^{'}

Lorentz–Lorenz: molar refraction (cm³/mol) is a measure of electronic polarisability; it is additive, built up from the refractions of atoms and bonds.

\frac{ε - 1}{ε + 2} \cdot \frac{M}{ρ} = \frac{N_{A}}{3 ε_{0}} (α + \frac{μ^{2}}{3 k T})

Clausius–Mossotti–Debye: polarisation has an electronic/atomic part (α) and an orientational part from permanent dipoles (μ²/3kT).

Theory

The covalent bond: saturability, directionality, hybridisation

A covalent bond is saturable — an atom forms only as many bonds as it has unpaired electrons (or an empty orbital and lone pairs) — and directional — the orbitals overlap in a definite direction. In valence bond (VB) theory a bond is the overlap of two atomic orbitals: σ along the axis, π side by side. The s and p orbitals of the central atom mix to give equivalent hybrid orbitals: sp³ (s-fraction 1/4, 109.5°, CH₄), sp² (1/3, 120°, ethene — the remaining p orbital gives the π bond), sp (1/2, 180°, ethyne — two p orbitals give two π bonds, which is why C≡C is a triple bond). The angle between equivalent hybrids is $\cos θ = - a / (1 - a)$ ( $a$ is the s-fraction). As the bond multiplicity (order) increases, the length decreases and the energy increases; a difference in electronegativity makes the bond polar, and the shift of the electron cloud in an external field is polarisability. VSEPR rules and molecular shapes are covered in the “Types of chemical bonding” experiment; here, in symmetry mode, we look at linear (CO₂, C₂H₂), planar (BF₃, C₆H₆) and three-dimensional (CH₄, SF₆) molecules.

Molecular orbital theory

In the MO LCAO method the electrons occupy orbitals that belong to the whole molecule. Combining two atomic orbitals gives two MOs: a bonding one (the phases match, density builds up between the nuclei, energy drops) and an antibonding one (a nodal plane between the nuclei, energy rises more). The electrons fill them according to the Pauli principle and Hund's rule. In period 2 molecules from Li₂ to N₂ the 2s and 2p energies are close, so they mix and lift σ2p above π2p — which is why B₂ is paramagnetic and C₂ is diamagnetic; in O₂ and F₂ the 2s–2p gap is large and the order is the usual one. O₂: (σ2s)²(σ*2s)²(σ2p)²(π2p)⁴(π*2p)² — two unpaired electrons, paramagnetic, bond order 2. This is a fact that the Lewis formula (O=O, all electrons paired) cannot explain, and MO theory predicts it. In heteronuclear CO, NO and HF the bonding orbitals are shifted towards the more electronegative atom. The HOMO–LUMO gap (the energy gap) determines reactivity and the absorption spectrum. The potential curve of H₂ and spectra are covered in the experiment “The molecule: bonding and spectra”, and the shapes of atomic orbitals in “Orbitals of the hydrogen atom”.

π-systems: the Hückel method and aromaticity

In molecules with conjugated double bonds the σ-framework and the π-electrons are treated separately. In the Hückel method (the simplest semi-empirical method) each carbon contributes one $p_{z}$ orbital, $H_{i i} = α$ (Coulomb integral), $H_{i j} = β$ for neighbouring atoms (resonance integral, $β < 0$ ), and the rest and the overlap are 0. Solving the secular equation gives: ethene $α \pm β$ ; butadiene $α \pm 1.618 β$ , $α \pm 0.618 β$ ; benzene $α + 2 β$ , $α + β$ (two levels), $α - β$ (two levels), $α - 2 β$ . Delocalisation energy: butadiene $0.472 β$ , benzene $2 β$ . The heat of hydrogenation of benzene is ≈ −208 kJ/mol and that of cyclohexene ≈ −120 kJ/mol: $3 \cdot (- 120) - (- 208) \approx 150$ kJ/mol of extra stability, so $| β | \approx 75$ kJ/mol. Hückel's rule: a planar ring with $4 n + 2$ π-electrons (benzene — 6) is aromatic and stable; with $4 n$ (cyclobutadiene — 4) it is antiaromatic: two electrons are left unpaired in two non-bonding levels. Computational methods (Hückel, extended Hückel, ab initio) give the bond order, charge distribution, ionisation potential and HOMO–LUMO gap; the MO levels on the stage are also drawn with such a simple model.

Symmetry elements, point groups and crystals

A symmetry operation maps the molecule onto itself; the geometric object that carries it out is an element: a $C_{n}$ axis (rotation by $360 ° / n$ ), a $σ$ plane (reflection; $σ_{h}$ is perpendicular to the principal axis, $σ_{v}$ and $σ_{d}$ contain the axis), a centre $i$ (inversion), $S_{n}$ (rotation + reflection in the perpendicular plane) and $E$ (identity). Operations can be multiplied (carried out one after another): $σ \cdot σ = E$ , $C_{3} \cdot C_{3} = C_{3}^{2}$ , $C_{2} \cdot σ_{h} = i$ ; all the operations of a molecule form a point group: H₂O $C_{2 v}$ (4 operations), NH₃ $C_{3 v}$ (6), BF₃ $D_{3 h}$ (12), CH₄ $T_{d}$ (24), SF₆ $O_{h}$ (48), C₆H₆ $D_{6 h}$ (24), CO₂ $D_{\infty h}$ . The group is found with a flowchart (on the board). Molecules with a centre of inversion, several $C_{n}$ axes or a $σ_{h}$ plane perpendicular to the principal axis are non-polar: a dipole is possible only in the groups $C_{1}$ , $C_{s}$ , $C_{n}$ and $C_{n v}$ . In complexes the geometry is also described by symmetry: octahedron $O_{h}$ , tetrahedron $T_{d}$ , square planar $D_{4 h}$ . In crystals translational symmetry is added to point symmetry: the unit cell repeats throughout the whole lattice; there are 7 crystal systems (cubic, tetragonal, orthorhombic, monoclinic, triclinic, trigonal, hexagonal) and 14 Bravais lattices. There are atomic, molecular, ionic and metallic lattices, and polytypes (SiC, ZnS), in which the layers of one substance are stacked in different ways. Structure is determined by X-ray diffraction — see the experiment “Crystalline and amorphous solids”.

Dipole, polarisability and refraction (lab)

The dipole moment $μ = q l$ (1 D = 3.336·10⁻³⁰ C·m) comes from the displacement of the centres of charge in a molecule; here the vector points from the negative end to the positive end. Bond dipoles add as vectors: in CO₂, BF₃, CCl₄ and trans-C₂H₂Cl₂ the sum is 0, although CO₂ does have a quadrupole moment. In a field a dielectric becomes polarised: electronic (the cloud shifts), atomic (the nuclei shift) and orientational (permanent dipoles turn; $μ^{2} / 3 k T$ — it decreases as temperature rises) polarisation — the Clausius–Mossotti–Debye equation. At the frequency of light only the electronic part can keep up: $ε \to n^{2}$ , and the Lorentz–Lorenz equation gives the molar refraction $R_{M}$ ; it is additive: $R (C H_{2}) \approx 4.62$ , C 2.418, H 1.100, O(OH) 1.525, Cl 5.967, C=C 1.733 cm³/mol. Laboratory: 2–3 drops of liquid are placed on the prism of an Abbe refractometer, the light–dark boundary in the eyepiece is made colourless with the compensator and brought to the cross with the knob, and $n_{D}$ (4 digits) is read from the scale (20 °C). $R_{M}$ is calculated and compared with the additive value; a calibration curve is drawn from $n$ of sugar solutions, and the concentration of an unknown solution is found. The band gap of solids and semiconductors — see the experiment “Bands in solids”.

Intermolecular forces; safety and the model

Van der Waals forces $U = - C_{6} / r^{6}$ : orientation (Keesom, $μ^{4} / 3 {(4 π ε_{0})}^{2} k T$ ), induction (Debye, $2 α^{'} μ^{2} / 4 π ε_{0}$ ) and dispersion (London, $\frac{3}{4} α^{' 2} I$ ). Calculation shows that even in HCl, HBr and HI the main contribution is dispersion; only in water does the orientation contribution dominate. The hydrogen bond (H on F, O or N and a lone pair of a neighbouring molecule) is stronger still: the boiling points of NH₃, H₂O and HF are higher than those of the heavier hydrides in their groups. Ion–dipole interaction dissolves salts in water; donor–acceptor molecular complexes (for example I₂·benzene) form a weak bond. Real gases and the critical state — see the experiment “Real gases”. Safety: liquid nitrogen and oxygen (−196 / −183 °C) cause cold burns on the skin — wear safety goggles and gloves; liquid O₂ forms explosive mixtures with oil, paper and organic substances, so keep it away from flames. Benzene, CCl₄ and chloroform are toxic — do the refractometry in a fume cupboard. The model: the MO levels are schematic (atomic orbital energies and a simple Hückel-type model), and the isosurfaces are drawn with nodeless Slater orbitals; bond dipoles are proportional to Δχ (calibrated on HCl) — lone pairs and induction are not taken into account, so the model μ differs from the experimental one; the magnetic force is shown qualitatively.

Did you know?

In 1966 Robert Mulliken received the Nobel Prize in Chemistry “for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method”.
The unit of dipole moment, the “debye”, is named after Peter Debye; in 1936 he received the Nobel Prize for his work on dipole moments and on X-ray and electron diffraction in gases.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Types of chemical bonding — Grade 8