Molecular geometry

Molecular geometry describes how a molecule's atoms are arranged in three-dimensional space. This includes the molecule's overall shape, along with specific measurements like bond lengths, bond angles, torsional angles, and any other parameters that define each atom's location. The geometry of a molecule affects many of its properties, such as how it reacts, its polarity, what state of matter it takes, its color, magnetism, and biological activity. The angles between bonds around a given atom are largely local features, meaning they don't depend much on the rest of the molecule and can be treated as roughly transferable.

To determine molecular geometry, scientists use spectroscopic and diffraction methods. Infrared, microwave, and Raman spectroscopy reveal geometry through the vibrational and rotational absorption patterns they detect. X-ray crystallography, neutron diffraction, and electron diffraction work on crystalline solids, measuring distances between nuclei and electron density concentrations. For small molecules in the gas phase, gas electron diffraction is used. NMR and FRET techniques provide complementary data, such as relative distances, dihedral angles, and connectivity. Low temperatures give the most reliable results, since higher temperatures cause the molecule to average over many accessible geometries. Larger molecules often have multiple stable geometries, or conformational isomers, that are close in energy on the potential energy surface. Geometries can also be calculated with high accuracy using ab initio quantum chemistry methods. A molecule's geometry may differ depending on whether it is in a solid, in solution, or in a gas. The position of each atom is set by the chemical bonds connecting it to its neighbors, and geometry is described by bond lengths (between two atoms), bond angles (among three atoms), and torsion or dihedral angles (across three consecutive bonds).

Because atomic motions follow quantum mechanics, "motion" must be defined quantum mechanically. External motions like translation and rotation barely change the molecule's geometry (rotation has a negligible effect via Coriolis forces and centrifugal distortion). The third type of motion is molecular vibration—internal movements like bond stretching and angle changes. These vibrations are approximately harmonic, and atoms oscillate around their equilibrium positions even at absolute zero, where they exhibit zero-point motion. At absolute zero, the wavefunction of a vibrational mode is not a sharp peak but roughly a Gaussian. At higher temperatures, vibrational modes can be thermally excited (classically, the molecules "vibrate faster"), but they still oscillate around the molecule's recognizable geometry. The probability of thermal excitation is given by the Boltzmann factor β = exp(−ΔE/kT), where ΔE is the excitation energy. At 298 K (25 °C), typical values are: β = 0.0890 for ΔE = 500 cm⁻¹, β = 0.0080 for ΔE = 1000 cm⁻¹, and β = 0.0007 for ΔE = 1500 cm⁻¹. (The reciprocal centimeter is an energy unit used in infrared spectroscopy; 1 cm⁻¹ equals 1.23984×10⁻⁴ eV.) So, for an excitation energy of 500 cm⁻¹, about 8.9% of molecules are thermally excited at room temperature. For context, water's lowest vibrational excitation—its bending mode—is about 1600 cm⁻¹, meaning less than 0.07% of water molecules vibrate faster at room temperature than at absolute zero.

Rotation, as noted, has little effect on geometry, but it is thermally excited at relatively low temperatures compared to vibration. Classically, more molecules rotate faster at higher temperatures, with higher angular velocity and momentum. Quantum mechanically, more rotational eigenstates with higher angular momentum become populated as temperature rises. Typical rotational excitation energies are a few cm⁻¹. Many spectroscopic experiments produce broadened results because they average over rotational states, and extracting geometries from high-temperature spectra is often difficult because more rotational states are involved in the averaging. Reliable molecular geometries from spectroscopy are best obtained near absolute zero, since higher temperatures populate too many rotational states.

Molecules are typically held together by covalent bonds—single, double, or triple bonds—where a bond is a shared pair of electrons. (Another bonding type, ionic bonding, involves a positive cation and a negative anion.) Molecular geometry is specified using bond lengths, bond angles, and torsional angles.