Understanding and visualizing the distribution of electrons in a molecule is essential for rationalizing molecular structure, reactivity, and other properties.
The electronic properties of a molecule are calculated using density-functional theory (DFT) to produce a wavefunction that describes the distribution and energies of the electrons in a system. From this wavefunction, we can derive various properties of the molecule and rationalize the structure and behavior.
Multipoles provide a systematic way to describe the charge distribution of a molecule.
The dipole is a vector in the direction of the polarization of the molecule (from positive to negative), with the magnitude indicating the strength of the total polarization. The quadrupole is a 3x3 tensor indicating the second-order distribution of charge within the molecule. The diagonal elements describe elongation or compression of charge along the principal axes, while the off-diagonal elements describe coupling or asymmetry between different axes.
The electron density can be partitioned among the atoms in various manners. Mulliken charges were initially popular in quantum chemistry, but have been surpassed by Löwdin and other charge partitioning methods that better account for basis set overlap.
Bond orders provide a quantitative measure of the bonding interaction between atoms, reflecting the number of effective electron pairs shared. Just like with atomic-charge partitioning, bond orders can be calculated in a variety of manners. Wiberg and Mayer assign bond orders differently, with the latter often being preferred due to better description of open-shell species.
Isosurfaces are used to visualize the distribution of a value across 3D space. An isosurface represents the boundary where a given value equals a specific cutoff value. Everything inside an isosurface has a scalar value greater than the cutoff. Everything outside an isosurface has a scalar value less than the cutoff.
Electron density isosurfaces indicate the region of space where electrons are most likely to be found. The electron density isosurface can be used to visualize the shape and extent of the electron cloud in a molecule. For open-shell species, it can also be useful to visualize the individual alpha and beta electron densities, and the difference between the two (referred to as the spin density). One can plot the electrostatic potential on top of the spin-density isosurface, showing regions of net negative and positive electrostatic potential. Areas of net negative electrostatic potential are most likely to be attacked by electrophiles, while regions of net positive electrostatic potential are most likely to be attacked by nucleophiles.
Molecular orbital isosurfaces indicate the region of space where an electron pair or individual electron is most likely to be found. Traditionally, the orbitals are colored to indicate the sign of the orbital, though the sign can trivially be inverted as needed without affecting the wavefunction.