The Non-Covalent Interactions index, commonly referred to as simply Non-Covalent Interactions (NCI) is a visualization index based in the Electron density (ρ) and the reduced density gradient (s). It is based on the empirical observation that Non-covalent interactions can be associated with the regions of small reduced density gradient at low electronic densities. In quantum chemistry, the non-covalent interactions index is used to visualize non-covalent interactions in three-dimensional space.^[1]

Its visual representation arises from the isosurfaces of the reduced density gradient colored by a scale of strength. The strength is usually estimated through the product of the electron density and the second eigenvalue (λ_H) of the Hessian of the electron density in each point of the isosurface, with the attractive or repulsive character being determined by the sign of λ_H. This allows for a direct representation and characterization of non-covalent interactions in three-dimensional space, including hydrogen bonds and steric clashes.^[2][3] Being based on the electron density and derived scalar fields, NCI indexes are invariant with respect to the transformation of molecular orbitals. Furthermore, the electron density of a system can be calculated both by X-ray diffraction experiments and theoretical wavefunction calculations.^[4]

The reduced density gradient (s) is a scalar field of the electron density (ρ) that can be defined as

${\displaystyle s(\mathbf {r} )={\frac {\left|\nabla \rho (\mathbf {r} )\right|}{2(3\pi ^{2})^{1/3}\rho (\mathbf {r} )^{4/3}}}}$

Within the Density Functional Theory framework the reduced density gradient arises in the definition of the Generalized Gradient Approximation of the exchange functional.^[5] The original definition is

${\displaystyle s(\mathbf {r} )={\frac {\left|\nabla \rho (\mathbf {r} )\right|}{2k_{F}\rho (\mathbf {r} )}}}$

in which k_F is the Fermi momentum of the free electron gas.^[6]

The NCI was developed by Canadian computational chemist Erin Johnson while she was a postdoctoral fellow at Duke University in the group of Weitao Yang.