Non-covalent interactions index

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]

NCI representation in 3D and 2D of a three water molecules cluster

NCI representation in 3D and 2D of a six water molecules cluster

NCI representation in 3D and 2D of an eight water molecules cluster

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]

References

1. Pastorczak, E. and Corminboeuf, C. (2017) ‘Perspective: Found in translation: Quantum chemical tools for grasping non-covalent interactions’, The Journal of Chemical Physics. AIP Publishing LLC , 146(12), p. 120901. doi: 10.1063/1.4978951.
2. Johnson, E. R. et al. (2010) ‘Revealing Noncovalent Interactions’, Journal of the American Chemical Society. American Chemical Society, 132(18), pp. 6498–6506. doi: 10.1021/ja100936w.
3. Contreras-García, J., Yang, W. and Johnson, E. R. (2011) ‘Analysis of Hydrogen-Bond Interaction Potentials from the Electron Density: Integration of Noncovalent Interaction Regions’, The Journal of Physical Chemistry A. American Chemical Society, 115(45), pp. 12983–12990. doi: 10.1021/jp204278k.
4. Saleh , G. et al. (2012) ‘Revealing Non-covalent Interactions in Molecular Crystals through Their Experimental Electron Densities’, Chemistry - A European Journal, 18(48), pp. 15523–15536. doi: 10.1002/chem.201201290.
5. Perdew, J. P., Burke, K. and Ernzerhof, M. (1996) ‘Generalized Gradient Approximation Made Simple’, 77. Available at: https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.77.3865
6. Hohenberg, P. and Kohn, W. (1964) ‘Inhomogeneous Electron Gas’, Physical Review. American Physical Society, 136(3B), pp. B864–B871. doi: 10.1103/PhysRev.136.B864.