The independent gradient model (IGM) originally proposed in Phys. Chem. Chem. Phys., 19, 17928 (2017) has been increasingly popular in visual analysis of intramolecular and intermolecular interactions in recent years, and it has many clear advantages over the widely employed noncovalent interaction (NCI) method, such as intrafragment and interfragment interactions can be elegantly isolated and thus separately studied, the isosurfaces are smoother and less jaggy. However, we frequently observed that there is an evident shortcoming of IGM map in graphically studying weak interactions, that is its isosurfaces are usually too bulgy; in these cases, not only the graphical effect is poor, but also the color on some areas on the isosurfaces is inappropriate and may lead to erroneous analysis conclusions. In addition, the IGM method was originally proposed based on promolecular density, which is quite crude and does not take actual electronic structure into account. In this article, we first present a detailed overview of the IGM analysis, and then propose our new variant of IGM, namely IGM based on Hirshfeld partition of molecular density (IGMH), which replaces the free-state atomic densities involved in the IGM method with the atomic densities derived by Hirshfeld partition of actual molecular electron density. This change makes IGM have more rigorous physical background. In addition, we describe some indices defined on the top of IGM or IGMH framework to quantify contributions from various atoms or atom pairs to interaction between specific fragments. A large number of application examples in this article, including molecular and periodic systems, weak and chemical bond interactions, fully demonstrate the important value of IGMH in intuitively understanding interactions in chemical systems. Comparisons also showed that the IGMH usually has markedly better graphical effect than IGM and overcomes known problems in IGM. Currently IGMH analysis has been efficiently supported in our freely available and user-friendly wavefunction analysis code Multiwfn (http://sobereva.com/multiwfn), and a detailed tutorial is presented. We hope that IGMH will become a new popular method among chemists for exploring interactions in wide variety of chemical systems.
Erratum: Steady state thermodynamics for homogeneous chemical systems [J. Chem. Phys. 101, 10 866 (1994)]
