The literature on this scalar field is enriched with the works of Professor Shridhar R. Gadre, Professor Jacopo Tomasi, Professor Peter Politzer, Jane Murray, Professor Robert G. Parr, P. L. A. Popeleier and their groups.

Molecular Electrostatic Potential (MESP), is a three-dimensional (3D) scalar field that has found an important place in chemistry with close connections to chemical concepts such as reactivity, weak intermolecular interactions, Hammet free energy relationships, aromaticity, hydration patterns etc. Several more complex problems involving protein-ligand interactions, drug binding sites, Bronsted acidity all find an interpretative and semi-quantitative explanation through molecular electrostatics. The MESP, V(**r**) is defined as the work done in bringing a unit test non-interacting positive charge from infinity to a point of reference with the molecule. It is expressed (in a.u.) by the following equation.

Here {ZA} are the discrete nuclear charges at position vectors {**R**A}. Rho(**r**) is the quantum mechanical one electron charge density which integrates to the total number of electrons N, that is defined as given below(summation over spins is implicit).

What makes this scalar field interesting, are its maximal and minimal characteristics,succinctly and systematically analyzable using the principles of topography. Topography of 3D scalar fields involves the identification and characterization of the four types of nondegenerate critical points (CP's), i.e. minima denoted as (3, +3), saddles (3, +1) and (3,-1) and maxima denoted as (3, -3) in the (rank, signature) notation.

**Atomic Electrostatic Potential: Nature of the Potential and Theorems**

The topographical nature of electrostatic potential for atoms has been scrutinized by Weinstein and Politzer,who have observed and proved the following characteristics for atomic ESP's. The ESP of a neutral atom is positive throughout and monotonically decays positively in the asymptotes. For a monoatomic cation the nature of the potential is still positive, however the die-off of the function is slower than that for the neutral atom. A monoatomic anion does provide an interesting feature of a minimum for the potential on every ray starting from the nuclear position. All these ray minima together form part of a neutral surface, inside of which the total charge integrates to zero, and the extra charge of the anion lies outside this surface.

**Characteristics of Molecular Electrostatic Potential**

**Nuclear Positions**Every nuclear position has a non-differentiable discontinuity and the electrostatic potential at a nuclear position is negative. A nuclear position is topographically characterized by a pseudomaximum of a homeomorphic function which could be defined at and around the nucleus. The high negative value of MESP at the nuclear position has been related to the total energy.**Absence of Nonnuclear Maxima**(Theorem due to Pathak and Gadre(JCP, 1995) Non-nuclear maxima are absent for MESP. This is understood as a consequence of the Poisson equation as given below (in a.u.).- The non-negativity of the charge density ensures that the sum of the eigenvalues of the Hessian matrix, given by the Laplacian in the RHS of the above equation,remains positive at all non-nuclear sites, thereby excluding the existence of a non-nuclear maximum where all 3 eigenvalues of the Hessian matrix have to be negative.
**Structural features**such as bonds and rings have a topographical manifestation in the MESP via the presence of positive valued (3,-1) and (3,+1) CP's.-
**Negative regions**Every neutral molecule has at least one negative valued minimum (3, +3), which is indicative of a site of electrophilic attack. For e.g. the so called ``rabbit eared'' lone pairs of water molecule have a topographical manifestation as two (3, +3) CP's. The negative region of a neutral molecule is of interest to a chemist and most of the chemical concepts that have been elucidated through an electrostatic viewpoint have connections with negative regions of MESP. The existence of at least one such negative region could be understood via the examination of a multipole expansion of the partitioned charge density centered at a nuclear postition with the assumption that the difference density has to be non-zero atleast at some points.

The above picture shows the negative region of water molecule, blue isosurfaces enclosing the negative valued minima at -0.099 a.u. The grey isosurface is the zero-valued one and the red XYZ 3D contours enclose the positive potential.