J. Chem. Sci. Vol. 128, No. 10, October 2016, pp. 1519–1526. c Indian Academy of Sciences. Special Issue on CHEMICAL BONDING DOI 10.1007/s12039-016-1160-7 PERSPECTIVE ARTICLE Bonding and Reactivity Patterns from Electrostatic Landscapes of Molecules SHRIDHAR R GADREa,b,∗ and ANMOL KUMARa aDepartment of Chemistry, Indian Institute of Technology, Kanpur, Uttar Pradesh 208 016, India bInterdisciplinary School of Scientific Computing, Savitribai Phule Pune University, Pune, Maharashtra 411 007, India e-mail: [email protected] MS received 28 July 2016; revised 11 August 2016; accepted 12 August 2016 Abstract. The topographical analysis of molecular electron density (MED) and molecular electrostatic poten- tial (MESP) offers insights into the bonding and reactivity patterns through the critical points (CPs) of these scalar fields. The MESP is found to be particularly useful for describing sites of electrophilic attack and weak intermolecular interactions. MESP is also shown to clearly distinguish between the lone pairs and π- delocalization. The concept of atoms in molecules (AIM) which has so far been primarily based on the gradients of MED, has recently been extended via the use of MESP. The portrayal of AIM through MESP clearly reveals the electron rich atoms in the molecule and also provides the details of the preferred direction of approach of an electrophile. This perspective briefly summarizes the prominent features of MESP topography and provides a future outlook. Keywords. Molecular electrostatic potential; molecular electron density; critical points; lone pairs; atoms- inmolecules. 1. Preamble interpretative tools is felt even more acutely. Quan- tum chemists have engaged themselves in developing A molecular system comprising of N electrons is such methods for the analysis of complicated molecular described by its wave function, dependent on 3N spa- wave functions. Here are some examples. tial co-ordinates apart from the spin variables. The dif- (i) The charge density bond order matrix3 in the ficulties in even listing out such a wave function as a Hückel theory function of 3N spatial co-ordinates numerically were (ii) Population analysis developed by Mulliken4 highlighted1 by Hartree’s historical remark that ‘if we (iii) The frontier orbitals introduced by Fukui5 were to print the wave function values for the ground (iv) Density matrices and natural orbitals systemati- state of the iron atom with sufficiently small intervals cally developed and popularised by Löwdin.6 in all the electronic coordinates, we should require a whole library to house the books in which they were printed’. There are much larger practical difficulties 2. Molecular Electron Density (MED) and its involved in the tabulation and analysis of the wave func- Topography tion of a large molecular system. However, as succinctly put by Coulson,2 the aim of a theoretical chemist is to The development of density functional theory (DFT) obtain ‘primitive patterns of understanding’ offered by by Hohenberg and Kohn7 introduced a revolutionary such a complicated, multi-dimensional wave function idea of employing the three-dimensional electron den- and not to get lost in the ‘horde of numbers gushing out sity ρ(r), as a basic variable, in place of the multi- of a computer’. With the advents in computational hard- dimensional wave function ψ(x1, x2, ......xN).Here ware and software, the computation of accurate molecu- xi denotes the combined space-spin co-ordinates of the lar wave function has become practically feasible in the electron i, with N being the number of electrons. Within last few decades. Due to this, the need for developing DFT, the properties of the ground state of a many- electron system emerge as functionals of the corre- sponding electron density, ρ(r). The three-dimensional ∗For correspondence scalar field of the molecular electron density (MED), Celebrating 100 years of Lewis Chemical Bond ρ(r), thus assumes a fundamental role. The entire 1519 1520 Shridhar R Gadre and Anmol Kumar edifice of density functional theory (DFT) was erected MED, viz.,(3,+1) and (3, −1). For a lucid introduction on the basis of the theorems of Hohenberg and Kohn,7 to this subject, see the book by Bader16 and a mono- followed up by the work of Kohn and Sham.8 Several graph by Gadre and Shirsat17 as well as a semi-popular energy density functionals have become available9 12 in article by Stewart.18 Bader and co-workers have car- the last 30 years or so for accurately describing the elec- ried out extensive topographical investigations13 16 on tronic structure of atoms and molecules. The search for MED since 1970. The highlights of Bader’s atoms-in- the elusive Hohenberg-Kohn functional is still continu- molecules (AIM) studies can be briefly summarized as ing, with a hope that highly accurate practical computa- follows. tion of the electronic structure of atoms and molecules would be feasible in the near future. (i) A pair of bonded atoms of a molecule always − Methods for the interpretation of the features of exhibits a (3, 1) saddle point between them. MED, such as chemical bonds, lone pairs, π delocalisa- Such a saddle point was called by Bader as a tion etc., were pioneered13 15 by Bader and co-workers. bond critical point (BCP). A unique gradient This work culminated into a treatise, viz., Atoms in path traversing through BCP connecting the two Molecules: A Quantum Theory by Bader.16 The MED, bonded atoms is termed as the “Atomic Inter- action Line” (AIL) or “Bond Path” in the AIM ρ(r), extracted for a many electron wave function ψ(x1, x ,......x ) is defined by the equation picture. A network of such AILs of the whole 2 N molecule is known as the molecular graph. 2 3 3 ρ(r) = N |(x1, x2, ..., xN )| d r2...d rN (ii) A ring structure in a molecule shows up as a = σ x1 x (3, +1) saddle point, called as a ring saddle point (1) (RCP). Here, xi stands for combined space-spin co-ordinates (iii) A cage structure appears as a (3, +3) cage of the electron i, N is the number of electrons and critical point (CCP), viz., a minimum in MED. denotes the summation over the spin co-ordinates of all (iv) The MED exhibits maxima at the nuclear sites. the electrons. The basic bonding features of a molecule However, there are some molecules exhibit- were brought out by Bader13 16 in terms of the critical ing non-nuclear maxima.16,19 Such maxima are points (CP) of the corresponding MED. A critical point shown by Li2,C2,Na4, etc. (CP) of a function f(x1, ···xn) is defined as a point, P, (v) Bader found that the value of MED, ρb at a bond where for i = 1ton, saddle point may be used to define the order n, ∂f of the corresponding bond by the relation ∇i f(x1,x2,...,xn)|P = = 0(2) ∂xi P n = exp[A(ρb − B)] (4) A further characterisation of the CP at the point P is done by inspecting the nature of the eigenvalues of the where A and B are fitting parameters. The eigen- values of the Hessian matrix at the BCP can be corresponding Hessian matrix, Hij where used to define the bond strain, bond ellipticity, ∂2f = etc.16 Hij ∂ ∂ (3) xi xj P (vi) Most fundamentally, AIM defines the existence of atoms in a molecule based on the properties The rank of a CP is the number of non-zero eigen- of gradient of MED. An atom in the molecule values. Thus, for the three-dimensional scalar field of is defined by a region of molecular space which MED, the maximum rank equals 3. A CP is said to be is limited by zero flux surface of gradient of degenerate if at least one of the eigenvalues is equal MED, which obeys the Equation ∇ρ(r).dS = to 0. Thus, for a nondegenerate CP, the rank, R equals 0. Such a region of molecular space is called 3. The signs of the eigenvalues determine the signa- the atomic basin. The set of gradient paths ture of the CP, for instance for a nondegenerate CP of MED originating in an atomic basin termi- that corresponds to a maximum, all the eigenvalues are nates at the enclosed nucleus.16 However, as the negative. Thus such a CP is labelled as (3, −3). In zero flux surface of ∇ρ(r) does not enclose an general, a CP is labelled as (R, σ), where σ , the signa- atom completely, Bader suggested use of low ture stands for the excess number of positive eigenval- valued-isosurface of MED for limiting the size ues over the negative ones. A nondegenerate minimum of atomic basins. carries a label of (3, +3). If some of the eigenvalues are positive and the others are negative, such a CP is termed Some of the features of MED topography are illus- as a saddle point. There are two types of nondegenerate trated below in Figure 1 with the examples of three saddle points for a 3-dimensional scalar field such as molecules, viz., methanol, cycopropane and cubane. Bonding Patterns and Electrostatic Landscapes 1521 Figure 1. Critical points (CPs) of the electron density for (a) CH3OH, (b) C3H6 and (c) C8H8 molecules. The grey, green and red dots denote (3,−1), (3,+1) and (3,+3) CPs. See text for details. The MED and molecular electrostatic potential (MESP) clusters. Contours of MED are projected in the planes values for all the illustrations in this perspective are containing the atoms of these clusters to bring out their computed using the Gaussian09 suite of programs20 unusual pattern. The two bond CPs can be observed 21 and DAMQT package, employing MP2/6-31+G(d) between the two Li atoms in the Li2 molecule (cf.