Exploring the Gradient Paths and Zero Flux Surfaces of Molecular

Article pubs.acs.org/JCTC

Exploring the Gradient Paths and Zero Flux Surfaces of Molecular Electrostatic Potential Anmol Kumar and Shridhar R. Gadre* Department of Chemistry, Indian Institute of Technology Kanpur, Kanpur 208016, India ABSTRACT: The gradient vector field of molecular electrostatic potential, ∇V(r), has remained relatively unexplored in molecular quantum mechanics. The present article explores the conceptual as well as practical aspects of this vector field. A three-dimensional atomic partition of molecular space has been achieved on the basis of zero flux surfaces (ZFSs) of ∇V(r). Such ZFSs may completely enclose some of the atoms in the molecule, unlike what is observed in density-based atomic partitioning. The demonstration of this phenomenon is elucidated through typical examples, e.g., N2, CO, H2O, H2CO, OF•, :CH2, and NH3BF3, where the electronegative atoms or group of atoms (group electronegativity) exhibits a closed ZFS of ∇V(r) around them. The present article determines an explicit reason for this phenomenon and also provides a necessary and sufficient condition for such a closed ZFS of ∇V(r) to exist. It also describes how the potential-based picture of atoms in molecules differs from its electron density-based analogue. This work further illustrates the manifestation of anisotropy in the gradient paths of MESP of some molecular systems, with respect to CO, •OH, H2O, and H2CO, and points to its potential in understanding the reactivity patterns of the interacting molecules.

■

INTRODUCTION The topographical study of real scalar fields of the molecules, e.g., molecular electron density (MED) and molecular electrostatic potential (MESP), has attracted much attention for scrutinizing chemical features of the molecular systems. The vast information stored in these scalar fields is succinctly understood in terms of the critical points (CPs) in the field. Although the typical analysis of these CPs provides the static information about the bonding situation in the molecules, the associated gradient vector field has the potential to bring out the vector properties of the molecules. Bader and co-workers have exploited MED, ρ(r), along with its gradient, ∇ρ(r), and Laplacian, ∇2ρ(r), to develop the concept of atoms in molecules (AIM).1 Similar exploration of the salient features of MESP2−4 have found profound applications,2,5,6 especially in chemical reactivity. Unfortunately, a detailed discussion on the gradient vector field of MESP is conspicuous by its absence in the literature. The gradient of MESP is equivalent to the internal electric field of the molecule, which portrays the path followed by a test charge to reach to the CPs in the molecule. Hence, a detailed general description of the electric field is expected to enrich the understanding of the dynamical aspects of the chemical reactivity. The gradient vector field of density has been utilized by Bader to develop the concept of partitioning the molecular space into atomic basins. Bader suggested1,7 the exhaustive partitioning of molecular space into nonoverlapping regions (atomic basins) based on the zero flux surfaces of ∇ρ(r). Each nucleus in the molecule is thus surrounded by a zero flux surface (ZFS), and all the trajectories of the gradient path in a © XXXX American Chemical Society

given atomic basin terminate at the corresponding enclosed nucleus. A unique feature of such a partition of MED is the openness of its zero flux surfaces, i.e., none of the atoms in the molecule are bounded by a zero flux surface in all directions. These MED-based atomic basins are shown1 to obey local virial theorem, confirming them to be reasonably isolated subsystems within the molecule for which average properties can be assigned. Because a MED-based partition is atom-centric, the integration of electron density inside an atomic basin provides point charges, termed as the Bader charges. The MED-based AIM has been further employed8,9 to characterize electronegativity, Lewis acidity, basicity, and so forth. The scalar field of MESP is a density-derived real scalar field that stands out for its ability to determine precise information on electron localization10,11 and reactive centers,12,13 apart from conveying the bonding situation in molecules. The MESP, V(r), at a point r, generated by a molecule having N nuclei with nuclear charges {ZA}, located at {RA}, and a continuous electron density ρ(r′), is given by N

V (r) =

∑ A

ZA − |r − RA|

∫

ρ(r′)d3r′ |r − r′|

(1)

As evident from eq 1, the values and features of MESP are governed by both the nuclear and electronic potential. Pioneering works of Tomasi2 and Pullman14,15 demonstrated the use of MESP in identifying the sites of electrophilic attack. Received: January 21, 2016

A

DOI: 10.1021/acs.jctc.6b00073 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation Later, the exhaustive work of Politzer and co-workers5,16−18 proposed and applied several methods for the use of MESP in describing a variety of chemical phenomena, such as bonding, chemical reactivity, inductive effect, resonance, and so forth. In the last two decades, Gadre and co-workers have established the rigorous quantitative applicability of MESP through topographical analysis.3,10−12,19−23 As mentioned previously, the gradient vector field of MESP (negative of the internal electric field due to molecular charge distribution) holds the key to the vector properties of the molecular reactivity pattern. The earlier study of Gadre et al.21 demonstrates the use of ∇V(r) in determining the shape and sizes of polyatomic anions. Sen and Politzer24 had previously developed an electrostatic potential (ESP)-based model for ionic radii of atomic anions. It was shown in their work that atomic anions exhibit a minimum in the MESP at a certain distance, such that the integration of charge in a sphere of radius r centered at the nucleus is equal to the nuclear charge. This idea was nontrivially extended by Gadre et al.,21 wherein the zero flux surface of the gradient of MESP was constructed for polyatomic anions. The directional minima encountered while moving outward from the nuclei of an anionic system were utilized for constructing the zero flux surface of the anions. It was further proven that the net charge possessed by the anion necessarily lies outside this surface as a consequence of Gauss’s theorem.19,21 The theoretical analysis and predictions based on MED and MESP are valuable because these scalar fields are amenable to direct experimental measurements.25 Although the experimental techniques are relatively new, it is now possible to evaluate the scalar field as well as their gradient and Laplacian through high resolution X-ray crystallography. Stewart26 initially designed the mathematical framework to relate the basis functions of self-consistent field atomic orbitals with the X-ray scattering factors. The current experimental determination of electron density is based on the multipolar expansion model, suggested by Hansen and Coppens,27 to fit the experimentally determined structure factor.

Although the above-mentioned experimental studies have indicated the existence of zero flux surface through the pattern of gradient paths, a detailed and general treatment of ∇V(r) is still warranted for providing a comprehensive understanding of the latter vector field. The current article provides a general repertoire of what the gradient vector field of MESP has to offer. We have achieved atomic partitioning of molecular space based on zero flux surfaces of ∇V(r), a concept similar to MED-based AIM. A detailed comparison of both partitioning methods leads us to new insights into the theoretical as well as practical aspects of the MESP-based partitioning scheme. Furthermore, the nonevident distribution of the gradient paths in space (as a consequence of zero flux surfaces) are studied to probe the reactivity patterns of some molecules.

■

METHODOLOGY AND COMPUTATIONAL DETAILS A range of molecular systems, namely neutral molecules, free radicals, and anions, are scrutinized for determining details regarding the gradient vector field of MESP ∇V(r). The density profile required initially for any topographical analysis is obtained from quantum mechanical calculations performed at the MP2/6-311++G(d,p) level of theory employing the Gaussian suite of programs.37 The minimum nature of the concerned structures is ensured through vibrational frequency analysis. A recently developed software package,38 DAMQT 2.1.0, is employed to carry out topographical analysis, such as mapping of CPs and creating atomic basins for MED and MESP of the molecules. The evaluation of MED in the DAMQT package is based on deformed atoms in molecule (DAM) method, wherein the electron density is partitioned and assigned to pseudoatomic centers. The DAM method facilitates the fast and accurate estimation of MED by expressing it in terms of spherical harmonics centered at nuclei and the radial factors. MESP values of the molecule are efficiently estimated employing the DAM procedure, wherein MESP is expressed in terms of effective atomic multipoles and effective inverse multipoles. A more detailed description of the evaluation of the scalar field and its properties within the DAMQT package can be found in a recent article by Kumar et al.38 The MESP of a stable molecule may display both nondegenerate and degenerate CPs, unlike MED wherein only nondegenerate CPs are possible.19 A CP is characterized with a label (R, σ), where R denotes the rank of the corresponding Hessian matrix, namely, the number of nonzero eigenvalues, and signature σ denotes the sum of signs of the eigenvalues. Chemical significance is associated with nondegenerate MESP CPs, such as a (3, −1) CP located between two atoms reflecting bonding,4 whereas a negative valued (3, +3) CP denotes electron localization in the molecule,10 a feature absent in MED. The generation of MED and MESP-based atomic basins is carried out by tracing the gradient path starting from the (3, −1) CPs in the molecule. The GUI-enabled DAMQT package implements MPI libraries for fast and efficient computation of all the topographical analysis mentioned above. The quest of partitioning the molecular space based on the gradient of MESP, ∇V(r), begins with identifying the MESP CPs. By its very definition, a zero flux surface means that the direction of nonzero ∇V(r) at every point on the surface is perpendicular to the surface normal, S, i.e., ∇V(r).dS = 0. It should be noted that the latter is a stricter criterion than ∮ S∇V(r).dS = 0, which may be true for a region of space,

ρat (r) = Pat,coreρat,core (r ) + Pat,valκ 3ρat,val (κ r) +

∑ l = 0, lmax

κ′3 R at, l(κ′3 r)

∑ m = 0, l

Pat, lm ±ylm ± (r/r )

(2)

where ρat,core and ρat,val are spherically averaged Hartree−Fock core and valence densities, P is the population parameters, κ and κ′ are radial scaling factors, Rat,l (r) is a Slater-type radial function, and ylm±(r/r) are spherical harmonic functions in real form. The higher order terms of spherical harmonics enable the aspherical refinement of the electron density. Such models were further devised by Stewart28 for determining electrostatic potential as well as the gradient of MED or MESP from the parametrized electron distribution. Recent advances29−33 have been made in modeling the MESP from the experimentally determined high-resolution single-crystal X-ray diffraction data. Mata et al.34−36 have used the gradients of the experimentally obtained potential to reveal the position of zero-flux surfaces and critical points. The trajectories of gradient path of both MED and MESP were studied in specific two-dimensional planes of different crystals, such as glycyl-aspartic acid dihydrate, L-argininium phosphate monohydrate, phosphoric acid, and so forth. They employed the magnitude of electric field lines at the van der Waals surface of these molecules to specify the nucleophilic and electrophilic influence zones. B


Article

Journal of Chemical Theory and Computation although ∇V(r).dS may not always be zero at the surface points. Because the ∇V(r).dS = 0 criterion is also satisfied trivially for the cases where ∇V(r) = 0, i.e., the CPs of MESP may fall on the zero flux surface. This fact is utilized for constructing the zero flux surface by tracing of ∇V(r) starting from a (3, −1) bond CP. Because the gradient at the CP is zero, the gradient paths are initiated from a slightly displaced location around the CP. The eigenvalues and associated eigenvectors will be denoted as λi and μi, where |λ1| ≥ |λ2| ≥ |λ3|. The initiation points are placed around the (3, −1) CP in the eigenplane (see Figure 1) defined by two eigenvectors (μ2

state Gauss’s theorem central to the discussion. Combining the divergence theorem and Poisson equation, the Gauss’s theorem for gradient of MESP19 may be written as

∮S ∇V . dS = ∫Ω ∇2 Vd3r = 4π ∫Ω ρ(r)d3r

(3)

where Ω denotes the volume enclosed by the surface S. Eq 3 implies that charge inside a surface S would integrate to zero for a zero flux surface. If, for example, the concerned zero flux surface of a molecule encloses a single nucleus, then the electronic charge inside the surface should be equal to the corresponding nuclear charge. The following section demonstrates the ZFS of ∇V(r) for various molecular systems, its interpretations, and its consequences in deciding the distribution of gradient paths.

■

RESULTS AND DISCUSSION Zero Flux Surfaces of MESP Gradients. For an initial orientation, we show three-dimensional zero flux surfaces for some simple molecular systems. Figure 2 shows MESP-based ZFS (ZFS of ∇V(r)) of some molecules (N2, CO, H2O, H2CO, and NH3BF3) and free radicals (OF•, :CH2) along with their CPs. The gray, green, and red dots denote (3, −1), (3, +1), and (3, +3) CPs, respectively. The region containing a nucleus and bounded by an appropriate ZFS may be termed as an MESPbased atomic basin. All the gradient paths originating inside an MESP-based atomic basin would terminate at the enclosed nucleus. The MESP-based ZFS can be seen to be completely enclosing certain nucleus/nuclei in the molecule as against their MED counterparts. It may be noted that no MESP isosurface has been put as a limit on the outer region of space, unlike what is commonly implemented in defining MED-based atomic basins (a density isosurface value of 0.001 au). The ZFS in the N2 molecule symmetrically bisects the bond with none of the nitrogen atoms having a closed atomic basin. This feature is common to all of the homonuclear diatomic molecules because by symmetry both atoms cannot simultaneously possess a closed ZFS. The converse of the latter would mean the absence of any density outside the zero flux surface of

Figure 1. A representation of an eigenplane constructed by two eigenvectors corresponding to two negative eigenvalues of a (3, −1) MESP bond CP located between atoms A and B.

and μ3) corresponding to the two negative eigenvalues of a (3, −1) CP. The sphere-headed arrows (blue and green arrows in Figure 1) denote the eigenvectors associated with negative eigenvalues. An infinitesimally small step in this eigenplane, away from the CP (represented by black dots in Figure 1) ensures the orthogonality of the surface normal to the gradient of MESP. Because the eigenvectors of negative eigenvalues signify the directional maxima (gradient paths terminate at the CP in the direction of such eigenvectors), the points on the zero flux surface are obtained by tracing of the negative of ∇V(r). Before proceeding further into the details of atomic partitioning based on the gradient of MESP, we would like to

Figure 2. MESP-based zero flux surfaces of N2, CO, OF•, H2O, NH3BF3, singlet carbene (:CH2), and H2CO. The color coded dots for the CPs are red: (3, +3), green: (3, +1), and gray: (3, −1). The zero flux surface arising between nitrogen and hydrogen atoms of NH3BF3 is not shown for clarity. See text for more details. C


Article

Journal of Chemical Theory and Computation

anionic systems NO3− and OH− depicting the ZFS of ∇V(r). As is evident from the figure, all three oxygen atoms in NO3− bear closed atomic basins. Moreover, the nitrogen atom as well is surrounded by a large ZFS, thus possessing a closed atomic basin. A similar phenomenon can be observed in the case of the OH− anion. Both the oxygen as well as the hydrogen atoms possess closed MESP-based atomic basins for the OH− anion, unlike the •OH radical, wherein only the oxygen atom possesses a closed ZFS. A degenerate ring of CPs around the oxygen atom in the OH− anion (represented by a ring of blue dots in Figure 3) also distinguishes it from the •OH radical. The latter shows two (3, +3) CPs pertaining to two lone pairs on the oxygen atom. In summary, one may state that all of the atoms in every anion bear closed atomic basins, and the overall ZFS of the anion (known in the literature)21 is a union of the outer boundaries of the ZFSs arising from the atomic partition. Before providing an explanation for the closed nature of MESP-based ZFSs in neutral molecular systems and anions, it would be instructive to compare and contrast the MESP-based atomic basins with their MED counterparts. A Comparison of MED and MESP-Based Zero Flux Surfaces. Figure 4 shows MED-based ZFS for the same set of molecules and radicals considered earlier in Figure 2. Because the MED as well as its gradient die off exponentially toward infinity, Bader suggested the use of a small finite value of density (0.001 au) to limit the size of the atomic basin.1 The limiting boundary of the low MED value has not been shown in the figure. The first noticeable difference of these MED-based ZFSs with their MESP counterparts is the open nature of the atomic basins of the former. Almost all MED-based zero flux surfaces extend to asymptote on all sides. The MED-based atomic partition of the N2 molecule shows a resemblance to that of MESP. However, in the MED-based atomic partition of heteronuclear systems, each atom is partitioned by a ZFS at the bonding site whereas it remains open toward the asymptote. A noteworthy difference is elucidated by examining the location of bond CPs (BCPs) in the cases of MED and MESP, as they necessarily lie on the ZFS, which is shared between bonded atoms. Politzer and co-workers4,39 have used the location of MESP BCP to define the covalent radii of various atoms and have shown them to be compatible with the literature definition of the covalent radius. It is a general observation that BCP in MED is located away from the

both atoms. A heteronuclear diatomic molecule, such as CO, on the other hand displays a closed ZFS around the oxygen atom. Another closed atomic basin of the oxygen atom is exemplified in the case of H2O in Figure 2. Although the fluorine atom is enveloped by a closed ZFS in the OF• radical, the oxygen atom is open. The selective closing of ZFS around one of the bonded atoms may be heuristically explained in terms of the corresponding electronegativity difference. However, three-dimensional stretch of ZFS of singlet carbene (:CH2) shows that none of the atoms possesses any closed ZFS (see Figure 2). Similarly, in the case of the H2CO molecule, the oxygen atom exhibits a closed atomic basin, whereas the carbon atom is not surrounded by ZFS on all sides. Yet another intriguing ZFS can be observed (Figure 2) for the molecular adduct NH3:BF3. The fluorine atoms being more electronegative than boron indeed bear closed atomic basins. However, with the bonding between the boron and nitrogen atom, the closed atomic basin, which has hitherto been observed for more electronegative atoms, interestingly encloses the less electronegative boron atom. Anions are known to possess an overall ZFS that encloses the whole anionic system, and the net charge on the anion resides outside this surface.21 Figure 3 shows the atomic partitioning of

Figure 3. MESP-based zero flux surfaces of NO3− and OH− anions. The color coded dots for the CPs are red: (3, +3), green: (3, +1), and gray: (3, −1). The ring of blue dots around the oxygen atom in OH− anion represents the degenerate CPs. See text for more details.

Figure 4. MED-based zero flux surfaces of N2, CO, OF•, H2O, NH3BF3, singlet carbene (:CH2), and H2CO. See text for details. D


Article


exploited to expedite the process of obtaining the Bader charges, at least for the electronegative atoms. Local PH Relation and the Existence of a Closed Zero Flux Surface. The critical points in the scalar field and the long-range behavior of the field are related to each other by the Poincaré−Hopf (PH) relation. It provides a necessary condition for verifying whether the nondegenerate CPs in a region of space have been satisfactorily mapped. The PH relation expressed in terms of the number of nondegenerate CPs is given as

electronegative atom, whereas MESP BCPs are shifted toward it. This may be checked by mapping MED and MESP BCPs in some covalent and ionic diatomic molecules. Table 1 provides Table 1. Distance Comparison of BCPs for Diatomic Moleculesa molecule

D1med

D1mesp

D2med

D2mesp

ΔEN

H−H F−F Cl−O Br−O O−F C−N C−O Li−H Na−H H−F Li−Cl Li−F Na−F

0.37 0.71 0.84 0.90 0.70 0.74 0.76 0.89 0.91 0.77 1.30 0.98 1.06

0.37 0.71 0.70 0.73 0.65 0.57 0.57 0.63 0.70 0.59 1.10 0.79 0.87

0.37 0.71 0.76 0.84 0.63 0.39 0.38 0.71 1.0 0.15 0.69 0.62 0.92

0.37 0.71 0.89 1.00 0.68 0.56 0.57 0.97 1.20 0.33 0.91 0.81 1.12

0 0 0.28 0.48 0.54 0.49 0.89 1.22 1.27 1.78 2.18 3.00 3.05

n+3 − n+1 + n−1 − n−3 = χ

(4)

where n+3 is the number of (3, +3) CPs, n+1, the number of (3, +1) CPs, n−1, the number of (3, −1) CPs, n−3, the number of (3, −3) CPs, and χ the topological invariant known as the Euler characteristic. In the case of MED, χ is always equal to unity irrespective of the molecule considered. However, for MESP, Leboeuf et al.40 offered an interpretation of χ in terms of positive and negative long-range behavior. The χ for MESP is defined as χ = n− − n+ (5)

a

D1med and D1mesp denote the distances (in Å) of MED-BCP and MESP-BCP from the electronegative atom, respectively. D2med and D2mesp are the distances of the BCPs from the electropositive atom. ΔEN is the difference of electronegativity of the bonded atoms according to the Pauling scale.

where n− is the number of negative MESP islands and n+ is the number of positive MESP islands on an asymptotically large sphere enclosing the concerned molecule. Identifying the fact that the negative value of MESP at asymptote represents asymptotic maxima and vice versa, Roy et al.41 formulated a local PH relation for MESP. The latter may be used for validating the number of CPs inside a spherical region of any size centered at the center of mass of the molecule. The χ for local PH relation of MESP is expressed as χ = no − ni (6)

the distance of BCP from the electronegative/electropositive atoms of the molecule in the case of both MED and MESP. Barring the homonuclear diatomic molecule, the MESP BCP always leans toward the electronegative atom in comparison to the correponding MED BCP. As a result of the difference in the BCP location of MED and MESP, the MESP-based atomic basin of the electronegative atom turns out to be a spatial subset of its MED-based atomic basin. A schematic representation of the situation is provided in Figure 5 showing a diatomic molecule AB, wherein atom B is

where no and ni are the number of islands showing outflux and influx, respectively, of ∇V(r) with respect to radial vector r of the sphere. The outflux of gradient (∇V(r).dr > 0) signifies increasing function value, whereas the influx of gradient (∇V(r).dr < 0) denotes a decrease in the function value. Bond formation between heteroatoms leads to asymmetric distribution of the electron cloud based on the electronegativity difference of the bonding atoms. The fact that electron density inside a closed MESP atomic basin numerically equals the atomic number of the enclosed nucleus, and the above comparison of ZFS in MED and MESP, conveys that the electronegative atom tends to gather a larger number of electrons around itself than its atomic number. Such a phenomenon can also be shown by a polyatomic moiety as a result of electron-withdrawing substituents, as explained normally in terms of the group electronegativity.42 In other words, such electronegative atoms or group of atoms behave locally as anions within the molecule. This reminds us of the theorem for anions,21 which states that they possess a closed ZFS and the net anionic charge resides outside the surface. It is now evident why electronegative atoms possess a closed MESPbased atomic basin. It also explains the closed ZFS around the boron atom in the NH3BF3 molecule, wherein the fluorine atoms pull such a large amount of electron density that the overall density around the boron atom increases with respect to that of the nitrogen atom. However, as can be seen from the cases of singlet carbene (:CH2) and formaldehyde (H2CO), it is not necessary that a closed ZFS appear in any heteronuclear bonding situation. Out of the bonded carbon and hydrogen atoms in both of the molecular systems, none shows a closed

Figure 5. A schematic comparison of ZFS of MED and MESP for a heteronuclear diatomic molecule AB with atom B being more electronegative than atom A. Very low-valued (0.001 au) isosurface of ρ(r) limits the size of MED-based atomic basins. See text for details.

more electronegative than atom A. MED BCP is closer to atom A, whereas the MESP BCP is shifted toward atom B. Because the heteronuclear diatomic molecules tend to possess a closed MESP-based atomic basin around the more electronegative atom, the ZFS of ∇V(r) is seen to envelope atom B. Figure 5 clearly illustrates the MESP atomic basin of atom B to be a subset of its MED counterpart. This general observation can be E


Article

Journal of Chemical Theory and Computation ZFS around them. We realize that the closeness of the zero flux surface in a given molecule is inherently related to the local Poincaré−Hopf relation. The local PH relation can be exploited to obtain necessary and sufficient conditions for the existence of closed ZFS in a region of molecular space. There is a correlation between the disposition of CPs in a given region of space and the existence of closed MESP-based ZFSs. The necessary condition for the existence of closed ZFS in a given region can be stated as follows: if a sphere is placed such that it contains a nucleus or a group of nuclei along with adjoining CPs, the combination of CPs enclosed within the sphere should be equal to unity according to the local PH relation. The sufficiency is provided if the gradients are always pointing outward on the surface of this sphere, i.e., no equals 1 and ni equals 0 in the local PH relation. The sufficiency condition is based on the fact that, if a sphere contains a nucleus (gradients of MESP near the nucleus point towards it), it is necessary to encounter a directional zero of gradient in all of the directions to obtain a change in the gradient direction on the sphere. Thus, the necessary and sufficient condition may be expressed as n+3 − n+1 + n−1 − n−3 = no − ni = 1 − 0 = 1

of neutral molecules have shown the existence of at least one negative-valued MESP minimum. Such a point may be chosen as the originating point of the MESP gradient paths, as the MESP value at these points correlate with the nucleophilicity6,23,43 of the molecule. We discuss below the three-dimensional spread of MESP gradient paths in the open region of the MESP-based atomic partition (outside the closed ZFS) starting from the nondegenerate minima, namely, (3, +3) CPs. Letting the three eigenvalues of the Hessian matrix at the CP be λ1, λ2, and λ3 (λ1 being numerically the largest) with the corresponding eigenvectors μ1, μ2, and μ3, we employ two eigenplanes created by the combination of three eigenvectors of the (3, +3) CPs, namely, (μ1μ2) and (μ1μ3), for observing the pattern of gradient paths. The eigenvector associated with the largest eigenvalue, namely μ1, is common in both eigenplanes. Because the gradient paths would run from negative to relatively positive potential, the asymptote acts as a sink for the gradient paths. Additionally, the nuclei not surrounded by closed zero flux surfaces offer a deeper sink for the gradient paths. However, the gradient paths starting from (3, +3) CPs and terminating at MESP saddle points are found to be generally unique. The pattern of gradient paths is examined below through scrutiny of a few molecular systems. Figure 6 depicts the gradient paths of the CO molecule emanating from (3, +3) CPs in the open region of the MESP

(7)

As an example, in the case of a heteronuclear diatomic molecule, e.g., CO (see Figure 2) the l.h.s. of eq 7 equals unity if a sphere contains the oxygen atom (n−3 = 1), its MESP minimum (n+3 = 1), and the adjoining BCP (n−1 = 1). Thus, there may exist a closed ZFS inside this sphere. However, the criterion of the l.h.s. of eq 7 being equal to unity is fulfilled even if the similar procedure is implemented around carbon, as a MESP minimum exists alongside the carbon atom as well. Furthermore, the oxygen atom being more electronegative tends to possess more electrons around itself than its atomic number. The excess of electron cloud would create the possibility that the gradients of MESP on the surface of the enclosing sphere are all pointing outwards. The latter provides a sufficient condition for the oxygen atom in CO to possess a closed ZFS around itself. If the contradictory example of singlet carbene (:CH2) is analyzed by putting a sphere containing the carbon atom (n−3 = 1), two of its BCPs (n−1 = 2), and the MESP minimum (n−3 = 1), the combination of CPs (on the l.h.s. of eq 7) would equate to two. This explains why the carbon atom in the singlet carbene (:CH2) cannot possess any closed zero flux surface around it. Similar considerations may be applied for the carbon atom in H2CO to explain the openness of the atomic basin of carbon. Gradient Paths of MESP. The previous section discussed the existence and construction of closed zero flux surfaces (ZFS) in MESP around the electron-withdrawing atoms in the neutral molecules. The existence of such a closed surface implies that the gradient paths originating inside such a surface will all lead to the enclosed nucleus. It is further interesting to scrutinize the pattern of gradient paths in the open region of the MESP-based atomic partition. One may remember here that the negative gradient of MESP represents the force acting on a test charge. Assuming the interacting species to be far from each other, the gradient paths can guide the initial direction of the approaching molecular entity. Thus, the study of the gradient paths of MESP may be expected to provide a deeper understanding into the reactivity patterns of the molecules at large distances. However, it may be borne in mind that the polarization and dispersion effects may be significant at closer separation between the interacting moieties. Studies on MESP

Figure 6. Illustration of gradient paths of MESP originating in an eigenplane of the (3, +3) CPs as well as zero flux surfaces (pink mesh) for the CO molecule. See text for details.

partition along with its ZFS. As mentioned before, the CO molecule possesses two (3, +3) CPs, wherein the MESP at the one representing lone pair of carbon atom is more negativevalued than of the oxygen atom. Interestingly, the (3, +3) CP denoting the lone pair of the oxygen atom lies on the ZFS enclosing the oxygen atom. The most interesting feature is that the gradient paths originating at the (3, +3) CP of the oxygen atom in the open region of MESP partition form loops and end at the carbon nucleus. On the other hand, gradient paths originating from the (3, +3) CP alongside the carbon atom traverse a large region of space before terminating at the carbon nucleus. Indeed, they could not terminate at the oxygen atom as a gradient path cannot cross the zero flux surface. Putting a limit on the concerned region of space, it is apparent that the gradient paths emanating from the (3, +3) CP at the carbon end are extended more toward asymptote than those from the (3, +3) CP of the oxygen atom, which shows more confined loops of gradient paths. This brings out the difference in the pattern of gradient paths originating from seemingly similar CPs of the CO molecule. The fact that the gradient paths F


Article


Figure 7. Gradient paths of MESP originating in the two eigenplanes of the (3, +3) CPs for •OH, H2CO, and H2O molecule. The MESP-based ZFSs are not shown for clarity. The gradient paths of molecules placed on the left-hand side originate in the eigenplane constructed by the combination of eigenvectors μ1 and μ2, whereas for the ones placed on the right-hand side, the eigenplane is constructed by the combination of eigenvectors μ1 and μ3. See text for details.

statement, we take up the example of the •OH radical, which contains two lone pair CPs ((3, +3) CPs) on the oxygen atom. Both of the (3, +3) CPs are identical in their MESP parameters and are disposed in the plane containing the •OH radical; thus, the topographical features of •OH follow C2v symmetry. All of the gradient paths originating outside the closed ZFS around the oxygen atom terminate at the H atom. The gradient paths in (μ1μ3) eigenplane, namely σv plane of the molecule with CPs, can be seen to form extended loops as compared to those in the (μ1μ2) plane. The eigenvalues (all the eigenvalues below are in a.u.) associated with the eigenvectors of the (3, +3) CP of •OH are λ1 = 0.0927, λ2 = 0.0237, and λ3 = 0.0167. Similar anisotropic distribution of gradient paths can be observed in H2CO, where the eigenvalues of the corresponding (3, +3) CP are λ1 = 0.0960, λ2 = 0.0151, and λ3 = 0.0123. In the case of H2O, the gradient path originating in the (μ1μ2) eigenplane at

originating from (3, +3) CP at the carbon end are extended more toward asymptote correlates with the reactivity pattern of the CO molecule. In other words, the lone pair of carbon atom can be sensed at a larger distance as compared to the one on the oxygen atom. Yet another interesting feature of anisotropy in threedimensional spread of gradient paths around the (3, +3) CP may be observed in molecular systems such as •OH, H2CO, and H2O. Figure 7 provides the different pattern of gradient paths originating in the two different eigenplanes of the (3, +3) CPs for each of these radicals/molecules. It is observed that the gradient paths originating in the eigenplane created by (μ1, μ3) eigenvectors extend rapidly toward asymptote as compared to those in the eigenplane created by (μ1, μ2). This anisotropy is also seen for those gradient paths that form a loop and terminate at the nucleus in the molecule. To illustrate the above G


Article


atoms to possess a completely closed ZFS. In light of this, a necessary and sufficient condition based on local PH relation has been suggested for a region of space to possess a closed ZFS. The necessary condition states that, if a sphere enclosing a nucleus and its adjoining CPs has to possess a closed MESPbased ZFS inside it, then the combination of CPs (given by the l.h.s. of eq 7) should be equal to unity. Furthermore, the sufficiency condition states that the gradient of MESP at the surface of such a sphere should always point outward. The work has also illustrated the manifestation of anisotropy in the gradient paths of MESP of some molecular systems and has pointed out its potential in understanding the reactivity patterns of the interacting molecules. A connection with the reactivity pattern of the CO molecule is made by noticing that the gradient paths emerging from the lone pair of the carbon atom extend rapidly toward asymptote, whereas the ones emerging from the lone pair of oxygen atom form confined loops. Thus, the effect of the lone pair at the carbon atom is felt at a larger distance by an approaching electrophile. It is further observed that the gradient paths in general emerge anisotropically from the (3, +3) CP as a consequence of different magnitudes of its eigenvalues. Such anisotropic distribution makes the effect of the lone pairs be felt differently in varying directions. In light of this, it is suggestive that the anisotropy in gradient paths of MESP could be employed as an auxiliary tool for understanding the stereodynamics of the interacting molecules. In conclusion, the gradient vector field of MESP is seen to possess enormous potential for understanding molecular properties and reactivity, which has hitherto been restricted to examining only the scalar features of MESP.

(3, +3) CP rapidly terminate at the H atom forming loops, whereas the ones originating in orthogonal plane (μ1μ3) extend to the asymptote. The three eigenvalues of (3, +3) CP of H2O (λ1 = 0.1227, λ2 = 0.0312, and λ3 = 0.0059) reveal a large difference between λ2 and λ3. However, the previously studied CO molecule shows isotropic emergence of the gradient paths around the (3, +3) CPs. The eigenvalues of (3, +3) CP at the carbon end of CO are λ1 = 0.0384, λ2 = 0.0094, and λ3 = 0.0094, whereas those for (3, +3) CP alongside the oxygen atom are λ1 = 0.0163, λ2 = 0.0021, and λ3 = 0.0021. The above details elucidate the role of eigenvectors and associated eigenvalues to understand the anisotropy of the gradient paths originating from the (3, +3) CPs. It is evident from the discussion above that even an isotropic molecular system may possess anisotropy in its gradient path distribution. Furthermore, the anisotropy in the distribution of gradient paths can be predetermined by simply observing the eigenvalues of the eigenvectors at (3, +3) CPs. It can be heuristically stated that an approaching electrophile might sense the host molecule at a farther distance in the plane of space, which has extended gradient paths of the host. This may have implications in determining the chemical dynamics of the molecules that are intrinsically anisotropic. It will form a subject for probing the preferred direction as well as orientation of approach of a molecule toward another. This is expected to provide pictorial insights based on the vector properties of the molecules central for understanding the stereodynamics of the molecular reactions.

■

CONCLUSIONS The current article has elucidated the conceptual as well as practical aspects of the gradient vector field of MESP, ∇V(r), equivalent to the internal electric field of molecules, which has hitherto remained mostly unexplored. A three-dimensional MESP-based atomic partition of molecular space has been achieved on the basis of zero flux surfaces (ZFS) of ∇V(r). Previously, the idea of zero flux surface of the electric field was employed to determine the shapes and sizes of mono- or polyatomic anions.21 It was also proven that the net charge of the anion lies outside the ZFS enclosing the anion, as the charge inside ZFS should be zero according to Gauss’s theorem. The present work has generalized the idea of ZFS of ∇V(r) to obtain an atomic partition in the case of neutral or charged molecular systems. Interestingly, the ZFS of ∇V(r) is found to completely enclose some of the atoms in the molecule, a feature lacking in its MED analogue. The chemical reason for this phenomenon may be attributed to the electronegativity of the atoms. A more electronegative atom tends to pull more electron cloud toward itself in a bonding situation and behaves locally as an anion. This leads to a closed ZFS of ∇V(r) around such atoms, a phenomenon similar to anions. A polynuclear moiety in the molecule may also show an electron pulling effect, causing the whole moiety to be enclosed by a common ZFS. The latter phenomenon has been exemplified by the nature of ZFS in NH3BF3. The boron atom, although less electronegative than nitrogen, shows a closed ZFS due to the large inductive effect of fluorine atoms. The present work has also illustrated that each atom in an anion possesses a closed ZFS, and the overall ZFS of the polyatomic anion is a union of the outer boundaries of ZFSs arising from atomic partitioning. However, some of the neutral molecular systems, such as singlet carbene (:CH2), may intriguingly display none of the

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS A.K. thanks the Council of Scientific and Industrial Research (CSIR), New Delhi for a research fellowship. S.R.G. extends his thankfulness to the Department of Science and Technology (DST), New Delhi for the award of the J. C. Bose National Fellowship.

■

REFERENCES

(1) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford University Press: Oxford, 1994. (2) Scrocco, E.; Tomasi, J. The Electrostatic Molecular Potential as a Tool for the Interpretation of Molecular Properties. In Topics in Current Chemistry, New Concepts II; Springer: Heidelberg, 1973; Vol. 42, pp 95−170. (3) Pathak, R. K.; Gadre, S. R. Maximal and Minimal Characteristics of Molecular Electrostatic Potentials. J. Chem. Phys. 1990, 93, 1770− 1773. (4) Wiener, J. J. M.; Edward Grice, M.; Murray, J. S.; Politzer, P. Molecular Electrostatic Potentials as Indicators of Covalent Radii. J. Chem. Phys. 1996, 104, 5109. (5) Sjoberg, P.; Politzer, P. Use of the Electrostatic Potential at the Molecular Surface to Interpret and Predict Nucleophilic Processes. J. Phys. Chem. 1990, 94, 3959−3961. (6) Suresh, C. H.; Koga, N.; Gadre, S. R. Revisiting Markovnikov Addition to Alkenes via Molecular Electrostatic Potential. J. Org. Chem. 2001, 66, 6883−6890. H


Article


Charge Densities. II. Total Potential. Acta Crystallogr., Sect. A: Found. Crystallogr. 1997, 53, 556−563. (32) Pichon-Pesme, V.; Lachekar, H.; Souhassou, M.; Lecomte, C. Electron Density and Electrostatic Properties of Two Peptide Molecules: Tyrosyl-Glycyl-Glycine Monohydrate and Glycyl-Aspartic Acid Dihydrate. Acta Crystallogr., Sect. B: Struct. Sci. 2000, 56, 728− 737. (33) Arputharaj, D. S.; Hathwar, V. R.; Guru Row, T. N.; Kumaradhas, P. Topological Electron Density Analysis and Electrostatic Properties of Aspirin: An Experimental and Theoretical Study. Cryst. Growth Des. 2012, 12, 4357−4366. (34) Mata, I.; Molins, E.; Alkorta, I.; Espinosa, E. Topological Properties of the Electrostatic Potential in Weak and Moderate N···H Hydrogen Bonds. J. Phys. Chem. A 2007, 111, 6425−6433. (35) Mata, I.; Molins, E.; Espinosa, E. Zero-Flux Surfaces of the Electrostatic Potential: The Border of Influence Zones of Nucleophilic and Electrophilic Sites in Crystalline Environment. J. Phys. Chem. A 2007, 111, 9859−9870. (36) Mata, I.; Espinosa, E.; Molins, E.; Veintemillas, S.; Maniukiewicz, W.; Lecomte, C.; Cousson, A.; Paulus, W. Contributions to the Application of the Transferability Principle and the Multipolar Modeling of H Atoms: Electron-Density Study of LHistidinium Dihydrogen Orthophosphate Orthophosphoric Acid. I. Acta Crystallogr., Sect. A: Found. Crystallogr. 2006, 62, 365−378. (37) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, revision A.02; Gaussian, Inc.: Wallingford, CT, 2009. (38) Kumar, A.; Yeole, S. D.; Gadre, S. R.; López, R.; Rico, J. F.; Ramírez, G.; Ema, I.; Zorrilla, D. DAMQT 2.1.0: A New Version of the DAMQT Package Enabled with the Topographical Analysis of Electron Density and Electrostatic Potential in Molecules. J. Comput. Chem. 2015, 36, 2350−2359. (39) Politzer, P.; Murray, J. S.; Lane, P. Electrostatic Potentials and Covalent Radii. J. Comput. Chem. 2003, 24, 505−511. (40) Leboeuf, M.; Köster, A. M.; Jug, K.; Salahub, D. R. Topological Analysis of the Molecular Electrostatic Potential. J. Chem. Phys. 1999, 111, 4893−4905. (41) Roy, D.; Balanarayan, P.; Gadre, S. R. An Appraisal of Poincaré−Hopf Relation and Application to Topography of Molecular Electrostatic Potentials. J. Chem. Phys. 2008, 129, 174103. (42) Suresh, C. H.; Koga, N. A Molecular Electrostatic Potential Bond Critical Point Model for Atomic and Group Electronegativities. J. Am. Chem. Soc. 2002, 124, 1790−1797. (43) Suresh, C. H.; Koga, N.; Gadre, S. R. Molecular Electrostatic Potential and Electron Density Topography: Structure and Reactivity of (substituted arene) Cr(CO) 3 Complexes. Organometallics 2000, 19, 3008−3015.

(7) Bader, R. F. W.; Keith, T. A.; Gough, K. M.; Laidig, K. E. Properties of Atoms in Molecules: Additivity and Transferability of Group Polarizabilities. Mol. Phys. 1992, 75, 1167−1189. (8) Boyd, R. J.; Edgecombe, K. E. Atomic and Group Electronegativities from the Electron-Density Distributions of Molecules. J. Am. Chem. Soc. 1988, 110, 4182−4186. (9) Popelier, P. L. A. On the Full Topology of the Laplacian of the Electron Density. Coord. Chem. Rev. 2000, 197, 169−189. (10) Kumar, A.; Gadre, S. R.; Mohan, N.; Suresh, C. H. Lone Pairs: An Electrostatic Viewpoint. J. Phys. Chem. A 2014, 118, 526−532. (11) Kumar, A.; Gadre, S. R. On the Electrostatic Nature of Electrides. Phys. Chem. Chem. Phys. 2015, 17, 15030−15035. (12) Gadre, S. R.; Kumar, A. Understanding Lone Pair-π Interactions from Electrostatic Viewpoint. In Noncovalent Forces; Springer, 2015; pp 391−418. (13) Balanarayan, P.; Gadre, S. R. Why Are Carborane Acids so Acidic? An Electrostatic Interpretation of Brønsted Acid Strengths. Inorg. Chem. 2005, 44, 9613−9615. (14) Pullman, A.; Pullman, B. Molecular Electrostatic Potential of the Nucleic Acids. Q. Rev. Biophys. 1981, 14, 289−380. (15) Zakrzewska, K.; Pullman, B. The Electrostatic Potential of a Model Phosphatidylcholine Mono- and Bilayer. FEBS Lett. 1981, 135, 268−272. (16) Politzer, P.; Murray, J. S. The Fundamental Nature and Role of the Electrostatic Potential in Atoms and Molecules. Theor. Chem. Acc. 2002, 108, 134−142. (17) Politzer, P.; Murray, J. S.; Clark, T. Halogen Bonding: An Electrostatically-Driven Highly Directional Noncovalent Interaction. Phys. Chem. Chem. Phys. 2010, 12, 7748−7757. (18) Murray, J. S.; Politzer, P. The Electrostatic Potential: An Overview. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2011, 1, 153−163. (19) Gadre, S. R.; Shirsat, R. N. Electrostatics of Atoms and Molecules; Universities Press: Hyderabad, 2000. (20) Gadre, S. R.; Bendale, R. D. On the Similarity between Molecular Electron Densities, Electrostatic Potentials and Bare Nuclear Potentials. Chem. Phys. Lett. 1986, 130, 515−521. (21) Gadre, S. R.; Shrivastava, I. H. Shapes and Sizes of Molecular Anions via Topographical Analysis of Electrostatic Potential. J. Chem. Phys. 1991, 94, 4384−4390. (22) Kumar, A.; Gadre, S. R.; Chenxia, X.; Tianlv, X.; Kirk, S. R.; Jenkins, S. Hybrid QTAIM and Electrostatic Potential-Based Quantum Topology Phase Diagrams for Water Clusters. Phys. Chem. Chem. Phys. 2015, 17, 15258−15273. (23) Sarmah, S.; Guha, A. K.; Phukan, A. K.; Kumar, A.; Gadre, S. R. Stabilization of Si (0) and Ge (0) Compounds by Different Silylenes and Germylenes: A Density Functional and Molecular Electrostatic Study. Dalton Trans. 2013, 42, 13200−13209. (24) Sen, K. D.; Politzer, P. Characteristic Features of the Electrostatic Potentials of Singly Negative Monoatomic Ions. J. Chem. Phys. 1989, 90, 4370−4372. (25) Bertaut, F. The electrostatic energy of ionic networks. J. Phys. Radium 1952, 13, 499−505. (26) Stewart, R. F. Generalized X-Ray Scattering Factors. J. Chem. Phys. 1969, 51, 4569−4577. (27) Hansen, N. K.; Coppens, P. Testing Aspherical Atom Refinements on Small-Molecule Data Sets. Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 1978, 34, 909−921. (28) Stewart, R. F. On the Mapping of Electrostatic Properties from Bragg Diffraction Data. Chem. Phys. Lett. 1979, 65, 335−342. (29) Lecomte, C.; Ghermani, N.; Pichon-Pesme, V.; Souhassou, M. Experimental Electron Density and Electrostatic Properties of peptides by High Resolution X-Ray Diffraction. J. Mol. Struct.: THEOCHEM 1992, 255, 241−260. (30) Ghermani, N.-E.; Bouhmaida, N.; Lecomte, C. Modelling Electrostatic Potential from Experimentally Determined Charge Densities. I. Spherical-Atom Approximation. Acta Crystallogr., Sect. A: Found. Crystallogr. 1993, 49, 781−789. (31) Bouhmaida, N.; Ghermani, N.-E.; Lecomte, C.; Thalal, A. Modelling Electrostatic Potential from Experimentally Determined I


Exploring the Gradient Paths and Zero Flux Surfaces of Molecular

Recommend Documents