Article pubs.acs.org/JPCA
Lone Pairs: An Electrostatic Viewpoint Anmol Kumar,† Shridhar R. Gadre,*,† Neetha Mohan,‡ and Cherumuttathu H. Suresh*,‡ †
Department of Chemistry, Indian Institute of Technology Kanpur, Kanpur 208016, India Inorganic and Theoretical Chemistry Section, Chemical Sciences and Technology Division, CSIR-National Institute for Interdisciplinary Science and Technology, Trivandrum 695019, India
‡
S Supporting Information *
ABSTRACT: A clear-cut definition of lone pairs has been offered in terms of characteristics of minima in molecular electrostatic potential (MESP). The largest eigenvalue and corresponding eigenvector of the Hessian at the minima are shown to distinguish lone pair regions from the other types of electron localization (such as π bonds). A comparative study of lone pairs as depicted by various other scalar fields such as the Laplacian of electron density and electron localization function is made. Further, an attempt has been made to generalize the definition of lone pairs to the case of cations.
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INTRODUCTION “Lone pair” is one of the most commonly used terms in chemistry. Lone pairs are known to play an indispensible role in describing the structure and reactivity of molecules. Classically, a lone pair is defined as a valence electron pair, bound to a nucleus, not utilized in chemical bonding.1 Despite being in vogue for the past 90 years, the term “lone pair” has escaped a quantitative description. In this light, one would like to address questions such as: Can a lone pair be defined in terms of physical observables? How can one describe properties of a lone pair? Description of Lone Pairs in Bonding Theories. An early attempt to scrutinize the chemistry of the lone pairs of electrons was made by Sidgwick,2 who built upon the octet theory of Lewis to describe the arrangement of electron pairs in multivalent atoms. Later, in a popular theoretical approach, known as valence shell electron pair repulsion (VSEPR) model, Gillespie and Nyholm3 delineated the importance of varying electronic repulsions among shared pair and lone pair of electrons in valence shell orbitals. VSEPR theory still dominates the chemical thinking and is useful for understanding the geometrical arrangement of atoms in molecules. Pauling4 introduced the concept of hybridization of orbitals in 1931, which is used to-date as an essential tool by chemists to explain bonding patterns in molecules including the probable location of lone pair orbitals. Molecular orbital (MO) theory, a popular bonding theory, in its purest form, is based on the maximum amount of delocalization of MOs. Nevertheless, localization methods need to be invoked for describing a lone pair as an electron pair localized in a nonbonded valence MO.5 However, orbitals are not observables. Experiments can only provide © XXXX American Chemical Society
information about observables, e.g., the electron density distribution and the entities derivable from it. Thus, characterization of lone pairs in terms of physical observables, such as the molecular electron density (MED) and molecular electrostatic potential (MESP) is more realistic and appropriate. Description of Lone Pairs in Scalar Fields. The theory of atoms in molecules (AIM),6 pioneered by Bader and coworkers, exploits the scalar field of MED to extract chemical insights. The topology of MED yields reliable depiction of fundamental chemical phenomena such as the chemical bond, molecular structure and the concept of an atom in a molecule. However, topological features of MED remain silent in the regime of nonbonded electrons; i.e., the MED does not exhibit a local maximum in the lone pair region. It is the Laplacian of the electron density, viz. ∇2ρ(r), that turns out to be a better descriptor of the distribution of electrons in molecules.6 The topological analysis of the Laplacian shows critical points (CPs) in real space, providing a physical basis to the concept of electron pairs in VSEPR model.7 Laplacian shows the local concentration of electrons in a region where ∇2ρ(r) < 0 and local depletion otherwise.7 A related function L(r) = −∇2ρ(r) is found to be more convenient as the concentration of electron density in a region of space shows a maximum in the function. Many theoretical chemists have utilized L(r) for studying a wide range of chemical phenomena such as bonding, reaction mechanism, lone pair description, delocalization of electrons, etc.7−9 Typically, the contour maps of L(r) exhibit shell Received: November 28, 2013 Revised: December 26, 2013
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structure,8,9 wherein each shell is characterized by a shell of charge concentration followed by one of charge depletion. The presence of a local maximum in valence shell charge concentration (VSCC) denotes bonded and nonbonded charge concentrations.10 Nevertheless, the full topology of even simple molecules such as water shows at least 43 CPs in L(r). To quote Popelier verbatim in reference to water molecules: “It will become clear that the Laplacian of this simple molecule already shows a bewildering complexity”. This poses the problem of identifying and interpreting CPs of chemical importance. In yet another study on protonation energies of nitrogen and oxygen in molecules, Popelier et al.11 observed that protonation energy shows poor correlation with L(r) value at nonbonded maxima of atoms, which are in competition with other atoms of different nuclear charge. Becke and Edgecombe12 introduced a one-electron density function, known as the electron localization function (ELF), to describe bonded and nonbonded regions of molecule. The scalar field of ELF, constructed from kinetic energy contribution, describes the bonding patterns and angular disposition of lone pairs on the basis of topological features of local quantum mechanical functions associated with Pauli’s exclusion principle.13 The topology of isovalued surface of ELF was characterized and applied for defining the localization domains of core, bonded, and nonbonded attractors by Savin, Silvi, and others.14−16 The topological analysis of ELF produces CPs that relate this scalar field to chemical properties. The local maxima of the function and volumes of the basins give the positions and sizes of lone pairs, respectively. However, the size of bonding and nonbonding domains do not differ much in ELF, contrary to L(r), which assigns larger sizes to nonbonded than bonded domains.8 Values of ELF, unlike L(r), just provide the measure of the absolute degree of localization associated with each attractor domain and fail to provide quantified values for the lone pair strength. A recently published work done in a similar spirit of defining the lone pair using ELF has come to our attention. Authors have come up with the idea of using average electron population closing to two (a pair of electrons) in high ELF domains to define lone pair domains.17 However, as pointed out by Chesnut,13 the nuances of rich ELF topography are yet to be fully understood. In an attempt not based on scalar fields, Weinhold and coworkers developed natural bond orbital (NBO) analysis18−20 which transforms a given wave function into NBOs corresponding to localized electron pair orbitals as in the idealized Lewis structure. NBO methods, generally regarded as a “chemist’s basis set”, are strongly orbital based and exhibit a connection to the traditional concepts of resonance, hybridization, and bonding. In contrast to AIM, based on QTAIMtype virial partitioning of ρ(r), NBO methods are based on partitioning in terms of natural atomic orbitals (NAOs). The bonding concepts of NBO differ radically from Bader’s AIM theory as well as the electron localization function (ELF) because it provides localized hybrid bond and lone pair orbitals instead of a delocalized description of electrons. NBO analysis makes good predictions about the number of lone pairs and the percentage participation of s and p orbitals in a particular lone pair, generally in agreement with the classical hybridization concepts. However, it fails to predict the geometrical location of the lone pairs. Rather, it shows the occupancy in localized and delocalized MO. Further, the NBO method cannot provide a quantitative measure of lone pair strength.
In short, it is evident that the MO theory as well as density based descriptors, viz. Laplacian of MED and ELF, have their own shortcomings in defining and describing lone pairs in molecules. A more detailed comparison of NBO, ELF, and MESP is provided in the Supporting Information to illustrate this point on some test molecules. Thus, a clear-cut quantitative treatment of lone pairs in terms of observable molecular scalar fields is still conspicuous by its absence. It is the aim of the present communication to bridge this gap. Molecular Electrostatic Potential and Lone Pairs. The scalar field of MESP and its topographical features21−24 are the basis of the present study. Molecular electrostatic potential is a widely employed analytical tool for interpretation and prediction of molecular structure and reactivity. 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)
Tomasi and Pullman25,26 in their early pioneering works have demonstrated the use of MESP for successfully locating the sites of electrophilic attack. As a part of early success in the use of MESP, Pullman presented a lucid explanation of the preferred site of protonation in guanine and related molecules,26 based on the positions of the deepest minima in their MESP distribution. Although the predictions were in agreement with the experimental results, the characterization and exact location of local minima in MESP was not achieved. In their exhaustive work, Politzer and co-workers exploited MESP to describe a variety of chemical phenomena27−29 such as bonding, chemical reactivity, inductive effect, resonance, etc. Politzer and co-workers developed methods to locate the positive and negative regions of MESP in a given molecule, generated by texturing the MESP on the corresponding molecular surface. Location of such regions has been used for identifying the sites of nucleophilic and electrophilic attack respectively. Although these studies are quite elaborate and informative, still this MESP based approach has remained qualitative in terms of the rigorous treatment of lone pairs. The uniqueness of MESP lies in the fact that its topographical features are just rich enough to bring out chemically salient features. As is evident from eq 1, the scalar field of MESP has contributions from both the nuclear and electronic distribution and hence MESP can attain positive as well as negative values. The presence of negative-valued (except in certain cationic species) local minima in MESP topography brings out the electron rich region, a feature absent in positivevalued scalar field of MED. As is known, the negative regions of MESP are mainly contributed by the valence electronic charge distribution, which indeed makes MESP a proper candidate for a detailed description of bonding features as well as lone pairs. The rich topographical features of MESP30 are succinctly brought out in terms of its critical points (CP), i.e., the points where all the first order partial derivatives of the function vanish. A CP is characterized as (R, σ), where R denotes the rank of the corresponding Hessian matrix, viz. the number of nonzero eigenvalues whereas, σ denotes the signature, viz. the sum of signs of the eigenvalues. If all the eigenvalues of a CP are nonzero, it is said to be nondegenerate. For a threedimensional scalar field (such as MED or MESP), nondegenerate CPs are classified as (3, +3) minimum, (3, +1) B
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saddle, (3, −1) saddle, and (3, −3) maximum. These nondegenerate CPs bring out chemically significant features of molecules, e.g., a (3, −3) CP in MESP explicitly shows the presence of nucleus, as MESP of a molecule cannot exhibit non-nuclear maxima.21 Occurrence of a saddle point of the type (3, +1) is a signature of a ring structure. A bond between two atoms is reflected by the presence of a saddle of type (3, −1) in both MED and MESP. As mentioned above, the presence of a negative valued (3, +3) minimum in MESP signifies a pocket of electron concentration in free radicals, anions, and neutral molecules. However, in the case of MESP topography of certain cations, the electron rich region can be signified by positive valued (3, +3) minimum. The value of the MESP at a local minimum has been observed to correctly predict the strength of various noncovalent interactions between molecules.23,31 A close observation of the MESP topography reveals that local minima of type (3, +3) are associated with lone pairs as well as with π-bonds or any other pockets of electron concentration.32,33 This indeed raises a question: Given a molecule, how one can specifically determine whether an MESP minimum corresponds to a lone pair? The insight is provided by the eigenvalues and eigenvectors of the Hessian at a critical point. The Hessian matrix at each nondegenerate CP in a three-dimensional scalar field is associated with three eigenvalues and their respective eigenvectors. These eigenvalues physically signify the curvature of the function at the CP; i.e., they indicate how sharply the function is varying at the CP. The corresponding eigenvector represents the axis of the curvature. On the basis of the studies performed herein, it is proposed that the existence of lone pairs in molecules is distinctly reflected by a presence of minima in MESP with a large magnitude of corresponding largest eigenvalue. In addition, the eigenvector associated with the largest eigenvalue points toward the atom that bears the lone pair. For historical reasons, we retain the term “lone pair” to denote localized nonbonded valence electron concentrations, although the MESP minimum in such a region does not necessarily signify a pair of electrons. A detailed description of the study has been presented in the following sections.
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Figure 1. Illustration of the direction of the eigenvector (EV) and position vector (PV) for H2O molecule. Directions of EV (red arrow) corresponding to the largest eigenvalue (LE) and PV (black dotted line) joining the nucleus to the minimum (green) for H2O molecule are shown. θ is the angle between EV and PV.
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RESULTS AND DISCUSSION To start with, a test set of molecules bearing only lone pairs and no other electron localizations, based on chemical intuition, is chosen. Figure 2 shows the (3, +3) CPs (green) in MESP topography of H2O, CH3NH•, and NO3−, where the MESP minima are joined to the nearest atom by dotted lines. MESP topography of the H2O molecule reveals two lone pairs for O atom; one above and one below the plane of the molecule forming a tetrahedral structure (Figure 2a). Similarly, MESP topography of CH3NH• shows one CP at a distance of 2.39 au from the N atom with the CP−N−H bond angle being 123.6° whereas NO3− has two CPs each on both sides of each O atom coplanar with respect to the molecular plane (Figure 2b,c). The upper half of Table 1 lists the number of distinct (3, +3) CPs obtained for such molecules, radicals, and anions, distances of these CPs from the nearest atom, the MESP value at the minimum (Vmin), the largest eigenvalue (LE) of the Hessian at the CP, and the angle made by the corresponding eigenvector with the position vector joining the nearest atom to the CP (θ). More cases are tabulated in the Supporting Information (Tables S1 and S2). It is evident that for most of the molecules in the upper half of Table 1, the CPs lie close to the nucleus and the magnitude of the corresponding largest eigenvalue is numerically higher than 0.025 au. This clearly brings out the sharp variation of MESP at the CPs related to lone pairs. It is found that the direction of eigenvector at the CP points toward the nucleus to which it is associated, with a small deviation of about 5°, for these cases. Further, we examine the molecules that show electron localization in the molecule that is not associated with lone pairs. Few of them, viz. benzene, methane, and cyclopropane, along with the corresponding (3, +3) CPs in the MESP topography are shown in Figure 3. The data obtained regarding the characteristics of CPs for this set of molecules are listed in the lower half of Table 1. All the MESP minima for these systems generally lie farther from the nucleus than those in the upper half of the table. More importantly, it can be observed from the table that the largest eigenvalue of the Hessian at the CPs that correspond to π-delocalization or hydridic hydrogen is in general numerically smaller than 0.025 au. This indeed shows the shallow nature of such electron localizations as compared to the lone pairs. In addition, unlike lone pair CPs, the eigenvector
COMPUTATIONAL METHODS
The topographical analysis of MESP has been carried out on a large variety of molecular systems comprising free radicals, neutral molecules, anions, and cations, for characterization of the CPs that correspond to lone pairs and differentiating them from those that signify π-bonds. All the molecules under study are optimized at the MP2/6-311++G(d,p) level of theory and the corresponding density matrix obtained is utilized for the determination of the MESP values and mapping of MESP topography. Vibrational frequency analysis is performed to ensure the minimal nature of all the structures. The MESP CPs are located and characterized using the rapid topography mapping code developed recently by Gadre and co-workers.34 Because it is known that lone pairs are highly directional, we probed the direction of the eigenvector corresponding to the largest eigenvalue of the Hessian at the (3, +3) CP. Figure 1 illustrates the angle made by eigenvector (EV) of the largest eigenvalue at the CP with the position vector (PV), joining the (3, +3) CP and the oxygen atom in H2O. C
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Figure 2. MESP minima of (a) H2O, (b) CH3NH•, and (c) NO3−. Minima shown by green dots indicate the location of lone pairs. The dotted lines from green dots suggest that the lone pair belongs to the connected atom. MESP values at the minima are −0.0932, −0.1088, and −0.2706 au for H2O, CH3NH•, and NO3−, respectively.
distinct (3, +3) CPs obtained for these molecules, their distances from nearest atom, the minimal MESP value, the largest eigenvalue (LE) of the Hessian at the CP, and the angle made by the corresponding eigenvector with the position vector joining the nearest atom to the CP. The lone paircontaining atoms, viz. nitrogen in imidazole, pyrazine, pyridine, pyrimidine, and borazine, oxygen in furan, and phosphorus in phosphine and P(CH3)2 all satisfy the proposed criteria for characterizing lone pairs. However, the CPs corresponding to the aromatic ring or hydridic hydrogens of the molecules show a smaller magnitude of the eigenvalue and a larger deviation of the eigenvector than lone pair CPs, as proposed. Thus, it is evident that the combination of the magnitude of the largest eigenvalue and angle made by the corresponding eigenvector is helpful for distinguishing a minimum in the lone pair region from other types of electron localizations. Hence it may be tentatively stated that for a minimum to be associated with the lone pair, the largest eigenvalue should be numerically greater than 0.025 and the corresponding eigenvector should point in the direction of the nearest nucleus, with a small deviation of about 5°. In the spirit of the above definition of lone pairs, we further investigate the existence of lone pairs in cationic species. The fact that cations are electrophiles raises a question, whether cationic species at all have electron-rich regions? Surprisingly, for cations, the minima in the MESP are found in the region of the lone pairs, only if the lone pair is residing on the atom with no formal positive charge. However, unlike anions and neutral molecules, the value of the potential at the minima can be positive, depending on the separation between the atom bearing a lone pair and the positively charged atom. As the separation is increased, MESP gradually attains a negative value at the minimum. But the largest eigenvalue at the concerned minima is not much affected and still shows a sharp change in the MESP function. In particular, cationic species containing two nitrogen atoms, one of which bears the positive charge, are investigated. Their neutral counterparts are also studied for the comparison. Table 3 lists the description of (3, +3) CPs that signify a lone pair on the nitrogen atom bearing a zero formal charge. The provided information is the distance of (3, +3) CP from the nitrogen atom, the MESP value at the minimum (Vmin), the largest eigenvalue (LE) of the Hessian at the CP, and the angle made by corresponding eigenvector with the position vector joining nitrogen atom to (3, +3) CP.
Table 1. MESP Minima of Molecules Bearing Lone Pairs and Those Containing Other Electron Localizationsa molecules
no. of distinct minima
H2O N2 H2S O(CH3)2 • OCH3 CH3NH• NO3− H2PO4−
1 1 1 1 1 1 1 2
ClO4− MnO4− SO42− PO43− CH4 C6H6 CH2CHCHCH2 (CH2)4
1 1 1 1 1 1 1 2
C3H6 B2H6
1 2
distance (au)
Vmin (au)
LE (au)
angle (deg)
2.31 2.97 3.40 2.28 2.37 2.39 2.22 2.26 2.20 2.30 2.39 2.15 2.10 3.77 3.86 3.36 4.18 3.06 3.25 3.18 3.02
−0.0932 −0.0153 −0.0411 −0.0939 −0.0688 −0.1088 −0.2706 −0.2378 −0.2841 −0.2294 −0.2167 −0.4532 −0.6582 −0.0038 −0.0305 −0.0330 −0.0048 −0.0053 −0.0250 −0.0041 −0.0087
0.1437 0.0272 0.0462 0.1557 0.1164 0.1532 0.1936 0.1751 0.2281 0.1565 0.1311 0.2639 0.3289 0.0055 0.0173 0.0294 0.0046 0.0081 0.0283 0.0041 0.0088
1.44 0.00 0.44 1.09 4.16 0.94 4.68 4.99 0.63 2.21 2.70 4.16 4.88 31.02 27.90 16.86 20.59 56.93 26.05 3.41 27.39
a
Listed are the number of distinct negative-valued minima, distance of the minimum from the nearest atom, value of MESP (Vmin), the largest eigenvalue (LE) (all values in au) along with the angle made by the eigenvector (corresponding to LE) with the position vector joining the atom nearest to CP. See text and Figure 1 for details.
of the CPs denoting π-delocalization makes a larger angle (>10°) with the position vector joining the CP to the nearest atom. In particular, cyclopropane is known to display πdelocalization in the ring plane. For the cases of butadiene and cyclopropane, the largest eigenvalues are somewhat larger than the proposed cutoff value; still the angle between EV and PV is substantially large. The rest of the cases match well with the proposed threshold of the largest eigenvalue and the angle between the EV and PV. We choose yet another set of molecules, which possess both π bonds and lone pairs, with a view to examine whether the minima corresponding to lone pairs and π-electrons can be distinguished in such systems. Table 2 lists the number of D
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Figure 3. MESP topography of (a) benzene, (b) cyclopropane, and (c) methane. Green dots are the minima. In benzene, six minima each of value −0.0305 au appear above and below the plane of the ring at a distance of 3.86 au from the nearest carbon atom.
value at CP becomes negative. Figure 4 shows the negative and positive valued MESP isosurfaces, −0.034 (blue) and +0.34
Table 2. MESP Topography Features of Molecules Bearing Both Lone Pairs and π Bonds (Notations Identical to That in Table 1) molecules
no. of distinct CPs
distance (au)
Vmin (au)
LE (au)
angle (deg)
imidazole
2
pyrazine furan
1 2
pyridine
2
pyrimidine
2
borazine
2
phosphine
2
P(CH3)3
2
2.37 3.58 2.39 2.39 3.41 2.36 3.83 2.38 4.05 2.76 2.78 3.54 5.78 3.33 3.68
−0.1206 −0.0323 −0.0890 −0.0575 −0.0285 −0.1082 −0.0159 −0.0960 −0.0023 −0.0300 −0.0084 −0.0430 −0.0012 −0.0709 −0.0019
0.1625 0.0282 0.1443 0.1061 0.0254 0.1603 0.0120 0.1514 0.0062 0.0546 0.0116 0.0434 0.0008 0.0705 0.0058
0.26 42.24 0.00 0.00 11.11 0.00 23.91 0.29 37.09 3.59 6.30 0.00 0.00 0.00 0.92
Figure 4. MESP isosurfaces of value −0.034 (blue) and 0.34 (red), surrounding the [5-(methylamino)pentyl]ammonium cation, illustrating the existence of the negative MESP region in cationic species.
(red) respectively, around the MAPA cation. On the other hand, the eigenvalues of the Hessian at the CPs are still quite larger than the proposed value of 0.025 au, in all the cations. The angle of deviation of eigenvector at the CP is found to be larger in cationic species than their neutral counterparts of the molecule, which is due to the close proximity of positive charge causing a lot of polarization in the electronic cloud. Further, we investigate the hydronium ion, in which the oxygen atom bearing a formal positive charge is expected to possess a lone pair according to Lewis theory. Although no MESP minimum is observed near the oxygen atom, the orbital picture of the hydronium ion shows the HOMO, representing a lone pair orbital. Nevertheless, the structural parameters and the orbital coefficients on atoms suggest that the HOMO is not purely “nonbonding” in nature. It provides some amount of bonding interaction with the three hydrogen atoms. To verify the presence of a lone pair in the hydronium ion, the reactive tendencies of a hydronium ion toward electron deficient species have been studied. Calculations show that H3O+ fails to donate the electron pair in HOMO to an electron deficient molecule like BH3. Instead, a dihydrogen bonded complex is formed. A similar attempt of formation of the H3O+···H3N complex, with the hydrogen of the ammonia interacting with the lone pair on oxygen, fails, and the optimization leads to the formation of H2O···NH4+ complex. These results indicate that H3O+ behaves as if it has no lone pair on oxygen, in agreement with the topographical study. Recent studies on such cations also lacked the signature of lone pairs in MESP topography.35 In general, it could be stated that the existence of a lone pair on the atom bearing the formal positive charge of the cation is overshadowed by the positive potential field. More details of the
Table 3. Cation and Their Neutral Counterparts with Corresponding Vmin, Largest Eigenvalues, and Angle Made by Eigenvector and Line Joining the Nitrogen Atom to (3, +3) CP cation
distance (au)
Vmin (au)
LE (au)
angle (deg)
H2NNH3+ H2NCH2NH3+ H2N(CH2)2NH3+ H2N(CH2)3NH3+ H2N(CH2)4NH3+ H2N(CH2)5NH3+ H2NNH2 H2NCH2NH2 H2N(CH2)2NH2 H2N(CH2)3NH2 H2N(CH2)4NH2 H2N(CH2)5NH2
2.56 2.47 2.38 2.35 2.34 2.34 2.33 2.33 2.33 2.33 2.33 2.33
0.0999 0.0954 0.0301 0.0007 −0.0208 −0.0357 −0.1113 −0.1086 −0.1124 −0.1178 −0.1154 −0.1179
0.079 0.119 0.153 0.168 0.172 0.177 0.177 0.182 0.180 0.181 0.183 0.183
17.51 18.00 8.29 5.55 4.13 3.25 0.87 2.62 1.64 1.45 1.53 1.44
It can be noticed that the value of the potential (Vmin) is positive in cations when the intervening hydrocarbon chain length between the two nitrogen atoms is small. However, when the two nitrogen atoms in the molecule are sufficiently separated by a long hydrocarbon chain such as in the [5(methylamino)pentyl]ammonium (MAPA) cation, the MESP E
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ACKNOWLEDGMENTS A.K. and N.M. thank CSIR for a research fellowship. S.R.G. is thankful to the Department of Science and Technology (DST) for the award of the J. C. Bose National Fellowship. C.H.S. is grateful to CSIR for funding though the CSC0129 project.
study on cationic species are provided in the Supporting Information. In their earlier work, Gadre and Bhadane36 found that the (3, +3) CPs are located near the van der Waals (vdW) surface of the electronegative atoms and show an excellent correlation with the corresponding hydrogen bond distances. One may expect from the present study that the minima for the π systems, being delocalized in nature, lie far from the nucleus, typically outside the corresponding van der Waals surface. On the other hand, lone pairs are highly localized onto a nucleus and thus corresponding CPs are expected to lie just inside or close to the vdW surface of the concerned atom. We find the above statements regarding the position of the CPs to be generally true (cf. Figure S1, Supporting Information). Further, because the mapping of MESP topography is not sensitive beyond the Hartree−Fock method applied to reasonably large basis, viz. 6-31G(d,p),37 findings of the present work are quite general in nature.
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ABBREVIATIONS MESP, molecular electrostatic potential; AIM, atoms in molecules; NBO, natural bond orbitals; ELF, electron localization function; LE, largest eigenvalue; MAPA, [5(methylamino)pentyl]ammonium (MAPA); EV, eigenvector; PV, position vector
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CONCLUSIONS In the present communication, we have developed two criteria that a minimum in the MESP should satisfy to be associated with a lone pair. It has been proposed that the magnitude of the eigenvalue at the CP that corresponds to the lone pair is numerically greater than 0.025 au and the eigenvector associated with it nearly points in the direction (angle ≤5°) of the atom on which it is localized. It is noteworthy that our criteria are based on the eigenvalues and eigenvectors of the Hessian rather than on the MESP value at the minimum. As the MESP minimum value largely depends on the nature of atoms and molecules, viz. electronegativity, charge on the molecule, etc., it cannot distinguish between minima corresponding to various types of electron localization in molecules. However, the MESP value at the minimum provides the strength of the lone pair, as it correlates extremely well with the interaction energies of the lone pair−π complexes.30 In summary, it is hoped that the present study has provided a clear description of lone pairs in terms of the critical points of a scalar field that is amenable to theoretical as well as experimental investigations. ASSOCIATED CONTENT
S Supporting Information *
Detailed MESP topography features of all molecules bearing lone pairs and those containing other electron localizations and π bonds. Figure showing MESP topography. Comparison of NBO and ELF features. Discussion of lone pairs in cations, including figures showing the HOMO of H3O+, hydronium ion complexes, and MESP isosurfaces. This material is available free of charge via the Internet at http://pubs.acs.org
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REFERENCES
(1) Lee, J. D. Concise Inorganic Chemistry, 4 ed.; Wiley-Blackwell: London, 1991. (2) Sidgwick, N. V. The Electronic Theory of Valency; Oxford University Press: New York, 1927. (3) Gillespie, R. J.; Nyholm, R. S. Inorganic Stereochemistry. Q. Rev. 1957, 11, 339−380. (4) Pauling, L. The Nature of the Chemical Bond. Application of Results Obtained from the Quantum Mechanics and from a Theory of Paramagnetic Susceptibility to the Structure of Molecules. J. Am. Chem. Soc. 1931, 53, 1367−1400. (5) Chesnut, D. B. The Contribution to Bonding by Lone Pairs. Chem. Phys. 2003, 291, 141−152. (6) Bader, R. F. W. Atoms in Molecules, A Quantum Theory; Clarendon: Oxford, U.K., 1990. (7) Bader, R. F. W.; Gillespie, R. J.; MacDougall, P. J. A Physical Basis for the VSEPR Model of Molecular Geometry. J. Am. Chem. Soc. 1988, 110, 7329−7336. (8) Bader, R. F. W.; Johnson, S.; Tang, T.-H.; Popelier, P. L. A. The Electron Pair. J. Phys. Chem. 1996, 100, 15398−15415. (9) Bader, R. F. W.; Essén, H. J. The Characterization of Atomic Interactions. J. Chem. Phys. 1984, 80, 1943−1960. (10) Popelier, P. L. A. On the Full Topology of the Laplacian of the Electron Density. Coord. Chem. Rev. 2000, 197, 169−189. (11) Popelier, P. L. A.; Smith, P. J. Protonation Energies and the Laplacian of the Electron Density: A Critical Examination. Phys. Chem. Chem. Phys. 2001, 3, 4208−4212. (12) Becke, A. D.; Edgecombe, K. E. A Simple Measure of Electron Localization in Atomic and Molecular Systems. J. Chem. Phys. 1990, 92, 5397−5403. (13) Chesnut, D. B. An Electron Localization Function Study of the Lone Pair. J. Phys. Chem. A 2000, 104, 11644−11650. (14) Savin, A.; Becke, A. D.; Flad, J.; Nesper, R.; Preuss, H.; von Schnering, H. G. A New Look at Electron Localization. Angew. Chem., Int. Ed. Engl. 1991, 30, 409−412. (15) Savin, A.; Silvi, B.; Colonna, F. Topological Analysis of the Electron Localization Function Applied to Delocalized Bonds. Can. J. Chem. 1996, 74, 1088−1096. (16) Silvi, B.; Savin, A. Classification of Chemical Bonds Based on Topological Analysis of Electron Localization Functions. Nature 1994, 371, 683−686. (17) Rahm, M.; Christe, K. O. Quantifying the Nature of Lone Pair Domains. Chem. Phys. Chem. 2013, 14, 3714−3725. (18) Reed, A. E.; Curtiss, L. A.; Weinhold, F. Intermolecular Interactions from a Natural Bond Orbital, Donor-Acceptor Viewpoint. Chem. Rev. 1988, 88, 899−926. (19) Weinhold, F. Natural Bond Orbital Methods; In Encyclopedia of Computational Chemistry; Schleyer, P. v. R., Allinger, N. L., Clark, T., Gasteiger, J., Kollman, P. A., Schaefer, H. F., Schreiner, P. R., Eds.; Wiley: Chichester, U.K., 1998; Vol. 3, 1792−1811. (20) Glendening, E. D.; Landis, C. R.; Weinhold, F. Natural Bond Orbital Methods. WIREs Comput. Mol. Sci. 2002, 2, 1−42.
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AUTHOR INFORMATION
Corresponding Authors
*S. R. Gadre: e-mail, [email protected]; tel, +91-512-259 6706; homepage, http://home.iitk.ac.in/∼gadre/. *C. H. Suresh: e-mail, [email protected]; tel, +91-4712515472; homepage, http://www.niist.res.in/chsuresh. Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest. F
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(21) Gadre, S. R.; Pathak, R. K. Nonexistence of Local Maxima in Molecular Electrostatic Potential Maps. Proc. Ind. Acad. Sci. (Chem. Sci.) 1990, 102, 189−192. (22) Gadre, S. R.; Shirsat, R. N. Electrostatics of Atoms and Molecules; Universities Press: Hyderabad, 2000. (23) Deshmukh, M. M.; Gadre, S. R.; Tonner, R.; Frenking, G. Molecular Electrostatic Potentials of Divalent Carbon(0) Compounds. Phys. Chem. Chem. Phys. 2008, 10, 2298−2301. (24) Mathew, J.; Suresh, C. H. Use of Molecular Electrostatic Potential at the Carbene Carbon as a Simple and Efficient Electronic Parameter of N-Heterocyclic Carbenes. Inorg. Chem. 2010, 49, 4665− 4669. (25) Scrocco, E.; Tomasi, J. Electronic Molecular Structure, Reactivity and Intermolecular Forces: An Euristic Interpretation by Means of Electrostatic Molecular Potentials. Adv. Quantum Chem. 1978, 11, 115−193. (26) Pullman, B. 30 Years of Sanibel Symposia - Structure and Activity of Biomolecules. Int. J. Quantum Chem. Quantum Biol. Symp. 1990, 17, 81−92. (27) Politzer, P.; Truhlar, D. G. Chemical Application of Atomic and Molecular Electrostatic Potential; Springer: New York, 1981. (28) Murray, J. S.; Paulsen, K.; Politzer, P. Molecular Surface Electrostatic Potentials in the Analysis of Non-Hydrogen-Bonding Noncovalent Interactions. Proc. Ind. Acad. Sci. (Chem. Sci.) 1994, 106, 267−275. (29) 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. (30) Gadre, S. R.; Pundlik, S. S.; Limaye, A. C.; Rendell, A. P. Electrostatic Investigation of Metal Cation Binding to DNA Bases and Base Pairs. Chem. Commun. 1998, 31, 573−574. (31) Mohan, N.; Suresh, C. H.; Kumar, A.; Gadre, S. R. Molecular Electrostatics for Probing Lone Pair-π Interactions. Phys. Chem. Chem. Phys. 2013, 15, 18401−18409. (32) Kulkarni, S. A.; Gadre, S. R. Electron Localization in Molecules: A Comparative Study of Scalar Fields. J. Mol. Struct (THEOCHEM) 1996, 361, 83−91. (33) 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. (34) Yeole, S. D.; López, R.; Gadre, S. R. Rapid Topography Mapping of Scalar Fields: Large Molecular Clusters. J. Chem. Phys. 2012, 137, 074116−074123. (35) 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. (36) Gadre, S. R.; Bhadane, P. K. Patterns in Hydrogen Bonding via Electrostatic Potential Topography. J. Chem. Phys. 1997, 107, 5625− 5626. (37) Gadre, S. R.; Kulkarni, S. A.; Suresh, C. H.; Shrivastava, I. H. Basis-Set Dependence of the Molecular Electrostatic Potential Topography - A Case-Study of Substituted Benzenes. Chem. Phys. Lett. 1995, 239, 273−281.
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Lone Pairs: An Electrostatic Viewpoint Anmol Kumar,† Shridhar R. Gadre,*,† Neetha Mohan,‡ and Cherumuttathu H. Suresh*,‡ †
Department of Chemistry, Indian Institute of Technology Kanpur, Kanpur 208016, India Inorganic and Theoretical Chemistry Section, Chemical Sciences and Technology Division, CSIR-National Institute for Interdisciplinary Science and Technology, Trivandrum 695019, India
‡
S Supporting Information *
ABSTRACT: A clear-cut definition of lone pairs has been offered in terms of characteristics of minima in molecular electrostatic potential (MESP). The largest eigenvalue and corresponding eigenvector of the Hessian at the minima are shown to distinguish lone pair regions from the other types of electron localization (such as π bonds). A comparative study of lone pairs as depicted by various other scalar fields such as the Laplacian of electron density and electron localization function is made. Further, an attempt has been made to generalize the definition of lone pairs to the case of cations.
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INTRODUCTION “Lone pair” is one of the most commonly used terms in chemistry. Lone pairs are known to play an indispensible role in describing the structure and reactivity of molecules. Classically, a lone pair is defined as a valence electron pair, bound to a nucleus, not utilized in chemical bonding.1 Despite being in vogue for the past 90 years, the term “lone pair” has escaped a quantitative description. In this light, one would like to address questions such as: Can a lone pair be defined in terms of physical observables? How can one describe properties of a lone pair? Description of Lone Pairs in Bonding Theories. An early attempt to scrutinize the chemistry of the lone pairs of electrons was made by Sidgwick,2 who built upon the octet theory of Lewis to describe the arrangement of electron pairs in multivalent atoms. Later, in a popular theoretical approach, known as valence shell electron pair repulsion (VSEPR) model, Gillespie and Nyholm3 delineated the importance of varying electronic repulsions among shared pair and lone pair of electrons in valence shell orbitals. VSEPR theory still dominates the chemical thinking and is useful for understanding the geometrical arrangement of atoms in molecules. Pauling4 introduced the concept of hybridization of orbitals in 1931, which is used to-date as an essential tool by chemists to explain bonding patterns in molecules including the probable location of lone pair orbitals. Molecular orbital (MO) theory, a popular bonding theory, in its purest form, is based on the maximum amount of delocalization of MOs. Nevertheless, localization methods need to be invoked for describing a lone pair as an electron pair localized in a nonbonded valence MO.5 However, orbitals are not observables. Experiments can only provide © XXXX American Chemical Society
information about observables, e.g., the electron density distribution and the entities derivable from it. Thus, characterization of lone pairs in terms of physical observables, such as the molecular electron density (MED) and molecular electrostatic potential (MESP) is more realistic and appropriate. Description of Lone Pairs in Scalar Fields. The theory of atoms in molecules (AIM),6 pioneered by Bader and coworkers, exploits the scalar field of MED to extract chemical insights. The topology of MED yields reliable depiction of fundamental chemical phenomena such as the chemical bond, molecular structure and the concept of an atom in a molecule. However, topological features of MED remain silent in the regime of nonbonded electrons; i.e., the MED does not exhibit a local maximum in the lone pair region. It is the Laplacian of the electron density, viz. ∇2ρ(r), that turns out to be a better descriptor of the distribution of electrons in molecules.6 The topological analysis of the Laplacian shows critical points (CPs) in real space, providing a physical basis to the concept of electron pairs in VSEPR model.7 Laplacian shows the local concentration of electrons in a region where ∇2ρ(r) < 0 and local depletion otherwise.7 A related function L(r) = −∇2ρ(r) is found to be more convenient as the concentration of electron density in a region of space shows a maximum in the function. Many theoretical chemists have utilized L(r) for studying a wide range of chemical phenomena such as bonding, reaction mechanism, lone pair description, delocalization of electrons, etc.7−9 Typically, the contour maps of L(r) exhibit shell Received: November 28, 2013 Revised: December 26, 2013
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structure,8,9 wherein each shell is characterized by a shell of charge concentration followed by one of charge depletion. The presence of a local maximum in valence shell charge concentration (VSCC) denotes bonded and nonbonded charge concentrations.10 Nevertheless, the full topology of even simple molecules such as water shows at least 43 CPs in L(r). To quote Popelier verbatim in reference to water molecules: “It will become clear that the Laplacian of this simple molecule already shows a bewildering complexity”. This poses the problem of identifying and interpreting CPs of chemical importance. In yet another study on protonation energies of nitrogen and oxygen in molecules, Popelier et al.11 observed that protonation energy shows poor correlation with L(r) value at nonbonded maxima of atoms, which are in competition with other atoms of different nuclear charge. Becke and Edgecombe12 introduced a one-electron density function, known as the electron localization function (ELF), to describe bonded and nonbonded regions of molecule. The scalar field of ELF, constructed from kinetic energy contribution, describes the bonding patterns and angular disposition of lone pairs on the basis of topological features of local quantum mechanical functions associated with Pauli’s exclusion principle.13 The topology of isovalued surface of ELF was characterized and applied for defining the localization domains of core, bonded, and nonbonded attractors by Savin, Silvi, and others.14−16 The topological analysis of ELF produces CPs that relate this scalar field to chemical properties. The local maxima of the function and volumes of the basins give the positions and sizes of lone pairs, respectively. However, the size of bonding and nonbonding domains do not differ much in ELF, contrary to L(r), which assigns larger sizes to nonbonded than bonded domains.8 Values of ELF, unlike L(r), just provide the measure of the absolute degree of localization associated with each attractor domain and fail to provide quantified values for the lone pair strength. A recently published work done in a similar spirit of defining the lone pair using ELF has come to our attention. Authors have come up with the idea of using average electron population closing to two (a pair of electrons) in high ELF domains to define lone pair domains.17 However, as pointed out by Chesnut,13 the nuances of rich ELF topography are yet to be fully understood. In an attempt not based on scalar fields, Weinhold and coworkers developed natural bond orbital (NBO) analysis18−20 which transforms a given wave function into NBOs corresponding to localized electron pair orbitals as in the idealized Lewis structure. NBO methods, generally regarded as a “chemist’s basis set”, are strongly orbital based and exhibit a connection to the traditional concepts of resonance, hybridization, and bonding. In contrast to AIM, based on QTAIMtype virial partitioning of ρ(r), NBO methods are based on partitioning in terms of natural atomic orbitals (NAOs). The bonding concepts of NBO differ radically from Bader’s AIM theory as well as the electron localization function (ELF) because it provides localized hybrid bond and lone pair orbitals instead of a delocalized description of electrons. NBO analysis makes good predictions about the number of lone pairs and the percentage participation of s and p orbitals in a particular lone pair, generally in agreement with the classical hybridization concepts. However, it fails to predict the geometrical location of the lone pairs. Rather, it shows the occupancy in localized and delocalized MO. Further, the NBO method cannot provide a quantitative measure of lone pair strength.
In short, it is evident that the MO theory as well as density based descriptors, viz. Laplacian of MED and ELF, have their own shortcomings in defining and describing lone pairs in molecules. A more detailed comparison of NBO, ELF, and MESP is provided in the Supporting Information to illustrate this point on some test molecules. Thus, a clear-cut quantitative treatment of lone pairs in terms of observable molecular scalar fields is still conspicuous by its absence. It is the aim of the present communication to bridge this gap. Molecular Electrostatic Potential and Lone Pairs. The scalar field of MESP and its topographical features21−24 are the basis of the present study. Molecular electrostatic potential is a widely employed analytical tool for interpretation and prediction of molecular structure and reactivity. 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)
Tomasi and Pullman25,26 in their early pioneering works have demonstrated the use of MESP for successfully locating the sites of electrophilic attack. As a part of early success in the use of MESP, Pullman presented a lucid explanation of the preferred site of protonation in guanine and related molecules,26 based on the positions of the deepest minima in their MESP distribution. Although the predictions were in agreement with the experimental results, the characterization and exact location of local minima in MESP was not achieved. In their exhaustive work, Politzer and co-workers exploited MESP to describe a variety of chemical phenomena27−29 such as bonding, chemical reactivity, inductive effect, resonance, etc. Politzer and co-workers developed methods to locate the positive and negative regions of MESP in a given molecule, generated by texturing the MESP on the corresponding molecular surface. Location of such regions has been used for identifying the sites of nucleophilic and electrophilic attack respectively. Although these studies are quite elaborate and informative, still this MESP based approach has remained qualitative in terms of the rigorous treatment of lone pairs. The uniqueness of MESP lies in the fact that its topographical features are just rich enough to bring out chemically salient features. As is evident from eq 1, the scalar field of MESP has contributions from both the nuclear and electronic distribution and hence MESP can attain positive as well as negative values. The presence of negative-valued (except in certain cationic species) local minima in MESP topography brings out the electron rich region, a feature absent in positivevalued scalar field of MED. As is known, the negative regions of MESP are mainly contributed by the valence electronic charge distribution, which indeed makes MESP a proper candidate for a detailed description of bonding features as well as lone pairs. The rich topographical features of MESP30 are succinctly brought out in terms of its critical points (CP), i.e., the points where all the first order partial derivatives of the function vanish. A CP is characterized as (R, σ), where R denotes the rank of the corresponding Hessian matrix, viz. the number of nonzero eigenvalues whereas, σ denotes the signature, viz. the sum of signs of the eigenvalues. If all the eigenvalues of a CP are nonzero, it is said to be nondegenerate. For a threedimensional scalar field (such as MED or MESP), nondegenerate CPs are classified as (3, +3) minimum, (3, +1) B
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saddle, (3, −1) saddle, and (3, −3) maximum. These nondegenerate CPs bring out chemically significant features of molecules, e.g., a (3, −3) CP in MESP explicitly shows the presence of nucleus, as MESP of a molecule cannot exhibit non-nuclear maxima.21 Occurrence of a saddle point of the type (3, +1) is a signature of a ring structure. A bond between two atoms is reflected by the presence of a saddle of type (3, −1) in both MED and MESP. As mentioned above, the presence of a negative valued (3, +3) minimum in MESP signifies a pocket of electron concentration in free radicals, anions, and neutral molecules. However, in the case of MESP topography of certain cations, the electron rich region can be signified by positive valued (3, +3) minimum. The value of the MESP at a local minimum has been observed to correctly predict the strength of various noncovalent interactions between molecules.23,31 A close observation of the MESP topography reveals that local minima of type (3, +3) are associated with lone pairs as well as with π-bonds or any other pockets of electron concentration.32,33 This indeed raises a question: Given a molecule, how one can specifically determine whether an MESP minimum corresponds to a lone pair? The insight is provided by the eigenvalues and eigenvectors of the Hessian at a critical point. The Hessian matrix at each nondegenerate CP in a three-dimensional scalar field is associated with three eigenvalues and their respective eigenvectors. These eigenvalues physically signify the curvature of the function at the CP; i.e., they indicate how sharply the function is varying at the CP. The corresponding eigenvector represents the axis of the curvature. On the basis of the studies performed herein, it is proposed that the existence of lone pairs in molecules is distinctly reflected by a presence of minima in MESP with a large magnitude of corresponding largest eigenvalue. In addition, the eigenvector associated with the largest eigenvalue points toward the atom that bears the lone pair. For historical reasons, we retain the term “lone pair” to denote localized nonbonded valence electron concentrations, although the MESP minimum in such a region does not necessarily signify a pair of electrons. A detailed description of the study has been presented in the following sections.
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Figure 1. Illustration of the direction of the eigenvector (EV) and position vector (PV) for H2O molecule. Directions of EV (red arrow) corresponding to the largest eigenvalue (LE) and PV (black dotted line) joining the nucleus to the minimum (green) for H2O molecule are shown. θ is the angle between EV and PV.
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RESULTS AND DISCUSSION To start with, a test set of molecules bearing only lone pairs and no other electron localizations, based on chemical intuition, is chosen. Figure 2 shows the (3, +3) CPs (green) in MESP topography of H2O, CH3NH•, and NO3−, where the MESP minima are joined to the nearest atom by dotted lines. MESP topography of the H2O molecule reveals two lone pairs for O atom; one above and one below the plane of the molecule forming a tetrahedral structure (Figure 2a). Similarly, MESP topography of CH3NH• shows one CP at a distance of 2.39 au from the N atom with the CP−N−H bond angle being 123.6° whereas NO3− has two CPs each on both sides of each O atom coplanar with respect to the molecular plane (Figure 2b,c). The upper half of Table 1 lists the number of distinct (3, +3) CPs obtained for such molecules, radicals, and anions, distances of these CPs from the nearest atom, the MESP value at the minimum (Vmin), the largest eigenvalue (LE) of the Hessian at the CP, and the angle made by the corresponding eigenvector with the position vector joining the nearest atom to the CP (θ). More cases are tabulated in the Supporting Information (Tables S1 and S2). It is evident that for most of the molecules in the upper half of Table 1, the CPs lie close to the nucleus and the magnitude of the corresponding largest eigenvalue is numerically higher than 0.025 au. This clearly brings out the sharp variation of MESP at the CPs related to lone pairs. It is found that the direction of eigenvector at the CP points toward the nucleus to which it is associated, with a small deviation of about 5°, for these cases. Further, we examine the molecules that show electron localization in the molecule that is not associated with lone pairs. Few of them, viz. benzene, methane, and cyclopropane, along with the corresponding (3, +3) CPs in the MESP topography are shown in Figure 3. The data obtained regarding the characteristics of CPs for this set of molecules are listed in the lower half of Table 1. All the MESP minima for these systems generally lie farther from the nucleus than those in the upper half of the table. More importantly, it can be observed from the table that the largest eigenvalue of the Hessian at the CPs that correspond to π-delocalization or hydridic hydrogen is in general numerically smaller than 0.025 au. This indeed shows the shallow nature of such electron localizations as compared to the lone pairs. In addition, unlike lone pair CPs, the eigenvector
COMPUTATIONAL METHODS
The topographical analysis of MESP has been carried out on a large variety of molecular systems comprising free radicals, neutral molecules, anions, and cations, for characterization of the CPs that correspond to lone pairs and differentiating them from those that signify π-bonds. All the molecules under study are optimized at the MP2/6-311++G(d,p) level of theory and the corresponding density matrix obtained is utilized for the determination of the MESP values and mapping of MESP topography. Vibrational frequency analysis is performed to ensure the minimal nature of all the structures. The MESP CPs are located and characterized using the rapid topography mapping code developed recently by Gadre and co-workers.34 Because it is known that lone pairs are highly directional, we probed the direction of the eigenvector corresponding to the largest eigenvalue of the Hessian at the (3, +3) CP. Figure 1 illustrates the angle made by eigenvector (EV) of the largest eigenvalue at the CP with the position vector (PV), joining the (3, +3) CP and the oxygen atom in H2O. C
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Figure 2. MESP minima of (a) H2O, (b) CH3NH•, and (c) NO3−. Minima shown by green dots indicate the location of lone pairs. The dotted lines from green dots suggest that the lone pair belongs to the connected atom. MESP values at the minima are −0.0932, −0.1088, and −0.2706 au for H2O, CH3NH•, and NO3−, respectively.
distinct (3, +3) CPs obtained for these molecules, their distances from nearest atom, the minimal MESP value, the largest eigenvalue (LE) of the Hessian at the CP, and the angle made by the corresponding eigenvector with the position vector joining the nearest atom to the CP. The lone paircontaining atoms, viz. nitrogen in imidazole, pyrazine, pyridine, pyrimidine, and borazine, oxygen in furan, and phosphorus in phosphine and P(CH3)2 all satisfy the proposed criteria for characterizing lone pairs. However, the CPs corresponding to the aromatic ring or hydridic hydrogens of the molecules show a smaller magnitude of the eigenvalue and a larger deviation of the eigenvector than lone pair CPs, as proposed. Thus, it is evident that the combination of the magnitude of the largest eigenvalue and angle made by the corresponding eigenvector is helpful for distinguishing a minimum in the lone pair region from other types of electron localizations. Hence it may be tentatively stated that for a minimum to be associated with the lone pair, the largest eigenvalue should be numerically greater than 0.025 and the corresponding eigenvector should point in the direction of the nearest nucleus, with a small deviation of about 5°. In the spirit of the above definition of lone pairs, we further investigate the existence of lone pairs in cationic species. The fact that cations are electrophiles raises a question, whether cationic species at all have electron-rich regions? Surprisingly, for cations, the minima in the MESP are found in the region of the lone pairs, only if the lone pair is residing on the atom with no formal positive charge. However, unlike anions and neutral molecules, the value of the potential at the minima can be positive, depending on the separation between the atom bearing a lone pair and the positively charged atom. As the separation is increased, MESP gradually attains a negative value at the minimum. But the largest eigenvalue at the concerned minima is not much affected and still shows a sharp change in the MESP function. In particular, cationic species containing two nitrogen atoms, one of which bears the positive charge, are investigated. Their neutral counterparts are also studied for the comparison. Table 3 lists the description of (3, +3) CPs that signify a lone pair on the nitrogen atom bearing a zero formal charge. The provided information is the distance of (3, +3) CP from the nitrogen atom, the MESP value at the minimum (Vmin), the largest eigenvalue (LE) of the Hessian at the CP, and the angle made by corresponding eigenvector with the position vector joining nitrogen atom to (3, +3) CP.
Table 1. MESP Minima of Molecules Bearing Lone Pairs and Those Containing Other Electron Localizationsa molecules
no. of distinct minima
H2O N2 H2S O(CH3)2 • OCH3 CH3NH• NO3− H2PO4−
1 1 1 1 1 1 1 2
ClO4− MnO4− SO42− PO43− CH4 C6H6 CH2CHCHCH2 (CH2)4
1 1 1 1 1 1 1 2
C3H6 B2H6
1 2
distance (au)
Vmin (au)
LE (au)
angle (deg)
2.31 2.97 3.40 2.28 2.37 2.39 2.22 2.26 2.20 2.30 2.39 2.15 2.10 3.77 3.86 3.36 4.18 3.06 3.25 3.18 3.02
−0.0932 −0.0153 −0.0411 −0.0939 −0.0688 −0.1088 −0.2706 −0.2378 −0.2841 −0.2294 −0.2167 −0.4532 −0.6582 −0.0038 −0.0305 −0.0330 −0.0048 −0.0053 −0.0250 −0.0041 −0.0087
0.1437 0.0272 0.0462 0.1557 0.1164 0.1532 0.1936 0.1751 0.2281 0.1565 0.1311 0.2639 0.3289 0.0055 0.0173 0.0294 0.0046 0.0081 0.0283 0.0041 0.0088
1.44 0.00 0.44 1.09 4.16 0.94 4.68 4.99 0.63 2.21 2.70 4.16 4.88 31.02 27.90 16.86 20.59 56.93 26.05 3.41 27.39
a
Listed are the number of distinct negative-valued minima, distance of the minimum from the nearest atom, value of MESP (Vmin), the largest eigenvalue (LE) (all values in au) along with the angle made by the eigenvector (corresponding to LE) with the position vector joining the atom nearest to CP. See text and Figure 1 for details.
of the CPs denoting π-delocalization makes a larger angle (>10°) with the position vector joining the CP to the nearest atom. In particular, cyclopropane is known to display πdelocalization in the ring plane. For the cases of butadiene and cyclopropane, the largest eigenvalues are somewhat larger than the proposed cutoff value; still the angle between EV and PV is substantially large. The rest of the cases match well with the proposed threshold of the largest eigenvalue and the angle between the EV and PV. We choose yet another set of molecules, which possess both π bonds and lone pairs, with a view to examine whether the minima corresponding to lone pairs and π-electrons can be distinguished in such systems. Table 2 lists the number of D
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Figure 3. MESP topography of (a) benzene, (b) cyclopropane, and (c) methane. Green dots are the minima. In benzene, six minima each of value −0.0305 au appear above and below the plane of the ring at a distance of 3.86 au from the nearest carbon atom.
value at CP becomes negative. Figure 4 shows the negative and positive valued MESP isosurfaces, −0.034 (blue) and +0.34
Table 2. MESP Topography Features of Molecules Bearing Both Lone Pairs and π Bonds (Notations Identical to That in Table 1) molecules
no. of distinct CPs
distance (au)
Vmin (au)
LE (au)
angle (deg)
imidazole
2
pyrazine furan
1 2
pyridine
2
pyrimidine
2
borazine
2
phosphine
2
P(CH3)3
2
2.37 3.58 2.39 2.39 3.41 2.36 3.83 2.38 4.05 2.76 2.78 3.54 5.78 3.33 3.68
−0.1206 −0.0323 −0.0890 −0.0575 −0.0285 −0.1082 −0.0159 −0.0960 −0.0023 −0.0300 −0.0084 −0.0430 −0.0012 −0.0709 −0.0019
0.1625 0.0282 0.1443 0.1061 0.0254 0.1603 0.0120 0.1514 0.0062 0.0546 0.0116 0.0434 0.0008 0.0705 0.0058
0.26 42.24 0.00 0.00 11.11 0.00 23.91 0.29 37.09 3.59 6.30 0.00 0.00 0.00 0.92
Figure 4. MESP isosurfaces of value −0.034 (blue) and 0.34 (red), surrounding the [5-(methylamino)pentyl]ammonium cation, illustrating the existence of the negative MESP region in cationic species.
(red) respectively, around the MAPA cation. On the other hand, the eigenvalues of the Hessian at the CPs are still quite larger than the proposed value of 0.025 au, in all the cations. The angle of deviation of eigenvector at the CP is found to be larger in cationic species than their neutral counterparts of the molecule, which is due to the close proximity of positive charge causing a lot of polarization in the electronic cloud. Further, we investigate the hydronium ion, in which the oxygen atom bearing a formal positive charge is expected to possess a lone pair according to Lewis theory. Although no MESP minimum is observed near the oxygen atom, the orbital picture of the hydronium ion shows the HOMO, representing a lone pair orbital. Nevertheless, the structural parameters and the orbital coefficients on atoms suggest that the HOMO is not purely “nonbonding” in nature. It provides some amount of bonding interaction with the three hydrogen atoms. To verify the presence of a lone pair in the hydronium ion, the reactive tendencies of a hydronium ion toward electron deficient species have been studied. Calculations show that H3O+ fails to donate the electron pair in HOMO to an electron deficient molecule like BH3. Instead, a dihydrogen bonded complex is formed. A similar attempt of formation of the H3O+···H3N complex, with the hydrogen of the ammonia interacting with the lone pair on oxygen, fails, and the optimization leads to the formation of H2O···NH4+ complex. These results indicate that H3O+ behaves as if it has no lone pair on oxygen, in agreement with the topographical study. Recent studies on such cations also lacked the signature of lone pairs in MESP topography.35 In general, it could be stated that the existence of a lone pair on the atom bearing the formal positive charge of the cation is overshadowed by the positive potential field. More details of the
Table 3. Cation and Their Neutral Counterparts with Corresponding Vmin, Largest Eigenvalues, and Angle Made by Eigenvector and Line Joining the Nitrogen Atom to (3, +3) CP cation
distance (au)
Vmin (au)
LE (au)
angle (deg)
H2NNH3+ H2NCH2NH3+ H2N(CH2)2NH3+ H2N(CH2)3NH3+ H2N(CH2)4NH3+ H2N(CH2)5NH3+ H2NNH2 H2NCH2NH2 H2N(CH2)2NH2 H2N(CH2)3NH2 H2N(CH2)4NH2 H2N(CH2)5NH2
2.56 2.47 2.38 2.35 2.34 2.34 2.33 2.33 2.33 2.33 2.33 2.33
0.0999 0.0954 0.0301 0.0007 −0.0208 −0.0357 −0.1113 −0.1086 −0.1124 −0.1178 −0.1154 −0.1179
0.079 0.119 0.153 0.168 0.172 0.177 0.177 0.182 0.180 0.181 0.183 0.183
17.51 18.00 8.29 5.55 4.13 3.25 0.87 2.62 1.64 1.45 1.53 1.44
It can be noticed that the value of the potential (Vmin) is positive in cations when the intervening hydrocarbon chain length between the two nitrogen atoms is small. However, when the two nitrogen atoms in the molecule are sufficiently separated by a long hydrocarbon chain such as in the [5(methylamino)pentyl]ammonium (MAPA) cation, the MESP E
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ACKNOWLEDGMENTS A.K. and N.M. thank CSIR for a research fellowship. S.R.G. is thankful to the Department of Science and Technology (DST) for the award of the J. C. Bose National Fellowship. C.H.S. is grateful to CSIR for funding though the CSC0129 project.
study on cationic species are provided in the Supporting Information. In their earlier work, Gadre and Bhadane36 found that the (3, +3) CPs are located near the van der Waals (vdW) surface of the electronegative atoms and show an excellent correlation with the corresponding hydrogen bond distances. One may expect from the present study that the minima for the π systems, being delocalized in nature, lie far from the nucleus, typically outside the corresponding van der Waals surface. On the other hand, lone pairs are highly localized onto a nucleus and thus corresponding CPs are expected to lie just inside or close to the vdW surface of the concerned atom. We find the above statements regarding the position of the CPs to be generally true (cf. Figure S1, Supporting Information). Further, because the mapping of MESP topography is not sensitive beyond the Hartree−Fock method applied to reasonably large basis, viz. 6-31G(d,p),37 findings of the present work are quite general in nature.
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ABBREVIATIONS MESP, molecular electrostatic potential; AIM, atoms in molecules; NBO, natural bond orbitals; ELF, electron localization function; LE, largest eigenvalue; MAPA, [5(methylamino)pentyl]ammonium (MAPA); EV, eigenvector; PV, position vector
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CONCLUSIONS In the present communication, we have developed two criteria that a minimum in the MESP should satisfy to be associated with a lone pair. It has been proposed that the magnitude of the eigenvalue at the CP that corresponds to the lone pair is numerically greater than 0.025 au and the eigenvector associated with it nearly points in the direction (angle ≤5°) of the atom on which it is localized. It is noteworthy that our criteria are based on the eigenvalues and eigenvectors of the Hessian rather than on the MESP value at the minimum. As the MESP minimum value largely depends on the nature of atoms and molecules, viz. electronegativity, charge on the molecule, etc., it cannot distinguish between minima corresponding to various types of electron localization in molecules. However, the MESP value at the minimum provides the strength of the lone pair, as it correlates extremely well with the interaction energies of the lone pair−π complexes.30 In summary, it is hoped that the present study has provided a clear description of lone pairs in terms of the critical points of a scalar field that is amenable to theoretical as well as experimental investigations. ASSOCIATED CONTENT
S Supporting Information *
Detailed MESP topography features of all molecules bearing lone pairs and those containing other electron localizations and π bonds. Figure showing MESP topography. Comparison of NBO and ELF features. Discussion of lone pairs in cations, including figures showing the HOMO of H3O+, hydronium ion complexes, and MESP isosurfaces. This material is available free of charge via the Internet at http://pubs.acs.org
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AUTHOR INFORMATION
Corresponding Authors
*S. R. Gadre: e-mail, [email protected]; tel, +91-512-259 6706; homepage, http://home.iitk.ac.in/∼gadre/. *C. H. Suresh: e-mail, [email protected]; tel, +91-4712515472; homepage, http://www.niist.res.in/chsuresh. Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest. F
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