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Electron–electron interactions between ELF basins
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Chemical Physics Letters 454 (2008) 396–403 www.elsevier.com/locate/cplett
Electron–electron interactions between ELF basins A. Martı´n Penda´s *, E. Francisco, M.A. Blanco Departamento de Quı´mica Fı´sica y Analı´tica, Facultad de Quı´mica, Universidad de Oviedo, 33006 Oviedo, Spain Received 11 January 2008; in final form 11 February 2008 Available online 15 February 2008
Abstract The electrostatic repulsions among the electrons that populate different attraction basins of the electron localization function (ELF) are numerically obtained for a number of simple test molecules. Differences in these interbasin repulsions are shown to provide a map onto the Gillespie–Nyholm rules of the valence shell electron pair repulsion model, and the reasons behind this correspondence are briefly analyzed. We think that these repulsion parameters, including the inter-basin exchange energy, which we relate to resonance and conjugation, open new avenues towards more quantitative ELF analyses. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Since its proposal in 1990 [1], the topology of electron localization function (ELF or g(r), a 3D scalar field) has been successfully used to map a wealth of chemical concepts (Refs. [2–4] provide some examples) onto real space parlance. The ability of ELF to provide vivid images of regions where electrons pair up, e.g., cores, bonds, lone pairs, etc, together with the change that its topology suffers when bonds break or form, allows following the course of chemical processes by means of what has been called bonding evolution theory (BET) [5–10]. BET has the advantage of using similar terms as those currently employed by conventional chemical reactivity theory, and has been welcomed by practicing chemists. Several interpretations [11–13] and generalizations [14–16] of ELF have extended its validity to general correlated wavefunctions, renewing the interest on localization and delocalization issues in chemistry. ELF attraction basins are, in general, 3D regions surrounded by the separatrices of the $g(r) gradient field (which may lie at infinity). One may thus in principle integrate operator densities over these ELF basins such that the expectation value of every global molecular observable *
Corresponding author. E-mail address: angel@fluor.quimica.uniovi.es (A. Martı´n Penda´s).
0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.02.029
A P may be partitioned R into basin contributions: hAi ¼ X hAiX , where hAiX ¼ X W AWdr, and A is the appropriate operator density corresponding to A. Unfortunately, only the attraction basins of the electron density field, q(r), provide hermitian-like behavior for kinetic energy densities, a fact that singles out the quantum theory of atoms in molecules (QTAIM) of Bader et al. [17] over other real space partitions. This precludes a rigourous partitioning of the total molecular energy, E, into ELF basin contributions. All common electrostatic potential terms, i.e., the electron–nucleus (en) and the electron–electron (ee) interactions, may however be safely decomposed through the topology induced by g(r). The mutual electron–nucleus attraction is then written as a sum ofPone-basin ELF terms, R only dependent on q(r): V en ¼ X;a X qðrÞ=jr Ra jdr, where a sums over nuclei. These integrals are easily obtained and behave as expected. Much more interesting is the decomposition of the interelectron into Rrepulsion P R 0 two-basin contributions: V ee ¼ 12 X;X0 P dr dr q ðr ;r 1 2 1 2 Þ= 2 X X 0 r12 . This may be easily recast as V ee ¼ XPX0 V X;X . ee Thanks to algorithmic advances [18,19], it is now feasible to compute the 6D numerical integrals needed to decompose Vee in real space partitionings. We have used these new computational tools to develop a theory of interacting quantum atoms [20–22] within the QTAIM, which has provided very interesting insights into the nature of the chemical bond [20–27]. Since we may always
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decompose q2 into a Coulombic contribution and a pure exchange–correlation term, q2 ¼ qC2 þ qxc [18], then 2 X;X0 X;X0 X;X0 V ee ¼ V C þ V xc . We have clearly shown [21–25] that 0 0 interbasin exchange–correlation energies, V X;X xc , X 6¼ X , provide the quantum mechanical glue associated to covalent interactions from the real space perspective. Actually, 0 V X;X may be seen as an energetic correlate of the xc covariance between the electron population distribution 0 functions associated to R two different basins, dX;X ¼ R 2covðX; X0 Þ ¼ 2 X dr1 X0 dr2 qxc 2 . This is a measure of the electron delocalization between the basins, a quantity which is alsoR known R as the delocalization index [4,28–31]. Similarly, X dr1 X dr2 qxc 2 is the basin localization index, kX. These quantities have also been reformulated and generalized in terms of statistical measures of the electron number distribution functions within basins [32,33]. In this Letter we report, for the first time, interbasin ELF electron–electron repulsions in a series of test molecules. To that end we have slightly modified our promolden code so that it can perform its integration tasks over ELF, instead of QTAIM basins. Although neither q’s nor q2’s appear spherical around the ELF attractors, and thus the ELF basins are not ideally suited to the bipolar spherical grids used in promolden, we have found no significant loss of precision in the ELF integrations reported here. We will undertake, however, a deeper, more specific revision of our algorithms to optimize their performance with ELF. Since the evaluation of ee repulsions needs a properly defined second-order matrix which is lacking in current density functional implementations, on the one hand; and our current promolden implementation of ELF uses just the classical formula by Becke and Edgecombe, on the otherhand, we have decided to examine simple Hartee–Fock (HF) wavefunctions. Therefore, all our calculations refer to HF//6-311G+(d,p) wavefunctions at theoretical stationary points, as calculated by the gamess code [34]. 0 Our results will show that a map of V X;X onto the Gillesee pie–Nyholm rules of the valence shell electron pair repulsion (VSEPR) model [35–37] appears in the cases studied. A correspondence between the ELF basins and the VSEPR domains is well known [38,39], and as recently summarized by Gillespie and Robinson [40]: ‘This (ELF) function exhibits maxima at the most probable positions of localized electron pairs and each maximum is surrounded by a basin in which there is an increased probability of finding an electron pair. These basins correspond to the qualitative electron pair domains of the VSEPR model and have the same geometry as the VSEPR domains’. We will also show that exchange–correlation energies between ELF basins do also provide a real space view of conjugation or hyperconjugation. We will first examine the tetrahedral arrangement of electron pairs in the AHnE4n (E lone pair) prototypical series formed by CH4, NH3, and H2O. We will then comment on the link of two of these tetrahedra, first in the edge and face sharing cases formed by ethylene and acetylene, which will exem-
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plify the effect of multiple bonds, and then in CH3– AHnE4n vertex sharing case. Finally, we will briefly comment on conjugation issues by comparing formaldehyde and formamide. Some conclusions will end the Letter. 2. The tetrahedral AHnE4n case: CH4, NH3, and H2O The optimized geometries of methane, ammonia, and water at our calculation level provide AH distances of ˚ respectively, with HAH angles 1.084, 1.001, and 0.941 A equal to 107.42° and 106.22° for NH3 and H2O. The ELF topologies of the three compounds are simple, with one core basin for the central A atom which is linked to the four V(A, H) basins in methane, to the three V(A, H) basins and the E lone pair in ammonia, or only to the two lone pairs in water. There are also (3, 1) critical points (bond interaction points in ELF parlance, bip’s) between each E pair and the V(A, H) basins, between the two equivalent E pairs in water, and between each pair of different V(A, H) basins, except in H2O, where the two V(A, H) basins are not directly linked. The AE distances ˚ in NH3 and H2O, respectively, and are 0.750 and 0.598 A the EAH angles 111.49° and 111.34°, in the same order. The EAE angle in water is 105.38°. Table 1 contains a summary of our ELF data. As intrabasin values are regarded, it is interesting to notice how the population of the A cores and lone pairs increase with the electronegativity of the central atom, and how this is coupled to a decrease in the number of electrons in the bonding pairs. Similarly, the localization of the electrons in the different pair basins decreases as A becomes more electronegative. Following a rather general rule, the degree of localization (k/N) follows the order A > V > E. 88% of the carbon core electrons are localized in the core basin of methane, this number decreasing to 85% in water. Similarly 55–60% of the electrons are localized in the V(A, H) bonding basins, while only 53–56% of them are localized in lone pairs. The g, d data may be similarly analyzed for the interbasin cases. As known, the ELF core–valence transition is quite abrupt, signalling the separation of those basins as localization is concerned, and the g values at the AV bip for the A atom are very low, although they again increase with the electronegativity of A. As another rule, g increases at the bips in the AV < AE < VV < VE < EE order for a given system. The same general trend is found for delocalization indices. All this is compatible with the intrabasin behavior just commented. Turning to the main topic of this Letter, the electrostatic interaction between the two electrons lying approximately in each pair basin follows an intuitive behavior. Even scaling to exactly 2.0 electron occupation (so that Vee(scaled) = Vee (2.0/N)2), the core repulsion increases from methane to water, as the core shrinks. The same occurs for the V(A, H) case, although the loss of electrons in H2O makes the final Vee value smaller than its analogues in ammonia. The scaled values, 581, 683, and 776 kcal/mol
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Table 1 ELF properties for CH4, NH3, and H2O CH4
A
V
E
AV
AE
VV
VE
EE
g k N Vee
1.000 1.845 2.094 2367
1.000 1.362 1.976 567
– – – –
g d Vee Vxc
0.076 0.123 1573 38.3
– – – –
0.670 0.368 867 60.9
– – – –
– – – –
NH3 g k N Vee
A 1.000 1.820 2.107 3298
V 1.000 1.195 1.939 642
E 0.964 1.158 2.077 782
g d Vee Vxc
AV 0.106 0.129 1800 45.8
AE 0.132 0.186 2314 82.5
VV 0.714 0.403 975 77.0
VE 0.844 0.551 1099 110.3
EE – – – –
H2O g k N Vee
A 1.000 1.796 2.121 3808
V 0.999 0.962 1.751 595
E 0.925 1.163 2.190 1045
g d Vee Vxc
AV – 0.121 1816 44.6
AE 0.141 0.203 2863 102.9
VV – 0.335 876 103.2
VE 0.826 0.566 1215 136.1
EE 0.913 0.721 1635 174.7
A, V, and E stand for the A core, the V(A, H) valence basin, and the A lone pair(s), respectively. AV, AE, VV, EV, and EE are the pairs of the above basin. ELF (g) values are given at the attractors in the A,V, and E cases, and at the bips when pairs are considered. Similarly, k refers to the intra-basin localization (A,V,E) and d to the inter-basin delocalization (AV,AE, etc) indices. Basin populations(N) are given in e, and electron interactions in kcal/ mol.
for CH4, NH3, and H2O, respectively, grow monotonously. As the lone pairs are concerned, even their Vee scaled values, 725 and 871 kcal/mol in ammonia and water, respectively, are slightly larger than the V(A, H) ones. This is basically due to the absence of a second nucleus which spreads the electron charge farther away from the A nucleus. As commented, the E ELF attractors are closer to the A nuclei than the V(A, H) ones. The interbasin repulsions show the expected behavior. Interactions between the core and the valence basins are large, greater in the AE case. Moreover, the VV, EV, and EE interactions follow the order prescribed by the Gillespie–Nyholm (GN) rules: two lone pairs repel more intensely than a lone pair and a bond basin, and the latter repel more than two bond basins. It is also clear that the delocalization of electrons, and thus the exchange contribution to the energy increases again in the AV < AE < VV < VE < EE order. Notice, for instance, that as many as 0.57 electrons are delocalized from each OH bond into each oxygen’s lone pair in water (and viceversa), or that each of these two lone pairs delocalizes 0.7 electrons over the other’s basin. As we will see, lone pairs are particularly efficient at delocalizing electrons with their neighbors. It must be stressed that Vee must not be understood as the origin of the geometrical disposition of the electron pairs that the VSEPR model tries to explain. The basic electron arrangement is controlled by the antisymmetry of the wavefunction, which strongly forces equal spin electrons to lie far apart. Upon this scenario, nuclear attractions effectively pair up unequal spin electrons by overcoming their Coulombic repulsion. This generates the tetrahedral arrangement of the valence pairs in our nondiatomic cases. Finally, the inter-pair repulsion modulates the tetrahedral motif [41]. It is at this last level that the Vee electrostatic repulsion provides an approximate map
between a physical quantity and the VSEPR rules. The advantage of our data lies in the quantification of the rules. It is rather clear that repulsions between pairs associated to different central atoms will differ greatly according to, for instance, the electronegativity of the central atom. 3. Multiple bonds: ethylene and acetylene We have also examined C2H4 and C2H2 as examples to test the validity of the multiple bond VSEPR rule within our Vee mapping. Ethylene shows two V(C, C) attractors above and below the molecular plane, while acetylene shows a toroidal attractor surrounding the CC axis. A plot of the ELF attractors (g = 0.85 isosurfaces) for these and all the other systems examined in this Letter is found in Fig. 1. We will refer from now on to the different ELF basins according to the numbering shown in the figure. For ethylene, the basin populations are 2.090, 2.100, and 1.709 e for the C, V(C, H), and V(C, C) basins (basins 1, 3, and 7 in Fig. 1). Similarly, these numbers are 2.090, 2.243, and 5.331 for acetylene. In order to simplify our discussion, we will present only interbasin electron interactions from now on, see Table 2. It is first interesting to compare the CH4, C2H4, and C2H6 series. The population of the V(C, H) basin increases clearly along the series, with a related decrease in the population of the carbon cores. This process is coupled with an increased electron delocalization between the core and the CH bond basin. Accordingly, their mutual repulsion also increases, passing from 1600 to 1900 kcal/mol on going from methane to acetylene. Similarly, the core–core delocalization and repulsion is greater for ethylene than for acetylene, an effect which is basically related to the CC distance shortening. Notice the comparatively large Vxc and d values for the 3–4 interaction in C2H2. They signal a certain amount of long-range electron
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Fig. 1. ELF isosurfaces corresponding to g = 0.85 for the following systems, ordered in a row-wise manner from left to right: first row, C2H4 and C2H2; second row, staggered and eclipsed C2H6; third row, CH3NH2, CH3OH, and CH3F; fourth row, H2CO, and HCONH2 in its ground and one of its rotational transition states. All the attractors are numbered and their approximate locations pointed out. Notice that some connected sections of these g = 0.85 isosurfaces encompass several attractors. In particular, the torus-like portion of the isosurface surrounding the F atom in CH3F encloses the three fluorine lone pair attractors 7, 8, and 9.
Table 2 ELF interbasin electron interactions for ethylene and acetylene C2H4
1–2
1–3
1–4
1–7
3–4
3–5
3–6
3–7
7–8
d Vee Vxc
0.004 1099 0.6
0.135 1709 43.5
0.012 771 1.2
0.095 1371 30.9
0.065 660 6.0
0.377 989 64.0
0.061 562 4.0
0.291 798 48.5
0.529 801 121
C2H2 d Vee Vxc
1–2 0.006 1223 0.9
1–3 0.158 1919 54.5
1–4 0.021 806 1.9
1–5 0.297 4154 93.7
3–4 0.092 622 6.5
3–5 0.985 2520 154.3
Numbering refers to Fig. 1. All energetic data in kcal/mol.
delocalization among the hydrogens. A similar effect is found on comparing the 3–4 and 3–6 delocalizations for ethylene. Given the difference in the H3–H4 and H3–H6
˚ , respectively) and the general distances (2.449 and 3.059 A exponential decay in d with distance, the 3–6 delocalization is larger than expected. This is a real space signature of
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sigma hyperconjugation between the trans CH bonds [42]. On the contrary, the difference in both repulsion and delocalization for the geminal 3–5 interaction when comparing methane and ethylene is basically accounted for by the larger basin population in the latter. As the multiple bonds are regarded, each pair comprising the ethylene double bond provides core-V(C, C) repulsion, d, and Vxc values clearly smaller than for the coreV(C, H) case. This is a result of diminished V(C, C) population, and an example of the GN rule that decreases the ‘effective’ size of a bond pair basin as the electronegativity of the bonded atoms increases. Interestingly, the core-V(C, C) parameters in C2H2 is almost triple than those in each of the two equivalent C–C bond basins in ethylene. A similar behavior is found for the 3–7 interaction in ethylene and the 3–5 repulsion in acetylene. Overall, the two C–C bond basins of C2H4 provide a pseudo-tetrahedral arrangement around each carbon. Since the 7–8 repulsion is smaller than the 3–5 one, one may also rationalize the opening of the H3–C1–H4 angle using tetrahedral repulsions. In any case, if the double or triple bond is understood as a monolytic object, the GN rule relative to multiple bonds is again compatible with our Vee mapping. 4. CH3–AHnE4n systems Let us turn our attention to a number of simple cases where two tetrahedra of electron pairs share one vertex (one bonding domain). First we will consider C2H6, both in its staggered (s) and eclipsed (e) stationary conformations. To our integration accuracy, the population of the C (2.092 e) and V(C, H) (1.996 e) basins differ in both conformations by less than 0.001 e. Their cumulative difference is however amplified, and the population of the V(C, C) basin is slightly smaller in the eclipsed (1.836 e) than in the staggered geometry (1.838 e). The interbasin electron interaction parameters are gathered in Table 3. There are several interesting points if Tables 1–3 are compared. Firstly, the C–V(C, H) interaction parameters are quite transferable between methane and ethane, although a slightly more compact electron distribution exists in the latter. This is in opposition to the C2H4 and C2H2 cases, so the core–valence repulsion clearly discriminates these CH bonds. The same is true for the geminal V(C, H)–V(C,
H0 ) repulsion parameters, e.g., the 3–5 or 3–4 interactions in ethylene or ethane, respectively. Moreover, the single bond V(C, C) basin in ethane behaves as a slightly weaker repeller than the V(C, H) one, in agreement with its comparable, but definitely smaller electron population. A CC single bond pair is thus almost equivalent, from our present viewpoint, to a CH one in saturated hydrocarbons. Secondly, the staggered–eclipsed comparison shows the sensitivity of our parameters to small perturbations. A QTAIM account of the role of interatomic Vxc in determining the conformational preference has already been presented [42] and will not be discussed here. Most of the relevant effects are related to changes in geometries. For instance, the H3–H5 and H3–H6 distances are 2.524 and ˚ in the s conformer, respectively, and 2.922 and 3.074 A ˚ 2.342 A in the e one, in the same order, and this fact justifies their tabulated repulsions. The total vicinal V(C, H) exchange contributions, i.e. the total exchange between all different V(C, H)–V(C0 ,H0 ) pairs that belong to different methyl groups, add to 24.6 kcal/mol in both geometries, but it should be noticed that the 3–6 Vxc value in the s conformer (1.8 kcal/mol) is larger than its 3–5 counterpart in the e geometry (1.5 kcal/mol), even when the H3–H6 distance in s is larger than the H3–H5 one in e. This is again a manifestation of higher delocalization (or hyperconjugation) for antiperiplanar configurations. Changing the second methyl to NH2, OH, or F does not alter the overall staggered tetrahedral arrangement of the pair basins, although the three lone pairs of the F atom in CH3F are not well resolved in Fig. 1 at the g = 0.85 value. Actually, the three attractors have g = 0.892, while the three bips interconnecting them lie at g = 0.891. Aside this minor point, these systems provide an interesting set to compare the behavior of V(C,A), V(A, H), and A lone pairs as the electronegativity of the A atom increases. Since the number of independent pair basins increases due to the lack of symmetry, we will restrict to valence–valence interactions. Core–core, and core–valence repulsions are of smaller chemical relevance, and their behavior follows the same valence–valence rules, as we have seen in the preceding examples. Electron populations in the ELF attraction basins of CH3NH2 are 2.091, 2.025, 2.009, 2.107, 1.998, 2.062, and 1.697 e for the attractors labelled as 1, 2, 3, 5, 6, 8, and 9, respectively. In methanol: 2.092, 2.127, 2.009,
Table 3 ELF interbasin electron interactions for the staggered (s) and eclipsed (e) conformations of ethane C2H6(s)
1–2
1–3
1–5
1–9
3–4
3–5
3–6
3–9
d Vee Vxc
0.002 951 0.3
0.006 695 0.6
0.124 1585 38.7
0.100 1537 34.3
0.366 887 61.0
0.030 580 3.2
0.032 496 1.8
0.282 851 52.0
C2H6(e) d Vee Vxc
1–2 0.002 943 0.3
1–3 0.006 690 0.6
1–5 0.125 1589 38.7
1–9 0.099 1525 33.8
3–4 0.371 890 61.9
3–5 0.022 518 1.5
3–6 0.044 610 5.2
3–9 0.283 848 51.7
Numbering refers to Fig. 1. All energetic data in kcal/mol.
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2.025, 1.764, 1.380, and 2.287 e for the 1, 2, 3, 4, 6, 7, and 8 basins. And 2.089, 2.034, 2.135, 1.013, and 2.220 e for the 1, 2, 5, 6, and 7 basins in CH3F. Notice how the V(C,F) basin (number 6), cannot be properly classified as a pair basin. This is a well known characteristic of C–F links. Table 4 contains a summary of the valence repulsions. As the methyl is concerned, we can now follow the evolution of the population and inter-basin repulsion for any of its three pairs of V(C, H)–V(C, H) interactions. Vee increases slightly but steadily along the CH4 < C2H6 < CH3NH2 < CH3OH < CH3F series, from 867 to 940 kcal/ mol, following the concomitant increase in the population of the basin. Similarly, the V(C, H)–V(C,A) repulsion decreases along the series (no value in methane) from 851 kcal/mol in ethane to 522 kcal/mol in fluoromethane. An important component in this change is again the decrease in electron population of the V(C,A) pair basin, but in the end, a rule appears relating the ‘effective’ GN size of a CA bond pair with the electronegativity of the A atom. Notice that the increase in the H–C–H angle of the methyl group from 107° to 110° on passing from methanamine to fluoromethane provides again a consistent mapping of the GN rules onto our Vee values. We can also compare our previous NH3 and H2O data with the numbers of Table 4. For instance, the V(N,H)– V(N,H) repulsion in the amino group of CH3NH2 (1038 kcal/mol) is larger than that found in ammonia (975 kcal/mol). The same is true for Vxc or d. This is coupled to an also larger E–V(N,H) repulsion (1124 versus 1099 kcal/mol) which, however, supplies a smaller Vxc term, and to similarly compatible E–V(C,N) and V(N,H)–V(C,N) repulsions. Since the relative weight of the bond–bond and bond–lone pair Vee has not changed
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much in methylamine with respect to ammonia, the H– N–H angle (107.5°) remains almost untouched. A similar analysis may be made in methanol. Let us just notice that, following the general trends, the E–E and E–V(O,H) repulsions are slightly larger than those in H2O. Another important point concerns the vicinal interactions. For instance, the gauge 3–5 repulsion in staggered ethane, 580 kcal/mol, is smaller by almost 50 kcal/mol than its equivalent 3–6 interaction in CH3NH2, but very similar to the 4–6 repulsion found in methanol. The V(C, H)–E gauge interaction, grows steadily from 683 to 829 kcal/ mol on going from CH3NH2 to CH3F. All this may be rationalized in terms of distances and basin populations, and it is relevant to point out that these interactions carry non-negligible electron delocalizations and exchange stabilizations associated with them ranging from 7.1 to 8.6 kcal/mol. Our Vee mapping also holds for the E–E interaction in methanol. The E–E (8–9) repulsion is significantly larger than that found in H2O, and the E–O–E angle (109.8°) increases. 5. Conjugation: formaldehyde versus formamide Let us conclude our analysis by comparing formaldehyde with the planar (p) geometry and one pyramidal (py) transition state (see Fig. 1) of formamide, 16 kcal/ mol above the planar situation. This large rotational barrier in HCONH2 has been traditionally related to p conjugation and is the subject of continuous revisions due to its relation to the structure of polypeptides and proteins [43]. Since even now the number of valence–valence interactions is too large, we will only comment on the O–C–A backbone. As shown in Fig. 1, the C–O double bond in
Table 4 ELF valence interbasin electron interactions for the CH3NH2, CH3OH, and CH3F systems at their theoretical equilibrium geometries CH3NH2
2–3
2–6
2–8
2–9
3–6
3–7
3–8
d Vee Vxc
0.366 911 61.4
0.036 624 4.0
0.050 590 3.3
0.220 811 42.6
0.031 627 3.5
0.033 539 2.1
0.056 683 7.1
d Vee Vxc
3–9 0.218 806 42.4
6–7 0.420 1038 83.8
6–8 0.552 1124 100.8
6–9 0.316 919 69.2
8–9 0.412 1010 93.8
CH3OH d Vee Vxc
3–4 0.366 918 61.7
3–6 0.030 500 2.0
3–8 0.055 803 7.4
3–7 0.162 680 33.3
4–5 0.367 922 61.8
4–6 0.028 579 3.3
4–8 0.048 707 3.6
d Vee Vxc
4–9 0.067 811 9.1
4–7 0.165 682 33.7
6–8 0.568 1280 138
6–7 0.228 722 52.8
8–9 0.741 1775 181
8–7 0.418 1057 114
CH3F d Vee Vxc
2–3 0.371 940 62.7
2–7 0.059 829 8.6
2–9 0.043 730 3.7
2–6 0.114 522 24.9
7–8 0.728 1970 53.7
7–6 0.318 845 18.2
Numbering refers to Fig. 1. Energetic data in kcal/mol.
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formaldehyde has two equivalent V(C,O) pair basins (the 5 and 6 basins in the figure) lying above and below the molecular plane, as the C–C link in ethylene. Accordingly, the two lone pairs of oxygen (basins 7 and 8 in Fig. 1) thus lie in the H–C–H plane. The populations of the 1, 2, 3, 5, and 7 basins are 2.087, 2.118, 2.103, 1.266, and 2.524 e, respectively. Each of the V(C,O) basins contains much less than 2 e, this being a characteristic of highly heteropolar bonding basins, as already commented. In planar formamide, the two V(C,O) basins have collapsed into one over the C–O axis, basin number 8 in Fig. 1, and the nitrogen’s lone pair (basin 7 in Fig. 1) is orthogonal to the plane, displaying two equivalent attractors which are not well resolved from the V(C,N) bonding pair. This is a direct demonstration of the ‘partial’ p character of the C–N bond, as it is well known. The basin populations of basins 1–4, 7–9, and 11 are 2.087, 2.119, 2.117, 2.109, 2.198, 2.408, 2.688, and 0.820 e, respectively. Rotation of the NH2 group around the C–N bond induces the pyramidalization of the N atom. In the transition state geometry the lone pair lies in the H–C–N plane, the C–O link goes back to its two equivalent V(C,O) basins above and below the plane, and most signs of p delocalization around the C–N bond have disappeared. The basin populations of basins 1-4,79, and 11 are 2.087, 2.119, 2.115, 2.107, 2.114, 1.871, 1.298, and 2.557 e, in that order. A simple look at the basin populations allows us to recover many traditional chemical concepts. For instance, the V(C,N) population in planar formamide, 2.198 e, is considerably larger than its equivalent in CH3NH2, 1.697 e. Similarly, the total V(C,O) population, 2.408 e, is significantly smaller than its equivalent in formaldehyde, 2.532 e. Coupled to this, the oxygen’s lone pairs bear greater population, and the complete lone pair on the N atom (1.639 e) has considerably less electrons than its analogue in CH3NH2, 2.062 e. This number is recovered in the pyramidal geometry, with a N lone pair populated with 2.114 e. Table 5 contains the basin-basin interaction parameters of the O–C–A backbone. Each V(C,O) basin of the C–O double bond in H2COH has a noticeably smaller repulsion with the V(C, H) basin, 650 kcal/mol, than its equivalent in
ethylene, 798 kcal/mol. This is compensated with the V(C, H)–E repulsion, 838 and 965 kcal/mol with the trans and cis lone pairs, respectively. The equivalent V(C, H)–V(C, H) figures in C2H4, 562 and 660 kcal/mol, in the same order, show how large the vicinal V(C, H)–E interaction is in formaldehyde. Notice also how the much larger population of the two oxygen’s lone pairs generates very large 7-8 interactions, with an huge delocalization between the two pairs (d = 0.865) that leads to a E–O–E angle of 122 degrees, much larger than that found in H2O. Let us now turn to conjugation in planar formamide with respect to the pyramidal transition state. Pyramidalization decreases the population of the O lone pairs, and increases that in the V(C,O) basin(s). These are relatively localized changes which may be envisioned as a flow of electrons from the O lone pairs into the double C–O bond. The 7-8 V(C,N)–V(C,O) interaction parameters in the planar geometry are similar to twice the 8-9 analogues in the pyramidal structure, even when the p7+p8 population is considerably larger than its py8+2py9 equivalent. A similar analysis holds for the V(C,N)–E(O) terms. r delocalization of the O–C–N backbone is thus not an important ingredient in the stabilization of planar CONH2. A completely different picture is obtained if we look at the behaviour of the nitrogen’s lone pair. In the planar geometry, the V(C,N)– E(N) repulsion is about 1090 kcal/mol (twice the 7-11 results from Table 5). This is similar (but smaller) to the value at the barrier, 1120 kcal/mol. However, the delocalization indices and exchange stabilizations associated to this interaction are completely different, d = 0.620 versus 0.445 in the p and py geometries, respectively. Or Vxc = 146 versus 99 kcal/mol. Notice that this difference is counteracted by the larger delocalization and exchange between the C–O bond basin(s) and the N lone pair in the pyramidal transition state: d = 0.120 and Vxc = 26 kcal/mol for py, while the figures are 0.050 and -5.6 kcal/mol for p. This may be interpreted as a more efficient anti hyperconjugative effect between the C–O bond and the N lone pair in the py conformation. Overall, the picture offered by our interaction analysis shows very clearly a charge transfer compatible with the
Table 5 ELF valence interbasin electron interactions for the O–C–A backbone in formaldehyde and the planar and pyramidal geometries of formamide H2CO d Vee Vxc
3–4 0.367 842 61.7
3–5 0.166 650 32.1
3–7 0.107 838 8.8
3–8 0.139 965 17.0
5–7 0.419 1020 102
7–8 0.865 2160 214
HCONH2(p) d Vee Vxc
7–8 0.234 1380 52.0
7–9 0.056 962 5.0
7–10 0.077 1060 10.3
7–11 0.320 547 78.1
8–9 0.840 2060 206
8–10 0.814 2000 200
8–11 0.025 341 2.8
9–10 0.986 2350 243
9–11 0.015 270 0.9
10–11 0.015 302 1.1
HCONH2(py) d Vee Vxc
7–8 0.445 1120 99.1
7–9 0.014 414 1.2
7–11 0.011 594 0.5
7–12 0.017 643 1.1
8–9 0.120 632 26.2
8–11 0.057 783 5.1
8–12 0.079 887 10.6
9–10 0.334 6200 768
9–11 0.430 1050 105
11–12 0.884 2180 41.4
Numbering refers to Fig. 1. Energetic data in kcal/mol.
A. Martı´n Penda´s et al. / Chemical Physics Letters 454 (2008) 396–403
classical resonance structures in planar formamide, with a C–O bond that has lost part of its double bond character by allowing bonding electrons to flow from the bonding basin into the oxygen’s lone pairs. This process is coupled to a C–N link gaining double bond character by delocalizing electrons of the N lone pair into the C–N bond basin. However, the C–O and C–N bonds seem to be quite independent, and no important delocalization occurs, for example, between the N lone pair and the V(C,O) or E(O) basins. Interestingly enough, the total DVxc of our O–C–N backbone between the p and py structures offers an stabilizing balance of 18.8 kcal/mol for the planar structure, a value very similar to the total rotation barrier. The latter may thus be interpreted as a result of a larger exchange (delocalization) contribution in the planar arrangement. 6. Conclusions We have presented in this Letter, for the first time, numerical electron–electron interaction parameters between real space basins of ELF attractors in a battery of test systems. Our preliminary Hartree–Fock data show that once the overall arrangement of the electron pairs is determined by antisymmetry requirements, i.e. tetrahedral in the four pairs case, there exists a faithful mapping of the electrostatic electron–electron repulsion between the pair basins and the Gillespie–Nyholm rules of the VSEPR model. We believe that these results will hold in more general cases, and further calculations in other systems with non tetrahedral pair distributions will be considered in the near future. The inter-basin parameters used in this work: electron– electron total repulsions, Vee’s; the covariance in the basin populations, d’s; and the exchange interaction energy, Vxc’s, open new avenues towards quantitative ELF analyses. A combination of the ability of ELF or ELF-like fields to provide appealing pictures of chemical processes with our more quantitative analyses may prove a useful tool. For instance, the flow of charge among basins for the rotation barrier problem in formamide, taken together with the inter-basin delocalizations and their associated covalentlike Vxc contributions provided by our parameters, depict an scenario akin to classical Pauling resonance, with quantitative estimations of the stabilization brought about by each relevant electron delocalization. Acknowledgements Financial support from the Spanish MEC, Project No. CTQ2006-02976 and the ERDF of the European Union, is acknowledged. References [1] A.D. Becke, K.E. Edgecombe, J. Chem. Phys. 92 (1990) 5397. [2] B. Silvi, A. Savin, Nature 371 (1994) 683.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
[40] [41] [42] [43]
403
B. Silvi, I. Fourre´, M.E. Alikhani, Monatsh. Chem. 136 (2005) 855. J. Poater, M. Duran, M. Sola`, B. Silvi, Chem. Rev. 105 (2005) 3911. X. Krokidis, S. Noury, B. Silvi, J. Phys. Chem. A 101 (1997) 7277. X. Krokidis, V. Goncalves, A. Savin, B. Silvi, J. Phys. Chem. A 102 (1998) 5065. X. Krokidis, R. Vuilleumier, D. Borgis, B. Silvi, Mol. Phys. 96 (1999) 265. L. Joubert, C. Adamo, J. Chem. Phys. 123 (2005) 211103. S. Berski, J. Andre´s, B. Silvi, L. Domingo, J. Phys. Chem. A 107 (2003) 6014. V. Polo, J. Andre´s, J. Chem. Theor. Comput. 3 (2007) 816. A. Savin, O. Jepsen, J. Flad, O. Andersen, Angew. Chem. Int. Ed. 31 (1992) 187. R. Nalewajski, A. Koster, S. Escalante, J. Phys. Chem. A 109 (2005) 10038. E. Matito, B. Silvi, M. Duran, M. Sola`, J. Chem. Phys. 125 (2006) 02430. B. Silvi, J. Phys. Chem. A 107 (2003) 3081. M. Kohout, K. Pernal, F.R. Wagner, Y. Grin, Theor. Chem. Acc. 112 (2004) 453. M. Kohout, Int. J. Quantum Chem. 97 (2004) 651. R.F.W. Bader, Atoms in Molecules, Oxford University Press, Oxford, 1990. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Chem. Phys. 120 (2004) 4581. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, J. Comput. Chem. 26 (2005) 344. M.A. Blanco, A. Martı´n Penda´s, E. Francisco, J. Chem. Theor. Comput. 1 (2005) 1096. E. Francisco, A. Martı´n Penda´s, M.A. Blanco, J. Chem. Theor. Comput. 2 (2006) 90. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Comput. Chem. 28 (2007) 161. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Chem. Phys. 125 (2006) 184112. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, Faraday Discuss. 135 (2007) 423. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, J. Phys. Chem. A 110 (2006) 12864. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, C. Gatti, Chem. Eur. J. 13 (2007) 9362. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, Chem. Phys. Lett. 437 (2007) 287. R.F.W. Bader, M.E. Stephens, J. Am. Chem. Soc. 97 (1975) 7391. ´ ngya´n, M. Loos, I. Mayer, J. Phys. Chem. 98 (1994) 5244. J.G. A X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1999) 304. X. Fradera, J. Poater, S. Simon, M. Duran, M. Sola`, Theor. Chem. Acc. 108 (2002) 214. B. Silvi, Phys. Chem. Chem. Phys. 6 (2004) 256. E. Francisco, A. Martı´n Penda´s, M.A. Blanco, J. Chem. Phys. 126 (2007) 094102. M.W. Schmidt et al., J. Comput. Chem. 14 (1993) 1347. R.J. Gillespie, R.S. Nyholm, Q. Rev. Chem. Soc. 11 (1957) 339. R.J. Gillespie, I. Hargittai, The VSEPR Model of Molecular Geometry, Allyn & Bacon, 1991. R.J. Gillespie, Coord. Chem. Rev. 197 (2000) 51. R.J. Gillespie, B. Silvi, Coord. Chem. Rev. 233–234 (2002) 53. B. Silvi, R.J. Gillespie, The Quantum Theory of Atoms, in: C. Matta, R.J. Boyd (Eds.), Molecules: From Solid State to DNA and Drug Design, Wiley, NY, 2007, p. 141. R.J. Gillespie, E.A. Robinson, J. Comput. Chem. 28 (2007) 87. R.J. Gillespie, P.L.A. Popelier, Chemical Bonding and Molecular Geometry, Oxford University Press, New York, 2001. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Comput. Chem., submitted for publication. J.I. Mujika, J.M. Matxain, L.A. Eriksson, X. Lopez, Chem. Eur. J. 12 (2006) 7215.
Chemical Physics Letters 454 (2008) 396–403 www.elsevier.com/locate/cplett
Electron–electron interactions between ELF basins A. Martı´n Penda´s *, E. Francisco, M.A. Blanco Departamento de Quı´mica Fı´sica y Analı´tica, Facultad de Quı´mica, Universidad de Oviedo, 33006 Oviedo, Spain Received 11 January 2008; in final form 11 February 2008 Available online 15 February 2008
Abstract The electrostatic repulsions among the electrons that populate different attraction basins of the electron localization function (ELF) are numerically obtained for a number of simple test molecules. Differences in these interbasin repulsions are shown to provide a map onto the Gillespie–Nyholm rules of the valence shell electron pair repulsion model, and the reasons behind this correspondence are briefly analyzed. We think that these repulsion parameters, including the inter-basin exchange energy, which we relate to resonance and conjugation, open new avenues towards more quantitative ELF analyses. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Since its proposal in 1990 [1], the topology of electron localization function (ELF or g(r), a 3D scalar field) has been successfully used to map a wealth of chemical concepts (Refs. [2–4] provide some examples) onto real space parlance. The ability of ELF to provide vivid images of regions where electrons pair up, e.g., cores, bonds, lone pairs, etc, together with the change that its topology suffers when bonds break or form, allows following the course of chemical processes by means of what has been called bonding evolution theory (BET) [5–10]. BET has the advantage of using similar terms as those currently employed by conventional chemical reactivity theory, and has been welcomed by practicing chemists. Several interpretations [11–13] and generalizations [14–16] of ELF have extended its validity to general correlated wavefunctions, renewing the interest on localization and delocalization issues in chemistry. ELF attraction basins are, in general, 3D regions surrounded by the separatrices of the $g(r) gradient field (which may lie at infinity). One may thus in principle integrate operator densities over these ELF basins such that the expectation value of every global molecular observable *
Corresponding author. E-mail address: angel@fluor.quimica.uniovi.es (A. Martı´n Penda´s).
0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.02.029
A P may be partitioned R into basin contributions: hAi ¼ X hAiX , where hAiX ¼ X W AWdr, and A is the appropriate operator density corresponding to A. Unfortunately, only the attraction basins of the electron density field, q(r), provide hermitian-like behavior for kinetic energy densities, a fact that singles out the quantum theory of atoms in molecules (QTAIM) of Bader et al. [17] over other real space partitions. This precludes a rigourous partitioning of the total molecular energy, E, into ELF basin contributions. All common electrostatic potential terms, i.e., the electron–nucleus (en) and the electron–electron (ee) interactions, may however be safely decomposed through the topology induced by g(r). The mutual electron–nucleus attraction is then written as a sum ofPone-basin ELF terms, R only dependent on q(r): V en ¼ X;a X qðrÞ=jr Ra jdr, where a sums over nuclei. These integrals are easily obtained and behave as expected. Much more interesting is the decomposition of the interelectron into Rrepulsion P R 0 two-basin contributions: V ee ¼ 12 X;X0 P dr dr q ðr ;r 1 2 1 2 Þ= 2 X X 0 r12 . This may be easily recast as V ee ¼ XPX0 V X;X . ee Thanks to algorithmic advances [18,19], it is now feasible to compute the 6D numerical integrals needed to decompose Vee in real space partitionings. We have used these new computational tools to develop a theory of interacting quantum atoms [20–22] within the QTAIM, which has provided very interesting insights into the nature of the chemical bond [20–27]. Since we may always
A. Martı´n Penda´s et al. / Chemical Physics Letters 454 (2008) 396–403
decompose q2 into a Coulombic contribution and a pure exchange–correlation term, q2 ¼ qC2 þ qxc [18], then 2 X;X0 X;X0 X;X0 V ee ¼ V C þ V xc . We have clearly shown [21–25] that 0 0 interbasin exchange–correlation energies, V X;X xc , X 6¼ X , provide the quantum mechanical glue associated to covalent interactions from the real space perspective. Actually, 0 V X;X may be seen as an energetic correlate of the xc covariance between the electron population distribution 0 functions associated to R two different basins, dX;X ¼ R 2covðX; X0 Þ ¼ 2 X dr1 X0 dr2 qxc 2 . This is a measure of the electron delocalization between the basins, a quantity which is alsoR known R as the delocalization index [4,28–31]. Similarly, X dr1 X dr2 qxc 2 is the basin localization index, kX. These quantities have also been reformulated and generalized in terms of statistical measures of the electron number distribution functions within basins [32,33]. In this Letter we report, for the first time, interbasin ELF electron–electron repulsions in a series of test molecules. To that end we have slightly modified our promolden code so that it can perform its integration tasks over ELF, instead of QTAIM basins. Although neither q’s nor q2’s appear spherical around the ELF attractors, and thus the ELF basins are not ideally suited to the bipolar spherical grids used in promolden, we have found no significant loss of precision in the ELF integrations reported here. We will undertake, however, a deeper, more specific revision of our algorithms to optimize their performance with ELF. Since the evaluation of ee repulsions needs a properly defined second-order matrix which is lacking in current density functional implementations, on the one hand; and our current promolden implementation of ELF uses just the classical formula by Becke and Edgecombe, on the otherhand, we have decided to examine simple Hartee–Fock (HF) wavefunctions. Therefore, all our calculations refer to HF//6-311G+(d,p) wavefunctions at theoretical stationary points, as calculated by the gamess code [34]. 0 Our results will show that a map of V X;X onto the Gillesee pie–Nyholm rules of the valence shell electron pair repulsion (VSEPR) model [35–37] appears in the cases studied. A correspondence between the ELF basins and the VSEPR domains is well known [38,39], and as recently summarized by Gillespie and Robinson [40]: ‘This (ELF) function exhibits maxima at the most probable positions of localized electron pairs and each maximum is surrounded by a basin in which there is an increased probability of finding an electron pair. These basins correspond to the qualitative electron pair domains of the VSEPR model and have the same geometry as the VSEPR domains’. We will also show that exchange–correlation energies between ELF basins do also provide a real space view of conjugation or hyperconjugation. We will first examine the tetrahedral arrangement of electron pairs in the AHnE4n (E lone pair) prototypical series formed by CH4, NH3, and H2O. We will then comment on the link of two of these tetrahedra, first in the edge and face sharing cases formed by ethylene and acetylene, which will exem-
397
plify the effect of multiple bonds, and then in CH3– AHnE4n vertex sharing case. Finally, we will briefly comment on conjugation issues by comparing formaldehyde and formamide. Some conclusions will end the Letter. 2. The tetrahedral AHnE4n case: CH4, NH3, and H2O The optimized geometries of methane, ammonia, and water at our calculation level provide AH distances of ˚ respectively, with HAH angles 1.084, 1.001, and 0.941 A equal to 107.42° and 106.22° for NH3 and H2O. The ELF topologies of the three compounds are simple, with one core basin for the central A atom which is linked to the four V(A, H) basins in methane, to the three V(A, H) basins and the E lone pair in ammonia, or only to the two lone pairs in water. There are also (3, 1) critical points (bond interaction points in ELF parlance, bip’s) between each E pair and the V(A, H) basins, between the two equivalent E pairs in water, and between each pair of different V(A, H) basins, except in H2O, where the two V(A, H) basins are not directly linked. The AE distances ˚ in NH3 and H2O, respectively, and are 0.750 and 0.598 A the EAH angles 111.49° and 111.34°, in the same order. The EAE angle in water is 105.38°. Table 1 contains a summary of our ELF data. As intrabasin values are regarded, it is interesting to notice how the population of the A cores and lone pairs increase with the electronegativity of the central atom, and how this is coupled to a decrease in the number of electrons in the bonding pairs. Similarly, the localization of the electrons in the different pair basins decreases as A becomes more electronegative. Following a rather general rule, the degree of localization (k/N) follows the order A > V > E. 88% of the carbon core electrons are localized in the core basin of methane, this number decreasing to 85% in water. Similarly 55–60% of the electrons are localized in the V(A, H) bonding basins, while only 53–56% of them are localized in lone pairs. The g, d data may be similarly analyzed for the interbasin cases. As known, the ELF core–valence transition is quite abrupt, signalling the separation of those basins as localization is concerned, and the g values at the AV bip for the A atom are very low, although they again increase with the electronegativity of A. As another rule, g increases at the bips in the AV < AE < VV < VE < EE order for a given system. The same general trend is found for delocalization indices. All this is compatible with the intrabasin behavior just commented. Turning to the main topic of this Letter, the electrostatic interaction between the two electrons lying approximately in each pair basin follows an intuitive behavior. Even scaling to exactly 2.0 electron occupation (so that Vee(scaled) = Vee (2.0/N)2), the core repulsion increases from methane to water, as the core shrinks. The same occurs for the V(A, H) case, although the loss of electrons in H2O makes the final Vee value smaller than its analogues in ammonia. The scaled values, 581, 683, and 776 kcal/mol
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Table 1 ELF properties for CH4, NH3, and H2O CH4
A
V
E
AV
AE
VV
VE
EE
g k N Vee
1.000 1.845 2.094 2367
1.000 1.362 1.976 567
– – – –
g d Vee Vxc
0.076 0.123 1573 38.3
– – – –
0.670 0.368 867 60.9
– – – –
– – – –
NH3 g k N Vee
A 1.000 1.820 2.107 3298
V 1.000 1.195 1.939 642
E 0.964 1.158 2.077 782
g d Vee Vxc
AV 0.106 0.129 1800 45.8
AE 0.132 0.186 2314 82.5
VV 0.714 0.403 975 77.0
VE 0.844 0.551 1099 110.3
EE – – – –
H2O g k N Vee
A 1.000 1.796 2.121 3808
V 0.999 0.962 1.751 595
E 0.925 1.163 2.190 1045
g d Vee Vxc
AV – 0.121 1816 44.6
AE 0.141 0.203 2863 102.9
VV – 0.335 876 103.2
VE 0.826 0.566 1215 136.1
EE 0.913 0.721 1635 174.7
A, V, and E stand for the A core, the V(A, H) valence basin, and the A lone pair(s), respectively. AV, AE, VV, EV, and EE are the pairs of the above basin. ELF (g) values are given at the attractors in the A,V, and E cases, and at the bips when pairs are considered. Similarly, k refers to the intra-basin localization (A,V,E) and d to the inter-basin delocalization (AV,AE, etc) indices. Basin populations(N) are given in e, and electron interactions in kcal/ mol.
for CH4, NH3, and H2O, respectively, grow monotonously. As the lone pairs are concerned, even their Vee scaled values, 725 and 871 kcal/mol in ammonia and water, respectively, are slightly larger than the V(A, H) ones. This is basically due to the absence of a second nucleus which spreads the electron charge farther away from the A nucleus. As commented, the E ELF attractors are closer to the A nuclei than the V(A, H) ones. The interbasin repulsions show the expected behavior. Interactions between the core and the valence basins are large, greater in the AE case. Moreover, the VV, EV, and EE interactions follow the order prescribed by the Gillespie–Nyholm (GN) rules: two lone pairs repel more intensely than a lone pair and a bond basin, and the latter repel more than two bond basins. It is also clear that the delocalization of electrons, and thus the exchange contribution to the energy increases again in the AV < AE < VV < VE < EE order. Notice, for instance, that as many as 0.57 electrons are delocalized from each OH bond into each oxygen’s lone pair in water (and viceversa), or that each of these two lone pairs delocalizes 0.7 electrons over the other’s basin. As we will see, lone pairs are particularly efficient at delocalizing electrons with their neighbors. It must be stressed that Vee must not be understood as the origin of the geometrical disposition of the electron pairs that the VSEPR model tries to explain. The basic electron arrangement is controlled by the antisymmetry of the wavefunction, which strongly forces equal spin electrons to lie far apart. Upon this scenario, nuclear attractions effectively pair up unequal spin electrons by overcoming their Coulombic repulsion. This generates the tetrahedral arrangement of the valence pairs in our nondiatomic cases. Finally, the inter-pair repulsion modulates the tetrahedral motif [41]. It is at this last level that the Vee electrostatic repulsion provides an approximate map
between a physical quantity and the VSEPR rules. The advantage of our data lies in the quantification of the rules. It is rather clear that repulsions between pairs associated to different central atoms will differ greatly according to, for instance, the electronegativity of the central atom. 3. Multiple bonds: ethylene and acetylene We have also examined C2H4 and C2H2 as examples to test the validity of the multiple bond VSEPR rule within our Vee mapping. Ethylene shows two V(C, C) attractors above and below the molecular plane, while acetylene shows a toroidal attractor surrounding the CC axis. A plot of the ELF attractors (g = 0.85 isosurfaces) for these and all the other systems examined in this Letter is found in Fig. 1. We will refer from now on to the different ELF basins according to the numbering shown in the figure. For ethylene, the basin populations are 2.090, 2.100, and 1.709 e for the C, V(C, H), and V(C, C) basins (basins 1, 3, and 7 in Fig. 1). Similarly, these numbers are 2.090, 2.243, and 5.331 for acetylene. In order to simplify our discussion, we will present only interbasin electron interactions from now on, see Table 2. It is first interesting to compare the CH4, C2H4, and C2H6 series. The population of the V(C, H) basin increases clearly along the series, with a related decrease in the population of the carbon cores. This process is coupled with an increased electron delocalization between the core and the CH bond basin. Accordingly, their mutual repulsion also increases, passing from 1600 to 1900 kcal/mol on going from methane to acetylene. Similarly, the core–core delocalization and repulsion is greater for ethylene than for acetylene, an effect which is basically related to the CC distance shortening. Notice the comparatively large Vxc and d values for the 3–4 interaction in C2H2. They signal a certain amount of long-range electron
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399
Fig. 1. ELF isosurfaces corresponding to g = 0.85 for the following systems, ordered in a row-wise manner from left to right: first row, C2H4 and C2H2; second row, staggered and eclipsed C2H6; third row, CH3NH2, CH3OH, and CH3F; fourth row, H2CO, and HCONH2 in its ground and one of its rotational transition states. All the attractors are numbered and their approximate locations pointed out. Notice that some connected sections of these g = 0.85 isosurfaces encompass several attractors. In particular, the torus-like portion of the isosurface surrounding the F atom in CH3F encloses the three fluorine lone pair attractors 7, 8, and 9.
Table 2 ELF interbasin electron interactions for ethylene and acetylene C2H4
1–2
1–3
1–4
1–7
3–4
3–5
3–6
3–7
7–8
d Vee Vxc
0.004 1099 0.6
0.135 1709 43.5
0.012 771 1.2
0.095 1371 30.9
0.065 660 6.0
0.377 989 64.0
0.061 562 4.0
0.291 798 48.5
0.529 801 121
C2H2 d Vee Vxc
1–2 0.006 1223 0.9
1–3 0.158 1919 54.5
1–4 0.021 806 1.9
1–5 0.297 4154 93.7
3–4 0.092 622 6.5
3–5 0.985 2520 154.3
Numbering refers to Fig. 1. All energetic data in kcal/mol.
delocalization among the hydrogens. A similar effect is found on comparing the 3–4 and 3–6 delocalizations for ethylene. Given the difference in the H3–H4 and H3–H6
˚ , respectively) and the general distances (2.449 and 3.059 A exponential decay in d with distance, the 3–6 delocalization is larger than expected. This is a real space signature of
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sigma hyperconjugation between the trans CH bonds [42]. On the contrary, the difference in both repulsion and delocalization for the geminal 3–5 interaction when comparing methane and ethylene is basically accounted for by the larger basin population in the latter. As the multiple bonds are regarded, each pair comprising the ethylene double bond provides core-V(C, C) repulsion, d, and Vxc values clearly smaller than for the coreV(C, H) case. This is a result of diminished V(C, C) population, and an example of the GN rule that decreases the ‘effective’ size of a bond pair basin as the electronegativity of the bonded atoms increases. Interestingly, the core-V(C, C) parameters in C2H2 is almost triple than those in each of the two equivalent C–C bond basins in ethylene. A similar behavior is found for the 3–7 interaction in ethylene and the 3–5 repulsion in acetylene. Overall, the two C–C bond basins of C2H4 provide a pseudo-tetrahedral arrangement around each carbon. Since the 7–8 repulsion is smaller than the 3–5 one, one may also rationalize the opening of the H3–C1–H4 angle using tetrahedral repulsions. In any case, if the double or triple bond is understood as a monolytic object, the GN rule relative to multiple bonds is again compatible with our Vee mapping. 4. CH3–AHnE4n systems Let us turn our attention to a number of simple cases where two tetrahedra of electron pairs share one vertex (one bonding domain). First we will consider C2H6, both in its staggered (s) and eclipsed (e) stationary conformations. To our integration accuracy, the population of the C (2.092 e) and V(C, H) (1.996 e) basins differ in both conformations by less than 0.001 e. Their cumulative difference is however amplified, and the population of the V(C, C) basin is slightly smaller in the eclipsed (1.836 e) than in the staggered geometry (1.838 e). The interbasin electron interaction parameters are gathered in Table 3. There are several interesting points if Tables 1–3 are compared. Firstly, the C–V(C, H) interaction parameters are quite transferable between methane and ethane, although a slightly more compact electron distribution exists in the latter. This is in opposition to the C2H4 and C2H2 cases, so the core–valence repulsion clearly discriminates these CH bonds. The same is true for the geminal V(C, H)–V(C,
H0 ) repulsion parameters, e.g., the 3–5 or 3–4 interactions in ethylene or ethane, respectively. Moreover, the single bond V(C, C) basin in ethane behaves as a slightly weaker repeller than the V(C, H) one, in agreement with its comparable, but definitely smaller electron population. A CC single bond pair is thus almost equivalent, from our present viewpoint, to a CH one in saturated hydrocarbons. Secondly, the staggered–eclipsed comparison shows the sensitivity of our parameters to small perturbations. A QTAIM account of the role of interatomic Vxc in determining the conformational preference has already been presented [42] and will not be discussed here. Most of the relevant effects are related to changes in geometries. For instance, the H3–H5 and H3–H6 distances are 2.524 and ˚ in the s conformer, respectively, and 2.922 and 3.074 A ˚ 2.342 A in the e one, in the same order, and this fact justifies their tabulated repulsions. The total vicinal V(C, H) exchange contributions, i.e. the total exchange between all different V(C, H)–V(C0 ,H0 ) pairs that belong to different methyl groups, add to 24.6 kcal/mol in both geometries, but it should be noticed that the 3–6 Vxc value in the s conformer (1.8 kcal/mol) is larger than its 3–5 counterpart in the e geometry (1.5 kcal/mol), even when the H3–H6 distance in s is larger than the H3–H5 one in e. This is again a manifestation of higher delocalization (or hyperconjugation) for antiperiplanar configurations. Changing the second methyl to NH2, OH, or F does not alter the overall staggered tetrahedral arrangement of the pair basins, although the three lone pairs of the F atom in CH3F are not well resolved in Fig. 1 at the g = 0.85 value. Actually, the three attractors have g = 0.892, while the three bips interconnecting them lie at g = 0.891. Aside this minor point, these systems provide an interesting set to compare the behavior of V(C,A), V(A, H), and A lone pairs as the electronegativity of the A atom increases. Since the number of independent pair basins increases due to the lack of symmetry, we will restrict to valence–valence interactions. Core–core, and core–valence repulsions are of smaller chemical relevance, and their behavior follows the same valence–valence rules, as we have seen in the preceding examples. Electron populations in the ELF attraction basins of CH3NH2 are 2.091, 2.025, 2.009, 2.107, 1.998, 2.062, and 1.697 e for the attractors labelled as 1, 2, 3, 5, 6, 8, and 9, respectively. In methanol: 2.092, 2.127, 2.009,
Table 3 ELF interbasin electron interactions for the staggered (s) and eclipsed (e) conformations of ethane C2H6(s)
1–2
1–3
1–5
1–9
3–4
3–5
3–6
3–9
d Vee Vxc
0.002 951 0.3
0.006 695 0.6
0.124 1585 38.7
0.100 1537 34.3
0.366 887 61.0
0.030 580 3.2
0.032 496 1.8
0.282 851 52.0
C2H6(e) d Vee Vxc
1–2 0.002 943 0.3
1–3 0.006 690 0.6
1–5 0.125 1589 38.7
1–9 0.099 1525 33.8
3–4 0.371 890 61.9
3–5 0.022 518 1.5
3–6 0.044 610 5.2
3–9 0.283 848 51.7
Numbering refers to Fig. 1. All energetic data in kcal/mol.
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2.025, 1.764, 1.380, and 2.287 e for the 1, 2, 3, 4, 6, 7, and 8 basins. And 2.089, 2.034, 2.135, 1.013, and 2.220 e for the 1, 2, 5, 6, and 7 basins in CH3F. Notice how the V(C,F) basin (number 6), cannot be properly classified as a pair basin. This is a well known characteristic of C–F links. Table 4 contains a summary of the valence repulsions. As the methyl is concerned, we can now follow the evolution of the population and inter-basin repulsion for any of its three pairs of V(C, H)–V(C, H) interactions. Vee increases slightly but steadily along the CH4 < C2H6 < CH3NH2 < CH3OH < CH3F series, from 867 to 940 kcal/ mol, following the concomitant increase in the population of the basin. Similarly, the V(C, H)–V(C,A) repulsion decreases along the series (no value in methane) from 851 kcal/mol in ethane to 522 kcal/mol in fluoromethane. An important component in this change is again the decrease in electron population of the V(C,A) pair basin, but in the end, a rule appears relating the ‘effective’ GN size of a CA bond pair with the electronegativity of the A atom. Notice that the increase in the H–C–H angle of the methyl group from 107° to 110° on passing from methanamine to fluoromethane provides again a consistent mapping of the GN rules onto our Vee values. We can also compare our previous NH3 and H2O data with the numbers of Table 4. For instance, the V(N,H)– V(N,H) repulsion in the amino group of CH3NH2 (1038 kcal/mol) is larger than that found in ammonia (975 kcal/mol). The same is true for Vxc or d. This is coupled to an also larger E–V(N,H) repulsion (1124 versus 1099 kcal/mol) which, however, supplies a smaller Vxc term, and to similarly compatible E–V(C,N) and V(N,H)–V(C,N) repulsions. Since the relative weight of the bond–bond and bond–lone pair Vee has not changed
401
much in methylamine with respect to ammonia, the H– N–H angle (107.5°) remains almost untouched. A similar analysis may be made in methanol. Let us just notice that, following the general trends, the E–E and E–V(O,H) repulsions are slightly larger than those in H2O. Another important point concerns the vicinal interactions. For instance, the gauge 3–5 repulsion in staggered ethane, 580 kcal/mol, is smaller by almost 50 kcal/mol than its equivalent 3–6 interaction in CH3NH2, but very similar to the 4–6 repulsion found in methanol. The V(C, H)–E gauge interaction, grows steadily from 683 to 829 kcal/ mol on going from CH3NH2 to CH3F. All this may be rationalized in terms of distances and basin populations, and it is relevant to point out that these interactions carry non-negligible electron delocalizations and exchange stabilizations associated with them ranging from 7.1 to 8.6 kcal/mol. Our Vee mapping also holds for the E–E interaction in methanol. The E–E (8–9) repulsion is significantly larger than that found in H2O, and the E–O–E angle (109.8°) increases. 5. Conjugation: formaldehyde versus formamide Let us conclude our analysis by comparing formaldehyde with the planar (p) geometry and one pyramidal (py) transition state (see Fig. 1) of formamide, 16 kcal/ mol above the planar situation. This large rotational barrier in HCONH2 has been traditionally related to p conjugation and is the subject of continuous revisions due to its relation to the structure of polypeptides and proteins [43]. Since even now the number of valence–valence interactions is too large, we will only comment on the O–C–A backbone. As shown in Fig. 1, the C–O double bond in
Table 4 ELF valence interbasin electron interactions for the CH3NH2, CH3OH, and CH3F systems at their theoretical equilibrium geometries CH3NH2
2–3
2–6
2–8
2–9
3–6
3–7
3–8
d Vee Vxc
0.366 911 61.4
0.036 624 4.0
0.050 590 3.3
0.220 811 42.6
0.031 627 3.5
0.033 539 2.1
0.056 683 7.1
d Vee Vxc
3–9 0.218 806 42.4
6–7 0.420 1038 83.8
6–8 0.552 1124 100.8
6–9 0.316 919 69.2
8–9 0.412 1010 93.8
CH3OH d Vee Vxc
3–4 0.366 918 61.7
3–6 0.030 500 2.0
3–8 0.055 803 7.4
3–7 0.162 680 33.3
4–5 0.367 922 61.8
4–6 0.028 579 3.3
4–8 0.048 707 3.6
d Vee Vxc
4–9 0.067 811 9.1
4–7 0.165 682 33.7
6–8 0.568 1280 138
6–7 0.228 722 52.8
8–9 0.741 1775 181
8–7 0.418 1057 114
CH3F d Vee Vxc
2–3 0.371 940 62.7
2–7 0.059 829 8.6
2–9 0.043 730 3.7
2–6 0.114 522 24.9
7–8 0.728 1970 53.7
7–6 0.318 845 18.2
Numbering refers to Fig. 1. Energetic data in kcal/mol.
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formaldehyde has two equivalent V(C,O) pair basins (the 5 and 6 basins in the figure) lying above and below the molecular plane, as the C–C link in ethylene. Accordingly, the two lone pairs of oxygen (basins 7 and 8 in Fig. 1) thus lie in the H–C–H plane. The populations of the 1, 2, 3, 5, and 7 basins are 2.087, 2.118, 2.103, 1.266, and 2.524 e, respectively. Each of the V(C,O) basins contains much less than 2 e, this being a characteristic of highly heteropolar bonding basins, as already commented. In planar formamide, the two V(C,O) basins have collapsed into one over the C–O axis, basin number 8 in Fig. 1, and the nitrogen’s lone pair (basin 7 in Fig. 1) is orthogonal to the plane, displaying two equivalent attractors which are not well resolved from the V(C,N) bonding pair. This is a direct demonstration of the ‘partial’ p character of the C–N bond, as it is well known. The basin populations of basins 1–4, 7–9, and 11 are 2.087, 2.119, 2.117, 2.109, 2.198, 2.408, 2.688, and 0.820 e, respectively. Rotation of the NH2 group around the C–N bond induces the pyramidalization of the N atom. In the transition state geometry the lone pair lies in the H–C–N plane, the C–O link goes back to its two equivalent V(C,O) basins above and below the plane, and most signs of p delocalization around the C–N bond have disappeared. The basin populations of basins 1-4,79, and 11 are 2.087, 2.119, 2.115, 2.107, 2.114, 1.871, 1.298, and 2.557 e, in that order. A simple look at the basin populations allows us to recover many traditional chemical concepts. For instance, the V(C,N) population in planar formamide, 2.198 e, is considerably larger than its equivalent in CH3NH2, 1.697 e. Similarly, the total V(C,O) population, 2.408 e, is significantly smaller than its equivalent in formaldehyde, 2.532 e. Coupled to this, the oxygen’s lone pairs bear greater population, and the complete lone pair on the N atom (1.639 e) has considerably less electrons than its analogue in CH3NH2, 2.062 e. This number is recovered in the pyramidal geometry, with a N lone pair populated with 2.114 e. Table 5 contains the basin-basin interaction parameters of the O–C–A backbone. Each V(C,O) basin of the C–O double bond in H2COH has a noticeably smaller repulsion with the V(C, H) basin, 650 kcal/mol, than its equivalent in
ethylene, 798 kcal/mol. This is compensated with the V(C, H)–E repulsion, 838 and 965 kcal/mol with the trans and cis lone pairs, respectively. The equivalent V(C, H)–V(C, H) figures in C2H4, 562 and 660 kcal/mol, in the same order, show how large the vicinal V(C, H)–E interaction is in formaldehyde. Notice also how the much larger population of the two oxygen’s lone pairs generates very large 7-8 interactions, with an huge delocalization between the two pairs (d = 0.865) that leads to a E–O–E angle of 122 degrees, much larger than that found in H2O. Let us now turn to conjugation in planar formamide with respect to the pyramidal transition state. Pyramidalization decreases the population of the O lone pairs, and increases that in the V(C,O) basin(s). These are relatively localized changes which may be envisioned as a flow of electrons from the O lone pairs into the double C–O bond. The 7-8 V(C,N)–V(C,O) interaction parameters in the planar geometry are similar to twice the 8-9 analogues in the pyramidal structure, even when the p7+p8 population is considerably larger than its py8+2py9 equivalent. A similar analysis holds for the V(C,N)–E(O) terms. r delocalization of the O–C–N backbone is thus not an important ingredient in the stabilization of planar CONH2. A completely different picture is obtained if we look at the behaviour of the nitrogen’s lone pair. In the planar geometry, the V(C,N)– E(N) repulsion is about 1090 kcal/mol (twice the 7-11 results from Table 5). This is similar (but smaller) to the value at the barrier, 1120 kcal/mol. However, the delocalization indices and exchange stabilizations associated to this interaction are completely different, d = 0.620 versus 0.445 in the p and py geometries, respectively. Or Vxc = 146 versus 99 kcal/mol. Notice that this difference is counteracted by the larger delocalization and exchange between the C–O bond basin(s) and the N lone pair in the pyramidal transition state: d = 0.120 and Vxc = 26 kcal/mol for py, while the figures are 0.050 and -5.6 kcal/mol for p. This may be interpreted as a more efficient anti hyperconjugative effect between the C–O bond and the N lone pair in the py conformation. Overall, the picture offered by our interaction analysis shows very clearly a charge transfer compatible with the
Table 5 ELF valence interbasin electron interactions for the O–C–A backbone in formaldehyde and the planar and pyramidal geometries of formamide H2CO d Vee Vxc
3–4 0.367 842 61.7
3–5 0.166 650 32.1
3–7 0.107 838 8.8
3–8 0.139 965 17.0
5–7 0.419 1020 102
7–8 0.865 2160 214
HCONH2(p) d Vee Vxc
7–8 0.234 1380 52.0
7–9 0.056 962 5.0
7–10 0.077 1060 10.3
7–11 0.320 547 78.1
8–9 0.840 2060 206
8–10 0.814 2000 200
8–11 0.025 341 2.8
9–10 0.986 2350 243
9–11 0.015 270 0.9
10–11 0.015 302 1.1
HCONH2(py) d Vee Vxc
7–8 0.445 1120 99.1
7–9 0.014 414 1.2
7–11 0.011 594 0.5
7–12 0.017 643 1.1
8–9 0.120 632 26.2
8–11 0.057 783 5.1
8–12 0.079 887 10.6
9–10 0.334 6200 768
9–11 0.430 1050 105
11–12 0.884 2180 41.4
Numbering refers to Fig. 1. Energetic data in kcal/mol.
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classical resonance structures in planar formamide, with a C–O bond that has lost part of its double bond character by allowing bonding electrons to flow from the bonding basin into the oxygen’s lone pairs. This process is coupled to a C–N link gaining double bond character by delocalizing electrons of the N lone pair into the C–N bond basin. However, the C–O and C–N bonds seem to be quite independent, and no important delocalization occurs, for example, between the N lone pair and the V(C,O) or E(O) basins. Interestingly enough, the total DVxc of our O–C–N backbone between the p and py structures offers an stabilizing balance of 18.8 kcal/mol for the planar structure, a value very similar to the total rotation barrier. The latter may thus be interpreted as a result of a larger exchange (delocalization) contribution in the planar arrangement. 6. Conclusions We have presented in this Letter, for the first time, numerical electron–electron interaction parameters between real space basins of ELF attractors in a battery of test systems. Our preliminary Hartree–Fock data show that once the overall arrangement of the electron pairs is determined by antisymmetry requirements, i.e. tetrahedral in the four pairs case, there exists a faithful mapping of the electrostatic electron–electron repulsion between the pair basins and the Gillespie–Nyholm rules of the VSEPR model. We believe that these results will hold in more general cases, and further calculations in other systems with non tetrahedral pair distributions will be considered in the near future. The inter-basin parameters used in this work: electron– electron total repulsions, Vee’s; the covariance in the basin populations, d’s; and the exchange interaction energy, Vxc’s, open new avenues towards quantitative ELF analyses. A combination of the ability of ELF or ELF-like fields to provide appealing pictures of chemical processes with our more quantitative analyses may prove a useful tool. For instance, the flow of charge among basins for the rotation barrier problem in formamide, taken together with the inter-basin delocalizations and their associated covalentlike Vxc contributions provided by our parameters, depict an scenario akin to classical Pauling resonance, with quantitative estimations of the stabilization brought about by each relevant electron delocalization. Acknowledgements Financial support from the Spanish MEC, Project No. CTQ2006-02976 and the ERDF of the European Union, is acknowledged. References [1] A.D. Becke, K.E. Edgecombe, J. Chem. Phys. 92 (1990) 5397. [2] B. Silvi, A. Savin, Nature 371 (1994) 683.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
[40] [41] [42] [43]
403
B. Silvi, I. Fourre´, M.E. Alikhani, Monatsh. Chem. 136 (2005) 855. J. Poater, M. Duran, M. Sola`, B. Silvi, Chem. Rev. 105 (2005) 3911. X. Krokidis, S. Noury, B. Silvi, J. Phys. Chem. A 101 (1997) 7277. X. Krokidis, V. Goncalves, A. Savin, B. Silvi, J. Phys. Chem. A 102 (1998) 5065. X. Krokidis, R. Vuilleumier, D. Borgis, B. Silvi, Mol. Phys. 96 (1999) 265. L. Joubert, C. Adamo, J. Chem. Phys. 123 (2005) 211103. S. Berski, J. Andre´s, B. Silvi, L. Domingo, J. Phys. Chem. A 107 (2003) 6014. V. Polo, J. Andre´s, J. Chem. Theor. Comput. 3 (2007) 816. A. Savin, O. Jepsen, J. Flad, O. Andersen, Angew. Chem. Int. Ed. 31 (1992) 187. R. Nalewajski, A. Koster, S. Escalante, J. Phys. Chem. A 109 (2005) 10038. E. Matito, B. Silvi, M. Duran, M. Sola`, J. Chem. Phys. 125 (2006) 02430. B. Silvi, J. Phys. Chem. A 107 (2003) 3081. M. Kohout, K. Pernal, F.R. Wagner, Y. Grin, Theor. Chem. Acc. 112 (2004) 453. M. Kohout, Int. J. Quantum Chem. 97 (2004) 651. R.F.W. Bader, Atoms in Molecules, Oxford University Press, Oxford, 1990. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Chem. Phys. 120 (2004) 4581. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, J. Comput. Chem. 26 (2005) 344. M.A. Blanco, A. Martı´n Penda´s, E. Francisco, J. Chem. Theor. Comput. 1 (2005) 1096. E. Francisco, A. Martı´n Penda´s, M.A. Blanco, J. Chem. Theor. Comput. 2 (2006) 90. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Comput. Chem. 28 (2007) 161. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Chem. Phys. 125 (2006) 184112. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, Faraday Discuss. 135 (2007) 423. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, J. Phys. Chem. A 110 (2006) 12864. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, C. Gatti, Chem. Eur. J. 13 (2007) 9362. A. Martı´n Penda´s, E. Francisco, M.A. Blanco, Chem. Phys. Lett. 437 (2007) 287. R.F.W. Bader, M.E. Stephens, J. Am. Chem. Soc. 97 (1975) 7391. ´ ngya´n, M. Loos, I. Mayer, J. Phys. Chem. 98 (1994) 5244. J.G. A X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1999) 304. X. Fradera, J. Poater, S. Simon, M. Duran, M. Sola`, Theor. Chem. Acc. 108 (2002) 214. B. Silvi, Phys. Chem. Chem. Phys. 6 (2004) 256. E. Francisco, A. Martı´n Penda´s, M.A. Blanco, J. Chem. Phys. 126 (2007) 094102. M.W. Schmidt et al., J. Comput. Chem. 14 (1993) 1347. R.J. Gillespie, R.S. Nyholm, Q. Rev. Chem. Soc. 11 (1957) 339. R.J. Gillespie, I. Hargittai, The VSEPR Model of Molecular Geometry, Allyn & Bacon, 1991. R.J. Gillespie, Coord. Chem. Rev. 197 (2000) 51. R.J. Gillespie, B. Silvi, Coord. Chem. Rev. 233–234 (2002) 53. B. Silvi, R.J. Gillespie, The Quantum Theory of Atoms, in: C. Matta, R.J. Boyd (Eds.), Molecules: From Solid State to DNA and Drug Design, Wiley, NY, 2007, p. 141. R.J. Gillespie, E.A. Robinson, J. Comput. Chem. 28 (2007) 87. R.J. Gillespie, P.L.A. Popelier, Chemical Bonding and Molecular Geometry, Oxford University Press, New York, 2001. A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Comput. Chem., submitted for publication. J.I. Mujika, J.M. Matxain, L.A. Eriksson, X. Lopez, Chem. Eur. J. 12 (2006) 7215.