A Bond Bundle Case Study of Diels-Alder Catalysis Using Oriented Electric Fields

Bond bundles are chemical bonding regions, analogous to Bader atoms, uniquely 1 defined according to the topology of the gradient bundle condensed charge density, 2 itself obtained by a process of infinitesimal partitioning of the three-dimensional 3 charge density into differential zero-flux surface bounded regions. Here we use 4 bond bundle analysis to investigate the response of the charge density to an oriented 5 electric field in general, and the catalytic effect of such a field on Diels-Alder reactions 6 in particular, which in this case is found to catalyze by allowing the transition state 7 valance bond bundle configuration to be achieved earlier along the reaction pathway. 8 Using precise numerical values, we arrive at the conclusion that chemical reactions 9 and electric field catalysis can be understood in terms of intra-atomic charge density 10 redistribution, i.e., that charge shifts within more so than between atoms account for 11 the making and breaking of bonds. 12

5 Previously, bond bundles were analyzed using a method of explicit bond bundle surface identification, rather than implicit identification using the gradient bundle condensed charge density as at present.  111 In addition to enabling the general identification of bond bundles, gradient bundle 112 decomposition enables two types of additional charge density analysis: i) analysis of 113 condensed deformation properties relative to a spherical atomic reference state, conceptually 114 similar to the chemical deformation densities of Schwarz et al. [14], though here applicable 115 beyond the charge density to any scalar field; and ii) local and global geometric analysis of the 116 charge density (gradient). Together with condensed properties that result from Equation

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(1)-that is, gradient bundle condensations of input three-dimensional scalar fields-there are 118 thus three major categories of gradient bundle condensed properties: condensed scalar fields 119 (category A); property fields derived therefrom, such as condensed deformation properties 120 (category B), and geometric charge density descriptors (category C).

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, however, is also a measure of total geometric Gaussian curvature within gradient 132 bundles [2,5], and can deviate from the behavior of more so than . In this case, note 133 that on the carbon atoms, and each have four minimum CPs, one each above and below 134 the molecular plane, and one each within the molecular plane on either side of the C=C bond.

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In , however, there is an additional minimum CP on the "back" of the carbon atom at the  The average curvature-scaled gradient bundle torsion 6 (KKK) shows, at a glance, which gradient 141 paths within an atomic basin will bend and curve primarily within the same plane. Here we 142 see that gradient paths within the carbon atomic basins will have very low torsion if they are 143 close to the molecular plane or the perpendicular plane-see the minimum CP representative 144 gradient bundles in Figure 2-and that those at approximately 45° to both planes will achieve 145 maximum torsion-see the saddle CP gradient bundle in the same figure. This is to be 146 expected, since both planes are symmetry planes and hence zero flux surfaces. The same 147 general behavior is also observed, however, on the H atoms, not only in the molecular plane, 148 but in the plane perpendicular to the molecular plane and also containing a C-H bond axis. ̅ is computed as the gradient path integral of the angles between neighboring line segments along a discretized path, ̅ %&' is computed as the angle between the originating and terminal ends of a gradient path, and 000 is computed as the gradient path integral of the of angles between planes defined by coincident pairs of line segments (three neighboring line segments, where the central segment is shared between the pairs) along the path, scaled at each line segment by the its value of path curvature. All three properties are computed for paths and then averaged over the gradient paths defining a particular gradient bundle to recover its values.
Investigations into the chemical significance of these pure geometric descriptors are ongoing, 150 but preliminary results show that the distribution and redistribution of charge density curvature 151 plays an important role in such fundamental chemical processes as carbonyl bond activation [5].

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Turning to the middle row of Figure 4, there are two derived condensed properties shown.

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The first is average kinetic energy per electron ( ⁄ ), which has been investigated by 154 Morgenstern The second derived condensed property shown in Figure 4 is the condensed deformation hence "equivalent". Each differential gradient bundle's atomic reference state value is simply 172 its share of the atomic basin condensed value as determined by its normalized solid angle ( ), 173 so the value of some condensed deformation property for differential gradient bundle is where 1-23 is the atomic basin condensed property value. In Figure 4 we see that kinetic  Condensed deformation properties may also be calculated by providing a reference 185 value to use in place of  . For example, if an accurate and comparable atomic energy is 186 already known, it may be used to define the spherical atomic reference state.  Hartree.
209 When reporting regional condensed values for geometric descriptors they should be 210 averaged over some other regional property. Here we've averaged over (divided by) volume, 211 resulting in units of angle per volume that gives a clear sense of one volumetric region being 212 more or less curved (i.e. less or more spherical; "atomic") than another. 213 We return briefly to Δ ./0 which, by definition, integrates to zero over an atomic basin-  found that an electric field oriented along the "reaction axis" pointing from the butadiene to 241 the ethylene (the negative z direction; electric field direction points positive to negative) 242 lowered the reaction barrier, but that a field in the opposite direction did not raise the reaction 243 barrier. Using this conventionally "simple" example of the charge density response to an 244 electric field, we will examine the inter-atomic and intra-atomic redistribution of electron 245 charge density that underly and accompany the response.

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All DFT calculations, including those used to produce the energy and distance values Theory Group at Colorado School of Mines [39].    For the field, charge density is "pushed" from H1 to H4 and from H2 to H3, leaving

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We can also visually inspect the condensed charge density response to the electric fields 325 by computing a difference condensed density, Δ , similar in concept to the deformation 326 kinetic energy density in Figure 4, but now instead of a spherical atomic reference state, the 327 difference is that between the no-electric-field system and those with applied fields, Δ 667 = 328 667 − 867 . Figure 6 shows the difference densities corresponding to the ethylene C atomic other. For all three, the region directly above and below the C=C bond path intersection with 332 the sphere-corresponding to the -bond-appear to be the among the most responsive, 333 making the C p-orbitals seem like sails catching electronic wind. The compliance of the p-334 orbital region is also evident in the high charge density shifts to/from the C=C bond wedge 335 regions resulting from the -field in Figure 5. The offsetting ± signs of Δ 9 and Δ : about 336 the C=C bond path in Figure 6 graphically show why only Δ ; achieves a (high) non-zero 337 regional value. Otherwise, the other observations from Figure 5   This key finding here is that picturing the response 345 of simple negative charges to an electric field can be a 346 helpful guide in many instances, such as in low-density 347 Van der Waals regions. This is significant because 348 the response of these regions appears to dictate the response of the inner valance molecular 349 charge redistribution. This interdependent behavior was not evident in the inter-atomic 350 charge transfer, but using the combined approach of atomic and bond regional property analysis, 351 we arrived at a clear picture of the ethylene electric field response.   the effect itself was observed as a lowering of the reaction barrier-i.e. of the TS energy-so were found to stabilize the TS relative to the reactant sum (R; also with the same field applied).

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The NEF barrier was lowered 5.9 kcal/mol by the -z field and 2.2 kcal/mol by the +z field, so the -z field has a stronger stabilizing effect, agreeing with the referenced investigation.
363 Table 3 lists the atomic basin and bond bundle condensed valence electron density ( < ) 364 values for the TS structure with/without the ± fields. The total valence density of an atom 365 is equal to its total density minus its core density, < = − =2>, . The gradient bundle 366 condensed valence density is then calculated, similar to the condensed deformation energy of

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When bond bundle properties are considered, we can interpret the regional electric field 382 responses in the context of valence bond theory. As an example and benchmark for the 383 electric field response, Table 4 lists regional valence electron counts for the NEF system in the    Table 4. So, the more strongly catalyzing 433 field for the forward reaction is that which shifts C-C bond orders in the TS to look more like 434 they will in the product state, while the effect of the +z field is accompanied by bond order 435 changes that are contradictory in this regard. This is again consistent with Hammond's 436 Postulate, and we see that the catalyzing field simply "pushes" the TS density in the direction 437 of the product state. Table 4 indicates that, for the NEF system, once the bonds in the system 438 reach that specific TS level of valance electron population ( <,?@ ), the TS has been reached.

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The remaining (majority) change in bond bundle valance density necessary to achieve the 440 product state thus occurs on the "downhill" side of the reaction. The -z field, by pushing and 441 pulling charge in the reaction direction, is causing <,?@ to be achieved-and thus making the Otherwise, we again see in Table 3 that the C intra-atomic charge redistribution can be 449 counter that expected from a simple Coulombic approach. In response to the -z field, the C1 450 atom shifts charge primarily from its C1-C2 to its C1-C5 bond wedges, opposite the direction 451 of the system as a whole. Its response to the +z field, however, is not to shift charge in the 452 opposite direction, but to instead shift charge from both C1-C2 and C1-C5 bond wedges into 453 its C-H bond wedges. The response of the ethylene C5 atom, however, is also opposite the 454 expected -z field response, shifting charge from its C bonds to its H bonds, but in this case its 455 response is equal and opposite with respect to field direction. On this basis we again conclude 456 that C atom intra-atomic charge redistribution is secondary to the primary low-lying density 457 electric field response, which here too is predicted by an electron gas approximation. Note 458 that, in this case, the newly formed C1-C5 bond, when analyzed as a standalone volume, has 459 charge shifting in the expected electron gas direction, opposite that of the applied field, so this 460 low-order bond region (with only ~0.3 valance electrons) responds as do other low-lying density regions.

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As in the analysis of the ethylene electric field response, the primary effect is the intra-463 atomic redistribution of C electron density. In the transition state context, changes in bond 464 bundle condensed valence density provided a straightforward chemical interpretation as to why 465 this charge density response should catalyze the reaction: The catalyzing -z field gives rise to 466 charge redistribution between C atom bond wedges such that the TS density more closely 467 resembles that of the product, effectively using Hammond's Postulate to shorten the charge 468 redistribution "distance" between the reactant and product states.

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The bridge between QTAIM and other branches of experimental and theoretical chemistry 471 and materials science rests largely on its ability to produce atomic (and therefore molecular 472 and crystalline) regional properties that can be readily compared to other results [1,3]. Well-473 defined regional energies are a quality of any region bounded by zero flux surfaces, and such 474 regions can be generated arbitrarily. Taking this property to the limit, the gradient bundle 475 decomposition method is the differential partitioning of into infinitesimal zero-flux surface Here we showed that the bond bundle decomposition method allows for the direct 480 qualitative and quantitative inspection of the distribution and redistribution of charge density 481 that accompany static and dynamic chemical bonding. Applied to the toy problem of electric 482 field catalysis, and the more general problem of the charge density response to an external 483 electric field, we observed that regional property shifts between bond wedges and bond bundles 484 are more dynamic in both sign and magnitude than are atomic basin regional shifts, and that 485 they allow for immediate chemical interpretation. In the analysis of both the ethylene and TS 486 electric field response, we saw that C atom intra-atomic redistribution can be counter that of , which involves countless chemical scenarios not unlike that treated in this work. 503 We hope to aid in the process of discovery with the dual ability of local direct inspection within 504 and large-scale correlation across enzyme active sites.