Overlapping between lobes

Figure 3.18e shows the effective force law (force versus displacement) between two parallel dimers with aspect ratio = 1.3 undergoing compression for the micromechanical model in which (1) lobe interactions are multiply counted or (2) the interaction potential is given by Equation 3.15. The two force laws are the same as long as overlaps between lobes have not merged, or < 0.021 for the configuration in Figure 3.18a. Beyond Sm, the two force laws differ. The force law based on the total area of overlap converges to linear behavior f 5 more quickly than the one that multiply counts lobe interactions, for example, it is not sensitive to the formation of the fourth lobe contact at 8/a = 84/a = 0.075. In future studies, these results can be compared to finite element analyses of linear elastic particles with complex shapes.

Together, the results in Figures 3.20 through 3.22 show that the low-frequency peak in D(w) for the interaction model in Equation 3.15 scales as < min ( d - 1) before the areas of overlap between lobes have merged, whereas it scales as min after the areas of overlap have merged. a> i for dimers... 

Conservation of orbital symmetry is a general principle that requires orbitals of the same phase (sign) to match up in a chemical reaction. For example, if terminal orbitals are to combine with one another in a cyclixation reaction as in pattern. A, they must rotate in the same dii ection (conrotatory ovei lap). but if they combine according to pattern H. they must rotate in opposite directions (disrotatory). In each case, rotation takes place so that overlap is between lobes of the it orbitals that are of the same sign. 

AO is antisymmetric to this reflection. In fact, we can easily see that any overlap between 2s and the positive lobe of 2p is exactly cancelled by that involving the negative lobe. 

Woodward and Hoffmann speeulated that the preferred motion was that whieh involved constructive (bonding) overlap between the terminal lobes of the highest-occupied molecular orbital (HOMO). 

In the interaction of the local 2pv orbitals, two more bonding molecular orbitals are formed against one less bonding. In all previous cases the opposite occurred. This is due to the negative overlap between adjacent 2py orbitals—whether, by convention, all positive lobes point in the clockwise direction, or whether all positive lobes point in the anticlockwise direction. The two bonding 2pv combinations in fact fall below the two antibonding (hybrid 2s, 2px) combinations. The former each have two electrons while the latter are empty. The six electrons of the three C—C bonds are nicely accounted for. The method creates simultaneously the acc and or c molecular orbitals of cyclopropane (note that the latter three lie relatively close in energy). 

As applied to cycloaddition reactions the rule is that reactions are allowed only when all overlaps between the HOMO of one reactant and the LUMO of the other are such that a positive lobe overlaps only with another positive lobe and a negative lobe only with another negative lobe. We may recall that monoalkenes have two n molecular orbitals (p. 9) and that conjugated dienes have four (p. 36), as shown in Figure 15.1. A concerted cyclization of two monoalkenes (a 2 -f- 2 reaction) is not allowed because it would require that a positive lobe overlap with a negative lobe (Fig. 15.2). On the other hand, the Diels-Alder reaction (a 2 -f 4 reaction) is allowed, whether considered from either direction (Fig. 15.3). 

The phase-twisted peak shapes (or mixed absorption-dispersion peak shape) is shown in Fig. 3.9. Such peak shapes arise by the overlapping of the absorptive and dispersive contributions in the peak. The center of the peak contains mainly the absorptive component, while as we move away from the center there is an increasing dispersive component. Such mixed phases in peaks reduce the signal-to-noise ratio complicated interference effects can arise when such lines lie close to one another. Overlap between positive regions of two different peaks can mutually reinforce the lines (constructive interference), while overlap between positive and negative lobes can mutually cancel the signals in the region of overlap (destructive interference). 

The antibonding tt MO results when orbital lobes of opposite sign overlap between adjacent carbon atoms => there is a node between each pair of carbon atoms. 

In order to discuss the density of states of metal surface atoms, we need to take a closer look at those orbitals that have a distinct orientation, the d-orbitals. Let us take an fee crystal and see how the d-orbitals combine to form bands. Figure A.7 schematically shows the shape of the d-orbitals. For clarity only the lobes in the (yz) plane are shown, while the toroidal component of the dz2 orbital has been left out. If these orbitals are placed within the fee structure as in Fig. A.7, one readily sees that the best overlap between nearest neighbors in the (yz) or (100) plane occurs between the dyz orbitals, whereas the orbitals along the y and z axes of the cube overlap to a much smaller extent. This is true in the other planes of an fee metal as well, the dtv, dxz, and d orbitals overlap more than the d,2 and the dx2.v2... 

This MO can be shown to have terminal lobes of the same phase, so that overlap between the hydrogen atom s Is orbital and both the terminal lobes of (38) s MO can be maintained in the T.S. (39) ... 

Front-side attack, corresponding to an attack on the big lobe of silicon, leads to retention. When unfavorable, out-of-phase overlap between the nucleophile and the orbitals of the leaving group predominates, nucleophilic attack occurs at the rear of the molecule, opposite X, leading to... 

Eq. (3.145)].222,1043 They gave instead the cyclopentenyl cation. The lack of formation of bishomoaromatic ions from cyclopentenyl derivatives is mainly due to steric reasons. The planar cyclopentene skeleton has to bend into the chair conformation to achieve any significant overlap between the empty p orbital and the 7i-p lobe of the olefinic bond, which is sterically unfavorable. However, such conformation already exists in ions 581 and 582. 

Overlap in the manner shown next is nonbonding. Both the positive lobe and the negative lobe of the p orbital overlap with the spherically symmetrical s orbital. The bonding overlap between the s orbital and one lobe of the p orbital is exactly canceled by an antibonding interaction between the s orbital and the lobe of opposite sign. 

The question of bending or not bending is determined by the trend in Pauli repulsion, which correlates with the overlap between the ft ho mo,x and the cJsomo (12). By symmetry, the overlap (S) between these orbitals is zero in the linear species. But on bending, the lobe of the oSOMO moves out of the nodal plane of the nHOMO x, and overlap begins to build up (12, right). 

It reproduces the ratio measured in the nozzle beam quantitatively as manifested in Figure 11.5. In the limit of large j, the coefficient Cj goes to 2 and the A-doublet ratio goes to infinity. The higher j the better defined is the rotation plane of OH and the higher is the alignment of the pn lobe of the unpaired electron perpendicular to this plane. Formally, the population of the two A-doublets can be interpreted as a Franck-Condon-type overlap between the wavefunction of the unpaired electron in the excited state of H2O and the two distinct wavefunctions of the unpaired electron in OH. 

A priori, we would expect disrotatory reactions to show poorer torquoselectivity than conrotatory reactions for two reasons. Consider, for example, the hexatriene cyclohexadiene interconversion. On the one hand, the overlap between R and the distal carbon C6 is similar for the in and out pathways, as in the in mode, the major lobe at C6 is oriented away from R ... 

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