Attractive Orbital Interactions

It was shown by van Leeuwen and Baerends56 that the energy change, when going from an initial density p to a final density pf, can be obtained from a path integral along a path in the space of densities that connects the initial and final densities. The path is arbitrary, since the initial and final energies depend only on the respective densities. In the present case, we may take the simple linear path measured by the parameter t, from p° = p(t = 0) to p = pexact = p(t = 1), 

we have used the fact that the derivative of the energy with respect to a density matrix element is the corresponding one-electron Hamiltonian matrix 

It is possible to use as the basis functions symmetry-adapted combinations of primitive basis functions. This affords a decomposition of the orbital interaction energy of Eq. [21] according to irreducible representations of the point group 

It is also possible to perform a basis set transformation from primitive basis functions to symmetry combinations of the KS MOs of the atoms or larger fragments that constitute a system. In that case the population matrix elements P v become more meaningful, because they reflect the involvement of the fragment MOs in the orbitals of the total system. A Mulliken population analysis in 

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Figure 5 Attractive orbital interaction terms (eV) between Mn2+ and 04 in Mn04 in various irreducible representations as functions of internuclear separation R (A).

Diastereoisomers have different relationships between nonbonded atoms, and as a consequence their energy content is different. It is generally found that due to steric effects the more extended (trans) isomer is more stable than the m-isomer by 1-10 kcal/mol. For example, (E)-2-butene (XLIV) is more stable than its (Z)-isomer by 1 kcal/mol [55]. However, through-bond and through-space attractive orbital interactions have been calculated in several cases to favor the c/s-isomer. Thus, (Z)-l-methoxypropene (XLV) is more stable than its (E)-diastereoisomer by about 0.5 kcal/mol [55]. 

Table 9 shows that the latter prediction is supported by the EDA results as far as the contributions of the orbitals with different symmetry to the AEo term is concerned. 90.4% of the latter come from the ti orbitals. The d orbitals of Xe do not play a role for the Xe F bonds. The contributions of the eg and t2g orbital interactions are neghgible. However, the EDA results show also that the attractive orbital interactions are compensated by the repulsive Pauli term. Xenon hexafluoride is stable because there is strong quasi-classical Coulomb attraction between Xe and Fe. The sum of the quantum theoretical expressions (AEorb and AEpauu) is destabilizing. 

In summary, it seems that for most Diels-Alder reactions secondary orbital interactions afford a satisfactory rationalisation of the endo-exo selectivity. However, since the endo-exo ratio is determined by small differences in transition state energies, the influence of other interactions, most often steric in origin and different for each particular reaction, is likely to be felt. The compact character of the Diels-Alder activated complex (the activation volume of the retro Diels-Alder reaction is negative) will attenuate these eflfects. The ideas of Sustmann" and Mattay ° provide an attractive alternative explanation, but, at the moment, lack the proper experimental foundation. 

Mataka and coworkers reported the studies of the Diels-Alder reactions of [3.3] orthoanthracenophanes 96 and 97, of which anthraceno unit, the potential diene, has two nonequivalent faces, inside and outside. The reactions of 96 with dien-ophiles gave the mixtures of inside and outside adducts with the ratios between 1 1 and 1 1.5. However, the ratio changes drastically, in favor of the inside adducts, when 97 reacts with dienophiles such as maleic anhydride, maleimide and naphto-quinone [55] (Scheme 46). Mataka suggested that the Jt-facial selectivity is controlled by an orbital interaction between the electron-poor dienophiles and the Jt-orbital of the facing aromatics, which would lead to a stabilization of the transition state, while Nishio suggested that the selectivity is due to the attractive k/k or CH/jt interaction [53]. 

The preference for the endo TS is considered to be the result of interaction between the dienophile substituent and the tt electrons of the diene. These are called secondary orbital interactions. Dipolar attractions and van der Waals attractions may also be involved.12 Some exo-endo ratios for thermal D-A reactions of cyclopentadiene are... 

For Li—F, the quantal ionic interaction can be qualitatively pictured in terms of the donor-acceptor interaction between a filled 2pf. orbital of the anion and the vacant 2su orbital of the cation. However, ionic-bond formation is accompanied by continuous changes in orbital hybridization and atomic charges whose magnitude can be estimated by the perturbation theory of donor-acceptor interactions. These changes affect not only the attractive interactions between filled and unfilled orbitals, but also the opposing filled—filled orbital interactions (steric repulsions) as the ionic valence shells begin to overlap. 

The endo selectivity in many Diels-Alder reactions has been attributed to attractive secondary orbital interactions. In addition to the primary stabilizing HOMO-LUMO interactions, additional stabilizing interactions between the remaining parts of the diene and the dienophile are possible in the endo transition state (Figure 3). This secondary orbital interaction was originally proposed for substituents having jr orbitals, e.g. CN and CHO, but was later extended to substituents with tt(CH2) type of orbitals, as encountered in cyclopropene57. 

Further examination of the results indicated that by invocation of Pearson s Hard-Soft Acid-Base (HSAB) theory (57), the results are consistent with experimental observation. According to Pearson s theory, which has been generalized to include nucleophiles (bases) and electrophiles (acids), interactions between hard reactants are proposed to be dependent on coulombic attraction. The combination of soft reactants, however, is thought to be due to overlap of the lowest unoccupied molecular orbital (LUMO) of the electrophile and the highest occupied molecular orbital (HOMO) of the nucleophile, the so-called frontier molecular orbitals. It was found that, compared to all other positions in the quinone methide, the alpha carbon had the greatest LUMO electron density. It appears, therefore, that the frontier molecular orbital interactions are overriding the unfavorable coulombic conditions. This interpretation also supports the preferential reaction of the sulfhydryl ion over the hydroxide ion in kraft pulping. In comparison to the hydroxide ion, the sulfhydryl is relatively soft, and in Pearson s theory, soft reactants will bond preferentially to soft reactants, while hard acids will favorably combine with hard bases. Since the alpha position is the softest in the entire molecule, as evidenced by the LUMO density, the softer sulfhydryl ion would be more likely to attack this position than the hydroxide. 

Van der Waals forces are very complex and manifest themselves even at distances at which it is unreasonable to assume that orbital interactions can occur. An explanation due to London in terms of the mutual attraction of induced dipoles (dispersion forces) accounts for the long-range behavior. The unoccupied-occupied orbital interactions will be the dominant component of van der Waals forces at short range. See Kauzmann, W., Quantum Chemistry, Academic, New York, 1957, Chapter 13, for a discussion of dispersion forces. 

The first term on the right is the operator for the electrons kinetic energy the second term is the operator for the potential energy of attraction between the electrons and the nucleus (r, being the distance between electron i and the nucleus) the third term is potential energy of repulsion between all pairs of electrons ru being the distance between electrons / and j) the last term is the spin-orbit interaction (discussed below). In addition, there are other relativistic terms besides spin-orbit interaction, which we neglect. 

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