Orbitals, exchanged electrons

In the interaction of a pair of atomic orbitals, two electrons form a bond and four electrons form no bond (Sect. 1.1). The snbstitnted carbocations are stabilized by the electron delocalization (hyperconjngation and resonance) through the interaction of the doubly occupied orbitals on the snbstitnents with the vacant p-orbital on the cation center. The exchange repulsion (Sect. 1.5) is cansed by four electrons. Now... 

As was mentioned previously, simple orbital products (electron configurations) must be converted into antisymmetrized orbital products (Slater determinants) in order to satisfy the Pauli principle. Thus, proper many-electron wavefunctions satisfy constraints of exchange antisymmetry that have no counterpart in pre-quantum theories. 

Percentage of "Exchanged Electron" Residing in Metal (M) and Ligand (L) Orbitals... 

The complex [(VO)2(dana)2] (dana = l,5-bis(p-methoxyphenyl)-l,3,5-pentanetrionato) was prepared and temperature dependent Xm measurements show antiferromagnetic behaviour with J = —80 cm 1.902 The / value is much lower than with [M2(dana)2(py)2] (M = Co, Cu) and this probably results from a different spatial orientation of the exchanging electrons if the unpaired electron is considered to be initially in a dxy orbital, a direct metal-metal interaction may be possible as the V—V distance is large (ca. 3.0-3.2 A V atoms are probably 0.5-0.6 A out of the plane and possibly one above and one below the ligand plane902), one would expect a weak exchange for the direct V—V interaction. 

For electron transfer to occur between reactants, an electronic interaction must exist which tends to delocalize the exchanging electron between sites. Neglecting overlap, the magnitude of the interaction is given by V (equation 17), where 0A and 0 are the electronic wavefunctions for the acceptor and donor orbitals and V is an electrostatic operator that describes the electronic perturbation between the electron donor and acceptor and causes electron transfer to occur,... 

The same phenomenon that leads to Hund s rule of maximum multiplicity in atoms (i.e., quantum-mechanical exchange stabilization) produces polarization of the electron spins in the C-H a bond. In a valence-bond treatment, the bond is comprised of one electron from a carbon sp2 orbital and another from a hydrogen Is orbital. Exchange forces act to polarize the sp2 electron so that its spin is parallel to the unpaired spin in the carbon 2p orbital this leaves the... 

A quite simple picture is obtained in the ET case if we go from many-particle theory to orbitals via Koopmans theorem. This treatment is correct in the limit case when donor and acceptor exchange electrons using well-separated MO s on donor and acceptor. The simplest example is the case studied by McConnell [7] (fig.l). Two identical n systems are connected via a hydrocarbon chain, which acts as a bridge. ET is possible in an open shell system. We may assume that either the whole ET system is either a negative ion or a positive ion and that the corresponding neutral molecule has closed shells. In the former case the electron occupies 7t LUMO. In the latter case there is a hole in rt HOMO. 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

Strategy Look for three-atom groupings that contain a multiple bond next to an atom with a p orbital. Exchange the positions of the bond and the electrons in the p orbital to draw the resonance form of each grouping. 

The electronic states on the metal are labeled by their quasi-momentum k they can exchange electrons with the adsorbate orbital the corresponding terms are ... 

Pericyclic reactions are the ones where the electrons rearrange through a closed loop of interacting orbitals, snch as in the electrocyclization of 1,3,5-hexatriene (88). Lemal pointed ont that a concerted reaction could also take place within a cyclic array, bnt where the orbitals involved do not form a closed loop. Rather, a disconnection occnrs at one or more atoms. At this disconnection, nonbonding and bonding orbitals exchange roles. Such a reaction has been termedpseudopericyclic. 

Figure 14). The Keggin structure can therefore accommodate as many as 24 additional electrons in this fashion. Indeed, an additional eight electrons (32 in all ) can be accommodated in nonbonding molecular orbitals, two electrons per trimeric unit. Each bears a terminal exchangeable water ligand (thereby keeping the anion charge low), and the browns have been shown to participate in atom-transfer processes, for example, equation (5).

The electron Hamiltonian (15) describes the so-called orbital exchange coupling in a three-dimensional (3D) crystal lattice. The Pauli matrices, cr O ), have the same properties as the z-component spin operator with S = As a i) represents not a real spin but orbital motion of electrons, it is called pseudo spin. For the respective solid-state 3D-exchange problem, basic concepts and approximations were well developed in physics of magnetic phase transitions. The key approach is the mean-fleld approximation. Similar to (8), it is based on the assumption that fluctuations, s(i) = terms quadratic in s i) can be neglected. We do not go into details here because the respective solution is well-known and discussed in many basic texts of solid state physics (e.g., see [15]). 

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