Orbital exchange

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... 

Excited states of Cr + complexes were explored by single crystal spectroscopy at low temperatures. In the dimeric [a Cr(0H)2Cra ] + the sharp 2E single excitations were used to determine orbital exchange parameters. Out-of-plane interactions are dominant. The complex CrCljt was studied in two exactly octahedral crystal environments. 

Table I. Structural and antiferromagnetic orbital exchange parameters...

Here 0, is the isospin describing the C, orbital, Iy is the orbital exchange constant, and Jq is the Heisenberg spin-exchange constant. This Hamiltonian can describe many of the magnetic properties of TDAE-C60. 

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. 

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. 

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]). 

Here is inverse of the respective dynamical matrix. In (17) it describes elastic coupling of tetragonal distortions, Qe(n) and Qs n), the ones active in the JT case E IS) e. Obviously, the only important contribution comes from the orbital exchange of close neighbors. In cubic symmetry, (17) simplifies to... 

The Hamiltonian of orbital exchange, (15), or (17), is invariant of the respective symmetry group of the undistorted crystal in its high-symmetry phase. Similarly to... 

In the OOA, one of its basic assumptions is reducing the intercell correlation to symmetry equivalent intersite orbital exchange coupling. This assumption simplifies the physical picture of the cooperative JT effect. Ligands and all the respective... 

In the simple case of corner-sharing octahedrons with first-long-period transition metals 3d elements) bridged by one second- or third-period ligand atom, the orbital exchange coupling can be approximated as a perturbation-theory energy correction. 

It includes Pauli matrix. Though nondiagonal, it can be transformed to a diagonal form with eigenvalues 1. Formally, it coincides with (1). The rest of the solution, including separation of the orbital exchange part (15), is the same as in Sect. 3 with no approximations involved. 

In (34), the effective Hamiltonian H2) is still a matrix in the orbital space of electron wave functions. Similar to (17), it describes intersite orbital exchange coupling. In (34) and (35), the factor J(i - ) is the parameter of orbital exchange coupling, same as in (3.7). This time it is supported by the assumption of weak JT coupling. The matrix elements (0 Qy (i) n) are determined with zero-coupling oscillator wave functions. They take a nonzero value for = 1 only. In other words, in the effective Hamiltonian H2), the virtual excited states are one-phonon states. Therefore, the effective Hamiltonian (34) describes phonon-mediated orbital exchange. 

Here Eji = lap-) is the JT stabilization energy. Compared to the parameters of direct orbital exchange, its magnitude is one or even two orders stronger [cf. (28)]. This explains unusually large values of parameters of orbital exchange in perovskites. Also, it is important to note that 0 directions are determined by orientation of the metal-ligand octahedron and may be different from directions of the primitive lattice. 

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