Orbital complex representation

Here the components of excited state J are expressed in a representation that diagonalizes the spin-orbit operator. In general, this will be a complex representation. The principle of spectroscopic stability can again be used to express the components of Jin a representation that we denote jM. This representation is made up of space and spin parts where the spin part diagonalizes the spin operator. 

Since ethylene oxide has a plane of symmetry bisecting the CC bond [marked by a dotted line in (81)], its MOs must be either symmetric or antisymmetric with respect to reflection in this plane. The MOs in the 7r-complex representation (78) satisfy this criterion but the bent-bond orbitals in (80) do not. We can, however, construct from the two bent-bond orbitals and 02 two linear combinations 0i + 02 02 which is... 

Molecular orbitals were one of the first molecular features that could be visualized with simple graphical hardware. The reason for this early representation is found in the complex theory of quantum chemistry. Basically, a structure is more attractive and easier to understand when orbitals are displayed, rather than numerical orbital coefficients. The molecular orbitals, calculated by semi-empirical or ab initio quantum mechanical methods, are represented by isosurfaces, corresponding to the electron density surfeces Figure 2-125a). 

FIGURE 16.36 I1ie tear-shaped objects are representations of the six ligand atomic orbitals that are used to build the molecular orbitals of an octahedral complex in ligand field theory. They might represent s- or p-orbitals on the ligands or hybrids of the two. 

Figure 2-2. Schematic representation of the radial waveforms for 3d, 45 and 4p orbitals in first row transition-metal ions of intermediate oxidation state (Werner-type complexes).

This procedure is strictly invalid, of course, since the symmetry of a six-coordinate complex with dissimilar ligands cannot be exactly octahedral. In this case, further splitting of the d orbitals takes place which is not representable by a single splitting parameter like 4oct-However, if the departure from Oh symmetry is slight, so that spectral bands are broadened rather than split, the law of average environments retains utility. 

Fig. 6.11. Representation of transition structure and die LUMO orbitals for three stereoisomeric complexes of A-acryloyloxazolidinone with a TADDOL model, Ti[0(CH2)40]Cl2. The LUMO energies (B3LYP/6-3111+G(d)) in kcal/mol. Reproduced from J. Org. Chem., 63, 2321 (1998), by permission of the American Chemical Society.

The effects of Lewis acids on the stereoselectivities can also be understood in terms of orbital interactions. The variation in charge at the respective basic centre gives rise to a change in the magnitude of the orbital coefficients of the entire interacting molecular orbital. These effects are visualized by the HOMO and LUMO representations of the Lewis acid-base complex of acrolein and trifluoroborane (Figure 3), and in an even more extreme case by the HOMO and LUMO representations of one of the simplest dienophile-Lewis acid complexes protonated acrolein92,93. 

Fig. 21 Schematic representation of strategies for spin alignment in D/A salts or complexes by application of spin conservation in different electron configurations of interacting molecular orbitals, (a) Typical D/A interaction between two closed-shell D and A, (b and b ) McConnell s proposal, (c) Breslow s extension, (d) Torrance s model, (e) Wudl s model, and (f) Chiang s model for further doping.

FIG. 5. Schematic representation of the orbital overlap between the chromium d, v orbital and the ligand protons Ha and Hb for oxo-Cr(V) complexes of 6 (left) and 7 (right). 

The proper way of dealing with periodic systems, like crystals, is to periodicize the orbital representation of the system. Thanks to a periodic exponential prefactor, an atomic orbital becomes a periodic multicenter entity and the Roothaan equations for the molecular orbital procedure are solved over this periodic basis. Apart from an exponential rise in mathematical complexity and in computing times, the conceptual basis of the method is not difficult to grasp [43]. Software for performing such calculations is quite easily available to academic scientists (see, e.g., CASTEP at www.castep.org CRYSTAL at www.crystal.unito.it WIEN2k at www.wien2k.at). 

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