Electronic coordinate system

If a hapto bond is formed, the metallic center is included in the delocalized coordination system, electrons and charge are now spread over the metallic center and the hydrocarbon system. Corresponding diagonal functions must be used. 

We shall pursue some of the consequences of a classical grazing model in which a repulsive force, due to electron-electron interactions, acts at small impact parameters and an attractive force, due to atomic polarisability, acts at large impact parameters. Four such classical paths are illustrated in figure 2 using a right handed coordinate system. Electrons scattered through 0 to the left by either a repulsive or attractive forces are detected by the electron detector at +0. 

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. 

Electron transfer reaction rates can depend strongly on tire polarity or dielectric properties of tire solvent. This is because (a) a polar solvent serves to stabilize botli tire initial and final states, tluis altering tire driving force of tire ET reaction, and (b) in a reaction coordinate system where the distance between reactants and products (DA and... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

For the alkali metal doped Cgo compounds, charge transfer of one electron per M atom to the Cgo molecule occurs, resulting in M+ ions at the tetrahedral and/or octahedral symmetry interstices of the cubic Cgo host structure. For the composition MaCgg, the resulting metallic crystal has basically the fee structure (see Fig. 2). Within this structure the alkali metal ions can sit on either tetragonal symmetry (1/4,1/4,1/4) sites, which are twice as numerous as the octahedral (l/2,0,0) sites (referenced to a simple cubic coordinate system). The electron-poor alkali metal ions tend to lie adjacent to a C=C double... 

To look ahead a little, there are properties that depend on the choice of coordinate system the electric dipole moment of a charged species is origin-dependent in a well-understood way. But not the charge density or the electronic energy Quantities that have the same value in any coordinate system are sometimes referred to as invariants, a term borrowed from the theory of relativity. 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

Here the eigenfunctions referred to are those for an electron in an atom, and r, 0 and

polar coordinates of the electron, the nucleus being at the origin of the coordinate system. 

Planar coordinated systems, you will recall from Chapter 1, formed a major group of exceptions to the otherwise very successful geometry modelling of Kepert. That model explicitly neglected any steric role for the non bonding electrons, however. Let us now recognize and incorporate the steric activity of the d shell in systems. 

The wave function of an electron corresponds to the expression used to describe the amplitude of a vibrating chord as a function of the position x. The opposite direction of the motion of the chord on the two sides of a vibrational node is expressed by opposite signs of the wave function. Similarly, the wave function of an electron has opposite signs on the two sides of a nodal surface. The wave function is a function of the site x, y, z, referred to a coordinate system that has its origin in the center of the atomic nucleus. 

The Jk-th wave function of the electrons in a chain of hydrogen atoms results in a similar way. From every atom we obtain a contribution 2 cosnka, i.e. the Is function %n of the n-th atom of the chain takes the place of A0. All atoms have the same function x, referred to the local coordinate system of the atom, and the index n designates the position of the atom in the chain. The k-th wave function is composed of contributions of all atoms ... 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

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