Electron-wave model

Small metal clusters are also of interest because of their importance in catalysis. Despite the fact that small clusters should consist of mostly surface atoms, measurement of the photon ionization threshold for Hg clusters suggest that a transition from van der Waals to metallic properties occurs in the range of 20-70 atoms per cluster [88] and near-bulk magnetic properties are expected for Ni, Pd, and Pt clusters of only 13 atoms [89] Theoretical calculations on Sin and other semiconductors predict that the stmcture reflects the bulk lattice for 1000 atoms but the bulk electronic wave functions are not obtained [90]. Bartell and co-workers [91] study beams of molecular clusters with electron dirfraction and molecular dynamics simulations and find new phases not observed in the bulk. Bulk models appear to be valid for their clusters of several thousand atoms (see Section IX-3). 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

Some more recent software uses the tensor LEED approximation of Rous and Pen-dry which can save a substantial amount of computer time [2.268-2.270]. In tensor LEED the amplitudes (0) of all escaping electron waves (spots) are first calculated conventionally as described above for a certain reference geometry. Then the derivatives of these amplitudes 5Ag/5ri with respect to small displacements of each atom i in this reference geometry are calculated. These derivatives are the constituents of the "tensor". The wave amplitude for a modified model geometry where atom i is displaced by the vector Aq is then approximately given by ... 

Semi-empirical models provide a method for calculating the electronic wave function, which may be used for predicting a variety of properties. There is nothing to hinder the... 

It is important to realize that whenever qualitative or frontier molecular orbital theory is invoked, the description is within the orbital (Hartree-Fock or Density Functional) model for the electronic wave function. In other words, rationalizing a trend in computational results by qualitative MO theory is only valid if the effect is present at the HF or DFT level. If the majority of the variation is due to electron correlation, an explanation in terms of interacting orbitals is not appropriate. 

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

Figure 6.13. The electron distribution in the model metal jellium gives rise to an electric double layer at the surface, which forms the origin of the surface contribution to the work function. The electron wave function reaches...

Hyperfine coupling constants provide a direct experimental measure of the distribution of unpaired spin density in paramagnetic molecules and can serve as a critical benchmark for electronic wave functions [1,2], Conversely, given an accurate theoretical model, one can obtain considerable information on the equilibrium stmcture of a free radical from the computed hyperfine coupling constants and from their dependenee on temperature. In this scenario, proper account of vibrational modulation effects is not less important than the use of a high quality electronic wave function. 

The free-electron gas was first applied to a metal by A. Sommerfeld (1928) and this application is also known as the Sommerfeld model. Although the model does not give results that are in quantitative agreement with experiments, it does predict the qualitative behavior of the electronic contribution to the heat capacity, electrical and thermal conductivity, and thermionic emission. The reason for the success of this model is that the quantum effects due to the antisymmetric character of the electronic wave function are very large and dominate the effects of the Coulombic interactions. 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

Figure 7. The occupation number densities as functions of wave vector for Na. The thick curves labeled (100), (110) and (111) represent the three principal directions within the first Brillouin zone, obtained by the FLAPW-GWA. The thin solid curve is obtained from an interacting electron-gas model [27]. The dash-dotted line represents the Fermi momentum. 

That light has a dual nature and behaves either like a wave or like a stream of particle-like photons is a fact we must accept, although it is nonintuitive. But remember, we have no direct experience of the behavior of very small particles such as electrons. Which model we use depends on the observations we are making. The wave model is appropriate when we are considering diffraction and interference experiments, but the particle (photon) model is essential when we are considering the interaction of light with individual atoms or molecules. 

As noted above, in the reference model the dependence of the electron wave functions A and B on the nuclear coordinates was entirely neglected and the wave functions A and B for the isolated ions or the wave functions calculated for corresponding equilibrium nuclear configurations Qk0i and Qkof according to Eqs. (8) were usually used in the calculations. 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

A simple model was considered above. A more refined theory taking into account the modulation of the electron wave function of the complex AL by fluctuations of the medium polarization is given in Ref. 35. 

---