Electrons in a Box

The underlying principle of the PEOE method is that the electronic polarization within the tr-bond skeleton as measured by the inductive effect is attenuated with each intervening o -bond. The electronic polarization within /r-bond systems as measured by the resonance or mesomeric effect, on the other hand, extends across an entire nr-system without any attenuation. The simple model of an electron in a box expresses this fact. Thus, in calculating the charge distribution in conjugated i -systems an approach different from the PEOE method has to be taken. 

Quantitative calculations can be made on the basis of the assumption that the density of levels in energy for the conduction band is given by the simple expression for the free electron in a box, and the interaction energy e of a dsp hybrid conduction electron and the atomic moment can be calculated from the spectroscopic values of the energy of interaction of electrons in the isolated atom. The results of this calculation for iron are discussed in the following section. 

The interaction energy of the valence electron with the two atomic 3d electrons, with parallel spins, is accordingly —0.67 ev, and the width of the energy band that would be occupied by uncoupled valence electrons is 1.34 ev. The number of orbitals in this band can be calculated from the equation for the distribution of energy levels for an electron in a box. The number of levels per atom is... 

Quantum Free-Electron Theory Constant-Potential Model, The simple quantum free-electron theory (1) is based on the electron-in-a-box model, where the box is the size of the crystal. This model assumes that (1) the positively charged ions and all other electrons (nonvalence electrons) are smeared out to give a constant background potential (a potential box having a constant interior potential), and (2) the electron cannot escape from the box boundary conditions are such that the wavefunction if/ is... 

The trapped electron provides a classic example of an electron in a box . A series of energy levels are available for the electron, and the energy required to transfer from one level to another falls in the visible part of the electromagnetic spectrum, hence the colour of the F-centre. There is an interesting natural example of this phenomenon The mineral... 

The average thermal energy is 4 x 10-21 J per molecule at room temperature, corresponding to a quantum number n around a hundred million. Hie difference in energy between successive levels is negligible for most purposes here. On the other hand, if we confine an electron in a box 0.2 nm long (the length of a typical chemical bond), we have... 

However Smith and Symons (43, 44, 45) found that the theory of Franck and Platzman did not account for environmental (temperature, solvents, added salts) effects on the spectrum. They proposed instead a theory based on an electron in a box of radius r. Absorption of hv causes the resulting atom to contract, the electron preserving its radius as in the ground state. Environmental effects change this radius. 

Let us reconsider the electron in a box but now reverse the variable. Let us hold the electron at a single energy and allow the box walls to move. Now the acceptable boxes are quantized in units of the selected half-wavelength. 

In each case the color is from a trapped electron. The color can be rationalized by using electron in a box arguments. As the box gets smaller the energy levels get further apart so the absorption moves further towards the blue... 

It is interesting that the electron cloud confined in the octahedron formed by the Na+ neighbors can be approximately treated as a free electron in a box of such dimensions. The straightforward calculation... 

This separation of the cr framework and the re bond is the essence of Hiickel theory. Because the re bond in ethylene in this treatment is self-contained, we may treat the electrons in it in the same way as we do for the fundamental quantum mechanical picture of an electron in a box. We look at each molecular wave function as one of a series of sine waves, with the limits of the box one bond length out from the atoms at the end of the conjugated system, and then inscribe sine waves so that a node always comes at the edge of the box. With two orbitals to consider for the re bond of ethylene, we only need the 180° sine curve for re and the 360° sine curve for re. These curves can be inscribed over the orbitals as they are on the left of Fig. 1.23, and we can see on the right how the vertical lines above and below the atoms duplicate the pattern of the coefficients, with both c and c2 positive in the re orbital, and c positive and c2 negative in re. ... 

With regard to the model of N electrons in a box, the Fermi energy as introduced previously does not say anything about how difficult it is to push an electron through the walls of the box. The minimum work required to bring an additional electron from infinity, push it through the walls of the box, and land it in an unoccupied level near the Fermi level is called... 

The luminescence of small particles, especially of semiconductors, is a fascinating development in the field of physical chemistry, although it is too early to evaluate the potential of these particles for applications. The essential point is that the physical properties of small semiconductor particles are different from the bulk properties and from the molecular properties. It is generally observed that the optical absorption edge shifts to the blue if the semiconductor particle size decreases. This is ascribed to the quantum size effect. This is most easily understood from the electron-in-a-box model. Due to their spatial confinement the kinetic energy of the electrons increases. This results in a larger band gap (84). 

The remarkable situation in which we find ourselves in modem materials science is that physics has for some time been sufficiently developed, in terms of fundamental quantum mechanics and statistical mechanics, that complete and exact ab initio calculations of materials properties can, in principle, be performed for any property and any material. The term ab initio" in this context means without any adjustable or phenomenological or calibration parameters being required or provided. One simply puts the required nuclei and electrons in a box and one applies theory to obtain the outcome of a specified measurement. The recipe for doing this is known but the execution can be tedious to the point of being impossible. The name of the game, therefore, has been to devise approximations and methods that make the actual calculations doable with limited computer resources. Thanks to increased computer power, the various approximations can be tested and surpassed and more and more complex materials can be modelled. This section provides a brief overview of the theoretical methods of solid state magnetism and of nanomaterial magnetism in particular. 

The state symbols 1Fb, xLa, lBb and xBa shown in Figure 4.27 were introduced in 1949 by Platt.297,298 He assumed a free-electron model, similar to the electron-in-a-box, in which the 7t-electrons of a cyclic system are confined to a one-dimensional loop of constant potential (a circular wire). The eigenvalues of a single electron in a perimeter of length / are given by Equation 4.37. 

Metals can be described by a model of free electrons in a box but this model does not explain the properties of non-metals or the change in conductivity with temperature for metals. 

Using the electron in a box notation, give the electronic structure of chlorine (Z = 17). 

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