Electron-in-a-box model

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 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 visible spectra of the cyanine dyes is a traditional physical chemistry spectroscopy experiment. In this exercise, students are asked to interpret these electronic spectra in terms of three quantum mechanical models the electron in a box model, the perturbed electron in a box model, and the Hiickel molecular orbital model. The students use the numerical methods of an earlier computer laboratory for the first and second model, and their classroom notes and Mathcad for the third. In this latter case, they use Mathcad s symbolic processor (MAPLE) to find eigenvalues for the Huckel matrices that emerge. 

The electron-in-a-box model (Figure 12.23) is a very simple quantum mechanical model that describes an electron in a covalent bond (Chapter 14). This simple model imagines an electron trapped between two infinitely high potential walls . The potential energy of the electron is zero inside the box but is infinite at the walls. The electron is trapped in a potential well and cannot escape. 

Figure 12.23 The electron-in-a-box model (1/represents the potential energy of the electron)... 

From the foregoing discussion it appears that the frontier orbital method is at once a simple, concise, and accurate method for assessing the stereochemical outcome of pericyclic reactions. Furthermore, it is a method that is equally applicable to symmetrical and to unsymmetrical systems. There are some disadvantages in the theory, however. Firstly, it is necessary to derive the general phase characteristics of the HOMO and LUMO levels. Hiickel molecular calculations can be used for tliis purpose, but there are available a number of approximate methods, for example the electron-in-a-box model, which are usually satisfactory even if they are more difficult to apply to more complex systems. Nevertheless, frontier orbital analysis is quicker and more simple than the formalized correlation diagram approach, and with a little practice one can intuitively arrive at the correct relative phase relationsliips in the HOMO and LUMO levels. 

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. 

Cadmium sulfide suspensions are characterized by an absorption spectrum in the visible range. In the case of small particles, a quantum size effect (28-37) is observed due to the perturbation of the electronic structure of the semiconductor with the change in the particle size. For the CdS semiconductor, as the diameter of the particles approaches the excitonic diameter, its electronic properties start to change (28,33,34). This gives a widening of the forbidden band and therefore a blue shift in the absorption threshold as the size decreases. This phenomenon occurs as the cristallite size is comparable or below the excitonic diameter of 50-60 A (34). In a first approximation, a simple electron hole in a box model can quantify this blue shift with the size variation (28,34,37). Thus the absorption threshold is directly related to the average size of the particles in solution. 

Atomic orbitals may be combined to form molecular orbitals. In such orbitals, there is a nonzero probability of finding an electron on any of the atoms that contribute to that molecular orbital. Consider an electron that is confined in a molecular orbital that extends over two adjacent carbon atoms. The electron can move freely between the two atoms. The C-C distance is 139 pm. (a) Using the particle in a box model, calculate the energy required to promote an electron from the n = 1 to n = 2 level assuming that the length of the box is equal to the distance between two carbon atoms, (b) To what wavelength of radiation does this correspond (c) Repeat the calculation for a linear chain of 1000 carbon atoms. 

The energy with n = 1 is 10 eV, and the next allowed one (n = 2) about 30 eV higher. These are typical energy magnitudes for electrons in atoms and molecules. Of course, the particle-in-a-box model is ridiculously oversimplified to apply seriously here (although it has been used to represent it electrons in systems with conjugated double bonds see Problem 5 below). It does, however, confirm our expectation that the quantum theory must be used for problems involving electrons in atoms and molecules. 

Using a particle in a box model with an infinite potential drop at the wall as the boundary condition, and taking into account that the exciton consists of an electron-hole pair, the Schrodinger equation can be solved yielding the energy of the lowest excited state (Brus 1983, 1984 Rossetti et al. 1984), i.e., the lower edge of the conduction band, as... 

The behavior of the polarizability of the n-alkanes may be rationalized with reference to the simple particle-in-a-box model. In this model, the polarizability increases as the fourth power in the length of the box, which implies that the polarizability per unit length (i.e., the differential polarizability) increases as the third power in the length of the box. Obviously, in the n-alkanes, the electrons do not at all behave according to this simple model, but we may still attribute the increasing differential polarizability of the n-alkanes to a more pronounced delocalization of the electrons in the longer chains, as the electrons become more loosely attached to the system. As the chain grows, this effect becomes less important and the differential polarizability becomes constant. 

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 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. 

Particle-in-a-box states for an electron in a 20 Pd atom linear chain assembled on a single crystal NIAI surface. The left set of curves shows the predictions of the one-dimensional parti-cle-in-a-box model, the center set of images is the 2D probability density distribution, and the right of curves is a line scan of the probability density distribution taken along the center of the chain. The chain was assembled and the probability densities measured using a scanning tunneling microscope. 

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