Electron in the box

Thus every value of W (provided W > F ) and therefore of X is possible, that is to say, the translational energy of a free electron is not quantized. For the electron in the box there are, however, boundary conditions that must be fulfilled... 

The next step up in complexity comes with four p orbitals conjugated together, with butadiene 1.5 as the parent member. There is a a framework 1.6 with 36 electrons and four p orbitals to house the remaining four. Using the electron in the box with four p orbitals, we can construct Fig. 1.31, which shows the four wave functions, inside which the p orbitals are placed at the appropriate regular intervals. We get a new set of orbitals, ipi, 3, and ip4, each described by Equation 1.11 with four terms. 

In the case of the electron in the box, we look at each energy level as a series of sine waves. We now do the same for conjugated systems, but this time the wave is seen in the coefficients, c. Thus the lowest-energy orbital of butadiene, il/1, reasonably enough, has a high concentration of electrons in the middle,... 

The picture at this point is that the total potential experienced by an electron in the box as a result of the presence of the ionic cores may be written as... 

The interaction of atomic orbitals giving rise to molecular orbitals is the simplest type of conjugation. Thus in ethylene the two p orbitals can be described as being conjugated with each other to make the n bond. The simplest extension to make longer conjugated systems is to add one p orbital at a time to the n bond to make successively the n components of the allyl system with three carbon atoms, of butadiene with four, of the pentadienyl system with five, and so on. Hiickel theory applies, because in each case we separate completely the n system from the a framework, and we can continue to use the electron-in-the-box model. 

Assume that S = 1.00 x 10 Vm (1.00 volt potential difference between the ends of the box). Using first-order perturbation theory, calculate the ground-state energy of the electron in the box and compare it to the result that you get in the absence of... 

Now we are ready to define the mathematics of the (quantized) PIB. Assume there is a potential V, which is zero along the box but that keeps the electrons in the box by rising straight up at the ends of the box to - -oo. 

In addition to the problem of the electron in the box, it must be taken into account that the amplitude function of the Bloch wave fulfils the periodicity and remains continuous and differentiable at the points at which the potential exhibits discontinuities [41-43]. 

Fig. 5.114 The increase of the bandgap (data in eV) in CdS with cluster size. The clusters the Cd-content of which are indicated in the top row (oo means bulk, the designation 40 A refers to a cluster for which the number of molecules is not precisely known but is above 32), are chemically complex in terms of the terminal groups the local environments, however, are comparable with pure CdS. The shifts of valence and conduction band (data in eV) agree with values predicted from an electron-in-the-box treatment (see Section 2.2.1) appropriately adjusted and extended (spherical geometry, finite energy threshold at the surface) [319].

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