Delta-well model

Since this model has the major defect of confining the electron too closely to the space between the nuclei, consider another one-dimensional model which avoids this defect, a delta-well model . A delta well —a well that is infinitesimally wide but infinitely deep—affords one bound state for a particle, t As the width / of the well approaches zero, and the potential energy V of a particle in the well becomes negatively infinite, in such a way that the product VI = —rj remains finite, then the energy of the electron in the bound state takes the form —E = rj2/4, and its wave function to the left and the right of the well is... 

Fig 5 11 The total energy, and its components, in the delta-well model of the hydrogen-molecule ion, when the electron is in the symmetric and antisymmetric bound states, plotted in atomic units... 

Fig 6 1 For the delta-well model of the hydrogen-molecule ion in Chapter 5, the two one-electron wave functions are spatially symmetrical (a) and anti-symmetrical (b)... 

Fig. 6 2 Specifying the positions of the two electrons in the one-dimensional delta-well model of the hydrogen molecule requires a plane instead of a line The regions marked + show the positions of the wells, and the dashed line shows the points at which the two electrons are coincident...

Fig. 6 3 The spatially symmetrical wave function for the two electrons in the delta-well model of the hydrogen molecule has two peaks, for electrons at different wells...

Fig. 8 12 Energy (vertically) versus bond length (horizontally) in the six wave functions of the one-dimensional cyclic delta-well model for benzene Dotted lines are the energies of the wave functions for two delta wells (Fig 5 8)...

Again a one-dimensional model, using delta wells to represent the atoms, provides a simple illustration of how the inquiry might proceed. The appendix to this chapter finds six independent wave functions, three bonding and three antibonding, for the problem. The six electrons can occupy, by spin-opposed pairs, the three bonding orbitals. It turns out that the resulting collection... 

Fig 8 11 A scheme of six similar delta wells, and local coordinates, to calculate the nonlocalized wave functions for a one-dimensional model of benzene... 

Now we check the above general results with Winter s delta-barrier model [63], see Figure 9.1. The initial state is an eigenstate of the infinite square well potential. 

Figure 6 shows this simple results agrees well with the exact splitting AE = E- — E+ for the delta function model [22] which is obtained by solving numerically a pair of transcendental equations. 

Figure 5.16. The four-environment model divides a homogeneous flow into four environments, each with its own local concentration vector. The joint concentration PDF is thus represented by four delta functions. The area under each delta function, as well as its location in concentration space, depends on the rates of exchange between environments. Since the same model can be employed for the mixture fraction, the exchange rates can be partially determined by forcing the four-environment model to predict the correct decay rate for the mixture-fraction variance. The extension to inhomogeneous flows is discussed in Section 5.10.

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