Electron wave-mechanical picture

Wave mechanical picture of diborane Consider each borane atom to be sp3 hybridised. Two terminal BH bonds are cr bonds involving a pair of electron each. This accounts for eight of the total twelve electrons available for bonding. Each of the bridging BHB linkage then involves a delocalized or three centre bond as follows. The appropriate combination of three orbital wave functions, fBj, fB2... 

The cyclopropane ring represents an interesting intermediate between alkanes and alkenes. Theoreticians have had great fun with this molecule, for which no simple (i.e. non-wave mechanical) picture is satisfactory. There is certainly some electron density in the middle of the ring, and it can be hydrogenated to propane quite easily, although less easily than propene. Alkyl- and alkenylcyclopropanes react in interesting ways (Chapters 7 and 11). 

In terms of the wave-mechanical picture, an atom may be looked upon as consisting of a positively charged nucleus surrounded by a negatively charged cloud, which is made up of contribution from electrons in various orbitals. Since the centers of positive and negative charges are coincident (see Figure 3.57a), the net dipole moment of the atom is zero. 

Schrodinger and de Broglie suggested a "wave-particle duality" for small particles—that is, if electromagnetic radiation showed some particle-like properties, then perhaps small particles might exhibit some wave-like properties. Explain. How does the wave mechanical picture of the atom fundamentally differ from the Bohr model How do wave mechanical orbitals differ from Bohr s orbits What does it mean to say that an orbital represents a probability map for an electron ... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

The density p(r) might also be described as the fractional probability of finding the (entire) electron at point r. However, chemical experiments generally do not probe the system in this manner, so it is preferable to picture p(r) as a continuous distribution of fractional electric charge. This change from a countable to a continuous picture of electron distribution is one of the most paradoxical (but necessary) conceptual steps to take in visualizing chemical phenomena in orbital terms. Bohr s orbits and the associated particulate picture of the electron can serve as a temporary conceptual crutch, but they are ultimately impediments to proper wave-mechanical visualization of chemical phenomena. 

A new quantum theory called wave mechanics (as formulated by Schrodinger) or quantum mechanics (as formulated by Heisenberg, Born and Dirac) was developed in 1926. This was immediately successful m accounting for a wide variety of experimental observations, and there is little doubt that, in principle, the theory is capable of describing any physical system. A strange feature of the new mechanics, however, is that nowhere does the path or velocity of the electron enter the description. In fact it is often impossible to visualize any classical motion that could be consistent with the quantum mechanical picture of the atom,... 

In the previous chapters, we developed an approach which can be used to put the process of developing mechanistic descriptions of PES (i.e. of developing MM force fields) on a rational basis. Deductive molecular mechanics [2-4] (DMM) allows us to develop a form of the MM force fields to analyze the form of the electronic wave function relevant to the physical picture of the electronic structure of the considered class of molecules. In this chapter we apply the previously developedDMM approach to analytical derivation of the QM based form of the force fields involving the nontransition metal atoms. 

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. ... 

If zero-point vibration amplitudes of the dot are comparable with the Fermi length of the electrons, the shuttling takes place at small bias voltage. This is the case for cold dots. The constructive interference of electron waves in the tunnel gap center effectively charges the dot. In the quantum limit, this charging requires a justification of the tunnel-term concept based on the Schrodinger equation. In next section we address a more rigorous quantum mechanical picture based on the "ab-initio" SET model. 

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