Localization properties of the wave function

The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of the atomic basins, and the associated electron densities, in a functional group are very similar in different molecules. The local properties of the wave function are therefore transferable to a very good approximation, which rationalizes the basis for organic chemistry, that functional groups react similarly in different molecules. It may be shown that any observable... 

For Ef near a band extremity, we find, using (49) for a , that directional properties of the wave functions. Thus [Pg.38]

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

There are two extreme views in modeling zeolitic catalysts. One is based on the observation that the catalytic activity is intimately related to the local properties of the zeolite s active sites and therefore requires a relatively small molecular model, including just a few atoms of the zeolite framework, in direct contact with the substrate molecule, i.e. a molecular cluster is sufficient to describe the essential features of reactivity. The other, opposing view emphasizes that zeolites are (micro)crystalline solids, corresponding to periodic lattices. While molecular clusters are best described by quantum chemical methods, based on the LCAO approximation, which develops the electronic wave function on a set of localized (usually Gaussian) basis functions, the methods developed out of solid state physics using plane wave basis sets, are much better adapted for the periodic lattice models. 

Several different kinds of quantum-chemical descriptors have been defined and these can be broadly divided into energy-based descriptors, orbital energies descriptors, local quantum-chemical properties, descriptors based on the analysis of the wave function, frontier orbital electron densities, superdelocalizability indices, polarizabilities, and derived from the Density Functional Theory [Cartier and Rivail, 1987 Bergmarm and Hinze, 1996 Karelson, Lobanov et al., 1996]. 

The total molecular susceptibility has now been expressed as a sum over operators localized on the various atomic nuclei. But they operate on wave functions that extend over the whole molecule. If the average values of these atomic operators are not greatly dependent on parts of the wave function far removed from the nucleus in question and if the relevant properties of the electron distribution around each nucleus are not much different for a given type of atom in different molecules, the terms within each sum over n in Eq. (41) will be independent and constant. They will, in short, be additive atomic susceptibilities that can be evaluated from measured molecules and used to predict the susceptibility of any desired molecule. We have already demonstrated the additivity of the diamagnetic susceptibilities [Eq. (37)]. 

The basic property that distinguishes the one-dimensional case from those of higher dimensionality and the one that is responsible for having all eigenstates localized in one dimension is the unique relation between the ratio of the amplitude of the wave function at two different points and the corresponding phases at these two points, i.e.. 

The filling of the / shell is a common feature of both lanthanides and actinides. However, there are remarkable differences in the properties of the 4/ and 5/ electrons. The 4/ orbitals of the lanthanides and the 5/ actinide orbitals have the same angular part of the wave function but differ in the radial part. The 5/ orbitals also have a radial node, while the 4/ orbitals do not. The major differences between actinide and lanthanide orbitals depend, then, on the relative energies and spatial distributions of these orbitals. The 5/ orbitals have a greater spatial extension relative to the Is and Ip than the 4/ orbitals have relative to the and 6/t. This allows a small covalent contribution from the 5/ orbitals, whereas no compounds in which 4/ orbitals are used exist. In fact, the 4/ electrons are so highly localized that they do not participate in chemical bonding, whereas the 5d and 6s valence electrons over-... 

After demonstrating the effect of disorder on the energy-band structure of a chain and on the localization properties of the corresponding wave functions, we shall describe in the subsequent sections several methods to determine the eneigy-level distribution (density of states) and wave functions of an aperiodic chain. 

The properties described imply an important message for [Ru(bpy)3], as is clearly displayed in the comparison of [Ru(bpy)3] with [Ru(i-biq)2(bpy)] (Table 7). The zfs and AEj.jjj increase by more than 70 and 30 %, respectively, according to the reduced spatial spread of the wave-functions to only one single ligand in the MLCT state of [Ru(i-biq)2(bpy)] " compared to the involvement of three ligands in [Ru(bpy)3]. In the situation of a localization in both compounds one would not expect to observe any obvious difference. Therefore, the results described in this section and in [248] clearly demonstrate the delocalized situation in [Ru(bpy)j] +. 

As is to be expected, inherent disorder has an effect on electronic and optical properties of amorphous semiconductors providing for distinct differences between them and the crystalline semiconductors. The inherent disorder provides for localized as well as nonlocalized states within the same band such that a critical energy, can be defined by distinguishing the two types of states (4). At E = E, the mean free path of the electron is on the order of the interatomic distance and the wave function fluctuates randomly such that the quantum number, k, is no longer vaHd. For E < E the wave functions are localized and for E > E they are nonlocalized. For E > E the motion of the electron is diffusive and the extended state mobiHty is approximately 10 cm /sV. For U <, conduction takes place by hopping from one localized site to the next. Hence, at U =, )J. goes through a... 

Two properties, in particular, make Feynman s approach superior to Benioff s (1) it is time independent, and (2) interactions between all logical variables are strictly local. It is also interesting to note that in Feynman s approach, quantum uncertainty (in the computation) resides not in the correctness of the final answer, but, effectively, in the time it takes for the computation to be completed. Peres [peres85] points out that quantum computers may be susceptible to a new kind of error since, in order to actually obtain the result of a computation, there must at some point be a macroscopic measurement of the quantum mechanical system to convert the data stored in the wave function into useful information, any imperfection in the measurement process would lead to an imperfect data readout. Peres overcomes this difficulty by constructing an error-correcting variant of Feynman s model. He also estimates the minimum amount of entropy that must be dissipated at a given noise level and tolerated error rate. 

A possible application for the formation of a-like condensates are selfconjugate 4n nuclei such as 8Be, 12C, 160,20Ne, 24Mg, and others. Of course, results obtained for infinite nuclear matter cannot immediately be applied to finite nuclei. However, they are of relevance, e.g., in the local density approximation. We know from the pairing case that the wave function for finite systems can more or less reflect properties of quantum condensates. 

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