Wave functions of class

The postulates of quantum mechanics, especially the probabilistic interpretation of the wave function given by Max Born, limits the class of functions allowed (to class Q , or quantum ). 

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Wave Functions of Class Q Boundary Conditions An Analogy... 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

Quantum chemical methods may be divided into two classes wave function-based techniques and functionals of the density and its derivatives. In the former, a simple Hamiltonian describes the interactions while a hierarchy of wave functions of increasing complexity is used to improve the calculation. With this approach it is in principle possible to come arbitrarily close to the correct solution, but at the expense of interpretability of the wave function the molecular orbital concept loses meaning for correlated wave functions. In DFT on the other hand, the complexity is built into the energy expression, rather than in the wave function which can still be written similar to a simple single-determinant Hartree-Fock wave function. We can thus still interpret our results in terms of a simple molecular orbital picture when using a cluster model of the metal substrate, i.e., the surface represented by a suitable number of metal atoms. 

In essence, jSel is the interaction between the electronic wave functions of A and B. Obviously there must be some spatial overlap between the two in order to give rise to a finite value. Furthermore, if 0A. and />B fall in the same symmetry class or, in complex molecules, have similar local symmetry, the value of j8 may be relatively large. This will depend upon whether or not the perturbation operators, H, have preferred symmetry properties. The Woodward-Hoffman rules suggest that these operators can... 

A semiempirical method can be developed for the arbitrary form of the trial wave function of electrons, which is predefined by the specific class of molecules to be described and by the physical properties and/or effects which have to be reproduced within its framework. Two characteristic examples will be considered in this section. One is the strictly local geminal (SLG) wave function the other is the somewhat less specified wave function of the GF form selected to describe transition metal complexes. 

The fundamental reasons for the difficulties faced by the MM methods when metal (both transition and nontransition) complexes are involved can be understood if one does not consider the MM as a purely empirical scheme (as it is frequently done), but think about them as of some reflection of specific features of molecular electronic structure, formalized by the form of the trial wave function of that class of compounds where such a parameterization might be possible. As shown in Chapter 3, organic compounds for which the MM methods are known to demonstrate significant successes can be described by the QC method, which directly leads to local and transferable two-center bonds. It is shown in Chapter 3 that the derivation of the MM method from the QC description is possible due to a common background of the MM and SLG description, which consists in the physical presence of two-center, two-electron bonds in organic molecules (in strict terms of Section 1.7 - numbers of electrons in each of the geminals weakly fluctuate). 

Davies[190] The problem is to explain why it is that in most situations. . . the wave function of a molecule seems to be not an eigenstate of the Hamiltonian, but one of a class of slowly time varying states which are more stable in some sense. 

An important qualitative description of the spectral behavior of class II compounds was presented by Robin and Day. This simple model has found apphcabihty to the discussion of the spectra of numerous mixed valence compounds in which some delocalization occurs. In this model, it is assumed that the ground-state wave function contains the function, a, which describes mixing of the wave function for site A with the wave function of site B. 

Fig. 2.6. Functions of class Q (i.e., wave functions allowed in quantum mechanics) — examples and counterexamples. A wave function (a) must not be zero everywhere in space (b) has to be continuous (c) carmot tend to infinity even at a single point (d) cannot tend to infinity (e.f,g) its first derivative cannot be discontinuous for infinite number of points th.i) must be square integrable (j Jr,I,m) has to be defined uniquely in space (for angular variable 6).

For energies of the system below the inner maxima of the fission barrier, the model wave functions are normally concentrated within one of the minima of the deformation potential. This process leads to a classification of the wave functions as class I (with major amplitude within the first well), class II (centered in the second well), and class HI (centered in the third well). A schematic illustration of this classification of vibrational states is shown inO Fig. 5.3. 

To explain isomerism we turn to Fermi s Golden Rule which relates the transition rate to the wave functions of the initial and any final states as well as the density of final states in a given energy interval. In short, a decay can only happen if a suitable final state exists, and, if it does, the transition rate is higher the more the wave function of the final state resembles that of the initial state. We therefore expect isomers to occur in several classes ... 

The second large class of computational methods that is most useful for predicting reactivity of zeolites is based on the quantum mechanical description of a chemical system. Quantum mechanics represents the highest level in the hierarchy of computational methods. By solving the electronic structure problem they provide us the energy and wave function of a system, from which all properties of all atoms in it can be derived. In practice, however, the exact solution of the electronic structure problem cannot be obtained for any realistic system. 

Each country adopts such and such a class as a function of its climatic conditions. France has chosen classes B, E, and F, respectively for the summer, winter, and cold wave periods. The first is from 1 May to 31 October, the second is from 1 November to 30 April, while the third has... 

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