Wave functions basic properties

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

Wfi cri you perform a single poin t sem i-cmpirical or ah initio ealeu-laliori, you obtain th c en ei gy and tli e first dci ivalives of the eu ei gy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a n urn ber of oth er molecti lar properties th at could be available to you. Molecularproperties arc basically an average over th e wave fun ction of certain operatorsdescribin g the property. For exam pie, the electron ic dipole operator is basical ly ju st the operator for the position of an electron and the electron ic con tribution to the dipole iTi otTi en t is... 

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

The basic idea of the pseudopotential theory is to replace the strong electron-ion potential by a much weaker potential - a pseudopotential that can describe the salient features of the valence electrons which determine most physical properties of molecules to a much greater extent than the core electrons do. Within the pseudopotential approximation, the core electrons are totally ignored and only the behaviour of the valence electrons outside the core region is considered as important and is described as accurately as possible [54]. Thus the core electrons and the strong ionic potential are replaced by a much weaker pseudopotential which acts on the associated valence pseudo wave functions rather than the real valence wave functions (p ). As... 

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

According to the basic principle of quantum mechanics, any measurable property can be computed ab initio if the total wave function y/ describing the quantum eigenstate of the system is known, since it contains the complete... 

The variable hardness in this work is the local hardness as given by the basic theory [2]. The electronic chemical potential in this work is a property if a given molecule (arrangement of nuclei) is also of the approximate wave function used to describe it. This does not represent an equilibrium system. The variation of the chemical potential is a consequence. 

Arriving subsequently at rigorous quantum mechanical descriptions, we have assumed that the reader has some preliminary knowledge of basic quantum mechanical formalism. We consider it methodologically important to illustrate the correspondence principle between quantum and classical concepts, in particular between the concept of coherence of the wave functions of magnetic sublevels, and the symmetry properties of spatial angular momenta distribution. 

The unique electronic structure of these (L-A3)MoO(dithiolene) complexes arises from two basic factors. The first is the strong axial a- and Ji-donor properties of the terminal oxo ligand, which dominates the ligand field and predetermines the energy of the Mo-based dxz, dyz, and dzi acceptor orbitals. The second is the equatorial dithiolene sulfur donors, from which the low-energy LMCT transitions arise. Dithiolene covalency contributions to the electroactive C, or redox, orbital can be directly probed via the relative oscillator strengths of the / —> ixy and /fp —> (/", transitions (see above). These three wave functions may be expanded in terms of Mo- and dithiolene sulfur-based functions ... 

The physics of condensed phases is commonly formulated as of infinite extent. However, solid and liquid objects in the laboratory are of finite size and terminate discontinuously in a surface (in vacuum) or an interface, under all other conditions. Atoms or molecules at the surface or interface of the condensed object find themselves in a completely different environment, compared to those in the interior of the body. They are less confined in at least one direction, which means that the wave function looks different in this direction - it is less classical. It is implied that surface or interfacial species show more quantum-mechanical behaviour, compared to the bulk. This is the basic reason for the special properties of surfaces and the origin of all interfacial phenomena. Surface chemistry should therefore be formulated strictly in terms of quantum theory, but this has never been attempted. In its present state of development it still is an empirical science, although many physico-chemical concepts are introduced to rationalize the behaviour of interfaces. 

This property is well-known for prolate spheroidal wave functions (the basic SVD functions in Fresnel or far-field approximation [2,7]), but, as it was shown [8], the double-orthogonality property is quite common for different physical Green fimctions. This property can be used for simple estimation of the noise (or stray light) impact on resolution enhancement [9]. 

The mechanisms of chemical reactions and the reactivity properties of the molecules involved started to be elucidated through the analysis of the wave functions defining the quantum state of molecular systems,5-7 for instance the Fukui s frontier molecular orbital (FMO) theory8,9 has been very successful in rationalizing organic reactions basically through the analysis of the in- and out-of-phase overlap between the highest occupied molecular orbital (HOMO) of the nucleophile and the lowest unoccupied molecular... 

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