Wavefunction localized electron

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

This expression is just the one which obtains for the Hartree product wave-function. The difference between this Hartree wavefunction and the Fock wavefunction of Eq. (1) is the absence of the antisymmetrizer j4 in that equation. This means that in the Hartree wavefunction each electron can be identified with a specific molecular orbital, whereas in the Fock wavefunction all electrons make use of all orbitals. The Hartree wavefunction is of course not a proper quantum mechanical wavefunction, since it is not antisymmetric in the electrons. Moreover, for the Fock wavefunction, it is in general not possible to reduce the interorbital exchange energy to zero. But the localized molecular orbitals, as defined here, represent that set of molecular orbitals for which the energy expression comes closest to the Hartree form, i.e. they come closest to being identifiable with electrons which are not exchanged among different orbitals. 

This effect can be illustrated by Fig. 14.2. The effective range of local modification of the sample states is determined by the effective lateral dimension 4ff of the tip wavefunction, which also determines the lateral resolution. In analogy with the analytic result for the hydrogen molecular ion problem, the local modification makes the amplitude of the sample wavefunction increase by a factor exp( — Vi) 1.213, which is equivalent to inducing a localized state of radius r 4tf/2 superimposed on the unperturbed state of the solid surface. The local density of that state is about (4/e — 1) 0.47 times the local electron density of the original stale in the middle of the gap. This superimposed local state cannot be formed by Bloch states with the same energy eigenvalue. Because of dispersion (that is, the finite value of dEldk and... 

Analysis of the radial pair distribution function for the electron centroid and solvent center-of-mass computed at different densities reveals some very interesting features. At high densities, the essentially localized electron is surrounded by the solvent resembling the solvation of a classical anion such as Cr or Br. At low densities, however, the electron is sufficiently extended (delocalized) such that its wavefunction tunnels through several neighboring water or ammonia molecules (Figure 16-9). 

Two different types of recombination are illustrated in Fig. 8.2. In the first example the electron and hole have almost complete spatial overlap and the recombination hfetime is of order 10 s. An exciton or a transition between an extended and a localized state are examples. In the second example, the transition is between localized electrons and holes that are spatially separated by a distance R which is larger than the localization radii R and R. The wavefunction envelope for r P R is approximately exp —r/R ) where r is the distance from the localized state. The overlap integral is (Thomas, Hopfield and Augustiniak 1965),... 

Energy is not the only property that is so determined by the electron density fragment pc. Since the (non-degenerate, ground state) local electron density pc(r) in any standard domain c fully determines the complete density pit), which in turn fully determines the molecular wavefunction P (up to a phase factor), all molecular properties P which can be expressed as expectation values of spin-free operators defined by the ground state wavefunction P are also determined by the local electron density pc(r) in the standard domain c. Consequently, any such property P is also a unique functional of the local electron density pc(r) within the standard domain c ... 

Fig. 16.6 A schematic representation of the potential surface for the electron between two centers (D—donor, A—acceptor). Shown also are the relevant diabatic electronic wavefunctions, localized on each center. The lower diagram corresponds to a stable nuclear configuration and the upper one— to a nuclear fluctuation that brings the system into the transition state where the diabatic electronic energies are equal. The electronic transition probability depends on the overlap between the electronic wavefunctions i/zp and tn this transition state.

The actinide element series, like the lanthanide series, is characterized by the filling of an f-electron shell. The chemical and physical properties, however, are quite different between these two series of f-electron elements, especially in the first half of the series. The differences are mainly due to the different radial extension of the 4f- and 5f-electron wavefunctions. For the rare-earth ions, even in metallic systems, the 4f electrons are spatially well localized near the ion sites. Photoemission spectra of the f electrons in lanthanide elements and compounds always show "final state multiplet" structure (3), spectra that result from partially filled shells of localized electrons. In contrast, the 5f electrons are not so well localized. They experience a smaller coulomb correlation interaction than the 4f electrons in the rare earths and stronger hybridization with the 6d- and 7s-derived conduction bands. The 5f s thus... 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

Let us mention the following general rule optical transitions between states which may be described by wavefunctions localized over distances of the order of the lattice constant are changed relatively little by disorder. We shall see several examples of this rule. This may also explain why the optical properties of amorphous insulators often differ little from those of crystalline insulators. On the other hand, if the bonds are very delocalized (i.e. valence electron wavefunctions spread over long distances) the changes are also small. Indeed, the optical properties of liquid metals differ little from those of crystalline metals. Semiconductors with covalent bonding are the materials most sensitive to disorder. Their transport properties are more drastically changed than their optical properties. 

Several approaches are available in the literature to generate and evaluate Hamiltonian matrix elements with wavefunctions of charge-localized, diabatic states. They differ in the level of theory used in the calculation and in the way localized electronic structures are created [15, 25, 26, 29-31]. When wavefunction-based quantum-chemical methods are employed, the framework of the generalized Mulliken-Hush method (GMH) [29, 32-34], is particularly successful. So far, it has been used in conjunction with accurate electronic structure methods for small and medium sized systems [35-37]. As an alternative to GMH and other derived methods [38, 39], additional methods have been explored for their applicability in larger systems such as constrained density functional method (CDFT) [25, 37, 40, 41], and fragmentation approaches [42-47], which also include the frozen density embedding (FDE) method [48, 49]. 

The usual approximations for this expression involve the Bom-Oppenheimer approximation and the assumption of localized electronic and nuclear wavefunctions. With this, the matrix element in Eq. (4) can be expressed as ... 

Local electron-correlation methods are ab-initio wavefunction-based electronic-structure methods that exploit the short-range nature of dynamic correlation effects and in this way allow linear scaling 0 N) in the electron-correlation calculations [128,129,131-135] to be attained. 0 N) methods are applied to the treatment of extended molecular systems at a very high level of accuracy and rehabihty as CPU time, memory and disk requirements scale hneaily with increasing molecular size N. 

The previous section points out that the Born-Oppenheimer approximation is useful in that electronic parts of wavefunctions can be separated from nuclear parts of wave-functions. However, it does not assist us in determining what the electronic wavefunctions are. Electrons in molecules are described approximately with orbitals just like electrons in atoms are described by orbitals. We have seen how quantum mechanics treats atomic orbitals. How does quantum mechanics treat molecular orbitals Molecular orbital theory is the most popular way to describe electrons in molecules. Rather than being localized on individual atoms, an electron in a molecule has a wave-function that extends over the entire molecule. There are several mathematical procedures for describing molecular orbitals, one of which we consider in this section. (Another perspective on molecular orbitals, called valence bond theory, will be discussed in Chapter 13. Valence bond theory focuses on electrons in the valence shell.)... 

Positrons diffusing through matter can be captured in special trapping sites. As shown in early studies, these trapping centres are crystal imperfections, such as vacancies and dislocations. The wavefunction of a positron captured in such a defect is localised until it annihilates with an electron of its immediate surroundings into y-rays. Since the local electron density and the electron momentum distribution are modified with respect to the defect-free crystal, the annihilation radiation can be utilised to obtain information on the localisation site. The different positron techniques are based on analysing the annihilation radiation. The principles of the basic positron methods are illustrated in Figure 4.27 [84]. 

Localized Orbital Methods for Wavefunction-based Electron Correlation Calculations... 

The perfect-pairing formula (7.3.7) has been widely employed in qualitative discussions of the interactions determining the shape and stability of polyatomic molecules and in the interpretation of empirical additivity rules etc., which apply in many instances and appear to support the validity of a wavefunction representing a single well-defined set of localized electron pair bonds. It must be remembered, however, that the derivation rests upon an orthogonality assumption that intro-... 

Since the pioneering cluster calculation on the KNiFs solid of Shulman and Sugano [6] there has been a wide variety of proposals of procedures to handle relatively localized electronic states of a solid with a molecule-like Hamiltonian that includes the relevant solid host effects, depending on the type of solids and on methodological flavors (Green s functions, wavefunctions, density functional, etc.). A recent summary of practical methods can be found in Huang and Carter [7]. Here we describe our choice of embedded-cluster method, particularly useful in ionic materials. 

Despite these many encouraging developments, it is appropriate to say that when it comes to lanthanides and actinides modern electronic stmcture theory currently can accomplish many things, but certainly not all. It is also clear that this book can only provide a snapshot of the current state of affairs. A number of promising computational approaches, e.g., local electron correlation schemes or FI 2-dependent wavefunctions, are currently developed and already successfiiUy applied to non-f-element systems. They will during the next years most... 

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