Atomic wavefunctions, effect overlapping

The second insight that can be obtained from the electronic band structure is the mobility of charge carriers, which is related to the width of the conduction and valence bands. For Si the bands are rather broad, spanning more than 10 eV. This is a direct consequence of the extensive overlap of the sp orbitals on neighboring atoms. More overlap between atomic wavefunctions results in broader bands and easier transport of free charge carriers through the material. This can be quantified via the curvature of the individual bands, which is directly related to the effective mass and mobility of the charge carriers ... 

To understand the oscillatory dependence of AE on d, it is necessary to look more closely at the interaction energy AW because, as (8.66) shows, AE is the sum of the two single-atom chemisorption energies (which are independent of d) plus AW. Hence, any effect of d on AE must arise due to AW. Alternatively, one may consider the situation in terms of the adatom wave-functions, which, as they spread out from each adatom, interfere in either a constructive or destructive fashion, thus creating oscillations in the electron density that are mirrored in the interaction. Since the wavefunctions are in or out of phase, depending on d, AE itself becomes a function of d. As d increases, the overlap of the wavefunctions decreases, and AE tends towards A eP. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

Of course, the SC orbitals associated with a particular fragment are completely free to overlap with one another, and with all of the SC orbitals of the other fragment(s). In particular, the < > are free to extend spatially over the nuclei of the B molecule, using the tails of the atomic functions %, thereby taking into account also effects connected to charge transfer interactions. Various calculations have confirmed that such a SC wavefunction is able to describe in a compact way the ground state of a weakly-bound system, without biasing the results with BSSE. 

In the electronic Hamiltonian t +i, is the transfer integral, i.e. the re-electron wavefunction overlap between nearest neighbour sites in the polymer chain, and is equivalent to the parameter /3 in Equation (4.20), and c 1+,s. and clhS are creation and annihilation operators that create an electron of spin s ( 1/2) on the carbon atom at site n-f 1 and destroy an electron of spin, s at the carbon atom on site n, i.e. in effect transfer an electron between adjacent carbon atoms in the polymer chain. The elastic term is just the energy of a spring of force constant k extended by an amount ( +1— u ), where the u are the displacements along the chain axis of the carbon atoms from their positions in the equal bond length structure, as indicated in Fig. 9.8(b). The extent of the overlap of 7i-electron wavefunction will depend on the separation of nearest neighbour carbon atoms and is approximated by ... 

The model is valid when the contribution of the overlap electron density to the cross section is neghgible and the effects of the anisotropy of the molecular ion potential on the outgoing electron wavefunction are small. Both of these conditions are most likely to hold when the photoelectron KE is high. At low photon energies, molecular effects on ionization cross sections are well documented, but even in these regions atomic effects are evident and the Gelius model provides a good basis for molecular cross section interpretation. 

The model of the photophysics we have advocated does not take into account spin-orbit coupling effects associated with silver as a heavy atom that might affect phosphorescent and nonradiative decay rates for the triplet state. The theoretical justification for this is that heavy atom effects are quite short range since they require wavefunction overlap. Effects of the silver are in any case likely to be much smaller than those of the Pt atom embedded in the porphyrin. Experimentally, we can rule out the importance of these effects since we do not observe phosphorescence enhancement on top of uniform evaporated silver films nor on films that become essentially continuous as for the thickest films in Figure 19.4. 

The physical picture on the Z -dependence of IAPbI can be qualitatively explained as follows. As described in eq. (4), the bond overlap population Pb is obtained from the sum of all overlap populations between the X and F or H atomic orbitals. Accordingly, APb depends on the relativistic variation of the radial wavefunctions of the atomic orbitals for each molecule. Because the relativistic effects on the atomic orbitals of fluorine or hydrogen atom are negligibly small, the relativistic effects on atomic orbitals of the atom X play a... 

The potential F(l, 2) is the total interaction energy between the two atoms, and includes contributions from electron-nuclear attraction terms as well as from electron-electron repulsion effects. As a result Xfls, Is ) is negative, and is primarily responsible for the stability of the H-H bond. Central to this description is the fact that the wavefunctions of the participating atoms, and overlap. The magnitude of K ls , Is,), and consequently the strength of the covalent bond, is determined by the degree of non-orthogonality between the two orbitals. 

If wavefunctions are calculated using semiempirical methods that assume zero overlap between atomic orbitals (AOs) on different atomic centers, then a MuUiken population analysis [84, 85] can be applied to the calculated TD to yield transition monopoles distributed over each atomic center. Such an approach has proven to be effective [82] an advantage being that the interaction between distributed monopoles can be computed considerably faster then that between TDCs. At the same time, the basic topology of the donor-acceptor... 

In a bulk semiconductor, photoexcitation generates electron-hole pairs which are weakly bounded by Coulomb interaction (called an exciton). Usually one can observe the absorption band of an exciton only at low temperature since the thermal energy at room temperature is large enough to break up the exciton. When the exciton is confined in an energy potential, the dissociation probability reduces and the overlap of the electron and hole wavefunction increases, which is manifested by a sharper absorption band observable at room temperature. This potential can be due to either a deformation in the lattice caused by an impurity atom or, in the present case, the surface boundary of a nanocluster. The confinement of an exciton by an impurity potential (called bound exciton) is well known in the semiconductor literature [16]. There is considerable similarity in the basic physics between confinement by an impurity potential and confinement by physical dimension. The confinement effects on the absorption cross section of a nanocluster are discussed in Section II. 

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