Envelope wave function

Presented experimental data reveal that for CdSe/ZnS quantum dots with two ZnS monolayers values follow a monotonous function drastically decaying with the QD core diameter. From the physico-chemical point of view, we conjecture that upon interaction of P with QD surface, the electron wave function may be locally modified (via inductive and/or mesomeric effects [9]) forming a surface local state capable to trap the electron of the photogenerated exciton (Fig. 2A). Thus, we will consider the behaviour of the electron wave function at the interface to the functional pyridyl group of the attached porphyrin. The single-carrier envelope wave functions y/a in a spherical core/shell QD are determined by the Schrodinger equation... 

The Hamiltonian of the electron-photon interaction will be used in a very simplified form taking into account only the simplest band structure of a semiconductor with parabolic electron and hole bands without complications related to heavy and light holes, spin-orbit splitted hole band or with the Dirac model of the band structure in the case of small band gap semiconductors. In the case of simple parabolic band after their size quantization in a spherical symmetry quantum dots the electrons and holes are characterized by envelope wave functions with the quantum numbers I, n, m. An essential simplification of the future calculations is the fact that in the selected simple model the band-to-band transitions under the influence of the electron-photon interaction Hamiltonian take place with the creation of an e-h pair with exactly the same quantum numbers for electron and for hole as follows e l,n,m), h l,n,m). ... 

The solutions of the Schrodinger equation show how j/ is distributed in the space around the nucleus of the hydrogen atom. The solutions for v / are characterized by the values of three quantum numbers and every allowed set of values for the quantum numbers, together with the associated wave function, strictly defines that space which is termed an atomic orbital. Other representations are used for atomic orbitals, such as the boundary surface and orbital envelopes described later in the chapter. 

Table 5 gives the errors for a DFT method, four hybrid DFT methods, and AMI. Although hybrid DFT is very affordable, it lacks the accuracy of multicoefficient semi-empirical methods based on explicitly correlated wave functions. Nevertheless the mPWlPW91/MG3S and AMI methods have performance/cost characteristics that put them near the envelope of best performance in Figure 1. AMI is valuable for larger systems where the other methods in the figure are not affordable. 

Quantum films represent a beautiful physical example of the particle in the box problem. The energy levels of the conduction band electrons in the electron well (MQW) can be easily calculated using the envelope function or the effective mass approximation [6]. The electron wave function is then [29-31]... 

We adopt a simplified microscopic quantum-mechanical model of a 2D Wannier-Mott exciton, in which the polarization (eqn C.3) can be taken to vanish for L > Lw/2 and inside the well to be given by the product of the Is-wave function of the relative motion of the electron and hole at the origin, with the lowest subband envelope functions for the electron and hole in the approximation of... 

Whereas the Frenkel tight-binding description is valid for excitons localized to atomic dimensions, the Wannier approximation describes excitons delocalized over many lattice distances. The hydrogen-like wave functions associated with these states serve as envelope functions describing average probability in spatial regions of lattice-constant dimensions. 

Each unsealed 0 at a specific R, in the case of diatomic molecules, gives rise to a scaled energy curve the envelope of a family of such curves can be taken to be the optimal intemuclear potential energy. Exact wave functions, minimal STO-SCF wave functions optmized with respect to coefficients and orbital exponents, and Hartree-Fock wave functions yield the optimal intemuclear potential-energy curves which satisfy the virial. At the minimum in the optimal PE-curve (—V /2 R ) will be unity. McLean uses the Dunham analysis (3b) to determine the spectroscopic constants through the potential curve... 

Hyperfine splitting due to interaction with ligand nuclei with 7 > 0 reflects the extent of spin delocalization onto neighboring atoms and can be used to characterize the types and numbers of such nuclei. In cases where these couplings are too small to be resolved in the EPR spectra, electron nuclear double resonance (ENDOR) or electron spin echo envelope modulation (ESEEM) can be used to measure the couplings as discussed in Chapter 2.3. Modern calculational tools are approaching the capabilities required to calculate g and A values from electronic wave functions. However, much of the spectroscopy that has been performed to date has used empirical correlations to interpret g and A values. 

Two methods aie used to represent the positive and negative values of the wave functions in atomic orbital envelope diagrams. Both are shown in Figure 2.7 for the 2p orbital. 

We see that the wave function ip2 resembles the function (multiplied with the transition dipole moment) but is weighted with the Fourier-tranform I t T) which determines the norm of the wave function. The integration is performed over a smooth envelope function f t,T) and an oscillating phase factor. According to the stationary phase approximation [14], the modulus of the integral will be largest at the stationary points Rg for which the exponent in Eq.(6) is zero ... 

---