Wannier equation

Here the Hamiltonian Ho describes the evolution of the polarization in an isolated IQW in the absence of electron-hole populations and corresponds to the Wannier equation (39). The resonant Wannier exciton wave function in momentum space Tk(q) is its eigenfunction with the eigenvalue Ew(k). The Hamiltonian H describes the nonlinear many-particle corrections. It is proportional... 

In tightly bound (Frenkel) excitons, the observed peaks do not respond to the hy-drogenic equation (4.39), because the excitation is localized in the close proximity of a single atom. Thus, the exciton radius is comparable to the interatomic spacing and, consequently, we cannot consider a continuous medium with a relative dielectric constant as we did in the case of Mott-Wannier excitons. 

Ashley, Moxom and Laricchia (1996) measured the positron impact-ionization cross section in helium and found that its energy dependence up to 10 eV beyond the threshold was quite accurately represented by a power law, as in equation (5.8), but with the exponent having the value 2.27 rather than Klar s value of 2.651. This discrepancy prompted Ihra et al. (1997) to extend the Wannier theory to energies slightly above the ionization threshold using hidden crossing theory. They derived a modified threshold law of the form... 

One has a separate operator for each spin orbital so the equation has to be solved several times and (controllable) problems of orthogonality have to be dealt with. Unlike the LSD energy, the LSD-SIC energy is not invariant to a unitary transformation among the occupied orbitals. For example, in a solid the SIC is zero for Bloch functions but not for Wannier functions. This clearly leads to arbitrariness in the application of LSD-SIC in situations involving wavefunctions which are delocalized by symmetry—a topic discussed further below. 

Here (pj(k, r) is the kth CO of the band j. Finally in equation (26) is a doubly excited Slater determinant in which was excited from the filled Wannier functions w,/f, wjjt, to the originally unfilled ones w, Wbj, . 

Projection of equation (29) on the space of doubly excited Slater determinants (expressed by Wannier functions) gives... 

We considered two approximate treatments of the DC field, i.e., one where we only included Z of Fig. 5 and equations (48)-(50), and another where the full sawtooth curve z was included. Some representative results are shown in Figs 7 and 8. Since the Wannier functions can be ascribed to individual unit cells, we show in Fig. 7 the number of electrons (relative to the number, 8, for the undistorted system) of each unit cell in the case that the field operator has the symmetry of z of Fig. 5. Not surprisingly, the electrons do show an asymmetric distribution, although the flow from one end of the Born von Karman zone to the other is small. The number of electrons inside the muffin-tin spheres also gives information on the electron redistributions. Thus, for e-E = 0.0002 hartree these numbers are 3.2403 and 3.2413 for the two carbon atoms per unit cell for the operator zi of Fig. 5, and 3.2217 and 3.2575 for the operator z- Here we also see a larger effect for z than for z However, for the z all atomic spheres show the same numbers, so that the charge redistribution of Fig. 6 is restricted to the interstitial region. 

E/N range." Peihaps the simplest approach (proposed by Wannier) is to add thermal and field energies, combining Equations 2.7 and 2.10 into... 

Eqn (6.19) is the Schrodinger equation for describing the Mott-Wannier ex-citon wavefunctions and energies. In more sophisticated treatments it is usually known as the Bethe-Salpeter equation. In the following two sections the solutions of this equation will be described. 

Applying an optical lattice provides a periodic structure for the polar molecules described by the Hamiltonian of Equation 12.1, with yjj) given by Equation 12.32. In the limit of a deep lattice, a standard expansion of the field operators i] (r) = w(r - Ri)b] in the second-quantized expression of Equation 12.1 in terms of lowest-band Wannier functions w(r) and particle creation operators b] [107] leads to the realization of the Hubbard model of Equation 12.9, characterized by strong nearest-neighbor interactions [85]. We notice that the particles are treated as hardcore because of the constraint Rq. The interaction parameters Uy and Vyk in Equation 12.9derive from theeffective interaction V ( ri ), and in the limit of well-localized Wannier functions reduce to... 

To solve the KS equations in 0 N) operations the localized, Wannier-like states are used as they are constrained by their own localization region. Each atom / is assigned a number of states equal to mt(Z ° /2 + 1) so that, if doubly occupied, they can contain at least one excess electron (they can also become empty during the minimization of the energy functional). These states are confined to a sphere of radius (common to all states) centered at nuclei I. Irrespective of whether the 0(N) functional or the standard diagonalization is used, an outer self-consistency iteration is required, in which the density matrix is updated. Even when the code is strictly 0 N), the CPU time may increase faster if the number of iterations required to achieve the solution increases with N. In fact, it is a common experience that the required number of self-consistency iterations increases with the size of the system. This is mainly because of the charge-sloshing effect, in which small displacements... 

The use of localized orbitals for the cluster calculation is an efficient approach for defective crystals. To connect the perfect crystal localized orbitals and molecular cluster one-electron states the molecular cluster having the shape of a superceU was considered [699]. Such a cluster differs from the cyclic cluster by the absence of PBC introduction for the one-electron states. Evidently, the molecular cluster chosen is neutral and stoichiometric but its point symmetry can be lower than that of the cyclic cluster. Let the locaUzed orthogonal crystalline orbitals (Wannier functions) be defined for the infinite crystal composed of snpercells. The corresponding BZ is L-times reduced (the snpercell is supposed to consist of L primitive unit cells). The Wannier functions W r — Ai) are now introdnced for the supercells with the translation vectors and satisfy the following equation ... 

This equation is written for the Wannier function of the ith superceU. Due to (10.9) n(0) can be considered as the average energy of the nth band (in the superceU classification of crystalline states). In fact, (10.10) corresponds to the molecular cluster having the shape of the supercell. The Bloch functions of the macrocrystal can be constructed from the cluster one-electron states ... 

If this expansion is substituted into equation (5.10), the 1 = 0 term (Ro = 0) selected from the sums over / on both sides, and use made of definition (5.5) of the Wannier functions, one obtains ... 

The form of phase AJ, even if it has to satisfy equation (5.18), is still undefined in the half of the BZ. It can be selected such that the Wannier functions should possibly be optimally localized. This requirement is not unique, but a physically meaningful condition is that w, should be maximal in a certain region of the crystal (for instance, in the reference cell and in a few of its neighbors). When the Bloch function is expressed in the form... 

In the cases of completely filled or empty bands (semiconductors or insulators), the numerical work can be reduced if one does not substitute the Bloch functions in integral (5.37) in their LCAO form [see equations (5.20) and (5.21)], but replaces the AOs by Wannier fiinctions (see Section 5.1). In this way matrix elements (5.37) retain their k-depen-dence (which is necessary for the correct treatment of the translational symmetry), but this procedure does not involve any special difficulties. By writing the Bloch functions in the form< )... 

The Bloch functions obtained from the HF CO calculation must be transformed to optimally localized Wannier functions (see Section 5.1) and the matrix elements of (equation 8.20) computed at selected points. 

Singlet and triplet exciton bands are distinguished by substituting in relation (8.20) not Wannier spin orbitals but only Wannier functions with spatial dependence. The only difference compared to equation (8.20) will be that, before its second term (the exchange term), one has to insert a factor 2Sm, where <5m = 1 in the singlet case and = 0 in the triplet case. 

Prove the expressions for the various terms in the energy of a Frenkel exciton represented by a Slater determinant of Wannier functions, given in Eq. (5.65). Derive the equations of motion for the electronic and ionic degrees of freedom from the Car-Parrinello lagrangian, Eq. (5.98). 

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