Wannier expression

Although Equation 3.22 was derived as early as 1953, there was some doubt about its validity until the mid-1970s, when careful experiments were carried out on ion motion in various buffer gases, which showed the Wannier expression to be a good approximation to the real mean ion kinetic energy [35]. 

A maximally localized Wannier function analysis84-86 was performed to better analyze the bonding in our simulations. The maximally localized Wannier functions express the quantum wave function in terms of functions localized at centers, rather than as delocalized plane waves. The positions of these centers give us insight into the localization of charge during the... 

In Section 2 we briefly summarize the basic mathematical expressions of the LCAO Hartree-Fock crystal orbital method both in its closed-shell and DODS (different orbitals for different spin) forms and describe the difficulties encountered in evaluating lattice sums in configuration space. Various possibilities for calculating optimally localised Wannier functions are also presented. They can be efficiently used in the calculation of excited states and correlation effects discussed in Section 3. 

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

The first two terms do not break the translational symmetry and can fairly simply be incorporated into the calculations. The matrix elements can be calculated using the expression of the Wannier functions in terms of the Bloch functions, and subsequently performing the required integrals analytically in the interstitial region and numerically inside the spheres with expressions that are very similar to those we need for the other lattice-periodic parts of the potential (see, e.g.. Refs. [2,31]). 

Williamson et al. used maximally localized Wannier functions to express the LMO s [160]. The LMOs were truncated by setting the value of the orbital to zero outside the sphere containing 99.9% of the orbital s density. The transformation from basis functions to MOs was sidestepped by tabulating the orbitals on a 3-D grid and using a spline procedure for orbital evaluation. 

This matrix element is evaluated by expressing H in terms of the valence and conduction Wannier orbital operators. Retaining terms that keep within the exciton subspace H becomes... 

An equivalent representation of the electronic structure is provided by Wannier functions [2], which are connected to the Bloch orbitals via a unitary transformation. Denoting the Wannier functions of band n of cell R by Wn v — R), we express the transformation as follows ... 

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... 

Probably the most powerful technique for sampling the conformation space of real polymers is the transfer matrix approach in combination with the RIS approximation discussed in the previous chapter. The approach was pioneered by Flory (e.g., [13]), but its potential for generating configurational averages was recognized very early (cf. the reference to E. Montrol in the seminal paper by Kramers and Wannier [14]). In the transfer matrix approach the partition function of a polymer is expressed in terms of a product of matrices, whose elements in the simplest case, e.g., polyethylene, have the form... 

The incremental scheme based on the wavefunction HF method was extended to the calculation of valence-band energies when the electron-correlation is taken into account. In [176,177] an effective Hamiltonian for the N — l)-electron system was set up in terms of local matrix elements derived from multireference configuration-interaction (MRCI) calcnlations for finite clnsters. This allowed correlation corrections to a HF band strnctnre to be expressed and rehable results obtained for the valence-band structure of covalent semicondnctors. A related method based on an efiective Hamiltonian in locahzed Wannier-type orbitals has also been proposed and applied to polymers [178,179]. Later, the incremental scheme was used to estimate the relative energies of valence-band states and also yield absolnte positions of snch states [180]. 

The local MP2 electron-correlation method for nonconducting crystals [109] is an extension to crystalline solids of the local correlation MP2 method for molecules (see Sect. 5.1.5), starting from a local representation of the occupied and virtual HF subspaces. The localized HF crystalline orbitals of the occupied states are provided in the LCAO approximation by the CRYSTAL program [23] and based on a Boys localization criterion. The localization technique was considered in Sect. 3.3.3. The label im of the occupied localized Wannier functions (LWF) Wim = Wj(r — Rm) includes the type of LWF and translation vector Rm, indicating the primitive unit cell, in which the LWF is centered (m = 0 for the reference cell). The index i runs from 1 to A i, the number of filled electron bands used for the localization procedure the correlation calculation is restricted usually to valence bands LWFs. The latter are expressed as a linear combination of the Gaussian-type atomic orbitals (AOs) Xfiif Rn) = Xfin numbered by index = 1,..., M M is the number of AOs in the reference cell) and the cell n translation vector... 

This result also demonstrates clearly that the Wannier functions are not eigenfunctions of the Hamiltonian of a periodic crystal. Therefore in many applications, when one applies the Wannier functions to describe local perturbations, it is more advantageous to start from a mixed representation in which some matrix elements are expressed with the help of Wannier functions and others with the aid of Bloch functions. 

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... 

As an example we examine some results obtained for trans-polyacetylene. The Wannier function of this polymer can be expressed in the LCAO form... 

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). 

Note the first two terms are in the Bloch functions and the last term is expressed in Wannier functions. Also, there are no finite matrix elements of different R s because of translation symmetry. Thus one looks for the linear combination of exciton functions with respect to the index 6 which diagonalizes the Hamiltonian matrix... 

In addition to a thermal translational energy, ions travelling along a drift tube are given an additional energy by the applied electric field. An expression for the total mean ion kinetic energy was derived by Wannier [33,34] and takes the following form ... 

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