Function Wannier

The protein field might be very important to the chemistry of the active site of this and other enzymes [11]. The effect of the environment was estimated by comparing the electronic structure of the complexes in vacuum with those in the presence of the protein. The Wannier functions [37], the centers (WFC) of which represent... 

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... 

Generalized Wannier Functions for Composite Energy Bands. 

III.B. In order to analyze the wavefunction in a chemically more intuitive way, it is useful to localize it. In the framework of AIMD this is, for example, done by calculating the maximally localized Wannier functions (MLWF) and the corresponding expectation values of the position operator for a MLWF basis the so-called maximally localized Wannier centers (MLWCs), see Fig. 1 (67-72). With the help of the MLWC it is possible to compute molecular dipole moments (72-82). Furthermore, it is possible with the MLWC to obtain molecular properties, e.g., IR spectra (75,76,82-85). 

An analysis of the Wannier functions in CPMD simulations of one dimethyl sulfoxide (DMSO) molecule dissolved in water was carried out by us in 2004 in order to gain more insight into the unusual properties of the DMSO-water mixture (72). In this special case, we have utilized MLWCs to calculate molecular dipole moments of the DMSO molecule in gas phase and aqueous solution. Comparing those two a large increase of the local dipole... 

For the description of a solution of alanine in water two models were compared and combined with one another (79), namely the continuum model approach and the cluster ansatz approach (148,149). In the cluster approach snapshots along a trajectory are harvested and subsequent quantum chemical analysis is carried out. In order to learn more about the structure and the effects of the solvent shell, the molecular dipole moments were computed. To harvest a trajectory and for comparison AIMD (here CPMD) simulations were carried out (79). The calculations contained one alanine molecule dissolved in 60 water molecules. The average dipole moments for alanine and water were derived by means of maximally localized Wannier functions (MLWF) (67-72). For the water molecules different solvent shells were selected according to the three radial pair distributions between water and the functional groups. An overview about the findings is given in Tables II and III. 

In deriving these formulae it is assumed that the wave functions fa are orthogonal to one another, as for instance are Wannier functions. If they are atomic functions falling off as e ar, a formula such as (9) must be modified by a term in the denominator to take account of the non-orthogonality. Reviews of the appropriate formulae are given in textbooks see e.g. Callaway (1964) and Wohlfarth (1953), who considered a linear chain of hydrogen atoms. 

Figure 1. Copper/oxygen layer of Superconducting cuprates. Black and white circles represent copper and oxygen atoms respectively. A pair of E-basis Wannier functions, which are a linear combination of Cu and O atomic orbitals, transforming as (x, y) are localised on every lattice point at the centre of the unit cell.

While the electronic structure calculations addressed in the preceding Section could in principle be used to construct the potential surfaces that are a prerequisite for dynamical calculations, such a procedure is in practice out of reach for large, extended systems like polymer junctions. At most, semiempirical calculations can be carried out as a function of selected relevant coordinates, see, e.g., the recent analysis of Ref. [44]. To proceed, we therefore resort to a different strategy, by constructing a suitably parametrized electron-phonon Hamiltonian model. This electron-phonon Hamiltonian underlies the two- and three-state diabatic models that are employed below (Secs. 4 and 5). The key ingredients are a lattice model formulated in the basis of localized Wannier functions and localized phonon modes (Sec. 3.1) and the construction of an associated diabatic Hamiltonian in a normal-mode representation (Sec. 3.2) [61]. 

This approach was developed originally as an approximate method, if the wave functions of isolated atoms are taken as a basis wave functions basis functions is possible. In principle, the TB model is reasonable only when local states can be orthogonalized. The method is useful to calculate the conductance of complex quantum systems in combination with ab initio methods. It is particular important to describe small molecules, when the atomic orbitals form the basis. 

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. 

Calculation of Wannier Functions.—In most further applications of the wave-functions obtained in HF CO studies (calculation of excitonic effects, CDW s, impurity and vacancy levels, etc.) the use of Wannier functions22 instead of the original Bloch functions seems to be very promising.23 The connection between the two basis sets is given by the transformation... 

The interesting point is that if we know the exact eigenstates of the crystal, we can substitute them into Eq. (3-38) to obtain localized Wannier functions s,> if substituted into Eq. (3-37), the localized Wannier functions will give exact eigenstates of the crystal. 

Although Wannier functions were conceived long ago (Wannier, 1937), they have not proved very useful except for formal analysis. They were first constructed for silicon by Callaway and Hughes (1967), but not in a form that has been helpful. Recently, Kane and Kane (1978) provided a set of Wannier functions for silicon that describes the valence bands very accurately and may provide a basis for accurate calculation of other properties. A similar approach has been made by Tejedor and Verges (1978). 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

A recent trend has been to use Wannier functions to calculate properties. Wannier functions are orthonormal localized functions spanning the same space as the eigenstates of a particular band or group of bands and are Fourier transforms of the Bloch eigenstates. For one band, i, Wannier functions, w, are given by... 

N is the number of bands considered in forming the Wannier function. 

Spontaneous polarisation, that is polarisation in the absence of an electric field, has been calculated using both a Wannier function approach and a Berry s phase approach. Berry s phase involves an adiabatic change around a closed loop which results in a change of phase without change in energy. A recent paper by Ferretti et alP used the PAW method with ultrasoft pseudopotentials and Wannier functions to calculate the spontaneous polarisation of AIN in its wurtzite phase. 

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