Wannier function localization properties

The idea of distributed dipole moments has also been transferred to the dynamic domain and we shall discuss recent work from our laboratory in this section in more detail. With the help of maximally localized Wannier functions local dipoles and charges on atoms can be derived. The Wannier functions are obtained by Boys localization scheme [217]. Thus, Wannier orbitals [218] are the condensed phase analogs of localized molecular orbitals known from quantum chemistry. Access to the electronic structure during a CPMD simulation allows the calculation of electronic properties. Through an appropriate unitary transformation U of the canonical Kohn-Sham orbitals maximally localized Wannier functions (MLWFs)... 

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... 

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... 

Direct d3mamics calculations with the BLYP exchange-correlation functional and electric properties computed from localized Wannier functions predicted an increase of the dipole moment from an equilibrium value of 1.87 D in the gas to an average value of 2.95 D in the liquid. 

Molina et al. [8] computed from first principles the dipole polarizabilities of a series of ions (e.g. Li+, Na+, Mg +, Ca " ") in aqueous solutions. The technique they employed is based on the linear response of the maximally localized Wannier functions to an externally applied electric field. They found that proton transfer leads to instantaneous switch of the molecular polarizability. Sin and Yang [9] employed DFT to compute the first hyperpolarizability and other properties (e.g. excitation energies) of 20 silalluorenes and spirobisilafluorenes. They found that the nonlinearity increases with (increasing) number of branches. This effect has been attributed to a cooperative enhancement of the charge-transfer. 

As the symmetrical orthogonalization procedure (3.122) leaves unchanged the reality and symmetry properties of the functions, the set of orthonormalized functions Wf r) satisfy all the requirements to the localized Wannier functions (reality, symmetry requirements and orthonormality) and (in the case when this set of functions is unique) has to coincide with the latter ... 

The character of the locahzation of Wannier functions depends on the analytical properties of Bloch states (as a function of the wavevector) that are essentially determined by the nature of the system under consideration. One can arbitrary change only the form of an unitary transformation of Bloch functions. It is just this arbitrariness that is used in the variational approach [42] to assure the best localization of Wannier functions. The accuracy of the Wannier functions obtained by the proposed method is determined solely by the accuracy of the Bloch functions and the size of the supercell used. As the calculations have shown, the proposed method is reUable and useful in the problem of generation of the locaUzed Wannier functions. In the two examples... 

Specialized orbitals tailored for analysis of phenomena such as bonding in molecular clusters and electron excitations are obtained by maximization of suitably chosen functionals. The so-called pseudo-Wannier orbitals are produced by a procedure that maximizes similarities between one-electron wavefunc-tions localized within analogous units of a given molecular cluster. These orbitals reveal terminal-group effects in linear polymers and provide systematic schemes for partitioning of the total energy and one-electron properties of finite clusters. 

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