Maximally localized Wannier functions

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... 

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). 

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. 

I. Souza, N. Marzari, and D. Vanderbilt (2002) Maximally localized Wannier functions for entangled energy bands. Phys. Rev. B 65, 035109... 

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)... 

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. 

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. 

The calculation of NMR parameter has been studied extensively see [3, 73] for general overviews. In 2001, Sebastian and Parrinello implemented the NMR chemical shift calculation in the plane wave AIMD code CPMD [74]. From this implementation it was possible to treat extended systems within periodic boundary conditions, i.e., the method was applicable to crystalline and amorphous insulators as well as to liquids. The problem of the position operator was solved by the use of maximally localized Wannier functions. Several benchmark calculations showed good agreement with experimental values. 

Many schemes were adapted to analyze the wavefunction (electronic structure) in AIMD simulations. The most important ones are the Wannier analysis based on maximally localized Wannier functions (MFWF) [83], the electron localization function (EFF)[84], the Fukui function [85], and the nucleus-independent chemical shift maps [74]. 

Computation of Maximally Localized Wannier Functions Using a Simultaneous Diagona-lization Algorithm. 

Me) intermediate, maximally localized Wannier functions were used to analyze the oxidation state of Pt. These localized bonding functions indicate that the Pt metal center remains Pt(II) throughout the reaction pathway without significant oxidation, which is in line with a highly electrophilic CH activation process. 

The exploitation of localized orbitals for dispersion energy calculations has already been proposed since the early works on local correlation methods [41 5]. In classical and semiclassical models most often the atoms are selected as force centers only a few works exploit the advantages related to the use of two-center localized orbitals and lone pairs. A notable exception is the recent work of Silvestrelh and coworkers [46-50], who adapted the Tkatchenko-Scheffler model [16] for maximally localized Wannier functions (MLWF), which are essentially Boys localized orbitals for solids. It is worthwhile to mention that one of the very first use of the bond polarizabilities as interacting units for the description of London dispersion forces has been suggested as early as in 1969 by Claverie and Rein [51] see also [52],... 

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... 

Marzari, N. and Vanderbilt, D. (1997) Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B Condens. Matter, 56, 12847. 

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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