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. 

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]. 

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. 

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). 

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)... 

By looking at the maximum localized Wannier functions and using different population analysis methods. Buhl et al. suggested that there is a donor-acceptor interaction between the lone pairs of Q and the anti-bonding population analysis revealed that the amount of charge transferred from anions to cations ( 0.2 5e) fluctuated very little during the M D trajectory. 

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. 

General Method to obtain Well Localized Wannier Functions for Composite Energy Bands in Linear Combination of Atomic Orbital Periodic Calculations. 

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]. 

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... 

Though the set of the functions 14 (r) is not orthogonal, these functions are close to the accurate localized Wannier functions — a ). They can be... 

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 practical methods of post-HF calculations for solids were discussed during the workshop Local correlation methods from molecules to crystals [156]. Computational strategies were considered and new developments suggested in this area of research (the texts of invited speaker talks are published on an Internet site [156]). In particular, it was stated that the development of post-HF methods for crystals is essentially connected with the progress in the localized Wannier-function (LWF)... 

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... 

Table 9.21. Localization indices of localized Wannier functions in SrTiOs and SrZrOs...

Localized Wannier functions (LWFs) have been calculated for three upper valence bands in SrTiOs and SrZrOs, represented mainly by O 2p, Sr 4p, and O 2s atomic states (in the case of SrZrOs the last two bands overlap considerably). A total of 15 crystalline orbitals have been used to generate, correspondingly, 15 LWFs per primitive unit cell in both crystals under consideration, three oxygen atoms occupy the same Wyckoff positions, and four LWFs can be attributed to each oxygen atom. It was found by calculations with CRYSTAL03 code [23] that the centroids of four functions are positioned near the center of one oxygen (at distances of about 0.3 A). 

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

There are basically two possible approaches to computing. In the case of insulators or semiconductors with a noncrossing completely filled valence band and empty conduction band, one can always use not the delocalized Bloch functions but localized Wannier functions (for their construction see Section 5.1). On the other hand, in the case of metallic polymers like (SN) with a partially filled valence band, one cannot construct Wannier functions and so one must resort to other methods (see Section 5.3). 

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. 

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. 

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