Maximally localized Wannier functions: Theory and applications

In the paper

“Inspection of the resulting MLWFs confirms their atomic d -like character. They are also significantly more localized than the ones obtained by projection and the corresponding disentangled bands are somewhat better behaved, displaying less spurious oscillations in the hybridization regions.”

In the paper

“We show as red circles in Fig. 5 the results obtained by forcing the entire valence manifold to be preserved, leading to a set of eight projected bands that reproduce exactly the four valence bands, and follow quite closely the four low-lying conduction bands.”

In the paper

“Projection techniques can work very well, and an application of this approach to graphene is shown in Fig. 6, where the π/π ⋆ manifold is disentangled with great accuracy by a simple projection onto atomic p z orbitals, or the entire occupied manifold together with π/π ⋆ manifold is obtained by projection onto atomic p z and sp 2 orbitals (one every other atom, for the case of the sp 2 orbitals”

In the paper

“The red circles show the results obtained using the iterative scheme to extract an optimally-smooth fivedimensional manifold. The maximal 'global smoothness of connection' is achieved by keeping the five d -like states and excluding the s -like state. This happens because the smoothness criterion embodied by Eqs. (48) and (49) implies that the orbital character is preserved as much as possible”

In the paper

“Goedecker et al. (2002) have predicted - entirely from first-principles - the existence of a new fourfold-coordinated defect that is stable inside the Si lattice (see Fig. 18).”

What the sources say

• Not checked yetcites Goedecker et al., 2002

In the paper

“In their study of β -rhombohedral boron, Ogitsu et al. (2009) were able to identify and study the relation between two-center and three-center bonds and boron vacancies, identifying the most electron-deficient bonds as the most chemically active.”

What the sources say

• Not checked yetcites Ogitsu et al., 2009

In the paper

“For example, Cangiani et al. (2004) discussed the case of TiO 2 polytypes, where bonding MLWFs associated with the O 2 s/ 2 p valence manifold are seen to be similar in the rutile and anatase form, with the third polytype (brookite) an average between the two (Posternak et al. , 2006).”

What the sources say

• Not checked yetcites Cangiani et al., 2004
• Not checked yetcites Posternak et al., 2006

In the paper

“This defect had not been considered before, but displays by far the lowest formation energy - at the DFT level - among all native defects in silicon.”

What the sources say

• Not checked yetcites Goedecker et al., 2002

In the paper

“While in the classical ionic picture of perovskites the B-cation (here Ti) is completely ionized in a 4+ state, the covalent nature of the bond becomes clearly apparent here, with the MLWFs showing a clear hybridization in the form of mixed p -d orbitals.”

In the paper

“The MLWFs also become significantly more delocalized at that pressure, hinting at a metallization transition similar to the one that occurs in Si in going from the diamond to the β -tin structure.”

What the sources say

• Not checked yetcites Martonak et al., 2000
• Not checked yetcites Martonak et al., 2001
• Not checked yetcites Molteni et al., 2001

In the paper

“Inspection of the relevant 'defective' MLWFs reveals that their spreads actually remain very close to those typical of crystalline silicon, and that the WFCs remain equally shared between the atoms in a typical covalent arrangement.”

What the sources say

• Not checked yetcites Goedecker et al., 2002

In the paper

“For the case of the silicon vacancy, MLWFs have been studied for all the charge states by (Corsetti and Mostofi, 2011), validating the canonical Watkins model (Watkins and Messmer, 1974).”

What the sources say

• Not checked yetcites Corsetti and Mostofi, 2011
• Not checked yetcites Watkins and Messmer, 1974

In the paper

“For a given spatial cutoff on the tight-binding Hamiltonian constructed from these MLWFs, it is found that the former give a qualitatively much better description of the DFT band structure than the latter, despite the counter-intuitive result that the 'off-bond' MLWFs are slightly more localized.”

What the sources say

• Not checked yetcites Corsetti, 2012

In the paper

“In the Ziegler-Natta reaction we have an interconversion of a double carbon bond into a single bond, and a characteristic agostic interaction between the C-H bond and the activated metal center. Both become immediately visible once the WFCs are monitored, greatly aiding the interpretation of the complex chemical pathways.”

What the sources say

• Not checked yetcites Boero et al., 2000b

In the paper

“For example, Zhong et al. (1994) reported values of +7 . 2 e for Ti displacements, and -5 . 7 e for O displacements along the Ti-O-Ti chains.”

What the sources say

• Not checked yetcites Zhong et al., 1994

In the paper

“Surprisingly, they find that the magnitude of this dipole vector is typically only about 2.1 D in liquid water, much less than the value obtained from the WF analysis.”

What the sources say

• Not checked yetcites Pasquarello and Resta, 2003

In the paper

“Using this definition, these authors found that the water molecule dipole is somewhat broadly distributed, but has an average value of about 3.0 D, about 60% higher than in the gas phase.”

What the sources say

• Not checked yetcites Silvestrelli and Parrinello, 1999a
• Not checked yetcites Silvestrelli and Parrinello, 1999b

In the paper

“Interestingly, it was found that the interstitial contribution - defined as the difference between the total orbital magnetization, Eq. (94), and the muffin-tin result - is not always negligible. In bcc Fe, for example, it amounts to more than 30% of the spontaneous orbital magnetization, and its inclusion improves the agreement with gyromagnetic measurements.”

What the sources say

• Not checked yetcites Ceresoli et al., 2010a

In the paper

“The spin contribution is typically dominant over the orbital one (e.g., by a factor of 10 or more in simple ferromagnets such as Fe, Ni and Co (Ceresoli et al. , 2010a)), but the orbital component is also of interest”

What the sources say

• Not checked yetcites Ceresoli et al., 2010a

In the paper

“While no examples are known to occur in nature, tight-binding models exhibiting such a behavior are not hard to construct (Haldane, 1988).”

What the sources say

• Not checked yetcites Haldane, 1988

In the paper

“The actual implementation of Wannier's vision in the context of first-principles electronic-structure calculations, such as those carried out in the Kohn-Sham framework of density-functional theory (Kohn and Sham, 1965), has instead been slower to unfold.”

What the sources say

• Not checked yetcites Kohn and Sham, 1965

In the paper

“As such, they have become an established tool in the post-processing of electronicstructure calculations.”

In the paper

“In a related development, Souza et al. (2001) generalized the method to handle the case in which one wants to construct a set of WFs that spans a subspace containing, e.g., the partially occupied bands of a metal.”

What the sources say

• Not checked yetcites Souza et al., 2001

In the paper

“This is a consequence of the phase indeterminacy that Bloch orbitals ψ n k have at every wavevector k - or, more generally, the 'gauge' indeterminacy associated with the freedom to apply any arbitrary unitary transformation to the occupied Bloch states at each k .”

In the paper

“Of broader interest to the entire condensed matter community is the use of WFs in the construction of model Hamiltonians for, e.g., correlated-electron and magnetic systems.”

In the paper

“The history is rather different in the chemistry literature, where 'localized molecular orbitals' (LMO's) (Boys, 1960, 1966; Edmiston and Ruedenberg, 1963; Foster and Boys, 1960a,b) have played a significant role in computational chemistry since its early days.”

What the sources say

• Not checked yetcites Boys, 1960, 1966
• Not checked yetcites Edmiston and Ruedenberg, 1963
• Not checked yetcites Foster and Boys, 1960a,b

In the paper

“WFs thus provide an attractive option for representing the space spanned by a Bloch band in a crystal, being localized while still carrying the same information contained in the Bloch functions.”

In the paper

“Since Bloch functions at different k have different envelope functions e i k · r , one can expect to be able to build a localized 'wave packet' by superposing Bloch functions of different k .”

In the paper

“These developments touched off a transformational shift in which the computational electronic-structure community started constructing maximally-localized WFs (MLWFs) explicitly and using these for different purposes.”

In the paper

“For example, MLWF's have been used in the theoretical analysis of phonons, photonic crystals, cold atom lattices, and the local dielectric responses of insulators.”

In the paper

“Electronic structure calculations are often carried out using periodic boundary conditions.”

In the paper

“The approach is similar in spirit to the construction of LMO's in chemistry”

In the paper

“For the case of periodic system, these one-particle orbitals are normally taken to be the Bloch functions ψ n k ( r ) that are labeled, according to Bloch's theorem, by a crystal momentum k lying inside the Brillouin zone and a band index n .”

In the paper

“In particular, to arrive at the Wannier representation (des Cloizeaux, 1963; Kohn, 1959; Wannier, 1937), one carries out a unitary transformation from the Bloch functions to a set of localized 'Wannier functions' (WFs) labeled by a cell index R and a band-like index n , such that in a crystal the WFs at different R are translational images of one another.”

What the sources say

• Not checked yetcites des Cloizeaux, 1963
• Not checked yetcites Kohn, 1959
• Not checked yetcites Wannier, 1937

In the paper

“The one-particle effective Hamiltonian H then commutes with the lattice-translation operator T R , allowing one to choose as common eigenstates the Bloch orbitals | ψ n k 〉 :”

In the paper

“Unlike Bloch functions, WFs are not eigenstates of the Hamiltonian; in selecting them, one trades off localization in energy for localization in space.”

In the paper

“an alternative representation in terms of localized 'Wannier functions' was introduced by Gregory Wannier in 1937.”

What the sources say

• Not checked yetcites Wannier, 1937

In the paper

“The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness.”

In the paper

“An alternative use of WFs as localized, transferable building blocks has taken place in the theory of ballistic (Landauer) transport, where Green's functions and self-energies can be constructed effectively in a Wannier basis”

In the paper

“This connection is embodied in the microscopic modern theory of polarization, alluded to above, and has led to important advances in the characterization and understanding of dielectric response and polarization in materials.”

In the paper

“The existence of exponentially localized WFs - i.e., WFs whose tails decay exponentially fast - is a famous problem in the band theory of solids, with close ties to the general properties of the insulating state (Kohn, 1964).”

What the sources say

• Not checked yetcites Kohn, 1964

In the paper

“This is typical of Fourier transforms: the smoother the reciprocal-space object, the more localized the resulting real-space object, and vice versa.”

In the paper

“These false minima either correspond to the formation of topological defects (e.g., 'vortices') in an otherwise smooth gauge field in discrete k space, or they can arise when the branch cuts for the complex logarithms in Eq. (28) and Eq. (29) are inconsistent, i.e., when at any given k -point the contributions from different b -vectors differ by random amounts of 2 π in the logarithm.”

In the paper

“This behavior derives from the finitedifference representation of the invariant part Ω I of the localization functional (essentially, a second derivative); while Ω I does not enter into the localization procedure, it does contribute to the spread, and in fact usually represents the largest term.”

In the paper

“Panati (2007) had demonstrated that this condition ensures the possibility of carrying out the gauge transformation of Eq. (8) in such a way that the resulting cell-periodic states | ˜ u n k 〉 are analytic functions of k across the entire BZ; 7 this in turn implies the exponential falloff of the WFs given by Eq. (10).”

What the sources say

• Not checked yetcites Panati (2007)

In the paper

“The most straightforward way to avoid local minima altogether is to initialize the minimization procedure with a gauge choice that is already fairly smooth.”

In the paper

“We note that in the limit of a single k point the distinction between Bloch orbitals and WFs becomes irrelevant, since no Fourier transform from k to R is involved in the transformation Eq. (10); rather, we want to find the optimal unitary matrix that rotates the ground-state selfconsistent orbitals into their maximally localized representation, making this problem exactly equivalent to the one of”

In the paper

“Such a construction has proved useful for a variety of purposes, from verifying numerically exponential localization in 1 dimension, to treating electric polarization or applied electric fields along a specific spatial direction (Giustino and Pasquarello, 2005; Giustino et al. , 2003; Murray and Vanderbilt, 2009; Stengel and Spaldin, 2006a; Wu et al. , 2006) or for analyzing aspects of topological”

What the sources say

• Not checked yetcites Giustino and Pasquarello, 2005
• Not checked yetcites Giustino et al., 2003
• Not checked yetcites Murray and Vanderbilt, 2009

In the paper

“The Wannier centers and the general shape of the MLWFs often converge rapidly as the cell size is increased, but even for the ideal case of an isolated molecule, the total spread Ω often displays much slower convergence.”

In the paper

“The localization procedure can then be entirely recast in real space, and corresponds exactly to determining Foster-Boys localized orbitals.”

In the paper

“and the subsequent removal of the requirement of inversion symmetry in the latter case by Nenciu (1983).”

What the sources say

• Not checked yetcites Nenciu (1983)

In the paper

“A simple yet often effective approach for constructing a smooth gauge in k , and a corresponding set of welllocalized WFs, is by projection - an approach that finds its roots in the analysis of des Cloizeaux (1964a).”

What the sources say

• Not checked yetcites des Cloizeaux (1964a)

In the paper

“In this approach, the Brillouin zone is considered to be sufficiently small that integrations over k -vectors can be approximated with a single k -point (usually at the Γ point, i.e., the origin in reciprocal space).”

In the paper

“It can be shown that not only ˜ Ω but also Ω I is positive definite , and moreover that Ω I is gauge-invariant , i.e., invariant under any arbitrary unitary transformation (10) of the Bloch orbitals (Marzari and Vanderbilt, 1997).”

What the sources say

• Not checked yetcites Marzari and Vanderbilt, 1997

In the paper

“Interestingly, it should also be mentioned that the local minima alluded to in the previous subsection are typically not found when using Γ sampling in large supercells.”

In the paper

“In such cases, therefore, the extra unitary mixing expressed in Eq. (8) is mandatory, even if it may be optional in the case of a set of discrete bands that do not touch anywhere in the BZ.”

In the paper

“Other milestones include the work of des Cloizeaux (1964b), who established the exponential localization in 1D crystals without inversion symmetry and in the centrosymmetric 3D case”

What the sources say

• Not checked yetcites des Cloizeaux (1964b)

In the paper

“proofs of exponential localization for the Wannier transform were obtained over the years only for limited cases, starting with the work of Kohn (1959), who considered a one-dimensional crystal with inversion symmetry.”

What the sources say

• Not checked yetcites Kohn (1959)

In the paper

“Heuristically, it is also found that the WFs corresponding to these local minima are intrinsically complex, while they are found to be real, a part from a single complex phase, at the desired global minimum (provided of course that the calculations do not include spin-orbit coupling).”

In the paper

“since we would not get well-localized WFs on the left-hand side otherwise.”

In the paper

“The asymptotic behavior of WFs was further clarified by He and Vanderbilt (2001), who found that the exponential decay is modulated by a power law.”

What the sources say

• Not checked yetcites He and Vanderbilt (2001)

In the paper

“Thus, once the M ( k , b ) mn have been calculated, no further interaction with the electronic-structure code that calculated the ground state wavefunctions is necessary - making the entire Wannierization procedure a code-independent post-processing step 5 .”

In the paper

“Instead, a straightforward projection approach ensures the similarity between the WF in the defect (supercell) and in the pristine (primitive cell) calculations, and this has been exploited to develop a scheme to unfold the band-structure of disordered supercells into the Brillouin zone of the underlying primitive cell, allowing a direct comparison with angular resolved photoemission spectroscopy”

What the sources say

• Not checked yetcites Ku et al., 2010

In the paper

“An extreme example is CO 2 (Brown et al. , 1977). In agreement with the O=C=O Lewis picture, ER gives two lone pairs on each oxygen and two τ orbitals between each carbon and oxygen. In contrast, the FB scheme gives a single lone pair on each oxygen and three highly polarized τ orbitals between the carbon and each oxygen, as shown in Fig. 10.”

What the sources say

• Not checked yetcites Brown et al., 1977

In the paper

“It is worth noting that the Γ-point formulation (Sec. II.F.2) appears to be less affected by these problems.”

In the paper

“The stationarity condition δ Ω I ( { ˜ u n k } ) = 0, subject to orthonormality constraints, is equivalent to solving the set of eigenvalue equations”

• Also stated in: § 3. Iterative minimization of Ω I · p. 15 · boosted

  “Clearly these equations, one for each k -point, are coupled, so that the problem has to be solved selfconsistently throughout the Brillouin zone.”

In the paper

“Here again, this approach may give useful physical and chemical intuition, but the one-electron quasiparticle wavefunctions do not have the physical interpretation of occupied states, and charge densities and other ground-state properties cannot be computed quantitatively from them.”

What the sources say

• Not checked yetcites Hamann and Vanderbilt (2009)
• Not checked yetcites Aryasetiawan and Gunnarsson, 1998

In the paper

“What they all have in common is that the resulting localized molecular orbitals (LMOs) φ i ( r ) can be expressed as linear combinations of a set of molecular eigenstates ψ i ( r ) (the 'canonical' MOs), typically chosen to be the occupied ones,”

In the paper

“One important difference is that the resulting states are not guaranteed to be individually smooth, and the minimization of Ω I must therefore be followed by a gauge-selection step, which is in every way identical to the one described earlier for isolated groups of bands.”

In the paper

“Problems associated with reaching local minima of the spread functional, and with obtaining Wannier functions that are not real-valued, are more pronounced in the case of entangled bands.”

In the paper

“While exponential localization has not been proven, both numerical evidence and the analogy with the isolated composite case suggest this might be the case.”

In the paper

“The goal here is to maximize the density overlap of the orbitals (von Niessen, 1972)”

What the sources say

• Not checked yetcites von Niessen, 1972

In the paper

“As indicated in Eq. (56), all of these schemes amount to unitary transformations among the canonical MOs and, as such, they give rise to representations of the electronic structure of the system that are equivalent to that provided by the original set of eigenstates.”

In the paper

“Since these are small matrices, each step of the iterative procedure is computationally inexpensive.”

In the paper

“The idea is to maximize the sum of the squares of the Mulliken atomic charges (Mulliken, 1955).”

What the sources say

• Not checked yetcites Mulliken, 1955

In the paper

“Empirical evidence and experience suggests that localized Wannier functions can be readily constructed also in the case of an entangled manifold of bands: Even for metals, smooth manifolds can be disentangled and 'wannierized' to give MLWFs.”

In the paper

“As we have seen, WFs are inherently non-unique and, in practice, some strategy is needed to determine the gauge.”

In the paper

“Projection methods have been extensively used to study strongly-correlated systems (Anisimov et al. , 2005; Berlijn et al. , 2011; Ku et al. , 2002), in particular to identify a 'correlated subspace' for LDA+U or DMFT calculations, as will be discussed in more detail in Sec. VII.”

What the sources say

• Not checked yetcites Anisimov et al., 2005
• Not checked yetcites Berlijn et al., 2011
• Not checked yetcites Ku et al., 2002

In the paper

“As illustrated in Fig. 9, they showed that by minimizing Ω = Ω I + ˜ Ω for different choices of J , one can find an optimal J such that the resulting 'partially-occupied' Wannier functions are most localized (provided enough bands are used to capture the bonding and anti-bonding combinations of those atomiclike WFs).”

What the sources say

• Not checked yetcites Thygesen et al. (2005a,b)

In the paper

“The latter approach is conceptually more satisfactory, as it does not depend on a particular choice of trial orbitals. However, it still does not uniquely solve the problem of choosing a gauge, as different localization criteria are possible and there is, a priori , no reason to choose one over another.”

In the paper

“MLWFs, which may be thought of as the solid-state equivalent of FB orbitals, have been widely used to examine chemical bonding, as will be discussed in detail in Sec. IV.”

In the paper

“It is now evident that T k , b measures the degree of mismatch (or 'spillage') between the neighboring Bloch subspaces at k and k + b , vanishing when they are identical, and that Ω I provides a BZ average of the local subspace mismatch.”

In the paper

“Since n ( r , r ′ ) is periodic in the sense of n ( r + R , r ′ + R ) = n ( r , r ′ ), its eigenvectors the so-called 'natural orbitals' - have the form of Bloch functions carrying a label n, k . If the insulator is essentially a correlated version of a band insulator having J bands, then at each k there will typically be J occupation eigenvalues ν n k that are close to unity, as well as some”

In the paper

“When applied to benzene, both schemes give alternating σ and τ orbitals. Trivially, there are two equivalent sets of these orbitals, reminiscent of the two Kekul´ e structures for benzene. In contrast to the FB and ER schemes, PM provides a clear separation of σ and π orbitals. For example, in benzene PM gives a framework of six σ orbitals and three π orbitals around the ring.”

In the paper

“In some situations the FB and ER orbitals have been found to be quite different. This has been observed when the bonding in a molecule can be represented as two distinct resonant structures. The ER scheme generally gives orbitals corresponding to one of the possible structures, whilst the FB orbitals correspond a hybrid of the structures.”

In the paper

“The above procedure achieves simultaneously the two goals of subspace selection and gauge selection, although neither of them is performed optimally.”

In the paper

“In many cases all three approaches lead to similar localized orbitals which agree with the simple Lewis picture of bonding. A notable exception is for systems that exhibit both σ and π bonding.”

In the paper

“While MLWFs for extended systems have been generated for the most part by minimizing the sum of quadratic spreads, Eq. (17), a variety of other localization criteria have been used over the years for molecular systems.”

In the paper

“The ER scheme has not been extensively examined, an isolated exception being the work of Miyake and Aryasetiawan (2008) who proposed a method to maximize the Coulomb self-interaction of the orbitals in periodic systems (see also Sec. VII.B.1). They examined a range of bulk transition metals and SrVO 3 and in each case found that the resulting WFs were essentially identical to MLWFs.”

What the sources say

• Not checked yetcites Miyake and Aryasetiawan (2008)

In the paper

“However, while such an approach may provide useful qualitative information, it cannot provide the basis for any exact theory. For example, the charge density, or expectation value of any other one-particle operator, obtained by tracing over these WFs will not match their exact many-body counterparts.”

What the sources say

• Not checked yetcites Koch and Goedecker (2001)

In the paper

“This is essentially the same as the Marzari-Vanderbilt criterion of minimizing the sum of the quadratic spreads, except that the sum runs over the orbitals in the molecule, rather than in one crystal cell,”

What the sources say

• Not checked yetcites Boys, 1960, 1966
• Not checked yetcites Foster and Boys, 1960a

In the paper

“From a computational point of view, the ER approach is more demanding (it scales as J 4 versus J 2 for FB), but recently more efficient implementations have been developed (Subotnik et al. , 2007).”

What the sources say

• Not checked yetcites Subotnik et al., 2007

In the paper

“The resulting many-body WF is a complex function of 3 M electron coordinates, and as such it is unwieldy to use in practice. However, it is closely related to Kohn's theory of the insulating state (Kohn, 1964), and in principle it can be used to formulate many-body versions of the theory of electric polarization and related quantities, as shall be mentioned in Sec. V.A.4.”

What the sources say

• Not checked yetcites Kohn, 1964
• Not checked yetcites Souza et al. (2000)

In the paper

“As a final comment, we mention that in the case of degeneracies or quasi degeneracies in the spreads of orbitals centered on the same site, the localization algorithm will perform rather arbitrary mixing of these (as can be the case, e.g., for the d or f electrons of a transition-metal ion, or for its t 2 g or e g groups). A solution to this problem is to diagonalize an operator with the desired”

What the sources say

• Not checked yetcites Posternak et al. (2002)

In the paper

“Here localization is achieved by maximizing the Coulomb selfinteraction of the orbitals (Edmiston and Ruedenberg, 1963)”

What the sources say

• Not checked yetcites Edmiston and Ruedenberg, 1963

In the paper

“The concept of WFs is closely tied to the framework of single-particle Hamiltonians. Only in this case can we define J occupied single-particle Bloch functions at each wavevector k and treat all J of them on an equal footing, allowing for invariance with respect to unitary mixing among them.”

In the paper

“As such, they are often very useful for extracting chemical trends and for allowing for statistics to be gathered on the nature of bonds (e.g covalent bonds vs. lone pairs), be it in the presence of structural complexity, as is the case of an amorphous solid, or following the intrinsic dynamics of a liquid or an unfolding chemical reaction.”

In the paper

“It is usually required that a subset of N ≤ J of the original eigenstates are identically preserved ('frozen in') in the disentangled subspace, in which case the optimization is performed with respect to the remaining J -N states φ n . J must be of sufficient size to capture all of the antibonding character of the AOs.”

In the paper

“Unlike WFs, however, the quasiatomic orbitals form a nonorthogonal basis of atom-centered functions, each with specified angular character. Their radial part, and hence their detailed local shape, depends on the bonding environment.”

In the paper

“Lu et al. (2004) proposed a more optimized procedure, based on maximizing the sum-over-square similarity measure 8”

What the sources say

• Not checked yetcites Lu et al. (2004)

In the paper

“Many of these methods also make use of nonorthogonal localized orbitals, referred to as 'support functions' (Hern´ andez and Gillan, 1995) or 'non-orthogonal generalized Wannier functions' (NGWFs) (Skylaris et al. , 2002), in contrast to canonical WFs, which are orthogonal.”

What the sources say

• Not checked yetcites Hernández and Gillan, 1995
• Not checked yetcites Skylaris et al., 2002

In the paper

“We emphasize once again that no choice of WFs, whether according to maximal localization or other criteria, can be regarded as 'more correct' than another.”

In the paper

“In Qian et al. (2008), QOs were obtained for simple metals and semiconductors. Later applications have used the orbitals for the study of quantum transport in nano-structures (Qian et al. , 2010).”

What the sources say

• Not checked yetcites Qian et al., 2008
• Not checked yetcites Qian et al., 2010

In the paper

“In the same way as was described for the MarzariVanderbilt scheme of Sec. II.C, the alternative localization criteria described above can be applied in a solidstate context to isolated groups of bands.”

In the paper

“This apparent simplification has introduced an energy dependence into the Hamiltonian. One practical way forward is to approximate this as an energy-independent Hamiltonian H eff 00 ( ε 0 ), choosing the reference energy ε 0 to be the average energy of the states of interest. This approach has been used to construct tight-binding Hamiltonians from full electronic structure calculations (Solovyev,”

What the sources say

• Not checked yetcites Solovyev, 2004

In the paper

“The maximally-localized Wannier functions (MLWFs) provide the natural generalization of this concept to the case of extended or solidstate systems.”

In the paper

“Finally, localized orbitals can embody the chemical concept of transferable functional groups, and thus be used to construct a good approximation for the electronic-structure of complex systems starting from the orbitals for the different fragments (Benoit et al. , 2001; Hierse and Stechel, 1994; Lee et al. , 2005), as will be discussed in Section VII.”

What the sources say

• Not checked yetcites Benoit et al., 2001
• Not checked yetcites Hierse and Stechel, 1994
• Not checked yetcites Lee et al., 2005

In the paper

“This arbitrariness seems less common in extended system, and often some of the most chemically meaningful information comes from the statistics of bonds as obtained in large-scale simulations, or in long first-principles molecular dynamics runs.”