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.”

In the paper

“In practice, linear-scaling methods target large systems, which means that Γ-point only sampling of the BZ is usually sufficient.”

In the paper

“Because WFs are intrinsically gauge-dependent, it is impossible, even in principle, to determine the WFs experimentally.”

In the paper

“By the same token, certain properties obtained from the WFs, such as the dipole moments of molecules in condensed phases (see Sec. V.B.3), must be interpreted with caution.”

In the paper

“One is often interested in the more general situation where the bands of interest are not isolated. The challenge then is to generate a 'disentangled' set of J localized WFs starting from some larger set of bands.”

In the paper

“An alternative scheme for obtaining a minimal basis representation is the perturbation approach introduced by L¨ owdin (1951).”

What the sources say

• Not checked yetcites Löwdin (1951)

In the paper

“A form of L¨ owdin partitioning has been widely used in connection with the linear-muffin-tin-orbital (LMTO) method, particularly in its most recent formulation, the N th -order muffin-tin-orbital (NMTO) approach (Andersen and Saha-Dasgupta, 2000; Zurek et al. , 2005).”

What the sources say

• Not checked yetcites Andersen and Saha-Dasgupta, 2000
• Not checked yetcites Zurek et al., 2005

In the paper

“Taking into account the energy dependence, this reduced set of orbitals spans the same space as the original full set of orbitals, and can be seen to be the original orbitals of the active set dressed by an energy-dependent linear combination of orbitals from the passive set.”

What the sources say

• Not checked yetcites Zurek et al., 2005

In the paper

“This manifests itself in the exponential localization of the WFs in insulators (discussed in Sec. II.G) and, more generally, in the localization properties of the single-particle density matrix”

In the paper

“The quasiatomic orbitals (QOs) scheme (Chan et al. , 2007; Lu et al. , 2004; Qian et al. , 2008) is a projectionbased approach that has the aim of extracting a minimal tight-binding basis from an initial first-principles calculation.”

What the sources say

• Not checked yetcites Chan et al., 2007
• Not checked yetcites Lu et al., 2004
• Not checked yetcites Qian et al., 2008

In the paper

“When using WFs to interpret the nature of chemical bonds, as in Sec. IV, the results may depend to some degree on the choice of WF construction method, and the optimal choice may in the end be a matter of taste.”

In the paper

“In general the angular dependence of the resulting QOs will no longer that of pure spherical harmonics, but only approximately so.”

In the paper

“Instead, just the knowledge of the positions of the WFCs and their spreads can capture most of the chemistry in the system and can identify the defects present.”

In the paper

“Such hybridization is at the origin of the ferroelectric instability, as argued by Posternak et al. (1994).”

What the sources say

• Not checked yetcites Posternak et al., 1994

In the paper

“Still, it is easy to see how MLWFs can make the connection between atomic constituents and solid-state bands, representing a formal derivation of 'atoms in solids.'”

In the paper

“One can then construct either a minimal basis of one p z MLWF per carbon, if interested only in the π/π ⋆ manifold around the Fermi energy, or a slightly larger set with an additional MLWF per covalent bond, if interested in describing both the partially occupied π/π ⋆ and the fully occupied σ manifolds.”

In the paper

“This has been explored further by AbuFarsakh and Qteish (2007), to introduce a formal definition of electronegativity, or rather of a bond-ionicity scale, based on the deviation of WFCs from their geometrical bond centers.”

What the sources say

• Not checked yetcites AbuFarsakh and Qteish, 2007

In the paper

“Following this definition, one can provide a more meaningful definition of atomic coordination number, argue for the presence (or absence) of bonds in defective configurations, and propose specific electronic signatures for identifying different defects (Silvestrelli et al. , 1998).”

What the sources say

• Not checked yetcites Silvestrelli et al., 1998

In the paper

“Rather than reflecting a 'true' chemical entity, these represent linear combinations of sp 3 orbitals that interfere constructively at the interstitial sites, thus providing the additional variational freedom needed to describe the entire occupied space.”

What the sources say

• Not checked yetcites Souza et al., 2001

In the paper

“It should be stressed that a 'chemical' interpretation of the MLWFs is most appropriate when they are formed from a unitary transformation of the occupied subspace.”

In the paper

“The reason is that the 'on-bond' MLWFs have a single dominant nearest neighbor interaction along a bond, whereas for the 'off-bond' MLWFs there are a larger number of weaker nearest-neighbor interactions that are not directed along the bonds (Corsetti, 2012).”

What the sources say

• Not checked yetcites Corsetti, 2012

In the paper

“It is worth mentioning that these qualitative features of Wannier functions tend to be robust, and often independent of the details of the method used to obtained them - maximally localized or not.”

In the paper

“For the case of amorphous silicon, for example, the existence of a bond between two ions can be defined by their sharing a common WFC within a distance that is smaller than the first minimum of the silicon-WFC pair correlation function.”

What the sources say

• Not checked yetcites Silvestrelli et al., 1998

In the paper

“These considerations suggest that the electronic configuration is locally almost indistinguishable from that of the perfect lattice, making this defect difficult to detect with standard electronic probes.”

What the sources say

• Not checked yetcites Goedecker et al., 2002

In the paper

“Once conduction bands are included via the disentanglement procedure, results depend on both the target dimensions of the disentangled manifold and on the states considered in the procedure (e.g., the 'outer window').”

In the paper

“The results given in this section are valid for any set of localized WFs; maximally localized ones do not play any special role.”

• Also stated in: § B. Local polar properties and dielectric response · p. 30 · plain

  “We emphasize again that MLWFs have no privileged role in Eq. (89); the expression remains correct for any WFs that are sufficiently well localized that the centers r n are well defined.”

In the paper

“Such approaches are particularly useful when combined with molecular-dynamics simulations, and most applications have taken place within the framework first proposed by Car and Parrinello (Car and Parrinello, 1985).”

What the sources say

• Not checked yetcites Car and Parrinello, 1985

In the paper

“Of course, such a definition in terms of the dipole of the molecular WF-center configuration remains heuristic at some level.”

In the paper

“Car-Parrinello molecular dynamics is particularly suited to the study of liquid systems, and applications have been numerous in all areas of physical chemistry.”

In the paper

“In these materials, partially covalent bonding associated with hybridization between O 2 p and Ti 3 d states plays a crucial role in stabilizing the ferroelectric state and generating anomalous dynamical effective charges (Posternak et al. , 1994; Zhong et al. , 1994).”

What the sources say

• Not checked yetcites Posternak et al., 1994
• Not checked yetcites Zhong et al., 1994

In the paper

“As a result, the center of charge of the WF shifts upward, and since electrons carry negative charge, this results in a negative anomalous contribution to the Z ∗ of the oxygen atom.”

In the paper

“When the O atom moves upward relative to the geometric center of the two neighboring Ti atoms as shown in Fig. 22(b), as it does in ferroelectrically distorted BaTiO 3 , the hybridization strengthens for the upper O-Ti bond and weakens for the lower one, endowing the WF with more Ti 3 d character on the top than on the bottom.”

In the paper

“Such efforts and the implementation in widely-available computer codes have given rise to an extensive literature dedicated to understanding and monitoring the nature of bonding in complex and realistic systems under varying thermodynamical conditions or during a chemical reaction.”

In the paper

“In practice one wants to solve for a longlived resonance in which the charge density and other properties of the insulator remain periodic, corresponding to what is meant experimentally by an insulator in a finite field.”

In the paper

“Since the extraction of Wannier centers in 1D is rather trivial, even in the multiband case (Bhattacharjee and Waghmare, 2005; Marzari and Vanderbilt, 1997; Sgiarovello et al. , 2001), it is not difficult to construct a 'Wannier center band structure' ¯ z ( k x , k y ), and use the planar-averaged values to assign dipole moments to layers.”

What the sources say

• Not checked yetcites Bhattacharjee and Waghmare, 2005
• Not checked yetcites Marzari and Vanderbilt, 1997
• Not checked yetcites Sgiarovello et al., 2001

In the paper

“While 〈 Ψ | ˆ x | Ψ 〉 is ill-defined in the extended many-body ground state | Ψ 〉 , the matrix element 〈 Ψ | ˆ X | Ψ 〉 is welldefined and can be used to obtain the electric polarization, up to the usual quantum.”

What the sources say

• Not checked yetcites Ortiz and Martin, 1994
• Not checked yetcites Resta, 1998

In the paper

“Thus, while the Wannier charge centers are related to the mean value of P under quantum fluctuations, their invariant quadratic spread Ω I is related to the corresponding variance of P .”

In the paper

“For example, this local measure of the dipole does not appear to be experimentally measurable even in principle, and clearly the use of one of the alternative measures of maximal localization discussed in Sec. III.A would give rise to slightly different values.”

In the paper

“Correspondingly, it can be shown that an arbitrary change of gauge can shift individual Wannier centers r n in arbitrary ways, except that the sum ∑ n r n is guaranteed to remain invariant (modulo a lattice vector).”

In the paper

“The theory of crystalline insulators in finite electric field E is a subtle one; the electric potential -E · r does not obey the conditions of Bloch's theorem, and moreover is unbounded from below, so that there is no well-defined ground state.”

In the paper

“Using such approaches, it is possible to compute the dynamical dipole-dipole correlations of polar liquids and compare the results with experimental infrared absorption spectra.”

What the sources say

• Not checked yetcites Kirchner, 2007
• Not checked yetcites Tse, 2002
• Not checked yetcites Tuckerman, 2002

In the paper

“While the dipole moment of an isolated H 2 O molecule is obviously well defined, a corresponding definition is not easy to arrive at in the liquid phase where molecules are in close contact with each other.”

In the paper

“Furthermore, we shall see that the use of MLWFs can provide a very useful, if heuristic, local decomposition of polar properties in a an extended system.”

In the paper

“Resta and coworkers have defined a measure of localization (Resta, 2002, 2006; Resta and Sorella, 1999; Sgiarovello et al. , 2001) that distinguishes insulators from metals quite generally, and have shown that this localization measure reduces, in the absence of two-particle interactions or disorder, to the invariant part of the spread functional Ω I given in Eq. (20).”

What the sources say

• Not checked yetcites Resta, 2002
• Not checked yetcites Resta and Sorella, 1999
• Not checked yetcites Sgiarovello et al., 2001

In the paper

“and clearly the use of one of the alternative measures of maximal localization discussed in Sec. III.A would give rise to slightly different values.”

In the paper

“If time-reversal symmetry is preserved at the surface, Z 2 -odd materials possess topologically-protected surface states which straddle the bulk gap, crossing the Fermi level an odd number of times.”

In the paper

“Most (i.e., 'normal') insulators are Z 2 -even, but strong spin-orbit effects can lead to the Z 2 -odd state, for which the surface-state dispersions are topologically required to display characteristic features that are amenable to experimental verification.”

What the sources say

• Not checked yetcites Hasan and Kane, 2010
• Not checked yetcites Hasan and Moore, 2011

In the paper

“The calculations on Bi 2 Se 3 revealed the existence of a single Dirac cone at the Γ point as shown in Fig. 30, in agreement with ARPES measurements (Xia et al. , 2009).”

What the sources say

• Not checked yetcites Xia et al., 2009

In the paper

“These surface states are doubly-degenerate at the time-reversal-invariant momenta of the surface BZ, and in the vicinity thereof they disperse linearly, forming Dirac cones.”

In the paper

“Several materials realizations of Z 2 -odd insulators have now been confirmed both theoretically and experimentally (Hasan and Kane, 2010; Hasan and Moore, 2011).”

What the sources say

• Not checked yetcites Hasan and Kane, 2010
• Not checked yetcites Hasan and Moore, 2011

In the paper

“This time, however, the skin contribution does not vanish. The problem is that 〈 φ j | v | φ j 〉 is nonzero for LMOs in the skin region, and the pattern of these velocity vectors is such as to describe a current circulating around the boundary of the sample, giving a second 'itinerant circulation' contribution”

In the paper

“In particular, the contribution of this band to the electric polarization of Eq. (90), and to the second term in the orbital magnetization expression of Eq. (94), are proportional to the Brillouin-zone integrals of A n k and E n k F n k , respectively.”

In the paper

“An insulator with a non-zero value would behave like an integer quantum Hall system, but without the need for an external magnetic field; such systems are usually denoted as 'quantum anomalous Hall' (QAH) insulators.”

In the paper

“In a ferromagnetic (or ferrimagnetic) material, the total magnetization has two components. One arises from electron spin and is proportional to the excess population of spin-up over spin-down electrons; a second corresponds to circulating orbital currents.”

In the paper

“By virtue of the exponential localization of the WFs within the periodic supercell (see Footnote 2), the magnitude of the matrix elements 〈 0 | H | R 〉 decreases rapidly with | R | , and this exponential localization is preserved even in the case of metals, provided a smooth subspace has been disentangled.”

In the paper

“Ordinary insulators are characterized by a zero value of the invariant.”

In the paper

“This is often the case for metals, as the presence of a Fermi surface introduces sharp discontinuities in k -space.”

In the paper

“Many of the deficiencies of the Kohn-Sham energy bands, such as the underestimation of the energy gaps of insulators and semiconductors, can be corrected using many-body perturbation theory in the form of the GW approximation (for a review, see Aryasetiawan and Gunnarsson (1998)).”

What the sources say

• Not checked yetcites Aryasetiawan and Gunnarsson, 1998

In the paper

“The advantage of such approach is that it does not require linear-response theory, and so it is amenable to large-scale calculations or complex exchange-correlation functionals (e.g., including Hubbard U corrections, or Hartree-Fock exchange), albeit at the cost of typically one self-consistent iteration for every nucleus considered.”

In the paper

“In the latter study the authors found that the simple scissors correction to the DFT-LDA bands is accurate near the Γ point only, and only for bands close to the highest valence and lowest conduction band.”

What the sources say

• Not checked yetcites Peelaers et al., 2011

In the paper

“Even if with this sampling the system has a pseudogap of 2eV, the metallic character of the bands is perfectly reproduced, and these are in excellent agreement with the bands calculated directly on the primitive cell by direct diagonalization of the Kohn-Sham Hamiltonian in the full plane-wave basis set (open circles).”

What the sources say

• Not checked yetcites Lee et al., 2005

In the paper

“Note that inclusion of the spin-orbit interaction is essential for any description of orbital magnetic effects.”

In the paper

“However, Zhang et al. (2011b) found that the topological surface states are essentially the same with and without this surface correction.”

What the sources say

• Not checked yetcites Zhang et al., 2011b

In the paper

“While the AHC of the cubic metals Fe and Ni is fairly isotropic, that of hcp Co was found to decrease by a factor of four as the magnetization is rotated from the c -axis to the basal plane.”

What the sources say

• Not checked yetcites Roman et al., 2009

In the paper

“A direct numerical differentiation of the eigenenergies calculated on a grid is cumbersome and becomes unreliable near band crossings.”

In the paper

“As shown by Graf and Vogl (1992) and Boykin (1995), both the first and the second derivatives are easily computed within tight-binding methods, even in the presence of band degeneracies, and the same can be done in a firstprinciples context using WFs.”

What the sources say

• Not checked yetcites Graf and Vogl, 1992
• Not checked yetcites Boykin, 1995

In the paper

“This so-called anomalous Hall effect occurs in ferromagnets from the combination of exchange splitting and spin-orbit interaction. For a recent review, see Nagaosa et al. (2010).”

What the sources say

• Not checked yetcites Nagaosa et al., 2010

In the paper

“A Wannier-based calculation free from such numerical instabilities was carried out by Yates et al. (2007), who obtained carefullyconverged values for σ xy in Pd and other cubic metals.”

What the sources say

• Not checked yetcites Yates et al., 2007

In the paper

“Different electron-phonon interpolation schemes have been put forth in the literature (Calandra et al. , 2010; Eiguren and Ambrosch-Draxl, 2008; Giustino et al. , 2007b)”

What the sources say

• Not checked yetcites Calandra et al. , 2010
• Not checked yetcites Eiguren and Ambrosch-Draxl, 2008
• Not checked yetcites Giustino et al. , 2007b

In the paper

“The similarity between the Wannier interpolation of energy bands and the Fourier interpolation of phonon dispersions was already noted. It suggests the possibility of interpolating the electron-phonon vertex in both the electron and the phonon momenta, once Eq. (116) has been calculated on a relatively coarse uniform ( k , q )-mesh.”