Maximally localized Wannier functions: Theory and applications

In the paper

“From these four bands, one obtains four equivalent WFs per unit cell, each localized on one of the four nearest-neighbor Si-Si or Ga-As bonds.”

In the paper

“The presence of a bond-centered inversion symmetry for Si, but not GaAs, is clearly reflected in the shapes of the WFs.”

In the paper

“While the overall agreement is quite good, significant deviations can be seen wherever higher unoccupied and unwanted states possessing some significant sp 3 character are admixed into the projected manifold.”

In the paper

“As an example, we plot in Fig. 5 the eight disentangled bands obtained by projecting the band structure of silicon, taken within an energy window that coincides with the entire energy axis shown, onto eight atomic-like sp 3 hybrids.”

In the paper

“in this case, bands up to 20 eV above the Fermi energy can be meaningfully described with a minimal basis set of seven orbitals, that would have been difficult to identify using only educated guesses for the projections.”

In the paper

“The bands that result from projecting onto five d -type atomic orbitals are shown as blue triangles in Fig. 7. They follow very closely the first-principles bands away from the s -d hybridization regions, where the interpolated bands remain narrow.”

In the paper

“By choosing different dimensions for the disentangled subspace, it was found that the composite set of bands is faithfully represented by seven MLWFs, of which five are the standard d -like orbitals, and the remaining two are s -like orbitals centered at the tetrahedral interstitial sites of the fcc structure.”

In the paper

“Such disentangled MLWFs, e.g., the p z MLWFs describing the π / π ∗ manifold of graphene shown in Fig. 6, are found to be strongly localized.”

In the paper

“In addition, it can be seen that in gallium arsenide this covalent bond and its WFC are shifted towards the more electronegative arsenic atom.”

In the paper

“A recent comparison of static polarizabilities for molecules, calculated using Eq. (89) with both MLWFs and NGWFs, demonstrates remarkable agreement between the two (O'Regan et al. , 2012).”

What the sources say

• Not checked yetcites O'Regan et al., 2012

In the paper

“Minimization of an appropriate energy functional with respect to the degrees of freedom present in the density matrix leads to ground-state non-orthogonal orbitals that are very similar in appearance to (orthogonal) MLWFs.”

In the paper

“These are obtained from the closed manifold of four valence bands, yielding four equivalent MLWFs that map into one another under the space-group symmetry operations of the crystal.”

In the paper

“Symmetric orthonormalization of the three NMTOs gives the familiar set of three symmetry related σ -bonding orbitals (compare with the MLWF of graphene in Fig. 6).”

In the paper

“It was found that both schemes gave essentially identical orbitals.”

What the sources say

• Not checked yetcites Lechermann et al., 2006

In the paper

“For these bands the agreement with the full calculation is perfect to within the resolution of the figure.”

What the sources say

• Not checked yetcites Zurek et al., 2005

In the paper

“For crystalline silicon, the four-dimensional manifold disentangled from the lowest part of the conduction bands gives rise to four identical antibonding orbitals (see Fig. 13) originating from the destructive interference of two atomic sp 3 orbitals centered on neighboring atoms, to be contrasted with the constructive interference shown in Fig. 2 for the valence WFs.”

In the paper

“For example, referring back to Fig. 6) we recall that the band structure of graphene can be described accurately by disentangling the partially occupied π manifold from the higher free-electron parabolic bands and the antibonding sp 2 bands.”

In the paper

“In this latter case, the bond-centered MLWFs come from the constructive superposition of two sp 2 orbitals pointing towards each other from adjacent carbons (Lee et al. , 2005).”

What the sources say

• Not checked yetcites Lee et al., 2005

In the paper

“For the doubly charged split-vacancy configuration, the ionic relaxation is such that one of the nearest neighbors of the vacancy site moves halfway towards the vacancy, relocating to the center of an octahedral cage of silicon atoms.”

What the sources say

• Not checked yetcites Corsetti and Mostofi, 2011

In the paper

“Somewhere in between, it is worth pointing out that the atom-centered sp 3 orbitals typical of group-IV or III-V semiconductors, that can be obtained in the diamond/zincblende structure by disentangling the lowest 4 conduction bands, can have a major lobe pointing either to the center of the bond or in the opposite direction (Lee, 2006; Wahn and Neugebauer, 2006).”

What the sources say

• Not checked yetcites Lee, 2006
• Not checked yetcites Wahn and Neugebauer, 2006

In the paper

“A classic example is shown in Fig. 15, showing for BaTiO 3 the three MLWFs per oxygen derived from the nine oxygen 2 p bands.”

In the paper

“This gives rise to six defect WFs, each corresponding to a bond between sp 3 d 2 hybrids on the central atom and dangling sp 3 orbitals on the neighbors, as shown in the last panel of Fig. 19.”

What the sources say

• Not checked yetcites Corsetti and Mostofi, 2011

In the paper

“In addition, an eight-dimensional manifold disentangled from an energy window including both the valence bands and the lowest part of the conduction bands gives rise to the atomic sp 3 tight-binding orbitals of crystalline silicon (see Fig. 14).”

In the paper

“On the contrary, as discussed in Sec. II.I.2 and shown in Fig. 8, a very good tight-binding basis for 3 d metals such as Cu can be constructed (Souza et al. , 2001) with 5 atom-centered d -like orbitals and two s -like orbitals in the interstitial positions.”

What the sources say

• Not checked yetcites Souza et al., 2001

In the paper

“Tang and IsmailBeigi (2009) were also able to study the evolution of 2D boron sheets as they were made more compact (from hexagonal to triangular), and showed that the in-plane bonding pattern of the hexagonal system was preserved, with only minor changes in the shape and position of the MLWFs.”

What the sources say

• Not checked yetcites Tang and IsmailBeigi, 2009

In the paper

“The analysis of the 'bond angles' formed by two WFCs and their common Si atom shows considerable departure from the tetrahedral rule at the transition pressure.”

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

“This work also demonstrated the importance of including the occupied defect levels in the gap when constructing the relevant WFs, which are shown in the first two panels of Fig. 19.”

What the sources say

• Not checked yetcites Corsetti and Mostofi, 2011

In the paper

“These features were shown to be in conflict with the behavior of widely-used empirical models.”

What the sources say

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

In the paper

“The figure illustrates this process for σ -oriented oxygen WFs, but a similar effect occurs for the π -oriented oxygen WFs, and the total anomalous dynamical charge can be accounted for quantitatively on the basis of the distortion-induced changes of each kind of WF in the crystal (Marzari and Vanderbilt, 1998).”

What the sources say

• Not checked yetcites Marzari and Vanderbilt, 1998

In the paper

“Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions”

What the sources say

• Not checked yetcites Marzari and Vanderbilt, 1997

In the paper

“but its implementation in the solid-state context required significant developments, due to the ill-conditioned nature of the position operator in periodic systems (Nenciu, 1991), that was clarified in the context of the 'modern theory' of polarization (King-Smith and Vanderbilt, 1993; Resta, 1994).”

What the sources say

• Not checked yetcites Nenciu, 1991
• Not checked yetcites King-Smith and Vanderbilt, 1993
• Not checked yetcites Resta, 1994

In the paper

“For example, the charge density obtained by summing the squares of the Bloch functions | ψ n k 〉 or the WFs | R n 〉 is identical; a similar reasoning applies to the trace of any one-particle operator.”

In the paper

“Marzari and Vanderbilt showed that the minimization of a localization functional corresponding to the sum of the second-moment spread of each Wannier charge density about its own center of charge was both formally attractive and computationally tractable.”

What the sources say

• Not checked yetcites Marzari and Vanderbilt, 1997

In the paper

“Thus, both sets of states provide an equally valid description of the band subspace, even if the WFs are not Hamiltonian eigenstates.”

In the paper

“It is easily shown that the | R n 〉 form an orthonormal set (see Sec. II.A.3)”

In the paper

“The transformations of Eqs. (3) and (4) constitute a unitary transformation between Bloch and Wannier states.”

In the paper

“and that two WFs | R n 〉 and | R ′ n 〉 transform into each other under translation by the lattice vector R -R ′ (Blount, 1962).”

What the sources say

• Not checked yetcites Blount, 1962

In the paper

“However, the theory of WFs is made more complex by the presence of a 'gauge freedom' that exists in the definition of the ψ n k .”

In the paper

“A major reason for this is that WFs are strongly non-unique.”

In the paper

“Therefore, even before one could attempt to compute the WFs for a given material, one had first to resolve the question of which states to use to compute WFs.”

In the paper

“But k lives in the periodic Brillouin zone, so the best we can do is to choose equal amplitudes all across the Brillouin zone.”

In the paper

“More interestingly, there are formal connections between the centers of charge of the WFs and the Berry phases of the Bloch functions as they are carried around the Brillouin zone.”

In the paper

“Bloch and WFs span the same Hilbert space.”

In the paper

“This second indeterminacy is all the more troublesome in the common case of degeneracy for the occupied bands at certain highsymmetry points in the Brillouin zone, making a partition into separate 'bands', that could separately be transformed in Wannier functions, problematic.”

In the paper

“An important development in this regard was the introduction by Marzari and Vanderbilt (1997) of a 'maximal localization' criterion for identifying a unique set of WFs for a given crystalline insulator.”

What the sources say

• Not checked yetcites Marzari and Vanderbilt, 1997

In the paper

“To get a very localized wave packet in real space, we need to use a very broad superposition in k space.”

In the paper

“At the minimum, the off-diagonal elements |〈 R m | r | 0 n 〉| 2 are as small as possible, realizing the best compromise in the simultaneous diagonalization, within the subspace of the Bloch bands considered, of the three position operators x , y and z , which do not in general commute when projected onto this space.”

In the paper

“However, in many applications the bands of interest are not isolated.”

In the paper

“These | ˜ ψ n k 〉 have now a smooth gauge in k , are related to the original | ψ n k 〉 by a unitary transformation, 3 and when substituted into Eq. (3) in place of the | ψ n k 〉 result in a set of well-localized WFs.”

In the paper

“We emphasize that the trial functions do not need to resemble the target WFs closely; it is often sufficient to choose simple analytic functions (e.g., spherical harmonics times Gaussians) provided they are roughly located on sites where WFs are expected to be centered and have appropriate angular character.”

In the paper

“These overlaps play a central role in the formalism, since all contributions to the localization functional can be expressed in terms of them.”

In the paper

“We also emphasize that there is no 'preferred gauge' assigned by the Schr¨ odinger equation to the Bloch orbitals.”

In the paper

“The problem of computing well localized WFs starting from entangled bands is thus broken down into two distinct steps, subspace selection and gauge selection.”

In the paper

“Once again, we emphasize that the gauge freedom expressed in Eq. (8) implies that the WFs are strongly nonunique.”

In the paper

“Traces over this band manifold are invariant with respect to any unitary transformation among the J occupied Bloch orbitals at a given wavevector”

In the paper

“It should be noted that the localization functional can display, in addition to the desired global minimum, multiple local minima that do not lead to the construction of meaningful localized orbitals.”

In the paper

“The difficulty in treating entangled bands stems from the fact that it is unclear exactly which states to choose to form J WFs, particularly in those regions of k -space where the bands of interest are hybridized with other unwanted bands.”

In the paper

“The problem arises when there are degeneracies among the bands in question at certain locations in the Brillouin zone.”

In the paper

“is invariant, even though the | ˜ ψ n k 〉 resulting from Eq. (8) are no longer generally eigenstates of H , and n is no longer a band index in the usual sense.”

In the paper

“In this sense, the WFs are 'more non-unique' than the Bloch functions, which only acquire a phase change.”

In the paper

“is invariant, even though the | ˜ ψ n k 〉 resulting from Eq. (8) are no longer generally eigenstates of H , and n is no longer a band index in the usual sense.”

In the paper

“is invariant, even though the | ˜ ψ n k 〉 resulting from Eq. (8) are no longer generally eigenstates of H , and n is no longer a band index in the usual sense.”

In the paper

“For this purpose, the projection method already described in Sec. II.B has been found to be extremely effective.”

In the paper

“As a result, band four is nonanalytic in k in the vicinity of Γ.”

In the paper

“As we will see, the same guiding principle can be used for both steps, namely, to achieve 'smoothness' in k -space.”

In the paper

“That is, if we simply insert the usual Bloch functions | ψ n k 〉 , defined to be eigenstates of H , into the right-hand side of Eq. (3), it is generally not possible to produce well-localized WFs.”

In the paper

“That is, different choices of smooth gauge correspond to different sets of WFs having in general different shapes and spreads.”

In the paper

“Thus, the non-uniqueness of the WFs resulting from Eq. (3) is unavoidable.”

In the paper

“These | ˜ ψ n k 〉 have now a smooth gauge in k , are related to the original | ψ n k 〉 by a unitary transformation, 3 and when substituted into Eq. (3) in place of the | ψ n k 〉 result in a set of well-localized WFs.”

In the paper

“We note in passing that the | ˜ ψ n k 〉 are uniquely defined by the trial orbitals g n ( r ) and the chosen (isolated) manifold, since any arbitrary unitary rotation among the | ψ n k 〉 orbitals cancels out exactly and does not affect the outcome of Eq. 16, eliminating thus any gauge freedom.”

In the paper

“Consider, for example, what happens if we try to construct a single WF from the highest occupied band n = 4 in GaAs. This would be doomed to failure, since this band becomes degenerate with bands two and three at the zone center, Γ, as shown in Fig. 3.”

In the paper

“The more complex problem of exponential localization of MLWFs for composite bands in 2D and 3D was finally proven by Panati and Pisante (2011).”

What the sources say

• Not checked yetcites Panati and Pisante (2011)

In the paper

“The early work on composite bands in 3D only established the exponential localization of the projection operator P , Eq. (13), not of the WFs themselves (des Cloizeaux, 1964a).”

What the sources say

• Not checked yetcites des Cloizeaux, 1964a

In the paper

“Marzari and Vanderbilt (1997) established this in 1D, by simply noting that the MLWF construction then reduces to finding the eigenstates of the projected position operator PxP , a case for which exponential localization had already been proven (Niu, 1991).”

What the sources say

• Not checked yetcites Marzari and Vanderbilt (1997)
• Not checked yetcites Niu, 1991

In the paper

“The question of exponential decay in 2D and 3D was finally settled by Brouder et al. (2007) who showed, as a corollary to a theorem by Panati (2007), that a necessary and sufficient condition for the existence of exponentially localized WFs in 2D and 3D is that the so-called 'Chern invariants' for the composite set of bands vanish identically.”

What the sources say

• Not checked yetcites Brouder et al. (2007)
• Not checked yetcites Panati (2007)

In the paper

“The method is successful as long as the inner-product matrix A k does not become singular (or nearly so) for any k , which can be ensured by checking that the ratio of maximum and minimum values of det S k in the Brillouin zone does not become too large.”

In the paper

“The Fourier transform of Eq. (3) would then result in a poorly localized object having power-law tails in real space.”

In the paper

“This turns out to be the solid-state equivalent of the Foster-Boys criterion of quantum chemistry (Boys, 1960, 1966; Foster and Boys, 1960a,b).”

What the sources say

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

In the paper

“The minimization procedure of Ω thus actually corresponds to the minimization of the non-invariant part ˜ Ω only.”

In the paper

“For example, spherical ( s -like) Gaussians located at the bond centers will suffice for the construction of well-localized WFs, akin to those shown in Fig. 2, spanning the four occupied valence bands of semiconductors such as Si and GaAs.”

In the paper

“These | ˜ ψ n k 〉 have now a smooth gauge in k , are related to the original | ψ n k 〉 by a unitary transformation, 3 and when substituted into Eq. (3) in place of the | ψ n k 〉 result in a set of well-localized WFs.”

In the paper

“Conversely, nonzero Chern numbers pose an obstruction to finding a globally smooth gauge in k -space.”

In the paper

“We know from Eq. (19) that the quadratic spread Ω of the WFs spanning a Bloch space of dimension J comprises two positive-definite contributions, one gauge-invariant (Ω I ), the other gaugedependent ( ˜ Ω).”

In the paper

“The projection technique discussed in Section II.B can be easily adapted to produce J smoothly-varying Bloch-like states starting from a larger set of Bloch bands (Souza et al. , 2001).”

What the sources say

• Not checked yetcites Souza et al., 2001

In the paper

“As it turns out, smoothness in k of a Bloch space is precisely what is measured by the functional Ω I introduced in Sec. II.C.1.”

In the paper

“Alternatively, it is possible to combine the two steps, and minimize Ω = Ω I + ˜ Ω simultaneously with respect to the choice of Hilbert subspace and the choice of gauge (Thygesen et al. , 2005a,b); this should lead to the most-localized set of J WFs that can be constructed from the initial J k Bloch states.”

What the sources say

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

In the paper

“For a double bond, both the FB and ER schemes mix the σ and π orbitals, giving rise to two bent 'banana bond' (or τ ) orbitals (Pauling, 1960) as shown in Fig 10.”

What the sources say

• Not checked yetcites Pauling, 1960 source

In the paper

“For example, it often results in 'symmetry-adapted' WFs which inherit the symmetry properties of the trial orbitals (Ku et al. , 2002).”

What the sources say

• Not checked yetcites Ku et al., 2002

In the paper

“We infer, therefore, that such problems are associated not with the wannierization part of the procedure, but rather with the initial selection of the smooth subspace from the full manifold of entangled bands.”

In the paper

“The reason for its success is easily understood: the localization of the trial orbitals in real space leads to smoothness in k -space.”

In the paper

“That choice ensures that at self-consistency the stationary point corresponds to the absolute minimum of Ω I .”

In the paper

“An iterative procedure is then used to extract selfconsistently across the BZ the J -dimensional subspaces having collectively the smallest possible value of Ω I (typically the minimization starts from an initial guess for the target subspaces given, e.g., by projection onto trial orbitals).”

What the sources say

• Not checked yetcites Souza et al., 2001

In the paper

“What Ω I manages to capture is the intrinsic smoothness of the underlying Hilbert space.”

In the paper

“This behavior can be avoided by forcing certain Bloch states to be preserved identically in the projected manifold - we refer to those as belonging to a frozen 'inner' window, since this is often the simplest procedure for selecting them.”

In the paper

“Viewed as function of k , the Bloch subspace obtained at the end of the minimization is 'optimally smooth' in that it changes as little as possible with k .”

In the paper

“Once the two-particle electron-electron interaction is formally included in the Hamiltonian, the many-body wavefunction cannot be reduced to any simple form allowing for the construction of WFs in the usual sense.”

In the paper

“First, it is no longer necessary to enforce explicit orthogonality constraints on the orbitals during the energy minimization procedure (Galli and Parrinello, 1992; Hern´ andez et al. , 1996; Hierse and Stechel, 1994; Mauri et al. , 1993; Ordej´ on et al. , 1993).”

What the sources say

• Not checked yetcites Galli and Parrinello, 1992
• Not checked yetcites Hernández et al., 1996
• Not checked yetcites Hierse and Stechel, 1994

In the paper

“In later work it was realized (Qian et al. , 2008) that the QOs can be constructed without explicit calculation of the eigenstates outside the frozen window.”

What the sources say

• Not checked yetcites Qian et al., 2008

In the paper

“Second, a nonorthogonal representation can be more localized than an essentially equivalent orthogonal one (Anderson, 1968; He and Vanderbilt, 2001).”

What the sources say

• Not checked yetcites Anderson, 1968
• Not checked yetcites He and Vanderbilt, 2001

In the paper

“We now see that the spatial localization of the density matrix is closely linked to that of the Wannier functions themselves.”

In the paper

“The common point between these variational methods is that the idempotency of the density operator or the orthogonality of the Wannier orbitals is not imposed explicitly during the minimization procedure. Instead, the energy functionals are constructed such that these properties are satisfied automatically at the minimum, which coincides with the true ground state of the system.”

In the paper

“It has also been shown to take the same form in metals at finite temperature, 9 but with γ determined by the ratio between the thermal energy k B T and the Fermi velocity (Goedecker, 1998).”

What the sources say

• Not checked yetcites Goedecker, 1998

In the paper

“It is clearly apparent that these MLWFs represent the intuitive chemical concept of a covalent bond, with each MLWF representing the bonding orbital created by the constructive interference of two atomic sp 3 orbitals centered on neighboring atoms.”

In the paper

“To give a specific example, accurate calculations on tetrahedral semiconductors will require the inclusion of d states in the MTO basis, i.e., nine states ( s , p , d ) per site.”

What the sources say

• Not checked yetcites Lambrecht and Andersen, 1986

In the paper

“The fundamental principle behind such approaches is the fact that electronic structure is inherently local (Heine, 1980), or 'nearsighted' (Kohn, 1996).”

What the sources say

• Not checked yetcites Heine, 1980
• Not checked yetcites Kohn, 1996

In the paper

“One of the most notable, albeit qualitative, characteristics of MLWFs is their ability to highlight the chemical signatures of different band manifolds.”

In the paper

“The quantity ρ ( r , r ′ ) has been shown to decay exponentially as exp( -γ | r -r ′ | ) in insulators and semiconductors, where the exponent γ can be heuristically related to the direct band gap of the system (des Cloizeaux, 1964a; Ismail-Beigi and Arias, 1999; Taraskin et al. , 2002).”

What the sources say

• Not checked yetcites des Cloizeaux, 1964a
• Not checked yetcites Ismail-Beigi and Arias, 1999
• Not checked yetcites Taraskin et al., 2002

In the paper

“Since MLWFs are uniquely defined for the case of insulators and semiconductors, they are particularly suited to discuss hybridization, covalency, and ionicity both in crystalline and disordered systems.”

In the paper

“In this way, properties computed from the WFs, such as interpolated bandstructures, will have the correct symmetry properties.”

In the paper

“This locality is exploited in linearscaling methods by retaining an amount of information in the density matrix that scales only linearly with system size.”

In the paper

“In addition, in the supercell approximation they can be used to describe any disordered bulk, amorphous, or liquid system (Payne et al. , 1992), providing a compact description of electronic states in terms of their Wannier centers, their coordination with other atoms, and the spatial distribution and symmetry of the MLWFs.”

What the sources say

• Not checked yetcites Payne et al., 1992

In the paper

“Downfolding will result in an MTO with either s or p character on a given site, with inclusion of d character on neighboring sites.”

In the paper

“In this case, it becomes riskier to draw conclusions from the qualitative features of the MLWFs.”

In the paper

“Whenever unoccupied states are included, MLWFs are more properly understood as forming a minimal tightbinding basis, and not necessarily as descriptors of the bonding.”

In the paper

“Moreover, a low activation energy is required for the selfannihilation of this defect; this consideration, in combination with the 'stealth' electronic signature, hints at the reason why such a defect could have eluded experimental discovery despite the fact that Si is one of the best studied materials in the history of technology.”

What the sources say

• Not checked yetcites Goedecker et al., 2002

In the paper

“The ability of Wannier functions to capture the electronic structure of complex materials has also been demonstrated in the study of boron allotropes.”

In the paper

“This purely geometrical analysis is completely blind to the actual electronic charge distribution, which ought to be important in any description of chemical bonding.”

In the paper

“The use of MLWFs to characterize electronic bonding in complex system has been greatly aided by the implementation of efficient and robust algorithms for maximal localization of the orbitals in the case of large, and often disordered, supercells in which the Brillouin zone can be sampled at a single point, usually the zone-center Γ point (Berghold et al. , 2000; Bernasconi and Madden, 2001; Silv”

What the sources say

• Not checked yetcites Berghold et al., 2000
• Not checked yetcites Bernasconi and Madden, 2001
• Not checked yetcites Silvestrelli, 1999

In the paper

“Besides its application to the study of disordered networks (Fitzhenry et al. , 2003; Lim et al. , 2002; Meregalli and Parrinello, 2001), the above analysis can also be effectively employed to elucidate the chemical and electronic properties accompanying structural transformations.”

What the sources say

• Not checked yetcites Fitzhenry et al., 2003
• Not checked yetcites Lim et al., 2002
• Not checked yetcites Meregalli and Parrinello, 2001

In the paper

“That is, it can reveal the atomic-like orbitals that conceptually lie behind any tight-binding formulation, but that can now be obtained directly from first principles according to a well-defined procedure.”

In the paper

“Interestingly, the MLWFs analysis can also point to structural defects that do not otherwise exhibit any significant electronic signature.”

In the paper

“Silvestrelli et al. (1998) were the first to argue that the WFCs can be a powerful tool for understanding bonding in low-symmetry cases, representing an insightful and an economical mapping of the continuous electronic degrees of freedom into a set of classical descriptors, i.e., the center positions and spreads of the WFs.”

What the sources say

• Not checked yetcites Silvestrelli et al., 1998

In the paper

“These can form the basis of the construction of Hamiltonians for model systems (e.g., strongly-correlated) or large-scale nanostructures, as will be discussed in Chap. VII.”

In the paper

“The same e R /V describes the quantization of charge transport under an adiabatic cycle (Thouless, 1983), and indeed the shifts of Wannier charge centers under such a cycle were recently proposed as a signature of formal oxidation state in crystalline solids (Jiang et al. , 2012).”

What the sources say

• Not checked yetcites Thouless, 1983

In the paper

“This latter case is a paradigmatic example of the chemical insight that can be gleaned by following the WFCs in the course of an first-principles simulation.”

What the sources say

• Not checked yetcites Boero et al., 2000b

In the paper

“Then, the change of polarization resulting from any physical change, such as the displacement of one atomic sublattice or the application of an electric field, can be related in a simple way to the displacements of the Wannier centers r n occurring as a result of this change.”

In the paper

“The essential observation is that, when studying a system that is layered or stacked along the z direction, one can still work with Bloch functions labeled by k x and k y while carrying out a Wannier construction or analysis only along z .”

In the paper

“These theories can be formulated either in terms of Berry phases and curvatures, or equivalently, by working in the Wannier representation.”

What the sources say

• Not checked yetcites King-Smith and Vanderbilt, 1993
• Not checked yetcites Resta, 1992
• Not checked yetcites Thonhauser et al., 2005

In the paper

“Moreover, Souza et al. (2000) have shown that this same quantity characterizes the root-mean-square quantum fluctuations of the macroscopic polarization.”

What the sources say

• Not checked yetcites Souza et al., 2000

In the paper

“is precisely the expression for the electronic contribution to the polarization in the Berry-phase theory (KingSmith and Vanderbilt, 1993; Resta, 1994; Vanderbilt and King-Smith, 1993), which was derived by considering the flow of charge during an arbitrary adiabatic change of the crystalline Hamiltonian.”

What the sources say

• Not checked yetcites KingSmith and Vanderbilt, 1993
• Not checked yetcites Resta, 1994
• Not checked yetcites Vanderbilt and King-Smith, 1993

In the paper

“In a related development, Stengel and Spaldin (2006a) introduced a Wannier-based method for computing polarizations along z and studying electric fields along z that work even in the case that the stacking includes metallic layers, as long as the system is effectively insulating along z (Stengel and Spaldin, 2007).”

What the sources say

• Not checked yetcites Stengel and Spaldin, 2006a
• Not checked yetcites Stengel and Spaldin, 2007

In the paper

“The Wannier representation provides an alternative method for computing such dielectric quantities by finite differences.”

In the paper

“This behavior can be understood (Posternak et al. , 1994; Zhong et al. , 1994) as arising from hybridization between neighboring O 2 p and Ti 3 d orbitals that dominate the valence and conduction bands, respectively.”

What the sources say

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

In the paper

“This approach is now standard for treating periodic insulators in finite electric fields in density-functional theory.”

What the sources say

• Not checked yetcites Nunes and Vanderbilt, 1994
• Not checked yetcites Nunes and Gonze, 2001
• Not checked yetcites Souza et al., 2002

In the paper

“A well-known feature of the Berry-phase theory is that the polarization is only well-defined modulo a quantum e R /V , where R is a real-space lattice vector.”

In the paper

“The problem is that matrix elements such as 〈 ψ n k | r | ψ n k 〉 and 〈 ψ n k | r × v | ψ n k 〉 are ill-defined for such extended states (Nenciu, 1991).”

What the sources say

• Not checked yetcites Nenciu, 1991

In the paper

“In fact, specialized algorithms have been developed to perform on-the-fly Car-Parrinello molecular dynamics in a Wannier representation (Iftimie et al. , 2004; Sharma et al. , 2003; Wu et al. , 2009).”

What the sources say

• Not checked yetcites Iftimie et al., 2004
• Not checked yetcites Sharma et al., 2003
• Not checked yetcites Wu et al., 2009

In the paper

“The Berry-phase theory can be regarded as providing a mapping of the distributed quantum-mechanical electronic charge density onto a lattice of negative point charges of charge -e , as illustrated in Fig. 21.”

In the paper

“It should be noted that the decomposition into Wannier dipoles is closer to the decomposition of the charge density into static (Szigeti) charges than to a decomposition into dynamical (Born) charges.”

In the paper

“In a related development, the orbital magnetoelectric coefficient α ij = ∂M orb ,j /∂ E i was found to contain an isotropic contribution having a topological character (the 'axion' contribution, corresponding to an E · B term in the effective Lagrangian).”

What the sources say

• Not checked yetcites Essin et al., 2009
• Not checked yetcites Qi et al., 2008

In the paper

“While it is possible to extract the needed information from the time-time correlation function of the total polarization P ( t ) of the entire supercell as calculated using the Berry-phase approach, methods which follow the time-evolution of local dipoles, as defined via WF-based methods, provide additional insight and efficiency (Bernasconi et al. , 1998; Chen et al. , 2008; McGrath et al. , 2007”

What the sources say

• Not checked yetcites Bernasconi et al., 1998
• Not checked yetcites Chen et al., 2008
• Not checked yetcites McGrath et al., 2007

In the paper

“Starting from this short-range real-space representation, the quantity f can now be accurately interpolated onto an arbitrary point k in reciprocal space by carrying out an inverse transformation (right panel).”

In the paper

“It is apparent that the interpolation procedure succeeds in resolving details of the true band structure on a scale much smaller than the spacing between q -points.”

What the sources say

• Not checked yetcites Yates et al., 2007

In the paper

“First, insulators having broken time-reversal ( T ) symmetry (e.g., insulating ferromagnets and ferrimagnets) can be classified by an integer 'Chern invariant' that is proportional to the Brillouin-zone integral of the Berry curvature F n k summed over occupied bands n .”

In the paper

“The ability to compute the orbital magnetization is also of use in obtaining the magnetic shielding of nuclei. This is responsible for the chemical shift effect observed in nuclear magnetic resonance (NMR) experiments.”

In the paper

“Zhang et al. (2010) used a simplified Wannier-based approach which succeeds in capturing the essential features of the topological surface states at a greatly reduced computational cost.”

What the sources say

• Not checked yetcites Zhang et al., 2010

In the paper

“Naively one might imagine computing the orbital magnetization M orb as the thermodynamic limit of Eq. (85) per unit volume for a large crystallite. However, as we discussed at the beginning of Sec. V, Bloch matrix elements of the position operator r and the circulation operator r × v are ill-defined. Therefore, such an approach is not suitable.”

In the paper

“By virtue of the spatial localization of the WFs, F ( R ) decays rapidly with | R | .”

In the paper

“As already mentioned briefly in Sec. II.J, Hamann and Vanderbilt (2009) proposed using Wannier interpolation to determine the GW QP bands very efficiently and accurately at arbitrary points in the BZ.”

What the sources say

• Not checked yetcites Hamann and Vanderbilt, 2009

In the paper

“The Chern-Simons magnetoelectric coupling has been evaluated from first-principles with the help of WFs for both topological and ordinary insulators (Coh et al. , 2011).”

What the sources say

• Not checked yetcites Coh et al., 2011

In the paper

“Note that the dashed lines pass through the open circles at the symmetry points, which denote exact (non-interpolated) GW results.”

What the sources say

• Not checked yetcites Hamann and Vanderbilt, 2009

In the paper

“It can be shown that a Wannier representation is not possible for a QAH insulator, and Thonhauser and Vanderbilt (2006) have explored the way in which the usual Wannier construction breaks down for model systems.”

What the sources say

• Not checked yetcites Thonhauser and Vanderbilt, 2006

In the paper

“The method is particularly useful in situations where a very fine sampling of the BZ is required to converge the quantity of interest.”

In the paper

“Wannier band-structure interpolation is extremely accurate.”

In the paper

“Localized Wannier functions are often introduced in textbooks as a formally exact localized basis spanning a band, or a group of bands, and their existence provides a rigorous justification for the tight-binding (TB) interpolation method (Ashcroft and Mermin, 1976; Harrison, 1980).”

What the sources say

• Not checked yetcites Ashcroft and Mermin, 1976
• Not checked yetcites Harrison, 1980

In the paper

“In the case of 3D TIs, the clearest experimental signature of the Z 2 -odd character is at present provided by ARPES measurements of the surface electron bands.”

In the paper

“This procedure will succeed in capturing variations in f ( k ) over reciprocal lengths smaller than the first-principles mesh spacing ∆ q , provided that the linear dimensions L = 2 π/ ∆ q of the equivalent supercell are large compared to the decay length of the WFs.”

In the paper

“The comparison with the first-principles bands (solid lines), reveals essentially perfect agreement within the inner window.”

What the sources say

• Not checked yetcites Wang et al., 2006

In the paper

“This feature can be exploited to calculate the diagonal elements of the Green's function matrix very efficiently using iterative schemes (Lopez-Sancho et al. , 1984, 1985).”

What the sources say

• Not checked yetcites Lopez-Sancho et al., 1984
• Not checked yetcites Lopez-Sancho et al., 1985

In the paper

“This term can be written as a Brillouin-zone integral of the Chern-Simons 3-form, defined in terms of multiband generalizations of the Berry connection A k and curvature F k introduced in the previous subsection (Essin et al. , 2009; Qi et al. , 2008).”

What the sources say

• Not checked yetcites Essin et al., 2009
• Not checked yetcites Qi et al., 2008

In the paper

“The nearest-neighbor coupling between principal layers means that for each k ‖ the Hamiltonian has a block tridiagonal form (the dependence of the Hamiltonian matrix on k ‖ is given by the usual Fourier sum expression).”

In the paper

“As the number of lattice vectors included in the summation in Eq. (98) equals the number of first-principles mesh points, beyond a certain mesh density the error incurred decreases exponentially (Yates et al. , 2007).”

What the sources say

• Not checked yetcites Yates et al., 2007

In the paper

“It should be noted that this approach contains no surface-specific information, being based exclusively on the bulk WFs.”

In the paper

“A careful generalization of Eq. (94) to the case in which a finite electric field is present has been carried out by Malashevich et al. (2010) in the Wannier representation using arguments similar to those in Secs. V.A.1 and V.C.2, and used to derive a complete expression for the orbital magnetoelectric response, of which the topological Chern-Simons term is only one contribution (Essin et al. , 2”

What the sources say

• Not checked yetcites Essin et al., 2010
• Not checked yetcites Malashevich et al., 2010

In the paper

“Second, depending on how their Bloch functions wrap the Brillouin zone, nonmagnetic ( T -invariant) insulators can be sorted into two classes denoted as ' Z 2 -even' and ' Z 2 -odd' (after the name Z 2 of the group { 0, 1 } under addition modulo 2).”

What the sources say

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

In the paper

“A number of efficient interpolation schemes, such as the method implemented in the BoltzTraP package (Madsen and Singh, 2006), have been developed for this purpose, but they are still prone to numerical instabilities near band degeneracies (Uehara and Tse, 2000).”

What the sources say

• Not checked yetcites Madsen and Singh, 2006
• Not checked yetcites Uehara and Tse, 2000

In the paper

“These pioneering firstprinciples calculations revealed that in ferromagnetic metals the Berry curvature displays strong and rapid variations in k -space (Fig. 32).”

What the sources say

• Not checked yetcites Yao et al., 2004

In the paper

“As a result, an ultradense BZ mesh containing millions of k -points is often needed in order to converge the calculation.”

In the paper

“Recalling that F n k = ∇ k × A n k [Eq. (96)] and using Stokes' theorem, Eq. (110) for the AHC can be recast in terms of the Berry phases of Fermi loops on planar slices of the Fermi surface.”

In the paper

“This approach has been implemented by Wang et al. (2007), using Wannier interpolation to sample efficiently the orbits with the very high density required near band-crossings.”

What the sources say

• Not checked yetcites Wang et al. (2007)

In the paper

“Both the k -space distribution of the Berry curvature and the integrated AHC were found to be in excellent agreement with the sum-over-states calculation of Yao et al. (2004).”

What the sources say

• Not checked yetcites Wang et al., 2006
• Not checked yetcites Yao et al., 2004

In the paper

“Such instabilities can be avoided by using a tight-binding parametrization to fit the first-principles band structure (Mazin et al. , 2000; Schulz et al. , 1992).”

What the sources say

• Not checked yetcites Mazin et al., 2000
• Not checked yetcites Schulz et al., 1992

In the paper

“Nevertheless, the Berry phase obtained by integrating over a closed loop in k -space, ϕ n = ∮ A n k · d l , is gauge-invariant (Xiao et al. , 2010).”

What the sources say

• Not checked yetcites Xiao et al. , 2010

In the paper

“Its localization in real-space ensures that Eq. (122) smoothly interpolates g W in the electron and phonon momenta.”

In the paper

“The second category includes methods in which WFs are used to identify and focus on a desired, physically relevant subspace of the electronic degrees of freedom that is singled out ('downfolded') for further detailed analysis or special treatment with a more accurate level of electronic structure theory”

In the paper

“The matrix element for scattering an electron from state ψ n k to state ψ m, k + q while absorbing a phonon q ν is proportional to the electron-phonon vertex”

In the paper

“The cost of evaluating Eq. (116) from first-principles over a large number of ( k , q )-points is quite high, however, and this has placed a serious limitation on the scope and accuracy of first-principles techniques for electron-phonon problems.”

In the paper

“This is possible because (a) WFs and bands are related by unitary transformations, and (b) WFs are sufficiently localized that any resulting tightbinding representation may be truncated with little loss of accuracy.”

In the paper

“The first category includes methods in which WFs are used to go up in the length scale of the simulations, using the results of electronic-structure calculations on small systems in order to construct accurate models of larger, often meso-scale, systems.”

In the paper

“Expressing the Hamiltonian H of the system in terms of a localized, real-space basis set enables it to be partitioned without ambiguity into sub-matrices that correspond to the individual subsystems.”

In the paper

“An issue that arises when combining matrix elements from more than one DFT calculation into a single tightbinding Hamiltonian is that a common reference potential must be defined.”

In the paper

“Such an approach has been applied to complex systems containing tens of thousands of atoms (Cantele et al. , 2009; Lee and Marzari, 2006; Li and Marzari, 2011; Li et al. , 2011; Shelley and Mostofi, 2011; Shelley et al. , 2011).”

What the sources say

• Not checked yetcites Cantele et al. , 2009
• Not checked yetcites Lee and Marzari, 2006
• Not checked yetcites Li and Marzari, 2011

In the paper

“Knowledge of the finite Hamiltonian sub-matrices in Eq. (126), therefore, is sufficient to calculate the conductance of the open leadconductor-lead system given by Eq. (124).”

In the paper

“As for the surface Green's functions g { L,R } of the semiinfinite leads, it is well-known that any solid (or surface) can be viewed as an infinite (semi-infinite in the case of surfaces) stack of principal layers with nearest-neighbor interactions (Lee and Joannopoulos, 1981).”

What the sources say

• Not checked yetcites Lee and Joannopoulos, 1981

In the paper

“Truncating the matrix elements of the Hamiltonian in this way incurs a small error which is systematically controlled by increasing the size of the PL.”

In the paper

“For many strongly-correlated electron problems, the essential physics of the system can be explained by considering only a subset of the electronic states.”

In the paper

“and thus greatly reduces self-interaction errors (Kulik et al. , 2006; Mori-S´ anchez et al. , 2008; Perdew and Zunger, 1981).”

What the sources say

• Not checked yetcites Kulik et al., 2006
• Not checked yetcites Mori-Sánchez et al., 2008
• Not checked yetcites Perdew and Zunger, 1981

In the paper

“The localized nature of MLWFs in real-space makes them a natural and appealing choice of basis for electronic structure calculations as the sparsity exhibited by operators that are represented in a localized basis may be exploited in order to achieve computational efficiencies and improved scaling of numerical algorithms with respect to system size.”

In the paper

“In quantum Monte Carlo (QMC) calculations (Foulkes et al. , 2001), a significant computational effort is expended in evaluating the Slater determinant for a given electronic configuration of N electrons. These Slater determinants are usually constructed from a set of extended single-particle states, obtained from, e.g., a DFT or Hartree-Fock calculation, represented in a basis set of, e.g.,”

What the sources say

• Not checked yetcites Foulkes et al. , 2001