Maximally localized Wannier functions: Theory and applications

From the four occupied valence bands of Si or GaAs, one obtains four equivalent Wannier functions per unit cell, each localized on one of the four nearest-neighbor Si-Si or Ga-As bonds.

“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.”

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

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

The projected silicon disentangled bands show significant deviations wherever higher unoccupied unwanted states with significant sp3 character are admixed into the projected manifold.

“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.”

Projecting the band structure of silicon onto eight atomic-like sp3 hybrids yields eight disentangled bands.

“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.”

For copper, bands up to 20 eV above the Fermi energy can be meaningfully described with a minimal basis set of seven orbitals.

“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.”

For crystalline copper, projecting onto five d-type atomic orbitals yields bands that follow closely the first-principles bands away from the s-d hybridization regions, where the interpolated bands remain narrow.

“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.”

For copper, the composite set of bands is faithfully represented by seven MLWFs: five standard d-like orbitals and two s-like orbitals centered at the tetrahedral interstitial sites of the fcc structure.

“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.”

Disentangled MLWFs, such as the pz MLWFs describing the π/π* manifold of graphene, are found to be strongly localized.

“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 gallium arsenide, the covalent bond and its WFC are shifted towards the more electronegative arsenic atom.

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

Static polarizabilities for molecules calculated with both MLWFs and NGWFs show remarkable agreement.

“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).”

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

“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.”

The closed manifold of four valence bands yields four equivalent MLWFs that map into one another under the space-group symmetry operations of the crystal.

“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.”

Symmetric orthonormalization of the three NMTOs gives the familiar set of three symmetry-related σ-bonding orbitals.

“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).”

MLWFs and NMTO orbitals for a set of t2g states around the Fermi level in SrVO3 are essentially identical.

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

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For the sp2 bonding bands of graphite, the downfolded calculation agrees with the full calculation perfectly to within the resolution of the figure.

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

• cites Zurek et al., 2005Not checked yet

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For crystalline silicon, the four-dimensional manifold disentangled from the lowest part of the conduction bands gives rise to four identical antibonding orbitals originating from destructive interference of two atomic sp3 orbitals centered on neighboring atoms.

“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.”

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 sp2 bands.

“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 larger set for graphene, the bond-centered MLWFs come from the constructive superposition of two sp2 orbitals pointing towards each other from adjacent carbons.

“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).”

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For the doubly charged split-vacancy configuration, ionic relaxation moves one nearest neighbor of the vacancy halfway towards the vacancy, relocating it to the center of an octahedral cage of silicon atoms.

“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.”

• cites Corsetti and Mostofi, 2011Not checked yet

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Atom-centered sp3 orbitals typical of group-IV or III-V semiconductors, obtained 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.

“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).”

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• cites Wahn and Neugebauer, 2006Not checked yet

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For BaTiO3, there are three MLWFs per oxygen derived from the nine oxygen 2p bands.

“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.”

This gives rise to six defect WFs, each corresponding to a bond between sp3d2 hybrids on the central atom and dangling sp3 orbitals on the neighbors.

“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.”

• cites Corsetti and Mostofi, 2011Not checked yet

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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 sp3 tight-binding orbitals of crystalline silicon.

“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).”

A very good tight-binding basis for 3d metals such as Cu can be constructed with 5 atom-centered d-like orbitals and two s-like orbitals in the interstitial positions.

“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.”

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In 2D boron sheets made more compact from hexagonal to triangular, the in-plane bonding pattern of the hexagonal system is preserved, with only minor changes in the shape and position of the MLWFs.

“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.”

• cites Tang and IsmailBeigi, 2009Not checked yet

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In silicon nanoclusters under pressure, the bond angles formed by two WFCs and their common Si atom show considerable departure from the tetrahedral rule at the transition pressure.

“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.”

• cites Martonak et al., 2000Not checked yet

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• cites Martonak et al., 2001Not checked yet

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• cites Molteni et al., 2001Not checked yet

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This work demonstrated the importance of including the occupied defect levels in the gap when constructing the relevant WFs.

“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.”

• cites Corsetti and Mostofi, 2011Not checked yet

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

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

• cites Silvestrelli and Parrinello, 1999aNot checked yet

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• cites Silvestrelli and Parrinello, 1999bNot checked yet

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The total anomalous dynamical charge in BaTiO3 can be accounted for quantitatively on the basis of the distortion-induced changes of each kind of WF in the crystal.

“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).”

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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.

“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”

• cites Marzari and Vanderbilt, 1997Not checked yet

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Implementation of the maximal localization approach in the solid-state context required significant developments due to the ill-conditioned nature of the position operator in periodic systems.

“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).”

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• cites King-Smith and Vanderbilt, 1993Not checked yet

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• cites Resta, 1994Not checked yet

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The charge density obtained by summing the squares of the Bloch functions |ψ n k> or the Wannier functions |R n> is identical.

“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.”

Marzari and Vanderbilt showed that 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.

“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.”

• cites Marzari and Vanderbilt, 1997Not checked yet

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• AssumptionThe authors state this as an assumption: The band of interest is isolated, remaining separated by a gap from the bands below and above at all k.

Both Bloch and Wannier sets of states provide an equally valid description of the band subspace, even if the Wannier functions are not Hamiltonian eigenstates.

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

The Wannier functions |R n> form an orthonormal set.

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

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

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

Two Wannier functions |R n> and |R' n> transform into each other under translation by the lattice vector R-R'.

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

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A gauge freedom exists in the definition of the ψ n k: one can replace them by a phase factor without changing the physical description of the system.

“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 .”

A major reason for the slow implementation of Wannier functions is that WFs are strongly non-unique.

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

Before computing WFs for a material, one had first to resolve which states to use to compute WFs.

“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.”

Since k lives in the periodic Brillouin zone, the best one can do is to choose equal amplitudes all across the Brillouin zone.

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

• AssumptionThe authors state this as an assumption: The band of interest is isolated, remaining separated by a gap from the bands below and above at all k.
• AssumptionThe authors state this as an assumption: The independent-particle approximation is valid for describing the electronic ground state.

There are formal connections between the centers of charge of WFs and the Berry phases of Bloch functions as they are carried around the Brillouin zone.

“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.”

Bloch and Wannier functions span the same Hilbert space.

“Bloch and WFs span the same Hilbert space.”

The gauge indeterminacy is all the more troublesome in the common case of degeneracy for occupied bands at certain high-symmetry points in the Brillouin zone, making a partition into separate bands problematic.

“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.”

Marzari and Vanderbilt (1997) introduced a maximal localization criterion for identifying a unique set of WFs for a given crystalline insulator.

“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.”

• cites Marzari and Vanderbilt, 1997Not checked yet

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• AssumptionThe authors state this as an assumption: The band of interest is isolated, remaining separated by a gap from the bands below and above at all k.

To get a very localized wave packet in real space, one needs to use a very broad superposition in k space.

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

• AssumptionThe authors state this as an assumption: The band of interest is isolated, remaining separated by a gap from the bands below and above at all k.
• AssumptionThe authors state this as an assumption: The independent-particle approximation is valid for describing the electronic ground state.

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.

“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 many applications the bands of interest are not isolated from all other bands by finite gaps throughout the Brillouin zone.

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

When the | ̃ψ n k> are substituted into Eq. (3) in place of the |ψ n k>, they result in a set of well-localized Wannier functions.

“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.”

The trial functions do not need to resemble the target Wannier functions closely; simple analytic functions (e.g., spherical harmonics times Gaussians) are often sufficient provided they are roughly located on sites where Wannier functions are expected to be centered and have appropriate angular character.

“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.”

All contributions to the localization functional can be expressed in terms of the overlaps M(k,b) mn between Bloch orbitals at neighboring k-points.

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

There is no 'preferred gauge' assigned by the Schrödinger equation to the Bloch orbitals.

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

The problem of computing well localized Wannier functions starting from entangled bands is broken down into two distinct steps: subspace selection and gauge selection.

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

The gauge freedom expressed in Eq. (8) implies that the Wannier functions are strongly nonunique.

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

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

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

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.

“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.”

The difficulty in treating entangled bands stems from the fact that it is unclear exactly which states to choose to form J Wannier functions, particularly where the bands of interest are hybridized with unwanted bands.

“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.”

The problem of producing well-localized Wannier functions arises when there are degeneracies among the bands in question at certain locations in the Brillouin zone.

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

n is no longer a band index in the usual sense for the | ̃ψ n k> resulting from Eq. (8).

“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.”

Wannier functions are 'more non-unique' than Bloch functions, which only acquire a phase change.

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

The | ̃ψ n k> resulting from Eq. (8) are no longer generally eigenstates of H.

“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.”

The projection operator onto the band manifold at wavevector k is invariant under the unitary transformation of Eq. (8).

“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.”

The projection method already described in Sec. II.B has been found to be extremely effective for initializing the minimization procedure with a smooth gauge.

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

Band four in GaAs is nonanalytic in k in the vicinity of Γ.

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

The same guiding principle, achieving smoothness in k-space, can be used for both subspace selection and gauge selection.

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

If one simply inserts 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 Wannier functions.

“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.”

Different choices of smooth gauge correspond to different sets of Wannier functions having in general different shapes and spreads.

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

The non-uniqueness of the Wannier functions resulting from Eq. (3) is unavoidable.

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

The | ̃ψ n k> are related to the original |ψ n k> by a unitary transformation.

“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.”

The | ̃ψ n k> are uniquely defined by the trial orbitals g n(r) and the chosen isolated manifold, eliminating any gauge freedom.

“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.”

Constructing a single Wannier function from the highest occupied band n=4 in GaAs would be doomed to failure, since this band becomes degenerate with bands two and three at the zone center, Γ.

“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.”

Exponential localization of maximally localized Wannier functions for composite bands in 2D and 3D was proven by Panati and Pisante (2011).

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

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The early work on composite bands in 3D only established the exponential localization of the projection operator P, Eq. (13), not of the Wannier functions themselves.

“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).”

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Maximally localized Wannier functions obtained by minimizing the quadratic spread functional are exponentially localized in 1D.

“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).”

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• cites Niu, 1991Not checked yet

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A necessary and sufficient condition for the existence of exponentially localized Wannier functions in 2D and 3D is that the Chern invariants for the composite set of bands vanish identically.

“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.”

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• cites Panati (2007)Not checked yet

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The projection method is successful as long as the inner-product matrix A k does not become singular (or nearly so) for any k.

“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.”

The Fourier transform of Eq. (3) for band four in GaAs would result in a poorly localized object having power-law tails in real space.

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

The localization functional Ω is the solid-state equivalent of the Foster-Boys criterion of quantum chemistry.

“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).”

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• cites Foster and Boys, 1960a,bNot checked yet

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The minimization procedure of Ω corresponds to the minimization of the non-invariant part ̃Ω only.

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

Spherical (s-like) Gaussians located at the bond centers will suffice for the construction of well-localized Wannier functions spanning the four occupied valence bands of semiconductors such as Si and GaAs.

“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.”

The Löwdin-orthonormalized Bloch-like states | ̃ψ n k> have a smooth gauge in k.

“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.”

Nonzero Chern numbers pose an obstruction to finding a globally smooth gauge in k-space.

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

The quadratic spread Ω of Wannier functions spanning a Bloch space of dimension J comprises two positive-definite contributions: a gauge-invariant part Ω_I and a gauge-dependent part ̃Ω.

“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).”

The projection technique can be adapted to produce J smoothly-varying Bloch-like states starting from a larger set of Bloch bands.

“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).”

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Smoothness in k of a Bloch space is precisely what is measured by the functional Ω_I.

“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.”

Combining subspace and gauge selection by minimizing Ω = Ω_I + ̃Ω simultaneously should lead to the most-localized set of J Wannier functions that can be constructed from the initial J_k Bloch states.

“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.”

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For a double bond, both the Foster-Boys and Edmiston-Ruedenberg schemes mix the σ and π orbitals, giving rise to two bent banana bond (τ) orbitals.

“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.”

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The one-shot projection procedure often results in symmetry-adapted Wannier functions that inherit the symmetry properties of the trial orbitals.

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

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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.

“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.”

The localization of trial orbitals in real space leads to smoothness in k-space.

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

Selecting the J eigenvectors with largest eigenvalues ensures that at self-consistency the stationary point corresponds to the absolute minimum of Ω_I.

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

An iterative procedure extracts self-consistently across the Brillouin zone the J-dimensional subspaces having collectively the smallest possible value of Ω_I.

“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).”

• cites Souza et al., 2001Not checked yet

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Ω_I captures the intrinsic smoothness of the underlying Hilbert space.

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

The deviations from unwanted states can be avoided by forcing certain Bloch states to be preserved identically in the projected manifold, a frozen inner window.

“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.”

The Bloch subspace obtained at the end of the minimization is optimally smooth in that it changes as little as possible with k.

“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 .”

Once the two-particle electron-electron interaction is formally included in the Hamiltonian, the many-body wavefunction cannot be reduced to a simple form allowing construction of Wannier functions in the usual sense.

“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.”

Allowing non-orthogonality removes the need to enforce explicit orthogonality constraints on the orbitals during energy minimization.

“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).”

• cites Galli and Parrinello, 1992Not checked yet

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• cites Hernández et al., 1996Not checked yet

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• cites Hierse and Stechel, 1994Not checked yet

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Qian et al. (2008) realized that quasiatomic orbitals can be constructed without explicit calculation of the eigenstates outside the frozen window.

“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.”

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A nonorthogonal representation can be more localized than an essentially equivalent orthogonal one.

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

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• cites He and Vanderbilt, 2001Not checked yet

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The spatial localization of the density matrix is closely linked to that of the Wannier functions themselves.

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

In variational linear-scaling methods, idempotency of the density operator or orthogonality of the Wannier orbitals is not imposed explicitly during minimization; instead, the energy functionals are constructed so that these properties are satisfied automatically at the minimum, which coincides with the true ground state.

“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.”

The density matrix takes the same exponential form in metals at finite temperature, with γ determined by the ratio between the thermal energy kBT and the Fermi velocity.

“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).”

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These MLWFs represent the intuitive chemical concept of a covalent bond, with each MLWF representing the bonding orbital created by constructive interference of two atomic sp3 orbitals centered on neighboring atoms.

“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.”

Accurate calculations on tetrahedral semiconductors require the inclusion of d states in the MTO basis, i.e., nine states (s, p, d) per site.

“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.”

• cites Lambrecht and Andersen, 1986Not checked yet

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The fundamental principle behind linear-scaling approaches is that electronic structure is inherently local or nearsighted.

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

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• cites Kohn, 1996Not checked yet

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One of the most notable characteristics of MLWFs is their ability to highlight the chemical signatures of different band manifolds.

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

The single-particle density matrix decays exponentially as exp(-γ|r-r′|) in insulators and semiconductors, with the exponent γ heuristically related to the direct band gap.

“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).”

• cites des Cloizeaux, 1964aNot checked yet

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• cites Ismail-Beigi and Arias, 1999Not checked yet

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• cites Taraskin et al., 2002Not checked yet

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Because MLWFs are uniquely defined for insulators and semiconductors, they are particularly suited to discuss hybridization, covalency, and ionicity in crystalline and disordered systems.

“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.”

Using symmetry-preserving Wannier functions ensures that properties computed from them, such as interpolated bandstructures, have the correct symmetry properties.

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

The locality of the density matrix is exploited in linear-scaling methods by retaining an amount of information that scales only linearly with system size.

“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 supercell approximation, MLWFs can be used to describe any disordered bulk, amorphous, or liquid system, providing a compact description in terms of Wannier centers, coordination, and spatial distribution and symmetry.

“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.”

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

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

When conduction bands are included, it becomes riskier to draw conclusions from the qualitative features of the MLWFs.

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

Whenever unoccupied states are included, MLWFs are more properly understood as forming a minimal tight-binding basis, and not necessarily as descriptors of bonding.

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

A low activation energy for self-annihilation of the defect, combined with its stealth electronic signature, hints at why such a defect could have eluded experimental discovery despite Si being one of the best studied materials.

“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.”

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

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

A purely geometrical coordination-number analysis is completely blind to the actual electronic charge distribution, which ought to be important in any description of chemical bonding.

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

The use of MLWFs to characterize electronic bonding in complex systems has been greatly aided by efficient and robust algorithms for maximal localization in large, often disordered, supercells with Brillouin zone sampling at a single point, usually the zone-center Γ point.

“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”

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• cites Bernasconi and Madden, 2001Not checked yet

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• cites Silvestrelli, 1999Not checked yet

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The WFC analysis can also be effectively employed to elucidate the chemical and electronic properties accompanying structural transformations.

“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.”

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• cites Lim et al., 2002Not checked yet

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• cites Meregalli and Parrinello, 2001Not checked yet

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MLWFs can reveal the atomic-like orbitals that conceptually lie behind any tight-binding formulation, obtained directly from first principles by a well-defined procedure.

“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.”

MLWFs analysis can point to structural defects that do not otherwise exhibit any significant electronic signature.

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

WFCs can be a powerful tool for understanding bonding in low-symmetry cases, representing an insightful and economical mapping of continuous electronic degrees of freedom into classical descriptors (center positions and spreads of WFs).

“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.”

• cites Silvestrelli et al., 1998Not checked yet

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These tight-binding orbitals can form the basis for constructing Hamiltonians for model systems or large-scale nanostructures.

“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.”

The same eR/V describes the quantization of charge transport under an adiabatic cycle.

“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).”

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The Ziegler-Natta polymerization is a paradigmatic example of the chemical insight that can be gleaned by following the WFCs in the course of a first-principles simulation.

“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.”

• cites Boero et al., 2000bNot checked yet

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The change of polarization resulting from any physical change can be related in a simple way to the displacements of the Wannier centers r_n occurring as a result of this change.

“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.”

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.

“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 .”

The modern theories of electric polarization and orbital magnetization can be formulated either in terms of Berry phases and curvatures, or equivalently, by working in the Wannier representation.

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

• cites King-Smith and Vanderbilt, 1993Not checked yet

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• cites Resta, 1992Not checked yet

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• cites Thonhauser et al., 2005Not checked yet

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Souza et al. (2000) have shown that the invariant quadratic spread Ω_I characterizes the root-mean-square quantum fluctuations of the macroscopic polarization.

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

• cites Souza et al., 2000Not checked yet

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Eq. (89) can be rigorously justified by noting that its second term is precisely the expression for the electronic contribution to the polarization in the Berry-phase theory.

“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.”

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• cites Resta, 1994Not checked yet

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• cites Vanderbilt and King-Smith, 1993Not checked yet

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Stengel and Spaldin introduced a Wannier-based method for computing polarizations along z and studying electric fields along z that works even when the stacking includes metallic layers, as long as the system is effectively insulating along z.

“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).”

• cites Stengel and Spaldin, 2006aNot checked yet

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• cites Stengel and Spaldin, 2007Not checked yet

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The Wannier representation provides an alternative method for computing dielectric quantities such as electric susceptibility and Born effective charges by finite differences.

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

The anomalous dynamical effective charges in BaTiO3 can be understood as arising from hybridization between neighboring O 2p and Ti 3d orbitals that dominate the valence and conduction bands.

“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.”

• cites Posternak et al., 1994Not checked yet

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• cites Zhong et al., 1994Not checked yet

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The electric-enthalpy approach is now standard for treating periodic insulators in finite electric fields in density-functional theory.

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

• cites Nunes and Vanderbilt, 1994Not checked yet

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• cites Nunes and Gonze, 2001Not checked yet

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• cites Souza et al., 2002Not checked yet

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The polarization is only well-defined modulo a quantum eR/V, where R is a real-space lattice vector.

“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.”

Matrix elements such as 〈ψnk|r|ψnk〉 and 〈ψnk|r×v|ψnk〉 are ill-defined for extended Bloch states.

“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).”

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Specialized algorithms have been developed to perform on-the-fly Car-Parrinello molecular dynamics in a Wannier representation.

“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).”

• cites Iftimie et al., 2004Not checked yet

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• cites Sharma et al., 2003Not checked yet

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• cites Wu et al., 2009Not checked yet

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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.

“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.”

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.

“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.”

The orbital magnetoelectric coefficient αij = ∂Morb,j/∂Ei contains an isotropic contribution having a topological character (the axion contribution).

“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).”

• cites Essin et al., 2009Not checked yet

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• cites Qi et al., 2008Not checked yet

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Methods that follow the time-evolution of local dipoles defined via WF-based methods provide additional insight and efficiency compared to extracting information from the time-time correlation function of the total polarization P(t) of the entire supercell.

“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”

• cites Bernasconi et al., 1998Not checked yet

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• cites Chen et al., 2008Not checked yet

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• cites McGrath et al., 2007Not checked yet

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Starting from the short-range real-space representation, a quantity f can be accurately interpolated onto an arbitrary point k in reciprocal space.

“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).”

The interpolation procedure succeeds in resolving details of the true band structure on a scale much smaller than the spacing between q-points.

“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.”

• cites Yates et al., 2007Not checked yet

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Insulators having broken time-reversal symmetry can be classified by an integer 'Chern invariant' that is proportional to the Brillouin-zone integral of the Berry curvature F_nk summed over occupied bands n.

“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 .”

The ability to compute the orbital magnetization is also of use in obtaining the magnetic shielding of nuclei, which is responsible for the chemical shift effect observed in NMR experiments.

“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.”

A simplified Wannier-based approach succeeds in capturing the essential features of topological surface states at greatly reduced computational cost.

“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.”

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Computing the orbital magnetization as the thermodynamic limit of Eq. (85) per unit volume for a large crystallite is not suitable because Bloch matrix elements of the position operator r and the circulation operator r×v are ill-defined.

“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.”

By virtue of the spatial localization of WFs, the Wannier-transformed quantity F(R) decays rapidly with |R|.

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

Wannier interpolation can determine GW quasiparticle bands very efficiently and accurately at arbitrary points in the BZ.

“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.”

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The Chern-Simons magnetoelectric coupling has been evaluated from first-principles with WFs for both topological and ordinary insulators.

“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).”

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The interpolated GW bands of SrTiO3 pass through the exact GW results at symmetry points.

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

• cites Hamann and Vanderbilt, 2009Not checked yet

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A Wannier representation is not possible for a QAH insulator.

“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.”

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Wannier interpolation is particularly useful in situations where a very fine sampling of the BZ is required to converge the quantity of interest.

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

Wannier band-structure interpolation is extremely accurate.

“Wannier band-structure interpolation is extremely accurate.”

Localized Wannier functions provide a rigorous justification for the tight-binding interpolation method.

“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).”

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• cites Harrison, 1980Not checked yet

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In 3D topological insulators, the clearest experimental signature of Z2-odd character is provided by ARPES measurements of surface electron bands.

“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.”

The interpolation procedure succeeds in capturing variations in f(k) over reciprocal lengths smaller than the first-principles mesh spacing Δq, provided the supercell linear dimensions are large compared to the decay length of the WFs.

“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.”

The interpolated bands obtained using an 8×8×8 q-grid for bcc Fe show essentially perfect agreement with first-principles bands within the inner window.

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

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This block tridiagonal form can be exploited to calculate the diagonal elements of the Green's function matrix very efficiently using iterative schemes.

“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).”

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• cites Lopez-Sancho et al., 1985Not checked yet

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The topological axion term can be written as a Brillouin-zone integral of the Chern-Simons 3-form.

“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).”

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• cites Qi et al., 2008Not checked yet

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The nearest-neighbor coupling between principal layers means the Hamiltonian has a block tridiagonal form for each k‖.

“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).”

Beyond a certain mesh density, the error incurred in Wannier interpolation decreases exponentially.

“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).”

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The Wannier-based approach for topological surface states contains no surface-specific information, being based exclusively on bulk WFs.

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

A complete expression for the orbital magnetoelectric response has been derived, of which the topological Chern-Simons term is only one contribution.

“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”

• cites Essin et al., 2010Not checked yet

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• cites Malashevich et al., 2010Not checked yet

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Nonmagnetic T-invariant insulators can be sorted into two classes, Z2-even and Z2-odd, depending on how their Bloch functions wrap the Brillouin zone.

“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).”

• cites Hasan and Kane, 2010Not checked yet

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• cites Hasan and Moore, 2011Not checked yet

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Efficient interpolation schemes like BoltzTraP are still prone to numerical instabilities near band degeneracies.

“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).”

• cites Madsen and Singh, 2006Not checked yet

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• cites Uehara and Tse, 2000Not checked yet

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In ferromagnetic metals, the Berry curvature displays strong and rapid variations in k-space.

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

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

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

The AHC can be recast in terms of the Berry phases of Fermi loops on planar slices of the Fermi surface.

“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.”

Wang et al. (2007) implemented the Berry-phase approach using Wannier interpolation to sample efficiently the orbits with the very high density required near band-crossings.

“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.”

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The Wannier-based calculation of AHC in bcc Fe was found to be in excellent agreement with the sum-over-states calculation.

“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).”

• cites Wang et al., 2006Not checked yet

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• cites Yao et al., 2004Not checked yet

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Such instabilities can be avoided by using a tight-binding parametrization to fit the first-principles band structure.

“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).”

• cites Mazin et al., 2000Not checked yet

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• cites Schulz et al., 1992Not checked yet

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The Berry phase obtained by integrating the Berry connection over a closed loop in k-space is gauge-invariant.

“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).”

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The localization in real-space of the electron-phonon vertex in the Wannier representation ensures that Eq. (122) smoothly interpolates g_W in the electron and phonon momenta.

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

WFs can be used to identify and focus on a desired, physically relevant subspace of 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.

“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”

The matrix element for scattering an electron from state ψ_nk to state ψ_m,k+q while absorbing a phonon qν is proportional to the electron-phonon vertex.

“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”

The cost of evaluating the electron-phonon vertex from first-principles over a large number of (k,q)-points is quite high, placing a serious limitation on the scope and accuracy of first-principles techniques for electron-phonon problems.

“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.”

Wannier functions are sufficiently localized that any resulting tight-binding representation may be truncated with little loss of accuracy.

“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.”

WFs can be used to go up in length scale, using results of electronic-structure calculations on small systems to construct accurate models of larger, often meso-scale, systems.

“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.”

Expressing the Hamiltonian in a localized, real-space basis set enables it to be partitioned without ambiguity into sub-matrices corresponding to individual subsystems.

“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.”

When combining matrix elements from more than one DFT calculation into a single tight-binding Hamiltonian, a common reference potential must be defined.

“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.”

The MLWF fragment approach has been applied to complex systems containing tens of thousands of atoms.

“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).”

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• cites Lee and Marzari, 2006Not checked yet

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• cites Li and Marzari, 2011Not checked yet

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Knowledge of the finite Hamiltonian sub-matrices is sufficient to calculate the conductance of the open lead-conductor-lead system.

“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).”

Any solid or surface can be viewed as an infinite or semi-infinite stack of principal layers with nearest-neighbor interactions.

“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).”

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Truncating the matrix elements of the Hamiltonian using principal layers incurs a small error which is systematically controlled by increasing the size of the principal layer.

“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.”

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

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

DFT+U greatly reduces self-interaction errors.

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

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• cites Mori-Sánchez et al., 2008Not checked yet

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• cites Perdew and Zunger, 1981Not checked yet

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The localized nature of MLWFs in real-space makes them a natural and appealing choice of basis for electronic structure calculations because sparsity exhibited by operators in a localized basis may be exploited to achieve computational efficiencies and improved scaling with system size.

“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 QMC calculations, evaluating the Slater determinant for a given electronic configuration of N electrons usually scales as O(N^3).

“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.,”

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DFT+U effectively corrects for the lack of piecewise linearity in approximate energy functionals.

“DFT+ U effectively corrects for the lack of piecewise linearity in approximate energy functionals (Cococcioni and de Gironcoli, 2005; Perdew et al. , 1982)”

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• cites Perdew et al., 1982Not checked yet

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Functionals that aim to correct directly for self-interaction effects, such as the Perdew-Zunger correction or Koopmans-compliant functionals, can lead naturally in a periodic system to Wannier-like localized orbitals that minimize the total energy.

“functionals that aim to correct directly for some effects of self-interaction -such as the Perdew-Zunger correction (Perdew and Zunger, 1981) or Koopmans-compliant functionals (Dabo et al. , 2010) - can lead naturally in a periodic system to Wannier-like localized orbitals that minimize the total energy (Park et al. , 2011; Stengel and Spaldin, 2008)”

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• cites Dabo et al., 2010Not checked yet

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• cites Park et al., 2011Not checked yet

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The required unitary transformation for lattice Wannier functions is the matrix of eigenvectors [e_q]_{μν}.

“from which we conclude that the LWFs do indeed correspond to individual atomic displacements δ RR ′ δ ss ′ and, furthermore, that the required unitary transformation is the matrix of eigenvectors [ e q ] µν .”

MLWFs have been successfully used as Hubbard projectors.

“In fact, MLWFs have been successfully used as Hubbard projectors (Anisimov et al. , 2007; Fabris et al. , 2005; Miyake and Aryasetiawan, 2008)”

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• cites Fabris et al., 2005Not checked yet

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• cites Miyake and Aryasetiawan, 2008Not checked yet

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The non-uniqueness of the transformation between Bloch states and Wannier functions caused difficulties for early work on Wannier functions in photonic crystals.

“Although of great formal importance for justifying the existence of a suitable localized basis, and hence the tight-binding approach, the non-uniqueness of the transformation between Bloch states and Wannier functions caused difficulties.”

Kohn (1973) first showed that a localized orthonormal basis for lattice vibrations spanning the same space as delocalized normal modes could be constructed for isolated phonon branches in one dimension.

“By analogy with the electronic case, Kohn (1973) first showed (for isolated phonon branches in one dimension) that a similar approach could be used for constructing a localized orthonormal basis for lattice vibrations that span the same space as the delocalized normal modes.”

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The proof of exponential localization of lattice Wannier functions follows the same reasoning as for electronic Wannier functions.

“As noted by Kohn (1973), the proof of exponential localization of LWFs follows the same reasoning as for electronic Wannier functions (Sec. II.G).”

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Leung (1993) first suggested that transforming to a basis of Wannier functions localized in real space would be advantageous for computational efficiency in photonic crystals, especially when dealing with defects.

“Leung (1993) first suggested that transforming to a basis of Wannier functions localized in real space would be advantageous for computational efficiency, especially when dealing with defects in photonic crystals which, using conventional methods, require very large supercells for convergence.”

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Maximally-localized lattice Wannier functions correspond to displacements of individual atoms.

“Finally, as first pointed out by Kohn (1973), and subsequently by Giustino et al. (2007a), maximally-localized lattice Wannier functions correspond to displacements of individual atoms.”

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• cites Giustino et al. (2007a)Not checked yet

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A basis of Wannier functions localized to each site of an optical lattice may be used to expand the condensate wavefunction.

“As shown, for example, by Alfimov et al. (2002), a basis of Wannier functions, localized to each site α of the optical lattice, may be used to expand the condensate wavefunction,”

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The Marzari and Vanderbilt approach has been incorporated as a feature into a large number of modern first-principles electronic-structure code packages including Quantum ESPRESSO, Abinit, FLEUR, WIEN2k, Siesta, and VASP.

“it has been incorporated as a feature into a large number of modern first-principles electronic-structure code packages including Quantum ESPRESSO (Giannozzi et al. , 2009), Abinit (Gonze et al. , 2009), FLEUR (Freimuth et al. , 2008), WIEN2k (Kuneˇ s et al. , 2010; Schwarz et al. , 2002), Siesta (Koryt´ ar et al. , 2010; Soler et al. , 2002), and VASP (Franchini et al. , 2011; Kresse and Furthm¨”

• cites Giannozzi et al., 2009Not checked yet

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• cites Gonze et al., 2009Not checked yet

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• cites Kresse and Furthmuller, 1996Not checked yet

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The Marzari and Vanderbilt (1997) approach is essentially the generalization of the Foster and Boys approach to periodic systems, in which gauge freedom is resolved by minimizing the sum of the quadratic spreads of the Wannier functions.

“our emphasis has been on the one of Marzari and Vanderbilt (1997), essentially the generalization of the approach of Foster and Boys (Boys, 1960, 1966; Foster and Boys, 1960a,b) to periodic systems, in which the gauge freedom is resolved by minimizing the sum of the quadratic spreads of the WFs.”

• cites Marzari and Vanderbilt (1997)Not checked yet

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• cites Boys, 1960, 1966Not checked yet

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• cites Foster and Boys, 1960a,bNot checked yet

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