Maximally localized Wannier functions: Theory and applications

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8 1. Real-space representation An interesting consequence stemming from the choice of (18) as the localization functional is that it allows a natural decomposition of the functional into gaugeinvariant and gauge-dependent parts. That is, we can write where and Ω I = ∑ n [ Ω = Ω I + ˜Ω (19) ⟨ 0n | r 2 | 0n ⟩ − ∑ Rm ˜Ω = ∑ n ∑ Rm≠0n ∣ ∣⟨Rm|r|0n⟩ ∣ ] 2 (20) ∣ ∣⟨Rm|r|0n⟩ ∣ ∣ 2 . (21) It can be shown that not only ˜Ω but also Ω I is positive definite, and moreover that Ω I is gauge-invariant, i.e., invariant under any arbitrary unitary transformation (10) of the Bloch orbitals (Marzari and Vanderbilt, 1997). This follows straightforwardly from recasting Eq. (20) in terms of the band-group projection operator P , as defined in Eq. (15), and its complement Q = 1 − P : Ω I = ∑ ⟨0n|r α Qr α |0n⟩ nα = ∑ α Tr c [P r α Qr α ] . (22) The subscript ‘c’ indicates trace per unit cell. Clearly Ω I is gauge invariant, since it is expressed in terms of projection operators that are unaffected by any gauge transformation. It can also be seen to be positive definite by ∑ using the idempotency of P and Q to write Ω I = α Tr c [(P r α Q)(P r α Q) † ] = ∑ α ||P r αQ|| 2 c. The minimization procedure of Ω thus actually corresponds to the minimization of the non-invariant part ˜Ω only. At the minimum, the off-diagonal elements |⟨Rm|r|0n⟩| 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. 2. Reciprocal-space representation As shown by Blount (1962), matrix elements of the position operator between WFs take the form ∫ V ⟨Rn|r|0m⟩ = i (2π) 3 dk e ik·R ⟨u nk |∇ k |u mk ⟩ (23) and ⟨Rn|r 2 |0m⟩ = − V ∫ (2π) 3 dk e ik·R ⟨u nk |∇ 2 k|u mk ⟩ . (24) These expressions provide the needed connection with our underlying Bloch formalism, since they allow to express the localization functional Ω in terms of the matrix elements of ∇ k and ∇ 2 k . In addition, they allow to calculate the effects on the localization of any unitary transformation of the |u nk ⟩ without having to recalculate expensive (especially when plane-wave basis sets are used) scalar products. We thus determine the Bloch orbitals |u mk ⟩ on a regular mesh of k-points, and use finite differences to evaluate the above derivatives. In particular, we make the assumption that the BZ has been discretized into a uniform Monkhorst-Pack mesh, and the Bloch orbitals have been determined on that mesh. 4 For any f(k) that is a smooth function of k, it can be shown that its gradient can be expressed by finite differences as ∇f(k) = ∑ b w b b [f(k + b) − f(k)] + O(b 2 ) (25) calculated on stars (“shells”) of near-neighbor k-points; here b is a vector connecting a k-point to one of its neighbors, w b is an appropriate geometric factor that depends on the number of points in the star and its geometry (see Appendix B in Marzari and Vanderbilt (1997) and Mostofi et al. (2008) for a detailed description). In a similar way, |∇f(k)| 2 = ∑ b w b [f(k + b) − f(k)] 2 + O(b 3 ) . (26) It now becomes straightforward to calculate the matrix elements in Eqs. (23) and (24). All the information needed for the reciprocal-space derivatives is encoded in the overlaps between Bloch orbitals at neighboring k- points M (k,b) mn = ⟨u mk |u n,k+b ⟩ . (27) These overlaps play a central role in the formalism, since all contributions to the localization functional can be expressed in terms of them. Thus, once the M mn (k,b) have been calculated, no further interaction with the electronic-structure code that calculated the ground state wavefunctions is necessary - making the entire Wannierization procedure a code-independent post-processing step 5 . There is no unique form for the localization functional in terms of the overlap elements, as it is possible 4 Even the case of Γ-sampling – where the Brillouin zone is sampled with a single k-point – is encompassed by the above formulation. In this case the neighboring k-points are given by reciprocal lattice vectors G and the Bloch orbitals differ only by phase factors exp(iG · r) from their counterparts at Γ. The algebra does become simpler, though, as will be discussed in Sec. II.F.2. 5 In particular, see Ferretti et al. (2007) for the extension to ultrasoft pseudopotentials and the projector-augmented wave method, and Freimuth et al. (2008); Kuneš et al. (2010); and Posternak et al. (2002) for the full-potential linearized augmented planewave method.
9 to write down many alternative finite-difference expressions for ¯r n and ⟨r 2 ⟩ n which agree numerically to leading order in the mesh spacing b (first and second order for ¯r n and ⟨r 2 ⟩ n respectively). We give here the expressions of Marzari and Vanderbilt (1997), which have the desirable property of transforming correctly under gauge transformations that shift |0n⟩ by a lattice vector. They are ¯r n = − 1 N k,b ∑ k,b w b b Im ln M (k,b) nn (28) (where we use, as outlined in Sec. II.A.3, the convention of Eq. (14)), and ⟨r 2 ⟩ n = 1 ∑ { [ w b 1 − |M nn (k,b) | 2] [ ] } 2 + Im ln M nn (k,b) . N (29) The corresponding expressions for the gauge-invariant and gauge-dependent parts of the spread functional are and Ω I = 1 N ∑ k,b ˜Ω = 1 ∑ ∑ w b N k,b k,b m≠n n ( w b J − ∑ |M mn (k,b) | 2) (30) mn |M (k,b) mn | 2 (31) + 1 ∑ ∑ ( ) 2 w b −Im ln M nn (k,b) − b · ¯r n . N As mentioned, it is possible to write down alternative discretized expressions which agree numerically with Eqs. (28)–(31) up to the orders indicated in the mesh spacing b; at the same time, one needs to be careful in realizing that certain quantities, such as the spreads, will display slow convergence with respect to the BZ sampling (see II.F.2 for a discussion), or that some exact results (e.g., that the sum of the centers of the Wannier functions is invariant with respect to unitary transformations) might acquire some numerical noise. In particular, Stengel and Spaldin (2006a) showed how to modify the above expressions in a way that renders the spread functional strictly invariant under BZ folding. D. Localization procedure In order to minimize the localization functional, we consider the first-order change of the spread functional Ω arising from an infinitesimal gauge transformation U (k) mn = δ mn + dW mn (k) , where dW is an infinitesimal anti-Hermitian matrix, dW † = −dW , so that |u nk ⟩ → |u nk ⟩ + ∑ (k) m dW mn |u mk ⟩ . We use the convention ( ) dΩ = dΩ (32) dW dW mn nm (note the reversal of indices) and introduce A and S as the superoperators A[B] = (B − B † )/2 and S[B] = (B + B † )/2i. Defining q (k,b) n R (k,b) mn = Im ln M (k,b) nn + b · ¯r n , (33) T (k,b) mn = M mn (k,b) M nn (k,b)∗ , (34) = M mn (k,b) q (k,b) M nn (k,b) n , (35) and referring to Marzari and Vanderbilt (1997) for the details, we arrive at the explicit expression for the gradient G (k) = dΩ/dW (k) of the spread functional Ω as G (k) = 4 ∑ ) w b (A[R (k,b) ] − S[T (k,b) ] . (36) b This gradient is used to drive the evolution of the U mn (k) (and, implicitly, of the | Rn ⟩ of Eq. (10)) towards the minimum of Ω. A simple steepest-descent implementation, for example, takes small finite steps in the direction opposite to the gradient G until a minimum is reached. For details of the minimization strategies and the enforcement of unitarity during the search, the reader is referred to Mostofi et al. (2008). We should like to point out here, however, that most of the operations can be performed using inexpensive matrix algebra on small matrices. The most computationally demanding parts of the procedure are typically the calculation of the selfconsistent Bloch orbitals |u (0) nk ⟩ in the first place, and then the computation of a set of overlap matrices M (0)(k,b) mn = ⟨u (0) mk |u(0) n,k+b ⟩ (37) that are constructed once and for all from the |u (0) nk ⟩. After every update of the unitary matrices U (k) , the overlap matrices are updated with inexpensive matrix algebra M (k,b) = U (k)† M (0)(k,b) U (k+b) (38) without any need to access the Bloch wavefunctions themselves. This not only makes the algorithm computationally fast and efficient, but also makes it independent of the basis used to represent the Bloch functions. That is, any electronic-structure code package capable of providing the set of overlap matrices M (k,b) can easily be interfaced to a common Wannier maximal-localization code. E. Local minima 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
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Page 3 and 4: 3 tition into separate “bands”,
Page 5 and 6: 5 such that ∇ k |u nk ⟩ is well
Page 7: 7 orbitals g n (r) corresponding to
Page 11 and 12: 11 To proceed further, we write dΩ
Page 13 and 14: 13 of the states in this larger man
Page 15 and 16: 15 self-consistently at each k-poin
Page 17 and 18: 17 Problems associated with reachin
Page 19 and 20: 19 FIG. 10 (Color online) Ethane (l
Page 21 and 22: 21 Energy (eV) -10 -15 Carbon s-orb
Page 23 and 24: 23 density operator in an LCAO basi
Page 25 and 26: 25 15 Energy (eV) 10 5 0 -5 Frozen
Page 27 and 28: 27 FIG. 19 (Color online) Contour-s
Page 29 and 30: 29 straightforward. The electric di
Page 31 and 32: 31 localization length near an insu
Page 33 and 34: 33 Giustino et al. (2003) and used
Page 35 and 36: 35 which is the desired k-space bul
Page 37 and 38: 37 FIG. 25 (Color online) Schematic
Page 39 and 40: 39 obtained by unfolding the Γ-poi
Page 41 and 42: 41 FIG. 31 (Color online) Surface d
Page 43 and 44: 43 One first evaluates certain obje
Page 45 and 46: 45 VII. WANNIER FUNCTIONS AS BASIS
Page 47 and 48: 47 the conductor to the leads (Datt
Page 49 and 50: 49 operators that are represented i
Page 51 and 52: 51 2. Self-interaction and DFT + Hu
Page 53 and 54: may be expressed as where c = (µ 0
Page 55 and 56: 55 1995; Bradley et al., 1995; Davi
Page 57 and 58: 57 Alfimov, G. L., P. G. Kevrekidis
Page 59 and 60:
59 Franchini, C., R. Kovacik, M. Ma
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61 sity Press, Cambridge). Marzari,
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63 Silvestrelli, P. L., and M. Parr
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Maximally localized Wannier functions: Theory and applications

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