Generating Wannier Function within OpenMX Hongming Weng ( ) - PowerPoint PPT Presentation

Generating Wannier Function within OpenMX Hongming Weng ( 翁红明 ) Institute of Physics, Chinese Academy of Sciences July. 2-12, 2018@ISSP

Wave-function in Solids periodical boundary condition 1, Bloch representation R ] = 0 ⇒ ψ n k (r) = u n k (r) e i k ⋅ r [H,T r R ψ n k (r) → e i ϕ n (k) ψ n k (r) phase factor of periodic good quantum number: 2, Wannier representation n — band index V k — crystal momentum ψ n k (r) e i ϕ n (k ) − i k ⋅ R d k w n (r − R) = R n = ∫ (2 π ) 3 R — Lattice BZ Equivalence of these two representations: span the same Hilbert space

Orthonormality & Completeness of Wannier Function 2 V ⎛ ⎞ − i ϕ m ( ʹ k ) + i ʹ k k k ⋅ R R d ʹ ψ n k (r) e i ϕ n (k ) − i k ⋅ R d k ⋅ d r R m R n = R k (r) e k k ∫ ∫ ʹ ∫ ʹ ψ m ʹ ⎜ ⎟ r k π ) 3 (2 BZ BZ ⎝ ⎠ 2 V ⎛ ⎞ − i ϕ m ( ʹ k ) + i ʹ k k ⋅ k R R d ʹ k ⋅ e i ϕ n (k ) − i k ⋅ R d k e k ∫ ∫ ʹ = ψ m ʹ k ψ n k ⎜ ⎟ k π ) 3 (2 BZ BZ ⎝ ⎠ 2 V ⎛ ⎞ e − i ϕ m ( ʹ k k ) + i ʹ k ⋅ k R d ʹ R k ⋅ e i ϕ n (k ) k − i k ⋅ R d k ∫ ∫ ʹ = δ m , n δ ʹ ⎜ ⎟ k ,k k π ) 3 (2 BZ BZ ⎝ ⎠ 2 V ⎛ ⎞ e i k( ʹ R − R ) d k R d k ∫ ∫ = ⎜ ⎟ π ) 3 (2 BZ BZ ⎝ ⎠ = δ m , n δ ʹ R R ,R

Arbitrariness of Wannier Function 1. ψ n k (r) → e i ϕ n (k) ψ n k (r) The arbitrary phase is periodic in reciprocal lattice translation G but not assigned by the Schrodinger equation. k 2. ψ n k (r) → U mn ψ n k (r) ∑ m Freedom of Gauge Choice For composite bands, choice of phase and “band-index labeling” at each k For entangling bands, the subspace should be optimized. Optimal Subspace

Maximally Localized Wannier Functions N. Marzari and D. Vanderbilt PRB56, 12847 (1997) • Localization criterion Minimizing the spread functional defined as 0 n r 2 0 n − 0 n r 0 n 2 [ ] [ ] r 2 n − r r 2 ∑ ∑ Ω = = n n n by finding the proper choice of U mn (k) for a given set of Bloch functions. • Optimization with the knowledge of Gradient dW = ε ⋅ − G ( ) (k ) u m (k ) ⇐ U mn (k ) + dW (k ) U mn u n k ⇐ u n k + dW ∑ mn mn k n N V w n (r − R) = R n = U mn (k ) ψ m k (r) e − i k ⋅ R d k ∑ ∫ (2 π ) 3 BZ m = 1 The equation of motion for U mn (k) . U mn (k) is moving in the direction opposite to the gradient to decrease the value of Ω , until a minimum is reached. A proper Gauge choice.

Spread functional in real-space 0 n r 2 0 n − 0 n r 0 n 2 [ ] ∑ Ω = n ⎡ ⎤ 0 n r 2 0 n − 2 2 R m r 0 n R m r 0 n ∑ ∑ ∑ = + ⎢ ⎥ ⎣ ⎦ n R m R m ≠ 0 n ˜ ⎡ ⎤ 0 n r 2 0 n − 2 2 R m r 0 n R m r 0 n ∑ ∑ ∑ ∑ = + = Ω I + Ω ⎢ ⎥ ⎣ ⎦ n R m n R m ≠ 0 n 2 + 2 R n r 0 n R m r 0 n ∑ ∑ ∑ ∑ = Ω I + n R ≠ 0 m ≠ n R = Ω I + Ω D + Ω OD Ω I , Ω D and Ω OD are all positive-definite . Especially Ω I is gauge-invariant , means it will not change under any arbitrary unitary transformation of Bloch orbitals. Thus, only Ω D + Ω OD should be minimized.

Spread functional in Reciprocal-space 2 + 2 R n r 0 n R m r 0 n ∑ ∑ ∑ ∑ Ω = Ω I + n R ≠ 0 m ≠ n R = Ω I + Ω D + Ω OD Using the following transformations, matrix elements of the position operator in WF basis can be expressed in Bloch function basis: = 1 ⎛ ⎞ ⎡ ⎤ 0 n r 2 0 n − (k,b) 2 2 R m r 0 n w b J − M mn ∑ ∑ ∑ ∑ Ω I = ⎜ ⎟ ⎢ ⎥ N ⎣ ⎦ ⎝ ⎠ n R m k,b m , n Ω OD = 1 (k,b) 2 w b M mn ∑ ∑ N w b b α b ∑ = δ αβ k,b m ≠ n β b Ω D = 1 2 (k,b) − b ⋅ r w b − Imln M nn r (k,b) = u n k u n k + b ∑ ∑ ( ) M nn n N k,b n

Gradient of Spread Functional (k,b) = u n k u n k + b M nn (k ) u m u n k ⇐ u n k + dW ∑ mn k m (k,b) = − (k ) M mn (k,b) (k,b) dW (k + b) dM nn dW M nl ∑ ∑ + nm ln m l = − dW (k ) M (k,b) ] nn + M (k,b) dW (k + b) [ [ ] nn d Ω I , OD = 1 ⎛ ⎞ = 4 * (k ) M mn (k,b) M nn (k,b) w b Retr dW (k ) R (k,b) w b 4Re dW ∑ ∑ ∑ [ ] ⎜ ⎟ nm N N ⎝ ⎠ k,b m , n k,b d Ω D = − 4 w b Retr dW (k ) T (k,b) ∑ [ ] N k,b d Ω G (k ) = w b A R (k, k,b) ] - S T (k,b) dW (k ) = 4 ( ) ∑ [ [ ] b

(k,b) Overlap Matrix M mn N 1 iR p k ( k ) ( r ) = e ikr u m ∈ win ( k ) ( k ) ( r ) = e C m ∈ win , i α φ i α ( r − τ i − R p ) ∑ ∑ ψ m ∈ win N p i , α k (r) u n k (r) e i k ⋅ r e (k,b) = u m ⋅ r ψ n k + b (r) = ψ m − i (k + b) k + b (r) M mn N = 1 − i R p k e i R q (k + b) (k + b) φ i α ( r − τ i − R p ) e − i b ⋅ r φ j β ( r − τ j − R q ) e C m , i α (k ) * C n , j β ∑ ∑ N p , q i , α j , β N = 1 ⋅ b φ j β ( r − τ j − R q ) − i (R p − R q )k (k + b) φ i α ( r − τ i − R p ) e − i (r − R q ) e C m , i α (k ) * C n , j β ∑ ∑ N p , q i , α j , β r ≡ r − τ i − R p ʹ N (k,b) = 1 ⋅ b φ j β ( ʹ (k + b) φ i α ( ʹ − i (R p − R q )k − i ( ʹ r + τ i + R p − R q ) r (k ) * C n , j β M mn e C m , i α r ) e r + τ i − τ j + R p − R q ) ∑ ∑ N p , q i , α j , β N ⋅ b φ j β ( ʹ (k + b) φ i α ( ʹ i R q ⋅ k − i ( ʹ r r + τ i − R q ) (k ) * C n , j β e C m , i α r ) e r + τ i − τ j − R q ) ∑ ∑ = q i , α j , β N − i b ⋅ τ i C m , i α (k + b) φ i α ( ʹ i R q ⋅ (k + b) (k ) * C n , j β − i b ⋅ ʹ r φ j β ( ʹ r e e r ) e r + τ i − τ j − R q ) ∑ ∑ = q i , α j , β

Initial guess for MLWF (k ) N win (k ) u m (k ) = u m A k g A ∑ φ n k = mn n mn k m The resulting N functions can be orthonormalized by Löwdin transformation N (k ) = φ m opt = S mn ≡ S mn k φ n k = ( A + A ) mn ( S − 1/2 ) mn φ m u n k ∑ k m = 1 (k ) N win N (k ) u p k Benefit from initial guess: ( S − 1/2 ) mn A pm ∑ ∑ = 1. to avoid the local minima and m = 1 p = 1 accelerate the convergence; (k ) N win 2. to eliminate the random phase factor − 1/2 ) pn u p k ( AS ∑ of Bloch function after diagonalizing = p = 1 Therefore, AS -1/2 is used as the initial guess of U (k )

Initial guess for MLWF in OpenMX N 1 (k ) = u m iR p k ( k ) ( r ) = e C m ∈ win , i α ( k ) φ i α ( r − τ i − R p ) A k g ∑ ∑ ψ m ∈ win mn n N p i , α In OpenMX, we use the pesudo-atomic orbital as initial trial functions. g n ( r ) = g j , β ( r ) = φ j , β ( r ) For selected Bloch function, the projection matrix element can be expressed as: N 1 * φ i α ( r − τ i − R p ) φ j , β ( r ) ( k ) = ψ m ∈ win − iR p k ( k ) ( k ) A ( r ) g n ( r ) = e C n ∈ win , i α ∑ ∑ mn N p i , α Advantages over other ab-initio program based on plane-wave basis: 1. Easier to calculate; 2. Can be tuned by generating new PAO; 3. Can be put anywhere in the unit cell; 4. Quantization axis and hybridizations can also be controlled.

Entangled Bands Case to optimize Ω I N win ( k ) Select bands locates in an energy window. These bands constitute a large F (k) space . The number of bands at each k inside the window should be larger or equal to the number of WF. Target is to find an optimized subspace S (k) , which gives the smallest Ω I N kp Ω I = 1 w b T ∑ ∑ k , b N kp k = 1 b (k,b) 2 T k , b = N − M mn = Tr P k Q k + b ∑ [ ] m , n P k is the operator which project onto a set of Inner window bands while Q k+b is projecting onto the left set of bands. Therefore, Ω I measures the mismatch between two sets of bands at neighboring k and k+b points, respectively.

Iterative minimization of Ω I Using Lagrange multipliers to enforce orthonormality and the stationary condition at i- th iteration is: N ( i ) ( i ) u nk δ Ω I δ ( i ) − δ m , n ( i ) [ u mk ] = 0 ∑ * + Λ nm , k ( i ) ( i ) * δ u δ u mk mk n = 1 N kp ( i ) = 1 ( i ) ( k ) ∑ Ω I ω I N kp k = 1 ( i ) ( k ) = ( i ) ( i ) w b T w b T ∑ ∑ ω I = k , b k , b b b N N ⎡ ⎤ ( i ) u n , k + b 2 ( i − 1 ) w b 1 − u mk ∑ ∑ ∑ = ⎢ ⎥ ⎣ ⎦ b m = 1 n = 1 is the subspace at k point S ( i ) If inner window is set, the space needed to be (k ) in the i- th iteration optimized is smaller.

Interpolation of band structure q is the grid of BZ used for constructing MLWF N q N q n R = 1 − i q ⋅ R u n q ( W ) = ( W ) e u n q U mn (q) φ m ∑ ∑ q N q q = 1 m = 1 ( W ) q ( W ) H(q) u m * (q) u i q H(q) ( W ) = H nm ( ) = u n q U in U jm ( q ) u j q ∑ ∑ q i j * (q) u i q H(q) u j q U jm ( q ) = U in U + H(q) U ∑ [ ] nm = i , j Hamiltonian in Wannier gauge can be diagonalized and the bands inside the inner window will have the same eigenvalues as in original Hamiltonian gauge. Other operators can be transferred into Wannier gauge in the similar way.

Interpolation of band structure Fourier transfer into the R space: N q ( W ) (R) = 1 − i q ⋅ R H nm ( W ) (q) H nm e ∑ N q q = 1 Here R denotes the Wigner-Seitz supercell centered home unit cell. To do the interpolation of band structure at arbitrary k point, inverse Fourier transform is performed: ( W ) (k) = e i k ⋅ R H nm ( W ) (R) H nm ∑ R Diagonalize this Hamiltonian, the eigenvalues and states will be gotten. This is directly related to Slater-Koster interpolation, with MLWFs playing the role of the TB basis orbitals.

Wannier in OpenMX a chain of V-Benzene 16

Wannier in OpenMX redefined local coordinate axis of z’ and x’. y’ is automatically checked to be satisfied in right- handed system. 17