Closest Wannier functions to a given
set of localized orbitals•
••
••

1/25

Motivations of the study
Maximally localized Wannier function
Closest Wannier functions
Benchmark calculations
Outlook
Refs:
arXiv:2306.15296
https://www.openmx-square.org/cwf/

Taisuke Ozaki
ISSP, the Univ. of Tokyo
The 4th OpenMX developer's meeting on Nov. 9-10, 2023

How to analyze DFT results?
⚫ The DFT-KS method is a versatile framework to calculate electronic
structures of real materials.

E    =  dr 3  (r )vext (r ) + T    + J    + Exc   Hˆ KSi =  ii

1 2ˆ
H KS = −  + veff
2The right structure isenergetically stable, being
consistent with the experiment.

⚫ However, it is always difficult to explain why one structure is more stable than theothers. Other cases: effective charge, XPS binding energy, effective spin-spininteraction.
⚫ We need to develop an analysis method to give a physically and chemicallyconvincing interpretation.

2/25

Purpose of the study⚫ Our goal in this study is to establish methods to analyze electronicstructures in a physically and chemically convincing manner.
⚫ To do that, I consider that it is important to develop an efficient androbust method to calculate Wannier functions (WFs), while keeping theshape of atomic orbitals as much as possible.
⚫ Such WFs can be utilized for calculations including⚫ Effective atomic charge⚫ Local decomposition of total energy⚫ Analysis of magnetic interaction by theLiechtenstein method⚫ Coherent potential method⚫ Speed-up of the SCF calculation

3/30

Intrinsic Atomic Orbital (IAO)
J. Chem. Theory Comput., 9, 4834 (2013).

The IAO charge seems to be agood descriptor to explain theXPS binding energies.

4/25

Polar decomposition and an inequality

5/25

Proc. Amer. Math. Soc. 6, 111 (1955).

In the paper, they showed a theorem on a matrix inequality:|| || is the Frobenius norm.

It is found that the inequality is very important for our formulation.

Maximally Localized Wannier Functions

6/25

Phys. Rev. B 56, 12847 (1997).

The MLWFs are obtained by minimizing the spread function, while
keeping the unitary transformation.

Wannier interpolated bands

Graphene caseMLWFs

6

Closest Wannier Functions (CWFs)
arXiv:2306.15296

The CWFs are obtained by minimizing the distance measure (DM) function,
while keeping the unitary transformation.

Graphene case

CWFs

7/25

Projection and window function
For a localized function of Q, the projection is considered with a window function:

Window function:

1 − e x0 + x1w( ) =+x0
x1
1 + e (1 + e )(x0 = k0B−T0)

x1 = kB−T11

The window function allows us tofocus on the targeted states in thewindow region.

8/25

Distance measure (DM) function
Projection function:

Wannier function:

Residual function:
We define the distance measure function as

where X is given by a squared Frobenius norm:

9/25

Minimization of the DM function #1
It is proven that the minimum is obtained when B=U which is calculated from thepolar decomposition of A as.

UP

The difference of X at U and B is given by

From the Cauchy-Schwarz inequality, the diagonal term of D is bounded by 1.
Thus, we obtain

10/25

Minimization of the DM function #2
Thus, we find the minimum of F[B] at U asmin F [ B] = F [U ]B
21= kp − 1)
(N BC kp

A = W V †U = WV †This means that the mean squared distance between L andW is related to the deviation of the singular values σ fromunity.

LRp

1− ik R=e
Rp ak , pN BC kWRp

1− ik R=e
Rp uk , pN BC k11/25

Algorithm
1. Determining the target region.
2. Choosing a set of localized orbitals {Q}.
3. Calculation of the projection matrix of A(k).
4. Performing the SVD of A(k) ⇒ WΣV † .

5. Calculation of U(k) as W(k)V †(k).
6. Summation over k and μ.
No iterative calculation is required.

12/25

Cancellation of nondeterministic phases
Nondeterministic phases appear in three places:WRp

1− ik R=e
Rp uk , p
N BC k

1. Calculation of Bloch functions at each k-point
2. SVD of A:

A = W V † = WX X †V † = W ' V '†U = W 'V '† = WV †X is a diagonal phase matrix.

3. Singular vectors for degenerate singular values
A = W V † = WK K †V † = W ' V '†K is a unitary matrix.
U = W 'V '† = WV †It is quite important to note that the three nondeterministic phases are
all canceled out through the polar decomposition.
Thus, it turns out that the proposed method is free from complications
arising from the choice of gauge.

13/25

14/25

Relation to the Löwdin orthogonalization
The CWF method is closely related to the Löwdin orthogonalization.
By Fourier-transforming overlap integrals of L, we haveik  Re
L0 p | LRq =  ak , p ak ,q A† (k ) A(k ) = S (k )By using the polar decomposition of A, we have

S (k ) = A† (k ) A(k ) = V † (k ) 2 (k )V (k )・・・ (A)

Comparing (A) with P (k ) = V † (k )(k )V (k ) , one obtains

P(k ) = S 1/2 (k )Remembering A(k ) = U (k ) P(k ) , and if P(k) is invertible, we have

U (k ) = A(k ) S −1/2 (k )Thus, we see that in some cases the CWF is equivalent to the Löwdin
orthogonalization.
The closest property is also shown in B.C. Carlson and J.M. Keller, PR 105, 102 (1957).

Guiding functions⚫ Atomic orbitals (AOs)
PAOs are used as the guiding functions.

⚫

Hybrid Orbitals (HOs)
By diagonalizing the on-site density matrix, one can obtain the
hybrid orbitals, which will be used as the guiding functions.

⚫

Embedded Molecular Orbitals (EMOs)
The EMOs can be obtained by the singular value decomposition for the
local trace of occupation projector.

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Embedded molecular orbitals (EMOs)

16/25

Embedded molecular orbitals are calculated by considering a partial traceof a projection operator and performing SVD for Λ for a group of atomic
basis orbitals.
Bloch functionk =  eRProjection operator for the occupied states

ik R c    k ,i1
ˆP=k f ( k ) k
N BC kRii

The total number of electrons is given by(0)N ele= 2tr  Pˆ where= 2 trg (  0 g | Pˆ |  0 g )  = 2 trg   g gg|  0 g ) = ( ,  0i1 ,  0i 2 ,)

 g =  0 g ,Rg ' sRg ',0 g
Rg 'Performing SVD for  g  g , we obtain Yg as EMOs.
† † g  g = Yg  2g Yg†

17/25OpenMX Open source package for Material eXplorer•

Software package for density functional calculations of molecules and bulks•

Norm-conserving pseudopotentials (PPs)•

Variationally optimized numerical atomic basis functions

Basic functionalities

Extensions•

SCF calc. by LDA, GGA, DFT+U•O(N) and low-order scaling diagonalization•

Total energy and forces on atoms•

Non-collinear DFT for non-collinear magnetism•

Band dispersion and density of states•

Spin-orbit coupling included self-consistently•

Geometry optimization by BFGS, RF, EF•

Electronic transport by non-equilibrium Green function•

Charge analysis by Mullken, Voronoi, ESP•

Electronic polarization by the Berry phase formalism•

Molecular dynamics with NEV and NVT ensembles•

Maximally localized Wannier functions•

Charge doping•

Effective screening medium method for biased system•

Fermi surface•

Reaction path search by the NEB method•

Analysis of charge, spin, potentials by cube files•

Band unfolding method•

Database of optimized PPs and basis funcitons•

STM image by the Tersoff-Hamann method

•

etc.

Benchmark calculation: SiI have performed the benchmark calculations for 30 systems, and found thatthe method works well.
https://www.openmx-square.org/cwf/
Silicon
CWFs

Wannier interpolated bands
Solid: conventional
Red circle: Wannier interpolated

Tight-binding hopping integrals as
a function of distance

18/25

Benchmark calculation: Copper

19/25

FCC copper
Wannier interpolated bands
Solid: conventional
Red circle: Wannier interpolated

CWFsTight-binding hopping integrals as
a function of distancehttps://www.openmx-square.org/cwf/

Benchmark calculation: FeBCC iron
Wannier interpolated bands
Solid: conventional
Red circle: Wannier interpolated

CWFs for up-spin states

20/25

Benchmark calculation: TTF-TCNQ

21/25

TTF-TCNQ
CWFs
Wannier interpolated bands
Solid: conventional
Red circle: Wannier interpolated

Tight-binding hopping integrals as
a function of distanceBenchmark calculation: Bi2Se3

22/25

Bi2Se3
Wannier interpolated bands
Solid: conventional
Red circle: Wannier interpolated

CWFs

Tight-binding hopping integrals asa function of distanceEffective charge
i = N(val)i−patomai

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W0 p Pˆ W0 p

• The calculated charges by CWF quickly converge as a function of basis functions.
• MLWF and Bader overestimate the charge transfer in a HCN molecule.

CWF vs. MLWF for HCN
CWFs

24/25

MLWFs

The number that follows is the population number considering the spin degeneracy.

The CWFs preserves the shape of atomic orbitals, while the MLWFs (j), (k), (l),(o), and (p) largely deform as a result of the maximally localization. Inparticularly, (l) and (p) are bond-centered, leading to no justification for
attributing the population to a single atom.

Summary

25/25⚫ We introduced a distance measure function of F[B], and defined
CWFs as WFs which minimize F[B].
⚫ The minimization of F[B] can be performed by a polar
decomposition of the projection matrix with a window function.
⚫ The disentanglement of bands is inherently addressed byintroducing a smoothly varying window function and a greaternumber of Bloch functions, even for isolated bands.
⚫ Wannier interpolated bands well reproduce the conventional
bands of a wide variety of systems.
⚫ We further showed the usefulness of the proposed method in
calculating effective atomic charges.
Refs: arXiv:2306.15296
https://www.openmx-square.org/cwf/