Localized orbitals, distribution of electron charge centers

and geometric phases, in ABINIT

Joydeep Bhattacharjee

JNCASR and Motorola Bangalore, India

OUTLINE

Introduction :• Motivation• History• Basic definitions: Geometric phases

New applications :• Wannier type localized orbitals (WLO) and Wannier functions in any dimensions 0-Dimension : molecules, MonSm cluster

2-Dimensions : AuS monolayer 3-Dimensions : Si,GaAs,ABO3 and metals Distribution of electron charge centres (DECC)

MOTIVATION

Summary of the applications : New methods to obtain precise and localized description of electrons in real space within the frame work of first principles DFT.

Advantages of localized description :• Bonding orbitals Nature of chemical bonds• Atomic orbital character charge population• Linear scaling methods for electronic structure• Model hamiltonian for many body calculations• LCAO based expansion of periodic wave function • Electron transport• Quantum molecular dynamics

HISTORY

Foster & Boys localization scheme (1960) for molecules :• Maximization of distances between centroids of orthonormal orbitals

Edmiston & Ruedenberg (1963):• Maximization of the sum of orbital self-repulsion energies Kohn (1973):

• Formal proof for existance of real, orthonormal and exponentially localized Wannier function• Variational mimization scheme from appropriate trial functions

Marzari & Vanderbilt (1997) : • Extension of Foster-Boys scheme to periodic systems• Maximally localized WF by variational minization of spread

Gregory Wannier (1937) : Wannier functions• The most widely used localized orbitals till date

O. K. Andersen (1974) : Muffin tin orbitals

BASIC SCHEME

A DFT code calculatescell periodic wave functions

Post processing codes Chemical

information

Microscopics of bonding

Various quantities interms of semi-pure orbitals

Cell periodicity Supercell periodicity Atomic orbitals

GEOMETRIC PHASE

In case of many bands :

(Pancharatnam connection)

In the continuum limit :

Geometric phase (GP) characteristic of any evolving system in any regime : classical as well as quantum.

GP between two single particle states for H(ξ) and H(ξ’) :

BERRY PHASE

Closed path in parameter space

gauge independent Berry phase (1984)

Total geometric phase:

Open path with symmetry related end points

where

Gauge independent Zak phase (1989)

Example : cell periodic wave functions of electrons at k and (k +G).

PARALLEL TRANSPORT

0 Parallel transport (PT) No random phases

and

Evolving wave function from ξ to (ξ+Δξ) through /)(nu

Generalized PT gauge :

If PT:

Total geometric phase matrix Γ:

ELECTRONS IN CRYSTAL

Bloch function(BF)

Wannier function(WF) :

Smoothly varying BF in k Highly localized WF in r

Identities:

Identities :

;

;

Physical length scale of electrons

Non unique due to gauge freedom !

Total spread

Total polarizationWannier center

POLARIZATION

Electronic polarization as Berry phase

Integrand geometric phase in continuum limit !

Using parallel transported wave functions :

Conventional definition of P is ill defined for infinitely extendedcell peridic wave functions. Only solution : Wannier centers

;

-vs-

Berry phase formulation of electronic polarization(1992).

LOCALIZATION

Perfect localization : Constant Berry connection

Smooth cell periodic wave function Localized WFs

Maximal localization in α direction : Diagonal Bα with constant diagonal elements w.r.t. Kα

This can be shown analytically.

Measure of smoothness : Berry connection (B )

A continuum formulation to extract gradual development of phase along kα :

Perfectly localizing wave functions are eigenstates of projected position operator in the reciprocal space.

WF in Any Dimensions

(A). Direct derivation of maximally localizing gauge(B). Construct smoothest possible BF in k space

To find the maximally localizing gauge : 2 possible approach

• Approach (A) is accomplished in 1D: based on parallel transport (PT) in k space.

We will take approach (B) in case of 3D

Constraint : WFs closest to eigen states of the position operators

• Direct generalization of this is not possible in 3D due to :

(1) Smooth periodic BF not possible through PT in D>1

(2) Non commutivity of <P(X,Y,Z)P>

By construction are smooth and periodic in k space

Where and

FORMALISM

Initial template of localized orbitals: not necessarily orthogonal

Auxiliary subspace of cell periodic wave functions :

with preferred center of mass and symmetry

may not span the same subspace.

Parallel transport in a different paradigm than in 1D : between Bloch subspace and auxiliary subspace:

• Randomness of phases with Bloch function removed• Unambiguous band index of Bloch functions

• maximally aligned with

For metals

WLO : UTILITIES

FT Wannier type local orbitals (WLO)

WLO maximally manifests Ylm symmetry for a given occupancy of bands For insulators :

Electronic polarization can be obtained as :

Born effective charge Z*:

Fat bands : Explicit contribution of each band to an orbital from M matrix A new tool within plane wave DFT

with

localized WFs in 3D can be obtained by joint diagonalizing projected position operators :

Optionally followed by symmetrization in case of lone pairs

WLO WF

1 (optionally 2) step process from WLOs to WFs

localized WFs can be obtained by joint diagonalizing projected position operators :

where maximally diagonalize the three position operators

Joint diagonalization maximally hybridize occupied orbitals

For ionic systems and covalent systems with lone pair:Symmetrization w.r.t. specific planes of reflection.

FLOW CHART

Template of initial localized nonorthogonal orbitals

Auxiliary subspace

Overlap matrix S with energy eigen states

Parallel transport (SVD)

Transformed periodic wave functions

WLO

Joint diagonalization of projected position operators

Localized WF in 3D

ALO, BLO Pel Z*

STE

P 1

STE

P 2

Exact centers for BLOs

Symmetrization

WLO : Facts

Ionic system : • ALOs are highly localized .• Dimension [auxiliary subspace] = Dimension [occupied subspace]

Covalent system :• BLOs are highly localized.• To construct ALOs: Dimension [auxiliary subspace] > Dimension [occupied subspace] to include the anti bonding subspace.

For an optimal choice of symmetry and center of mass: Diagonal elements of Berry connection matrix are constant.

Maximal localization of individual Φκ in α direction.

Unrealistic choice of orbitals : Immediately reflected in all zero rows in S matrix. zero singular values

WF in 0D : Molecules

Localization length squared (Bohr2)C-C bond : WF : 2.73, BLO : 2.97C-H bond : WF : 2.68, BLO : 2.65

B-H-B 3 centered bond has centroid close to H.

C2H6 B2H6

Localization length squared: WF: 3.18, ALO : 3.44 Bohr2

WF in 0D : Clusters

Inorganic fulerene

Mo14S25

Experiment :Abundant closed cageMonSm clusters in gas phaseobtained throughlaser desorptionionization of UVirradiated MoS2

nano flakes.

Theory :Study of stabilitythrough energetics and chemical bonding derived from electronicstructure calculation.

In collaboration with Prof. T. Pradeep’s group in IIT Chennai

WCs

Rich bondingscenario evenIn smallclusters

• Mo-S triple bond in MoS• Mo-S double bond in MoS3• Mo-S single bond in Mo2S3• Mo-Mo triple bond in Mo2S3• Mo-S-Mo tri-centred bond in MoS4

WF in clustersMo-Mo

Multicentred bonds in cage clusters :

Similar to MoS2 layer in bulk : S prefers tetrahedral coordinationPolar covalent σ bonds make the stable outer shell

WF : C4H4

Mixing of σ and π bonds : Same as in C2H2

C-C σ

C-H

C-C σ + C-C π

WF in 2D

AuS self assembled monolayer

In collaboration with research groups of Prof. E. Kaxirus and Prof. C.M. Friend in Harvard

Upon sulfur deposition and annealing at 450 K in ultra-high vacuum Au atoms are etched from the ‘inert’ Au(111) surface to form a robust gold sulfide layer with rich coordination chemistry .

We studied the robustness of the structuresIn terms of bonding

WF in 2D

A stable AuS monolayer

• Polar covalent bonds confined to the plane.• Bonds do not change in shape in the presence of substrate• Their normalization increases in the presence of substrate

WF in 2D monolayer + substrate

Bonds within the AuS monolayer remainlargely unchanged even in the presence of the substrate

SILICON ALOs

Bonding + anti-bonding subspace : localized ALOs

Constructed from wave functions in: (1) the occupied subspace

(2) Occupied + unoccupied subspace

All isovalues are at 60% of max

WF: Si & GaAs

No lone pair : No symmetrization required

Si GaAs

Homopolar Heteropolar

AsGa

WF: SiO2

Lone pair : Symmetrization required

2s + 2p 2s – 2p π-like 2p lone pair

Plane of symmetrization : ┴ to Si-O-Si at O

SiO

Z* in PbTiO3 & BaTiO3

Inter atomic charge transfer through π like orbitals

O

Ti

e e

|ALO|2 integrated in YZ plane

TidisplacedTi notdisplaced

PbTiO3 : 7.02 a.u. BaTiO3 : 7.13 a.u.

Zn, n: nominalZa, a: anomalousZa = Z* - Zn, Zn(Ti) = 4 a.u.

X

Pb:[Xe]4f145d106s26p2

Ti:[Ne]3s23p63d24s2

O:[He]2s22p4

Ba:[Kr]4d105s25p66s2

+1.60+1.63

+0.82+0.81

O 2py/pz O2px

Z*x(Ti)

Ti

PbTiO3 fat bands

PbTiO3 WFs

2s+2px hybridization : Enhanced localization in x

2s + 2px+ 3dx22s - 2px+ 3dx2

2py+ 3dxy

Pb

OTi

Symmetrization w. r. t. PbO plane

ALLUMINIUM

Highly directional metallic bonding in Al

BLO in Al centered at tetrahedral hole

Power law decay of BLO

ALO: BLO:

Al: [Ne] 3s2 3p1

Normalization:ALO : 0.3BLO : 0.6 X 2Total: 1.5

COPPER

Extended BLO : Weak covalent character

BLO centered at octahedral hole:

ALO

Normalization:4s : 0.3453d : 0.93 X 5BLO : 0.505Total: 5.5

• 3d states are highly localized• 4s and BLO have large spread.

Cu : [Ar] 3d10 4s1

CONCLUSIONS : WF in Any D

Semi analytic and mostly non iterative method to construct localized orbitals WLO for any energy window.

Electronic polarization and related quantities like Born effective charge can be easily obtained from WLOs.

Localized WFs in 3D are obtained in a single step of joint diagonalization of the three position operators projected in the WLO basis, occasionally followed by symmetrization.

The formalism involves independent procedures at each k efficient implementation in parallel computers.

Easy to implement as a post-processing module to any standard DFT package.

DECC

Most often in 3D perfectly localized WFs are not possible.Available WFs are only approximate description due to non-commutivity of the three position operators projected on to the occupied subspace.

A possible way out : Collect WCs corresponding to different directions of perfect localization and somehow link them !

A map of WCs in real space based on joint quantum probability distribution function

Wisdom from our 1D Wannier function work : Coordinate of Wannier centers(WC) along any direction of perfect localization are exactly known !

Distribution of Electron Charge Centers

DECC : FORMULATION

Total geometric phase matrix Γ obtained from PT wave functions

Eigen states of Γ in k space

DECC functional form :

Joint probability distribution

; ;

Joint probability function for more than two non-commuting operators are known not to be always positive. We find it -ve mostly in the antibonding regions.

DECC normaizes to

DECC : Utility

Isolated features easily integrable Accurate estimation static charge

Exact calculation of polarization

Exact estimate of Born dynalical charge Z*

Quantitative understanding of charge distribution Precise characterization of bonding

DECC : Ionic insulators

Z* = 7 Z* = 4

DECC : Metallic bonds

Metallic bond : Multiatomic sharing of eletrons

Very weak 1st neighbour bond

Effect of stacking fault :much vigorous in Al than Cu

9.01.7

1.4

10.4 0.6

2.61.4

Al=[*]3s23p2

Cu=[*]3d104s1

Mo=[*]4s24p64d55s1

Pb=[*]6s26p2

DECC : Covalent bonds

CONCLUSIONS : DECC

Exact estimation polarization, static charge, dynamic charge, bond order

Concept of real space electronic lattice

Precise quantitative understanding of bonding

Metallic bonds : Multiatomic sharing of electrons very weak nearest neighbour bond. Bond centers are mostly placed at the centers of bonding polyhedron Cage critical points in Bader’s language.

REFERENCES

Published work: Geometric phases and Wannier functions of Bloch electrons in one

dimension. J. Bhattacharjee and U. V. Waghmare Rhys. Rev. B 71, 045106 (2005)

Localized orbital description of electronic structures of extended periodic

metals, insulators and confined systems: Density functional theory calculations

J. Bhattacharjee and U. V. Waghmare Phys. Rev. B 73 , (R)121102 (2006)

Novel Cage Clusters of MoS2 in Gas Phase D.M.D.J. Singh, T. Pradeep, J.Bhattacharjee and U.V.Waghmare J. Phys. Chem. A 109, 7339-7342 (2005)

Submitted : Distribution of Electron Charge Centers: A Picture of Bonding Based on Geometric phases J. Bhattacharjee, S. Narasimhan and U.V. Waghmare

Rich coordination chemistry of Au adatoms in gold sulphide monolayer on Au(111) S.Y. Quek, M.M.Biener, J.Biener, J.Bhattacharjee,C.M. Friend, U.V.Waghmare and E. Kaxiras

REFERENCES

Gas phase closed-cage clusters derived from MoS2 nanoflakes D.M.D.J. Singh, T. Pradeep, J.Bhattacharjee and U.V.Waghmare

Manuscript under prepation: Orbital picture of dynamical charge J Bhattacharjee and U.V. Waghmare

Codes are interfaced with Abinit: DECC : Abinit-4.1.4 (not available in public domain) WLO in 3D : Abinit--5.x.x (somewhat hidden)

Thanks for attention

Acknowledgement

My parents and my Ph.D advisor.

My relatives and friends back home and here in Bangalore

My teachers inJNCASR, CU, RKMVCC, KPAHHS

My lab mates and seniors

bose & cat2

Complab, Academic section, Admin

CSIR, DST, Govt of India

||-TRANSPORT SCHEME

Continuum approach : DFT linear response.

evaluated by minimizing the functional

Discrete approach

maximally aligned with

within parallel transport gauge.

Linking unk abd vnk using parallel transport :

Take overlap:

Singular value decomposition

where

No phase difference

FLOW CHART

An easy way to localized WFs in 3D

Template of initial localized nonorthogonal orbitals

Auxiliary subspace

Overlap matrix S with energy eigen states

Parallel transport (SVD)

Transformed periodic wave functions

WLO

Joint diagonalization of projected position operators

Localized WF in 3D

ALO, BLO Pel Z*

STE

P 1

STE

P 2

Exact centers for BLOs

Symmetrization

WLO WF

1 (optionally 2) step process from WLOs to WFs

localized WFs can be obtained by joint diagonalizing projected position operators :

where maximally diagonalize the three position operators

Joint diagonalization maximally hybridize occupied orbitals

For ionic systems and covalent systems with lone pair:Symmetrization w.r.t. specific planes of reflection.

WF : C4H4

Mixing of σ and π bonds : Same as in C2H2

C-C σ

C-H

C-C σ + C-C π

WCs in clusters

Polar covalent σ bonds make the stable outer shell

Similar to MoS2 layer in bulk : S prefers tetrahedral coordination

WF: SiO2

Lone pair : Symmetrization required

2s + 2p 2s – 2p π-like 2p lone pair

Plane of symmetrization : ┴ to Si-O-Si at O

SiO

DECC : G selection

Number of G directions increases with order of G shell

BOOP & BOLD

BOOP BOLD calculation based on WLO or WF

(A) Construct WLOs(or WFs) for each configurations

Corresponding to the AWLOs construct atom centered WLOs for isolated atoms with same unit cell and k-mesh pure atomic orbitals PWLOs within the pseudopotential used.

(B) Construct atom centred WLOs (AWLO) without FD

Compare the fat bands of (A) and (B) to find overlapping set ofWLOs(or WFs) from (A) and AWLOs from (B)

Calculate S and X matrices from the nonorthogonal set of PWLOs

Calculate BOOP BOLD by specifying desired PWLO subsets

BOOP & BOLD

• Choice of l and m :