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
 Gregory Wannier (1937) : Wannier functions
• The most widely used localized orbitals till date
 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
 O. K. Andersen (1974) : Muffin tin orbitals
 Marzari & Vanderbilt (1997) :
• Extension of Foster-Boys scheme to periodic systems
• Maximally localized WF by variational minization of spread
BASIC SCHEME
A DFT code calculates
cell periodic wave functions
Post processing
codes
Chemical
information
Microscopics
of bonding
Various quantities in
terms of semi-pure orbitals
Cell periodicity  Supercell periodicity Atomic orbitals
GEOMETRIC PHASE
 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’) :
(Pancharatnam connection)
In the continuum limit :
 In case of many bands :
BERRY PHASE
 Closed path in parameter space
Total geometric phase:
where
 gauge independent Berry phase (1984)
 Open path with symmetry related end points
 Gauge independent Zak phase (1989)
Example : cell periodic wave functions of electrons at k and (k +G).
PARALLEL TRANSPORT
Evolving wave function from ξ to (ξ+Δξ) through
un ( ) / 

0  Parallel transport (PT)  No random phases
Generalized PT gauge :
If PT:
and
Total geometric phase matrix Γ:
ELECTRONS IN CRYSTAL
 Bloch function(BF)
;
Identities:
 Wannier function(WF) :
Non unique due to gauge freedom !
Identities :
Total polarization
;
Wannier center
Total spread
 Physical length scale of electrons
Smoothly varying BF in k  Highly localized WF in r
POLARIZATION
Conventional definition of P is ill defined for infinitely extended
cell peridic wave functions.
-vsOnly solution : Wannier centers
;
Integrand  geometric phase in continuum limit !
 Berry phase formulation of electronic polarization(1992).
Using parallel transported wave functions :
Electronic polarization as Berry phase
LOCALIZATION
 Smooth cell periodic wave function  Localized WFs
 Measure of smoothness : Berry connection (B )
A continuum formulation to extract gradual development
of phase along kα :
 Maximal localization in α direction :
Diagonal Bα with constant diagonal elements w.r.t. Kα
This can be shown analytically.
 Perfectly localizing wave functions are eigenstates of
projected position operator in the reciprocal space.
Perfect localization : Constant Berry connection
WF in Any Dimensions
To find the maximally localizing gauge : 2 possible approach
(A). Direct derivation of maximally localizing gauge
(B). Construct smoothest possible BF in k space
• Approach (A) is accomplished in 1D: based on parallel
transport (PT) in k space.
• 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>
 We will take approach (B) in case of 3D
Constraint : WFs closest to eigen states of the position operators
FORMALISM
 Initial template of localized orbitals:
not necessarily orthogonal
with preferred center of mass
and symmetry
 Auxiliary subspace of cell periodic wave functions :
Where
By construction
and
may not span the same subspace.
are smooth and periodic in k space
 Parallel transport in a different paradigm than in 1D :
between Bloch subspace and auxiliary subspace:
For metals 
•
maximally aligned with
• Randomness of phases with Bloch function removed
• Unambiguous band index of Bloch functions
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*:
with
 Fat bands : Explicit contribution of each band to an orbital
from M matrix  A new tool within plane wave DFT
 localized WFs in 3D can be obtained by joint diagonalizing
projected position operators :
 Optionally followed by symmetrization in case of lone pairs
WLO  WF
 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.
1 (optionally 2) step process from WLOs to WFs
FLOW CHART
STEP 1
Template of initial localized nonorthogonal orbitals
Auxiliary subspace
Overlap matrix S with energy eigen states
Parallel transport (SVD)
Transformed periodic wave functions
STEP 2
WLO
ALO, BLO Pel Z*
Joint diagonalization of projected position operators
Symmetrization
Localized WF in 3D
Exact centers
for BLOs
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.
 Unrealistic choice of orbitals :
Immediately reflected in all zero rows in S matrix.
 zero singular values
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.
WF in 0D : Molecules
C 2 H6
B2H6
B-H-B 3 centered bond
has centroid close to H.
Localization length squared:
WF: 3.18, ALO : 3.44 Bohr2
Localization length squared (Bohr2)
C-C bond : WF : 2.73, BLO : 2.97
C-H bond : WF : 2.68, BLO : 2.65
WF in 0D : Clusters
Experiment :
Inorganic fulerene
Abundant closed cage
MonSm clusters
in gas phase
obtained through
laser desorption
ionization of UV
irradiated MoS2
nano flakes.
Theory :
Study of stability
through energetics
and chemical bonding
derived from electronic
structure calculation.
In collaboration with Prof. T. Pradeep’s group in IIT Chennai
Mo14S25
WCs
Rich bonding
scenario even
In small
clusters
• 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 clusters
Mo-Mo
Multicentred bonds in cage clusters :
Similar to MoS2 layer in bulk : S prefers tetrahedral coordination
Polar covalent σ bonds make the stable outer shell
WF : C4H4
C-C σ + C-C π
C-C σ
C-H
Mixing of σ and π bonds : Same as in C2H2
WF in 2D
AuS self assembled monolayer
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 structures
In terms of bonding
In collaboration with research groups of Prof. E. Kaxirus and Prof. C.M. Friend in Harvard
WF in 2D
• 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
A stable AuS monolayer
WF in 2D monolayer + substrate
Bonds within the AuS monolayer remain
largely unchanged even in the presence of the substrate
SILICON ALOs
 Constructed from wave functions in:
(1) the occupied subspace
(2) Occupied +
unoccupied subspace
All isovalues are at 60% of max
Bonding + anti-bonding subspace : localized ALOs
WF: Si & GaAs
GaAs
Si
As
Homopolar
Ga
Heteropolar
No lone pair : No symmetrization required
WF: SiO2
O
Si
2s + 2p
2s – 2p
π-like 2p lone pair
Plane of symmetrization : ┴ to Si-O-Si at O
Lone pair : Symmetrization required
Z* in PbTiO3 & BaTiO3
|ALO|2 integrated in YZ plane
Pb:[Xe]4f145d106s26p2
Ti:[Ne]3s23p63d24s2
O:[He]2s22p4
Ba:[Kr]4d105s25p66s2
e
Ti
displaced
Ti not
displaced
e
O
Ti
Ti
X
O 2py/pz
O2px
Zn, n: nominal
Za, a: anomalous
Za = Z* - Zn, Zn(Ti) = 4 a.u.
Z*x(Ti)
PbTiO3 : 7.02 a.u.
BaTiO3 : 7.13 a.u.
+1.60
+1.63
+0.82
+0.81
Inter atomic charge transfer through π like orbitals
PbTiO3 fat bands
PbTiO3 WFs
Pb
Ti
O
2s - 2px+ 3dx2
2s + 2px+ 3dx2
Symmetrization w. r. t. PbO plane
2py+ 3dxy
2s+2px hybridization : Enhanced localization in x
ALLUMINIUM
Al: [Ne] 3s2 3p1
Power law decay of BLO
BLO in Al centered at
tetrahedral hole
ALO:
BLO:
Normalization:
ALO : 0.3
BLO : 0.6 X 2
Total: 1.5
Highly directional metallic bonding in Al
COPPER
 ALO
 BLO centered at octahedral hole:
Cu : [Ar] 3d10 4s1
• 3d states are
highly localized
• 4s and BLO have
large spread.
Normalization:
4s : 0.345
3d : 0.93 X 5
BLO : 0.505
Total: 5.5
Extended BLO : Weak covalent character
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.
 Wisdom from our 1D Wannier function work :
Coordinate of Wannier centers(WC) along any direction of
perfect
localization are exactly known !
 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
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
9.0
Al=[*]3s23p2
Cu=[*]3d104s1
Mo=[*]4s24p64d55s1
Pb=[*]6s26p2
1.4
1.7
2.6
10.4
0.6
1.4
Effect of stacking fault :
much vigorous in Al than Cu
Very weak 1st neighbour bond
Metallic bond : Multiatomic sharing of eletrons
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.
bose & cat2
My teachers in
JNCASR, CU, RKMVCC, KPAHHS
My lab mates and seniors
My relatives and friends back home
and here in Bangalore
Complab, Academic section, Admin
CSIR, DST, Govt of India
||-TRANSPORT SCHEME
 Continuum approach : DFT linear response.
evaluated by minimizing the functional
within parallel transport gauge.
 Discrete approach
Linking unk abd vnk using parallel transport :
Take overlap:
Singular value decomposition
where
maximally aligned with
 No phase difference
FLOW CHART
STEP 1
Template of initial localized nonorthogonal orbitals
Auxiliary subspace
Overlap matrix S with energy eigen states
Parallel transport (SVD)
Transformed periodic wave functions
STEP 2
WLO
ALO, BLO Pel Z*
Joint diagonalization of projected position operators
Symmetrization
Localized WF in 3D
Exact centers
for BLOs
An easy way to localized WFs in 3D
WLO  WF
 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.
1 (optionally 2) step process from WLOs to WFs
WF : C4H4
C-C σ + C-C π
C-C σ
C-H
Mixing of σ and π bonds : Same as in C2H2
WCs in clusters
Similar to MoS2 layer in bulk : S prefers tetrahedral coordination
Polar covalent σ bonds make the stable outer shell
WF: SiO2
O
Si
2s + 2p
2s – 2p
π-like 2p lone pair
Plane of symmetrization : ┴ to Si-O-Si at O
Lone pair : Symmetrization required
DECC : G selection
Number of G directions increases with order of G shell
BOOP & BOLD
(A) Construct WLOs(or WFs) for each configurations
(B) Construct atom centred WLOs (AWLO) without FD
Compare the fat bands of (A) and (B) to find overlapping set of
WLOs(or WFs) from (A) and AWLOs from (B)
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.
Calculate S and X matrices from the nonorthogonal set of PWLOs
Calculate BOOP BOLD by specifying desired PWLO subsets
BOOP BOLD calculation based on WLO or WF
BOOP & BOLD
• Choice of l and m :