Mössbauer parameters from DFT-based WIEN2k calculations

for extended systems

Peter BlahaInstitute of Materials Chemistry

TU Vienna

Main Mössbauer parameters:

The main (conventional) Mössbauer spectroscopy parameters which we want to calculate by theory are:

Isomer shift: = (0Sample – 0

Reference); it is proportional to the electron density at the nucleus the constant is proportional to the change of nuclear radii during

the transition (we use =-.291 au3mm s-1)

Magnetic Hyperfine fields: Btot = Bcontact + Borb + Bdip these fields are proportional to the spin-density at the nucleus and

the orbital moment of the probed atom as well as the spin moment distribution in the crystal

Quadrupole splitting: e Q Vzz given by the product of the nuclear quadrupole moment Q times the

electric field gradient Vzz. The EFG is proportional to an integral over the non-spherical charge density (weighted by 1/r3)

Schrödinger equation

From the previous slide it is obvious, that we need an accurate knowledge of the electron (and magnetization) density, which in principle can be obtained from the solution of the many-body Schrödinger equation in the corresponding solid.

H = E

However, a many-body problem with ~1023 particles is not solvable at all and we must create models for the real material and rely on an approximate solution of the Schrödinger equation. (This will be briefly discussed in the next slides and my preferred options are marked in red.)

Concepts when solving Schrödingers-equation

non relativisticsemi-relativisticfully-relativistic

“Muffin-tin” MTatomic sphere approximation (ASA)pseudopotential (PP)Full potential : FP

Hartree-Fock (+correlations)Density functional theory (DFT) Local density approximation (LDA) Generalized gradient approximation (GGA) Beyond LDA: e.g. LDA+U

Non-spinpolarizedSpin polarized(with certain magnetic order)

non periodic (cluster, individual MOs)periodic (unit cell, Blochfunctions,“bandstructure”)

plane waves : PWaugmented plane waves : APWatomic oribtals. e.g. Slater (STO), Gaussians (GTO), LMTO, numerical basis

Basis functions

Treatment of spin

Representationof solid

Form ofpotential

exchange and correlation potentialRelativistic treatment

of the electrons

Schrödinger - equationki

ki

kirV

)(

2

1 2

Representation of the solid

• cluster model: • approximate the solid by a finite (small) number of atoms. This can be a good approximation for “molecular crystals”.

• periodic model: • use a “unit cell”, which is repeated infinitely. This approximates the “real” solid (finite with surfaces, imperfect) by an infinite “ideal” solid. • with “supercells” also surfaces, vacancies or impurities can be modelled.

BaSnO3 SnS

Exchange and correlation

The electron-electron interaction can be approximated by:

• Hartree-Fock • exact exchange, but no correlation at all (in solids often very important !)• correlation can approximately be added at various levels: MP2, CC, CI, ... • HF (and even more adding correlation) is very expensive in solids.

• Density functional theory: • the exact functional is unknown, thus one must approximate exchange + correlation. Fast method, fairly accurate in solids.

• LDA: local density approximation, “free electron gas”• GGA: generalized gradient approximation, various functionals• hybrid-DFT: mixing of HF + GGA, various functionals• LDA+U, DMFT: explicit (heuristic) inclusion of correlations

• a brief summary of DFT follows

DFT Density Functional Theory

Hohenberg-Kohn theorem: (exact)

The total energy of an interacting inhomogeneous electron gas in the presence of an external potential Vext(r ) is a functional of the density

][)()( FrdrrVE ext

• This (exact) statement states, that we need to know only the electron density , but we do NOT need to know the wavefunctions (solution of Schrödinger equation not necessary).

• However, the exact functional form of F[] is unknown and any approximation so far is too crude.

DFT - Kohn Sham (still exact)

][

||

)()(

2

1)(][ xcexto Erdrd

rr

rrrdrVTE

Ekinetic non interacting

Ene Ecoulomb Eee Exc exchange-

correlation

Lets decompose the total functional F[] into parts whichcan be calculated „exactly“ and some „unknown, but small“rest:

• Ekinetic is the kinetic energy of non-interacting particles• Ecoulomb is the “classical” electrostatic interaction between electrons (including the self-interaction energy)• Exc is the exchange-correlation energy and should correct for the self-interaction and approx. kinetic energy

• The exact Exc is again unknown, but now an approximation for the

“small correction” may be accurate enough.

rdrr

r

||

)(

)(xcE

][

||

)()(

2

1)(][ xcexto Erdrd

rr

rrrdrVTE

1-electron equations (Kohn Sham)

)()())}(())(()(2

1{ 2 rrrVrVrV iiixcCext

FEi

ir

2||)(

vary

Exc and Vxc are unknown and must be approximated

LDA or GGA treat both, exchange and correlation effects

approximately

-Z/r

Kohn-Sham equations

Walter Kohn, Nobel Prize 1998 Chemistry

Approximations to EXC

Local density approximation (LDA):

xc is the exchange-correlation energy density of the homogeneous electron gas at density .

Second order gradient expansion (GEA):

The GEA XC-hole nXC(r,r’) is not a hole of any physical system and violates

nX(r,r’) ≤ 0 exchange hole must be negative ∫ nX(r,r’) dx = -1 must contain charge -1 ∫ nC(r,r’) dx = 0

drrrCEE LDAxc

GEAxc )()()( 3

2

,

drrdrrrExc

LDAxc

34.hom

)()]([)(

e- at r

nXC(r,r’)

Generalized gradient approximations (GGA)

“construct” GGAs by obeying as many known constraints as possible (Perdew)

recover LDA for slowly varying densities obey sum rules and properties of XC-holes long range limits: lim(r -> ∞): exc=-1/2r ; vxc=C-1/r scaling relations Lieb-Oxford bound

fitting some parameters to recover “exact” energies of small systems (set of small molecules) or lattice parameters in solids (Becke, Handy, Hammer, ..)

Perdew-Burke-Enzerhof – GGA (PRL 1996): well balanced GGA; equally “bad” for “all” systems

drrrFrEGGAxc )](),([)(

better approximations are constantly developed

meta-GGAs:

Perdew,Kurth,Zupan,Blaha (PRL 1999):

use laplacian of , or the kinetic energy density

analytic form for Vxc not possible (Vxc = dExc/dr) , SCF very difficult

better meta-GGAs under constant development …

drrrrrFrE GGAmxc )](),(),(),([)( 2

i

i rr2

)(2

1)(

more “non-local” functionals (“beyond LDA”)

Self-Interaction correction (Perdew,Zunger 1981; Svane+ Temmermann) vanishes for Bloch-states, select “localized states” by hand

LDA+U, DMFT (dynamical mean field theory) approximate HF for selected “highly-correlated” electrons

(3d,4f,5f) empirical parameter U

Exact exchange (similar to HF but DFT based, misses correlation)

Hybrid functionals (mixing of LDA (GGA) + HF)

DFT ground state of iron

LSDA NM fcc in contrast to

experiment

GGA FM bcc Correct lattice

constant Experiment

FM bcc

GGAGGA

LSDA

LSDA

LDA: Fe is nonmagnetic and in fcc structureGGA correctly predicts Fe to be ferromagnetic and in bcc structure

“Everybody in Austria knows about the importance of DFT”

“The 75th GGA-version follows the 52nd LDA-version”

(thanks to Claudia Ambrosch (TU Leoben))

basis set for the wave functions

Even with an approximate e--e- interaction the Schrödinger equation cannot be solved exactly, but we must expand the wave function into a basis set and rely on the variational principle.

• “quantum chemistry”: LCAO methods• Gauss functions (large “experience” for many atoms, wrong asymptotic, basis set for heavier atoms very large and problematic, .. )• Slater orbitals (correct r~0 and r~ asymptotic, expensive)• numerical atomic orbitals

• “physics”: plane wave (PW) based methods• plane waves + pseudo-potential (PP) approximation

• PP allow fast solutions for total energies, but not for Hyperfine parameters • augmented plane wave methods (APW)

• spatial decomposition of space with two different basis sets:• combination of PW (unbiased+flexible in interstitial regions)• + numerical basis functions (accurate in the atomic regions, correct cusp)

Computational approximations

• relativistic treatment:• non- or scalar-relativistic approximation (neglects spin-orbit, but includes Darvin s-shift and mass-velocity terms)• adding spin-orbit in “second variation” (good enough)• fully-relativistic treatment (Dirac-equation, very expensive)• point- or finite-nucleus

• restricted/unrestricted treatment of spin• use correct long range magnetic order (FM, AFM)

• approximations to the form of the potential• shape approximations (ASA)• pseudopotential (smooth, nodeless valence orbitals)• “full potential” (no approximation)

Concepts when solving Schrödingers-equation

• in many cases, the experimental knowledge about a certain system is very limited and also the exact atomic positions may not be known accurately (powder samples, impurities, surfaces, ...) • Thus we need a theoretical method which can not only calculate HFF-parameters, but can also model the sample:

• total energies + forces on the atoms: • perform structure optimization for “real” systems• calculate phonons• investigate various magnetic structures, exchange interactions

• electronic structure:• bandstructure + DOS • compare with ARPES, XANES, XES, EELS, ...

• hyperfine parameters • isomer shifts, hyperfine fields, electric field gradients

WIEN2k software package

An Augmented Plane Wave Plus Local Orbital

Program for Calculating Crystal Properties

 Peter Blaha

Karlheinz SchwarzGeorg Madsen

Dieter KvasnickaJoachim Luitz

November 2001Vienna, AUSTRIA

Vienna University of Technology

WIEN2k: ~1530 groupsmailinglist: 1800 users http://www.wien2k.at

APW based schemes

APW (J.C.Slater 1937) Non-linear eigenvalue problem Computationally very demanding

LAPW (O.K.Andersen 1975) Generalized eigenvalue problem Full-potential (A. Freeman et al.)

Local orbitals (D.J.Singh 1991) treatment of semi-core states (avoids ghostbands)

APW+lo (E.Sjöstedt, L.Nordstörm, D.J.Singh 2000) Efficience of APW + convenience of LAPW Basis for

K.Schwarz, P.Blaha, G.K.H.Madsen,Comp.Phys.Commun.147, 71-76 (2002)

PW:

APW Augmented Plane Wave method

The unit cell is partitioned into:

atomic spheresInterstitial region

rKkie

).(

Atomic partial waves m

mKm rYruA

)ˆ(),(

join

Rmt

unit cell

Basisset:

ul(r,) are the numerical solutionsof the radial Schrödinger equationin a given spherical potential for a particular energy Alm

K coefficients for matching the PW

Ir

Slater‘s APW (1937)

Atomic partial waves

Energy dependent basis functions lead to

Non-linear eigenvalue problem

m

mKm rYruA

)ˆ(),(

One had to numerically search for the energy, for which the det|H-ES| vanishes. Computationally very demanding. “Exact” solution for given (spherical) potential!

H HamiltonianS overlap matrix

Linearization of energy dependence

LAPW suggested by

)ˆ()],()(),()([ rYrEukBrEukA mnm

mnmkn

Atomic sphere

PW

O.K.Andersen,Phys.Rev. B 12, 3060 (1975)

expand ul at fixed energy El

and add

Almk, Blm

k: join PWs in value and slope

General eigenvalue problem (diagonalization)

additional constraint requires more PWs than APW

LAPW

/ll uu

APW

bonding

antibonding

center

Full-potential in LAPW (A.Freeman etal.)

The potential (and charge density) can be of general form (no shape approximation)SrTiO3

Fullpotential

Muffin tinapproximation

Inside each atomic sphere a local coordinate system is used (defining LM)

LM

LMLM rYrV )ˆ()( Rr

K

rKiKeV

.

Ir

TiO2 rutile

TiO

)(rV {

Problems of the LAPW method

LAPW can only treat ONE principle quantum number per l. Problems with high-lying “semi-core” states

Extending the basis: Local orbitals (LO)

LO: contains a second ul(E2) is confined to an atomic

sphere has zero value and slope at R can treat two principal QN n

for each azimuthal QN (3p and 4p)

corresponding states are strictly orthogonal (no “ghostbands”)

tail of semi-core states can be represented by plane waves

only slight increase of basis set(matrix size)

D.J.Singh,Phys.Rev. B 43 6388 (1991)

)ˆ(][ 211 rYuCuBuA mE

mE

mE

mLO

Ti atomic sphere

New ideas from Uppsala and Washington

)ˆ(),()( rYrEukA mm

nmkn

E.Sjöstedt, L.Nordström, D.J.Singh,An alternative way of linearizing the augmented plane wave method,

Solid State Commun. 114, 15 (2000)

• Use APW, but at fixed El (superior PW convergence)• Linearize with additional lo (add a few basis functions) )ˆ(][ 11 rYuBuA m

Em

Emlo

optimal solution: mixed basis• use APW+lo for states which are difficult to converge: (f or d- states, atoms with small spheres)• use LAPW+LO for all other atoms and ℓ

basis for

Relativistic treatment Valence states

Scalar relativistic mass-velocity Darwin s-shift

Spin orbit coupling on demand by second variational treatment

Semi-core states Scalar relativistic on demand

spin orbit coupling by second variational treatment

Additional local orbital (see Th-6p1/2)

Core states Fully relativistic

Dirac equation

For example: Ti

Relativistic semi-core states in fcc Th

additional local orbitals for 6p1/2 orbital in Th Spin-orbit (2nd variational

method)

J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz,Phys.Rev.B. 64, 153102 (2001)

Quantum mechanics at work

w2web GUI (graphical user interface)

Structure generator spacegroup selection import cif file

step by step initialization symmetry detection automatic input generation

SCF calculations Magnetism (spin-polarization) Spin-orbit coupling Forces (automatic geometry

optimization) Guided Tasks

Energy band structure DOS Electron density X-ray spectra Optics

An example:

In the following I will demonstrate on one example, which kind of problems you can solve using a DFT simulation with WIEN2k.

Of course, also Mössbauer parameters will be calculated and interpreted.

Verwey Transition and Mössbauer Parameters

in YBaFe2O5 by DFT calculations

Peter Blaha, C. Spiel, K.SchwarzInstitute of Materials Chemistry

TU Wien

Thanks to: P.Karen (Univ. Oslo, Norway)

C.Spiel, P.B., K.Schwarz, Phys.Rev.B.79, 085104 (2009)

“Technical details”:

WIEN2k (APW+lo) calculations

Rkmax=7, 100 k-points spin-polarized, various spin-structures+ spin-orbit couplingbased on density functional theory:

LSDA or GGA (PBE) Exc ≡ Exc(ρ, ∇ρ)

description of “highly correlated electrons”using “non-local” (orbital dep.) functionals

LDA+U, GGA+U hybrid-DFT (only for correlated electrons)

mixing exact exchange (HF) + GGA

WIEN2KAn Augmented Plane Wave

Plus Local Orbital Program for Calculating

Crystal Properties

 Peter Blaha

Karlheinz SchwarzGeorg Madsen

Dieter KvasnickaJoachim Luitz

http://www.wien2k.at

Verwey-transition: E.Verwey, Nature 144, 327 (1939)

Fe3O4, magnetitephase transition between a mixed-valence and a charge-ordered configuration with temp.

2 Fe2.5+ Fe2+ + Fe3+

cubic inverse spinel structure AB2O4

Fe2+A (Fe3+,Fe3+)B O4

Fe3+A (Fe2+,Fe3+)B O4

B

A

small, but complicated coupling between lattice and charge order

Double-cell perovskites: RBaFe2O5

ABO3 O-deficient double-perovskite

Y (R)

Ba

square pyramidalcoordination

Antiferromagnet with a 2 step Verwey transition around 300 KWoodward&Karen, Inorganic Chemistry 42, 1121 (2003)

experimental facts: structural changes in YBaFe2O5

• above TN (~430 K): tetragonal (P4/mmm)

• 430K: slight orthorhombic distortion (Pmmm) due to AFMall Fe in class-III mixed valence state +2.5;

• ~334K: dynamic charge order transition into class-II MV state, visible in calorimetry and Mössbauer, but not with X-rays

• 308K: complete charge order into class-I MV state (Fe2+ + Fe3+)

large structural changes (Pmma)due to Jahn-Teller distortion;

change of magnetic ordering: direct AFM Fe-Fe coupling vs. FM Fe-Fe exchange above TV

structural changes

charge ordered (CO) phase: valence mixed (VM) phase: Pmma a:b:c=2.09:1:1.96 (20K) Pmmm a:b:c=1.003:1:1.93

(340K)

Fe2+ and Fe3+ form chains along b contradicts Anderson charge-ordering conditions with minimal

electrostatic repulsion (checkerboard like pattern) has to be compensated by orbital ordering and e--lattice coupling

ab

c

antiferromagnetic structure

CO phase: G-type AFM VM phase: AFM arrangement in all directions, AFM for all Fe-O-Fe

superexchange paths also across Y-layer FM across Y-layer (direct Fe-Fe

exchange) Fe moments in b-direction

4 8independent Fe atoms

results of GGA-calculations:

Metallic behaviour/No bandgap Fe-dn t2g states not splitted at EF

overestimated covalency between O-p and Fe-eg

Magnetic moments too small Experiment:

CO: 4.15/3.65 (for Tb), 3.82 (av. for Y) VM: ~3.90

Calculation:

CO: 3.37/3.02 VM: 3.34

no significant charge order charges of Fe2+ and Fe3+ sites nearly identical

CO phase less stable than VM

LDA/GGA NOT suited for this compound!

Fe-eg t2g eg* t2g eg

“Localized electrons”: GGA+U

Hybrid-DFT Exc

PBE0 [] = ExcPBE [] + (Ex

HF[sel] – ExPBE[sel])

LDA+U, GGA+U ELDA+U(,n) = ELDA() + Eorb(n) – EDCC()

separate electrons into “itinerant” (LDA) and localized e- (TM-3d, RE 4f e-)

treat them with “approximate screened Hartree-Fock” correct for “double counting”

Hubbard-U describes coulomb energy for 2e- at the same site

orbital dependent potential

)21)(( ,',,', mmmm nJUV

Determination of U

Take Ueff as “empirical” parameter (fit to experiment) or estimate Ueff from constraint LDA calculations

constrain the occupation of certain states (add/subtract e-) switch off any hybridization of these states (“core”-states) calculate the resulting Etot

we used Ueff=7eV for all calculations

DOS: GGA+U vs. GGA GGA+U GGA

insulator, t2g band splits metallic single lower Hubbard-band in VM splits in CO with Fe3+ states lower than Fe2+

insulator metalGGA+U

magnetic moments and band gap

magnetic moments in very good agreement with exp. LDA/GGA: CO: 3.37/3.02 VM: 3.34 B

orbital moments small (but significant for Fe2+)band gap: smaller for VM than for CO phase

exp: semiconductor (like Ge); VM phase has increased conductivity

LDA/GGA: metallic

Charge transfer (in GGA+U)

Charges according to Baders “Atoms in Molecules” theory Define an “atom” as region within a zero flux surface Integrate charge inside this region

0 n

Structure optimization (GGA+U)

CO phase: Fe2+: shortest bond in y (O2b) Fe3+: shortest bond in z (O1)

VM phase: all Fe-O distances similar theory deviates along z !!

Fe-Fe interaction different U ?? finite temp. ??

O1O1

O2aO2b

O3

strong coupling between lattice and electrons !

Fe2+ (3d6) CO Fe3+ (3d5) VM Fe2.5+ (3d5.5)

majority-spin fully occupied strong covalency effects very localized states at lower energy than Fe2+

in eg and d-xz orbitals minority-spin states d-xz fully occupied (localized) empty d-z2 partly

occupied short bond in y short bond in z (missing O) FM Fe-Fe;

distances in z ??

Difference densities =cryst-at

sup

CO phase VM phase

Fe2+: d-xzFe3+: d-x2

O1 and O3: polarized toward Fe3+

Fe: d-z2 Fe-Fe interactionO: symmetric

dxz spin density (up-dn) of CO phase

Fe3+: no contribution

Fe2+: dxz

weak -bond with O

tilting of O3 -orbital

Mössbauer spectroscopy:

Isomer shift: = (0Sample – 0

Reference); =-0.291 au3mm s-1 proportional to the electron density at the nucleus

Magnetic Hyperfine fields: Btot=Bcontact + Borb + Bdip

Bcontact = 8/3 B [up(0) – dn(0)] … spin-density at the nucleus

… orbital-moment

… spin-moment

S(r) is reciprocal of the relativistic mass enhancement

nuclear quadrupole interaction

Nuclei with a nuclear quantum number I≥1 have an electric quadrupole moment Q, which describes the asphrericity of the nucleus

Nuclear quadrupole interaction (NQI) can aid to determine the distribution of the electronic charge surrounding such a nuclear site

Experiments NMR NQR Mössbauer PAC

electric field gradient (EFG)

Qh

edxxxx

xx

Vij

ijji

ji

2)(

)0(

2

1 2

heQ /

jiij xx

VV

)0(2

VV ijijij2

3

1

Electric field gradients (EFG)

EFG traceless tensorVV ijijij

2

3

1

jiij xx

VV

)0(2

0 zzyyxx VVVwith traceless

cc

bc

ac

cb

bb

ab

ca

ba

aa

V

V

V

V

V

V

V

V

V

zz

yy

xx

V

V

V

0

0

0

0

0

0|Vzz| |Vyy| |Vxx|

//

////

zz

yyxx

V

VV

„EFG“: Vzz asymmetry parameter

… 0 - 1principal component similarity

transformation

directions of Vxx, Vyy, Vzz

cubic: no EFG; hex, tetragonal (axial symm.): only Vzz

theoretical EFG calculations

LMLMLMc

jiij rYrVrd

rr

rrV

xx

VV )ˆ()(

)()(

)0(2

2220

2220

20

21

21

)0(

VVV

VVV

rVV

xx

yy

zz

EFG is tensor of second derivatives of VC at the nucleus:

222 )(211

)(211

)(

3

3

320

zyzxzyxxyd

dzz

zyxp

pzz

dzz

pzzzz

dddddr

V

pppr

V

VVdrr

YrV

Cartesian LM-repr.

EFG is proportial to differences of orbital occupations

Mössbauer spectroscopy

CO

VM

Isomer shift: charge transfer too small in LDA/GGA

CO

VM

Isomer shifts = (0Sample –

0Ref.)

The observed IS is proportional to the charge transfer between the different sites (source and absorber).

The 4s (valence) and 3s (semicore) contributions provide 110-120% of the effect, and are reduced slightly by opposite 1s,2s contributions.

is proportional to the change of nuclear radii during the transition, (in principle a “known nuclear constant” ). However, a meaningful depends on details of the calculations (we use =-0.291 au3mm s-1):

radial mesh for numerical basis functions (first radial mesh point, RMT) relativistic treatment (NREL, scalar/fully relativistic, point/finite nucleus)

Statements for YBa2Fe2O5

standard LDA/GGA lead to much too small charge transfer and thus the IS are too similar for all 3 Fe sites. need better XC-treatment

with increased on-site correlation U , the “localization” and thus the charge transfer is larger leading to more different IS in agreement with experiment. Fe site charge IS

Fe2+ +1.36 0.96

Fe3+ +1.84 0.28

Fe2.5+ +1.62 0.51

Hyperfine fields: Fe2+ has large Borb and Bdip

CO

VM

contact hyperfine fields

The contact HFF dominates in many cases. It is proportional to the “spin-density” at the nucleus.

Bcontact = 8/3 B [up(0) – dn(0)] It is also proportional to the spin-magnetic moment (but with

opposite sign) because the (3d) magnetic moment polarizes the core s-electrons (see below), provided the valence (4s) contribution is small

In many cases the “core-polarization” dominates (the core electrons are polarized due to the 3d magnetic moment):

1s - 1.4 T 2s,p - 113.7 T opposite sign; exchange effect with

3s wf. 3s,p + 60.7 T same sign as 3d moment, strong overlap

of 3d and 3s,p wf. 4s,p + 16.3 T often even smaller, but can

be large due to ligand total - 38.1 T effects (transferred

hyperfine fields) Typically, in DFT the contact HFF is about 10-20% too small

Orbital and dipolar - hyperfine fields

Borb is proportional to the “orbital moment” (an effect due to spin-orbit coupling)

In metals and for high-spin Fe3+ (3d5-up) compounds (closed shell ions) it is usually small.

In insulators/semiconductors with partially filled 3d-shells it can be very large (high-spin Fe2+: 3d5-up, 3d1-dn)

LDA/GGA usually underestimates this contribution

Bdip is proportional to the anisotropy of the spin-moments around the nucleus In many cases it is fairly small

EFG: Fe2+ has too small anisotropy in LDA/GGA

CO

VM

theoretical EFG calculations

• The coulomb potential Vc is a central quantity in any theoretical calculation (part of the Hamiltonian) and is obtained from all charges (electronic + nuclear) in the system.

• The EFG is a tensor of second derivatives of VC at the nucleus:

• Since we use an “all-electron” method, we have the full charge distribution of all electrons+nuclei and can obtain the EFG without further approximations. • The spherical harmonics Y20 projects out the non-spherical (and non-cubic) part of . The EFG is proportional to the differences in orbital occupations (eg. pz vs. px,py)

• We do not need any “Sternheimer factors” (these shielding effects are included in the self-consistent charge density)

drr

YrV

xx

VV zz

jiij 3

202 )(

;)0(

LM

LMLMc rYrVrdrr

rrV )ˆ()(

)()(

theoretical EFG calculations

drr

YrVzz 3

20)(The charge density in the integral can be decomposed in various ways for analysis:

according to energy (into various valence or semi-core contributions)

according to angular momentum l and m (orbitals) spatial decomposition into “atomic spheres” and the “rest”

(interstital)

Due to the 1/r3 factor, contributions near the nucleus dominate.

EF

theoretical EFG calculations

alinterstiti.....

.)(;;ˆ)(

:iondecomposit orbital

alinterstiti)(

)()(

)(

:iondecomposit spatial

,,,.,2020

320

int

33

2033

203

320

*

ddzz

ppzzzz

mmllnk

nkml

nklm

sphere

zz

erstitalsphere

LMLMLM

zz

VVV

contrdsddpprdYr

drr

rV

rdr

Yrrd

r

YYrrd

r

YrV

We write the charge density and the potential inside the atomic spheres in an lattice-harmonics expansion

)ˆ()()()ˆ()()( rYrvrVrYrrLM

LMLMLM

LMLM

theoretical EFG calculations

222 )(211

)(211

.....

3

3

zyzxzyxxyd

ddzz

zyxp

ppzz

ddzz

ppzzzz

dddddr

V

pppr

V

VVV

• EFG is proportial to differences of orbital occupations , e.g. between px,py and pz.• if these occupancies are the same by symmetry (cubic): Vzz=0• with “axial” (hexagonal, tetragonal) symmetry (px=py): =0

In the following various examples will be presented.

interstitial

Nuclear Quadrupole moment Q of 56Fe

zzQ eQV2

1

(exp.)

FeF2

FeZr3

Fe2P

Fe2O3FeNi

FeSi

FeS2Fe4N

YFe2

FeCl2FeBr2

Compare theoretical and experimental EFGs Q

( previous value)

testing various DFT approximations: GGA works surprisingly well in FeF2:

• There is ONE e- in the three Fe d-t2g states.• LDA: wrong metallic state, e- is distributed in all 3 orbitals wrong charge density and EFG• GGA splits Fe d-t2g states into a1g and eg’

correct charge density and EFG

Fe-EFG in FeF2:

LSDA: 6.2GGA: 16.8exp: 16.5

LSDA

GGA

Ba3Al2F12

Rings formed by four octahedra sharing

corners

-BaCaAlF7

Isolated octahedra

Ca2AlF7, Ba3AlF9-Ib, -Ba3AlF9

-BaAlF5

Isolated chains of octahedra linked

by corners

-CaAlF5, -CaAlF5, -BaAlF5, -BaAlF5

EFGs in fluoroaluminatesEFGs in fluoroaluminates 10 different phases of known structures from CaF2-AlF3,

BaF2-AlF3 binary systems and CaF2-BaF2-AlF3 ternary system

QQ and and Q Q calculationscalculations using XRD data using XRD data

Attributions performed with respect to the proportionality between |Vzz| and Q for the multi-site compounds

Calculated Q

0,0 0,2 0,4 0,6 0,8 1,0

Exp

erim

enta

l Q

0,0

0,2

0,4

0,6

0,8

1,0

AlF3

-CaAlF5

-CaAlF5

Ca2AlF7

-BaAlF5

-BaAlF5

-BaAlF5

Ba3Al2F12

Ba3AlF9-Ib

-Ba3AlF9

-BaCaAlF7

RegressionQ,mes. =Q, cal.

Calculated Vzz (V.m-2)

0,0 1,0e+21 2,0e+21 3,0e+21

Exp

erim

enta

l Q (

Hz)

0,0

2,0e+5

4,0e+5

6,0e+5

8,0e+5

1,0e+6

1,2e+6

1,4e+6

1,6e+6

1,8e+6AlF3

-CaAlF5

-CaAlF5

Ca2AlF7

-BaAlF5

-BaAlF5

-BaAlF5

Ba3Al2F12

Ba3AlF9-Ib

-Ba3AlF9

-BaCaAlF7

Régression

Q,exp = 0,803 Q,cal R2 = 0,38

Q = 4,712.10-16 |Vzz| with R2 = 0,77

Important discrepancies when structures are used which were determined from X-ray powder

diffraction data

M.Body, et al., J.Phys.Chem. A 2007, 111, 11873 (Univ. LeMans)

Very fine agreement between experimental and calculated values

Calculated Vzz (V.m-2)

0,0 5,0e+20 1,0e+21 1,5e+21 2,0e+21 2,5e+21 3,0e+21

Exp

erim

enta

l Q (

Hz)

0,0

2,0e+5

4,0e+5

6,0e+5

8,0e+5

1,0e+6

1,2e+6

1,4e+6

1,6e+6

1,8e+6

AlF3

-CaAlF5

-CaAlF5

Ca2AlF7

-BaAlF5

-BaAlF5

-BaAlF5

Ba3Al2F12

Ba3AlF9-Ib

-Ba3AlF9

-BaCaAlF7

Regression

Q = 5,85.10-16 Vzz

R2 = 0,993

Calculated Q

0,0 0,2 0,4 0,6 0,8 1,0

Exp

erim

enta

l Q

0,0

0,2

0,4

0,6

0,8

1,0

AlF3

-CaAlF5

-CaAlF5

Ca2AlF7

-BaAlF5

-BaAlF5

-BaAlF5

Ba3Al2F12

Ba3AlF9-Ib

-Ba3AlF9

-BaCaAlF7

RegressionQ,exp. =Q, cal.

Q, exp = 0,972 Q,cal

R2 = 0,983

QQ and and QQ after structure optimizationafter structure optimization

EFG (1021 V/m2) in YBa2Cu3O7

Site Vxx Vyy Vzz Y theory -0.9 2.9 -2.0 0.4 exp. - - - - Ba theory -8.7 -1.0 9.7 0.8 exp. 8.4 0.3 8.7 0.9 Cu(1) theory -5.2 6.6 -1.5 0.6 exp. 7.4 7.5 0.1 1.0 Cu(2) theory 2.6 2.4 -5.0 0.0 standard LDA calculations give exp. 6.2 6.2 12.3 0.0 good EFGs for all sites except

Cu(2) O(1) theory -5.7 17.9 -12.2 0.4 exp. 6.1 17.3 12.1 0.3 O(2) theory 12.3 -7.5 -4.8 0.2 exp. 10.5 6.3 4.1 0.2 O(3) theory -7.5 12.5 -5.0 0.2 exp. 6.3 10.2 3.9 0.2 O(4) theory -4.7 -7.1 11.8 0.2 exp. 4.0 7.6 11.6 0.3

K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Phys.Rev. B42, 2051 (1990) D.J.Singh, K.Schwarz, K.Schwarz, Phys.Rev. B46, 5849 (1992)

Interpretation of O-EFGs in YBa2Cu3O7

px py pz Vaa Vbb Vcc

O(1) 1.18 0.91 1.25 -6.1 18.3 -12.2

O(2) 1.01 1.21 1.18 11.8 -7.0 -4.8

O(3) 1.21 1.00 1.18 -7.0 11.9 -4.9

O(4) 1.18 1.19 0.99 -4.7 -7.0 11.7

pppzz

rnV

3

1

)(2

1yxzp pppn

Asymmetry count

EFG (p-contribution)

EFG is proportional to the asymmetric charge distribution around a given nucleus

Cu1-d

O1-py

EF

partly occupied

y

x

z

O(1)O(4)

O(2),O(3)

difference density

Cu(2) and O(4) EFG as function of r

EFG is determined by the non-spherical charge density inside sphere

Cu(2)

O(4)

drrrdrr

YrVYrr zz

LMLMLM /)(

)()()( 203

20

final EFG

r

r r

rr

semicore and valence Cu-EFG contributions

semicore Cu 3p-states have very little importance valence 3d-states: large contribution due to smaller d-x2-y2

occupation valence 4p-states: large contribution of opposite sign.

Originates from the tails of the O-2p orbitals (“re-expanded” as Cu 4p in the Cu atomic sphere, “off-site” contribution)

usually only contributions within the first node or within 1 bohr are important.

general statements for EFG contributions

Depending on the atom, the main EFG-contributions come from anisotropies in

occupations of different orbitals or the radial wave functions of different orbitals)

semicore p-states: they are of course always fully occupied, but due to an anisotropic neighborhood they may slightly contract/expand in different directions. The effect depends on:

the energy and spatial localization: Ti 3p more important than Cu 3p (almost inert)

the distance, type (atom) and geometry of the neighbors (a small octahedral distortion will produce a much smaller effect than a square-planar coordination)

valence p-states : always important (“on-site” O 2p or “off-site” Cu 4p)

valence d-states : in ionic or covalent TM compounds. In metals usually “small”.

valence f-states : very large contributions for “localized” 4f,5f systems unless the f-shell is “half-filled” (or full)

LDA/GGA problems in correlated TM oxides

As shown before, the approximations for exchange-correlation (LDA, GGA) can have significant influence on the quality of the results for the class of compounds with so called “correlated electrons”. Such electrons are in particular TM-3d electrons in TM-oxides (or, more general, in most ionic TM-compounds), but also the 4f and 5f electrons of lanthanides or actinides.

For these systems, “beyond-GGA” schemes like “LDA+U” or “hybrid-functionals” (mixing of Hartree-Fock+GGA) are necessary, as was demonstrated before for YBa2Fe2O5.

Another example of very bad EFGs within GGA would be the class of the undoped Cuprates (La2CuO4, YBa2Cu3O6), which are nonmagnetic metals instead of antiferromagnetic insulators in GGA.

Both, doped and undoped cuprates have a planar Cu – EFG in GGA-calculations, which is by a factor of 2-3 too small compared to experiments.

Cuprates: La2CuO4

LDA: nonmagnetic metal LDA+U: AFM insulator (in agreement with experiment)

lower Hubbard-band upper HB

Magn. moments and EFG in La2CuO4

LDA+U gives AF insulator with reasonable moment

U of 5-6 eV gives exp. EFG

GGAs “mimic” a U of 1-2 eV(EV-GGA more “effective”than PBE !!)

AMF FLL

exp.

Cu-EFG Vzz (1021 V/m2) in YBa2Cu3O6

for LDA, AMF, FLL and DFT-double counting corrections. NM and AF refers to non-magnetic and antiferromagnetic solutions. U and J of 8 and 1 eV is applied to both Cu sites, except for LDA+U* where U is applied only to the Cu(2) site.

Type Vzz -Cu(1) Vzz -Cu(2)

LDA (NM) -8.1 -3.7

AMF-LDA+U (NM) -4.6 -7.3

AMF-LDA+U*(NM) -8.0 -7.2

AMF-LDA+U (AF) -4.5 -13.3

AMF-LDA+U*(AF) -8.0 -13.3

DFT-LDA+U(AF) -4.8 -12.0

FLL-LDA+U (AF) -8.3 -12.3

FLL-LDA+U*(AF) -8.0 -13.3

Experiment 11.8 9.0

Cu(2) too small, Cu(1) ok

Cu(2) ok, Cu(1) wrong

DFT similar to AMF

Cu(2) ok,Cu(1) remains ok

Antiferromagn. FLL-calc. with U=6eV give again best results

Cu(2) still too small, Cu(1) wrong

EFG analysis in YBa2Cu3O6:

EFG contributions (for both spins and p-p and d-d contributions) in LSDA and LDA+U(DFT)

Cu(1) Cu(2)NM-LDA AF-LDA+U(DFT) NM-LDA AF-LDA+U(DFT)

p-p (up) -12.2 -12.2 5.7 5.8p-p (dn) -12.2 -12.2 5.7 6.3d-d (up) 7.1 8.8 -7.4 5.5d-d (dn) 7.1 8.8 -7.4 -31.1Semicore 2.1 2.0 -0.1 1.7Total -8.1 -4.8 -3.7 -12.0Exp. 11.8 9.0

Partial charges and anisotropy counts n in LDA and antiferromagnetic LDA+U(DFT) Cu(1) Cu(2)

NM-LDA AF-LDA+U(DFT) NM-LDA AF-LDA+U(DFT)4pz 0.103 0.104 0.027 0.0284px+py 0.041 0.042 0.131 0.140np -0.083 -0.083 0.039 0.042dx

2-y

2 1.776 1.782 1.433 1.228dz

2 1.474 1.398 1.757 1.784daver 1.802 1.818 1.815 1.841nd 0.257 0.324 -0.294 -0.529

zyxp

pzz ppp

rV )(2

113

Cu(2) has O in xy plane

Cu(1) has O only in z

large charge in z large negative EFG

large charge in xy large positive EFG

Cu(1) anisotropy increased inDFT and AMF schems: wrong!

Cu(2) dx2-y

2 depopulated: ok

Summary

EFGs can routinely be calculated for all kinds of solids. “semi-core” contribution large for “left”-atoms of the periodic

table p-p contribution always large (on-site (eg. O-2p) vs. off-site

(Fe-4p)) d-d (f-f) contributions for TM (lanthanide/actinide) compounds EFG stems from different orbital occupations due to covalency

or crystal field effects forget “point-charge models” and “Sternheimer anti-shielding

factors”EFG is very sensitive to

correct structural data (internal atomic positions) correct theoretical description of the electronic structure

“highly correlated” transition metal compounds (oxides, halides) 4f and 5f compounds “beyond” LDA (LDA+U, Hybrid-DFT, …)

Literature

WIEN2k and APW-based methods: P.Blaha, K.Schwarz, P.Sorantin and S.B.Trickey: Full-potential, linearized

augmented plane wave programs for crystalline systems , Comp.Phys.Commun. 59, 399 (1990)

P.Blaha and K.Schwarz: WIEN93: An energy band-structure program for ab initio calculations of electric field gradients in solids, NQI Newsletter Vol 1 (3), 32 (1994)

K.Schwarz and P.Blaha: Description of an LAPW DF Program (WIEN95), in: Lecture notes in chemistry, Vol. 67, p.139, Ed. C.Pisani, Springer (Berlin 1996)

H.Petrilli, P.E.Blöchl, P.Blaha, and K.Schwarz: Electric-field-gradient calculations using the projector augmented wave method, Phys.Rev. B57, 14690 (1998)

G.Madsen, P.Blaha, K.Schwarz, E.Sjöstedt and L.Nordström: Efficient linearization of the augmented plane-wave method, Phys.Rev. B64, 195134 (2001)

P.Blaha, K.Schwarz, G.Madsen, D.Kvasnicka and J.Luitz: WIEN2k: An augmented plane wave plus local orbitals program for calculating crystal properties. K.Schwarz, TU Wien, 2001 (ISBN 3-9501031-1-2) (http://www.wien2k.at)

K.Schwarz, P.Blaha and G.K.H.Madsen: Electronic structure calculations of solids using the WIEN2k package for material sciences, Comp.Phys.Commun. 147, 71 (2002)

D. J. Singh and L. Nordstrom: Planewaves, Pseudopotentials and the LAPW Method, Springer, Berlin 2006.

Literature

DFT and beyond: P. Hohenberg and W. Kohn: Inhomogeneous electron gas, Phys. Rev. 136, B864

1964. W. Kohn and L. J. Sham: Selfconistent equations including exchange and

correlation effects, Phys. Rev. 140, A1133 1965. V. I. Anisimov, J. Zaanen, and O. K. Andersen: Band theory and Mott

insulators: Hubbard U instead of Stoner I, Phys. Rev.B 44, 943 (1991). J. P. Perdew, K. Burke, and M. Ernzerhof: Generalized gradient approximation

made simple, Phys. Rev. Lett. 77, 3865 (1996); 78, 1396 (1997). A Primer in Density Functional Theory, edited by C. Fiolhais, F. Nogueira, and M.

Marques, Springer, Berlin, 2003. F. Tran, P. Blaha, K. Schwarz, P. Novak: Hybrid exchange-correlation

energy functionals for strongly correlated electrons: Applications to transition-metal monoxides; Phys.Rev B, 74 (2006), 155108.

P. Haas, F. Tran, P. Blaha: Calculation of the lattice constant of solids with semilocal functionals; Physical Review B, 79 (2009), 085104, Physical Review B, 79 (2009), 209902(E)

F.Tran and P.Blaha: Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential, Physical Review Letters, 102, (2009) 226401.

Literature

Hyperfine parameters using WIEN2k: P.Blaha, K.Schwarz and P.Herzig: First-principles calculation of the electric

field gradient of Li3N , Phys.Rev.Lett. 54, 1192 (1985) P.Blaha, K.Schwarz and P.H.Dederichs: First-principles calculation of the

electric field gradient in hcp metals , Phys.Rev. B37, 2792 (1988) P.Blaha and K.Schwarz: Theoretical investigation of isomer shifts in Fe, FeAl, FeTi and

FeCo , J.Phys.France C8, 101 (1988) P.Blaha, P.Sorantin, C.Ambrosch and K.Schwarz: Calculation of the electric field

gradient tensor from energy band structures , Hyperfine Interactions 51, 917 (1989) P.Blaha and K.Schwarz: Electric field gradient in Cu2O from band structure calculations,

Hyperfine Interact. 52, 153 (1989)   P.Blaha: Calculation of the pressure dependence of the EFG in bct In, hcp Ti and Zn

from energy band structures , Hyperfine Interact. 60, 773 (1990) C.Ambrosch-Draxl, P.Blaha and K.Schwarz: Calculation of EFGs in high Tc

superconductors , Hyperfine Interactions 61, 1117 (1990) K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Charge distribution and electric-field

gradients in YBa2Cu3O7-x , Phys.Rev. B42, 2051 (1990) P.Blaha, K.Schwarz and A.K.Ray: Isomer shifts and electric field gradients in Y(Fe1-xAlx)2,

J.Magn.Mag.Mat. 104-107, 683 (1992) D.J.Singh, K.Schwarz, K.Schwarz: Electric-field gradients in YBa2Cu3O7: Discrepancy

between experimental and local-density-approximation charge distributions, Phys.Rev. B46, 5849 (1992)

Literature

P.Blaha, D.J.Singh, P.I.Sorantin and K.Schwarz: Electric field gradient calculations for systems with giant extended core state contributions, Phys.Rev. B46, 1321 (1992)

W.Tröger, T.Butz, P.Blaha and K.Schwarz: Nuclear quadrupole interaction of 199mHg in mercury(I) and mercury(II) halides, Hyperfine Interactions 80, 1109 (1993)

P.Blaha, P.Dufek, and K.Schwarz: Electric field gradients, isomer shifts and hyperfine fields from band structure calculations in NiI2, Hyperfine Inter. 95, 257 (1995)

P.Dufek, P.Blaha and K.Schwarz: Determination of the nuclear quadrupole moment of 57Fe, Phys.Rev.Lett. 75, 3545 (1995)

P.Blaha, P.Dufek, K.Schwarz and H.Haas: Calculations of electric hyperfine interaction parameters in solids, Hyperfine Int. 97/98, 3 (1996)

K.Schwarz, H.Ripplinger and P.Blaha: Electric field gradient calculations of various borides; Z.Naturforsch. 51a, 527 (1996)

 B.Winkler, P.Blaha and K.Schwarz: Ab initio calculation of electric-field-gradient tensors of forsterite, American Mineralogist, 81, 545 (1996)

H.Petrilli, P.E.Blöchl, P.Blaha, and K.Schwarz: Electric-field-gradient calculations using the projector augmented wave method, Phys.Rev. B57, 14690 (1998)

P.Blaha, K.Schwarz, W.Faber and J.Luitz: Calculations of electric field gradients in solids: How theory can complement experiment, Hyperfine Int. 126, 389 (2000)

M.Divis, K.Schwarz, P.Blaha, G.Hilscher, H.Michor , S.Khmelevskyi: Rare earth borocarbides: Electronic structure calculations and electric field gradients, Phys.Rev. B62, 6774 (2000)

Literature

G.Principi, T.Spataru, A.Maddalena, A.Palenzona, P.Manfrinetti, P.Blaha, K.Schwarz, V.Kuncser and G.Filotti: A Mössbauer study of the new phases Th4Fe13Sn5 and ThFe0.22Sn2, J.Alloys and Compounds 317-318, 567 (2001)

R. Laskowski, G.K.H. Madsen, P. Blaha, K. Schwarz: Magnetic structure and electric-field gradients of uranium dioxide: An ab initio study; Physical Review B, 69 (2004), S. 140408(R).

P. Palade, G. Principi, T. Spataru, P. Blaha, K. Schwarz, V. Kuncser, S. Lo Russo, S. Dal Toe, V. Yartys: Mössbauer study of LaNiSn and NdNiSn compounds and their deuterides; Journal of Radioanalytical and Nuclear Chemistry, 266 (2005), 553 - 556.

P. Blaha, K. Schwarz, P. Novak: Electric Field Gradients in Cuprates: Does LDA+U give the Correct Charge Distribution ?; International Journal of Quantum Chemistry, 101 (2005), 550

M. Body, C. Legein, J. Buzare, G. Silly, P. Blaha, C. Martineau, F. Calvayrac: Advances in Structural Analysis of Fluoroaluminates Using DFT Calculations of 27Al Electric Field Gradients; Journal of Physical Chemistry A, 111 (2007), S. 11873 - 11884.

Seung-baek Ryu, Satyendra K. Das, Tilman Butz, Werner Schmitz, Christian Spiel, Peter Blaha, and Karlheinz Schwarz: Nuclear quadrupole interaction at 44Sc in the anatase and rutile modifications of TiO2 : Time-differential perturbed-angular-correlation measurements and ab initio calculations, Physical Review B, 77 (2008), S. 094124

C. Spiel, P. Blaha, K. Schwarz: Density functional calculations on the charge-ordered and valence-mixed modification of YBaFe2O5; Phys. Rev. B 79 (2009), 115123.