Transcript 1. Mössbauer parameters from DFT-based - Extra Materials

Mössbauer parameters from DFTbased WIEN2k calculations for
extended systems
Peter Blaha
Institute of Materials Chemistry
TU Vienna
Main Mössbauer parameters:


The main (conventional) Mössbauer spectroscopy parameters which we
want to calculate by theory are:
Isomer shift: d = a (r0Sample – r0Reference);



Magnetic Hyperfine fields: Btot = Bcontact + Borb + Bdip


it is proportional to the electron density r0 at the nucleus
the constant a is proportional to the change of nuclear radii during the
transition (we use a=-.291 au3mm s-1)
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:

D  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 manybody Schrödinger equation in the corresponding solid.
HY=EY

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
Treatment of
spin
Non-spinpolarized
Spin polarized
(with certain magnetic order)
Form of
potential
“Muffin-tin” MT
atomic sphere approximation (ASA)
pseudopotential (PP)
Full potential : FP
Relativistic treatment
of the electrons
exchange and correlation potential
non relativistic
semi-relativistic
fully-relativistic
 1 2
 k
k k



V
(
r
)



i i
 2
 i
Representation
of solid
non periodic
(cluster, individual MOs)
periodic
(unit cell, Blochfunctions,
“bandstructure”)
Hartree-Fock (+correlations)
Density functional theory (DFT)
Local density approximation (LDA)
Generalized gradient approximation (GGA)
Beyond LDA: e.g. LDA+U
Schrödinger - equation
Basis functions
plane waves : PW
augmented plane waves : APW
atomic oribtals. e.g. Slater (STO), Gaussians (GTO),
LMTO, numerical basis
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 r

 
E   Vext ( r ) r ( r )dr  F [ r ]
• This (exact) statement states, that we need to know only the
electron density r, but we do NOT need to know the
wavefunctions (solution of Schrödinger equation not necessary).
• However, the exact functional form of F[r] is unknown and
any approximation so far is too crude.
DFT - Kohn Sham (still exact)
Lets decompose the total functional F[r] into parts which
can be calculated „exactly“ and some „unknown, but small“
rest:


)  
1
r
(
r
)
r
(
r
 
E  To [ r ]   Vext r ( r )d r     dr dr   E xc [ r ]
2 | r r |
Ekinetic
Ene
Ecoulomb Eee
Exc exchange-correlation
non interacting
• 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.
Kohn-Sham equations


  1 r ( r ) r ( r )  
E  To [ r ]   Vext r ( r )d r  

 dr dr   E xc [ r ]

2
|r r |
vary r
1-electron equations (Kohn Sham)
1 2





{   Vext ( r )  VC ( r ( r ))  Vxc ( r ( r ))} i ( r )   i  i ( r )
2

E xc ( r )
r (r ) 
-Z/r

2
d
r
r
(
r
)

|

|


 i

| r  r |
r
i  EF
Exc and Vxc are unknown and must be approximated
LDA or GGA
treat both, exchange and correlation effects approximately
Walter Kohn, Nobel Prize 1998 Chemistry
Approximations to EXC

Local density approximation (LDA):
E


LDA
xc
  r (r) 
hom .
xc
[ r ( r )] dr

 r (r)
4
3
dr
xc is the exchange-correlation energy density of the homogeneous
electron gas at density r.
Second order gradient expansion (GEA):
E

GEA
xc
E
LDA
xc
   C ( r ) r (r)
2
3
r ( r ) dr
 , 
The GEA XC-hole nXC(r,r’) is not a hole of any physical system and
violates



nX(r,r’) ≤ 0
∫ nX(r,r’) dx = -1
∫ nC(r,r’) dx = 0
exchange hole must be negative
must contain charge -1
e- at r
nXC(r,r’)
Generalized gradient approximations (GGA)
GGA
E xc
  r ( r ) F [ r ( r ), r ( r )] dr

“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
better approximations are constantly developed

meta-GGAs:

Perdew,Kurth,Zupan,Blaha (PRL 1999):
m GGA
Exc
  r ( r ) F [ r ( r ), r ( r ), 2 r ( r ), ( r )] dr

1
2
 i ( r )

2 i

use laplacian of r, or the kinetic energy density  ( r ) 

analytic form for Vxc not possible (Vxc = dExc/dr) , SCF very difficult
better meta-GGAs under constant development …
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



GGA

LSDA
GGA

GGA



LSDA
NM
fcc
in contrast to
experiment
FM
bcc
Correct lattice
constant
Experiment


FM
bcc
LDA: Fe is nonmagnetic and in fcc structure
GGA 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 HFFparameters, 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 Schwarz
Georg Madsen
Dieter Kvasnicka
Joachim Luitz
November 2001
Vienna, AUSTRIA
Vienna University of Technology
WIEN2k: ~1530 groups
mailinglist: 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)
APW Augmented Plane Wave method
The unit cell is partitioned into:
atomic spheres
Interstitial region
unit cell
Rmt
Basisset:
  
i ( k  K ).r
PW:
ul(r,) are the numerical solutions
e
Atomic partial waves
K
A
 mu (r,  )Ym (rˆ)
m
rI
join
of the radial Schrödinger equation
in a given spherical potential for a
particular energy 
AlmK coefficients for matching the PW
Slater‘s APW (1937)
Atomic partial waves
K
A
 mu (r,  )Ym (rˆ)
m
Energy dependent basis
functions lead to 
Non-linear eigenvalue problem
H Hamiltonian
S overlap matrix
One had to numerically search for the energy, for which
the det|H-ES| vanishes. Computationally very demanding.
“Exact” solution for given (spherical) potential!
Linearization of energy dependence
LAPW suggested by
center
O.K.Andersen,
Phys.Rev. B 12, 3060
(1975)
 kn 
[ A
m
antibonding
bonding
 ( E , r )]Ym ( rˆ)
(kn )u ( E , r )  Bm (kn )u
m
expand ul at fixed energy El and
add
ul  ul / 
Almk, Blmk: join PWs in
value and slope
 General eigenvalue problem
(diagonalization)
 additional constraint requires
more PWs than APW
Atomic sphere
LAPW
APW
PW
Full-potential in LAPW (A.Freeman etal.)

SrTiO3
The potential (and charge density)
can be of general form
(no shape approximation)
VLM ( r )YLM ( rˆ)
{
V (r ) 
Full
potential
LM
VK e
 
iK . r
rI
K

Inside each atomic sphere a
local coordinate system is used
(defining LM)
Muffin tin
approximation
TiO2 rutile
r  Ra
Ti
O
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  [ AmuE1  BmuE1  CmuE2 ]Ym (rˆ)

LO: contains a second ul(E2)



Ti atomic sphere



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)
New ideas from Uppsala and Washington
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)
kn   Am (kn )u ( E , r )Ym (rˆ)
lo  [ AmuE1  BmuE1 ]Ym (rˆ)
m
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
For example:
Ti

Valence states

Scalar relativistic




Spin orbit coupling on demand by
second variational treatment
Semi-core states


Scalar relativistic
on demand



mass-velocity
Darwin s-shift
spin orbit coupling by second
variational treatment
Additional local orbital (see Th-6p1/2)
Core states

Fully relativistic

Dirac equation
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



step by step initialization



symmetry detection
automatic input generation
SCF calculations

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

spacegroup selection
import cif file
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.Schwarz
Institute 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 coupling
 based 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
WIEN2K
An Augmented Plane Wave
Plus Local Orbital
Program for Calculating
Crystal Properties
Peter Blaha
Karlheinz Schwarz
Georg Madsen
Dieter Kvasnicka
Joachim Luitz
http://www.wien2k.at
Verwey-transition:
Fe3O4, magnetite
E.Verwey, Nature 144, 327 (1939)
phase 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
Ba
Y (R)
square pyramidal
coordination
Antiferromagnet with a 2 step Verwey transition around 300 K
Woodward&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 AFM
all 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:
Pmma a:b:c=2.09:1:1.96 (20K)
valence mixed (VM) phase:
Pmmm a:b:c=1.003:1:1.93 (340K)
c
b
a



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
antiferromagnetic structure
CO phase: G-type AFM


AFM arrangement in all directions,
also across Y-layer
Fe moments in b-direction
VM phase:
AFM for all Fe-O-Fe superexchange paths
FM across Y-layer (direct Fe-Fe exchange)
4
8
independent 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:



Calculation:



CO: 4.15/3.65 (for Tb), 3.82 (av. for Y)
VM: ~3.90
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


ExcPBE0 [r] = ExcPBE [r] + a (ExHF[sel] – ExPBE[rsel])
LDA+U, GGA+U

ELDA+U(r,n) = ELDA(r) + Eorb(n) – EDCC(r)



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
Vm,m',  (U  J )( 1  nm,m', )
2
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
singleinsulator,
lower Hubbard-band
t2g band splits
in VM splits in CO with Fe3+ states
metallic
lower than Fe2+
insulator
GGA+U
metal
magnetic moments and band gap

magnetic moments in very good agreement with exp.



LDA/GGA: CO: 3.37/3.02
VM: 3.34 mB
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
r  n  0

Integrate charge inside this region
Structure optimization (GGA+U)

O2a
CO phase:
Fe2+: shortest bond in y (O2b)
3+: shortest bond in z (O1)
 Fe
O2b


VM phase:
all Fe-O distances similar
 theory deviates along z !!




Fe-Fe interaction
different U ??
finite temp. ??
O3
O1
O1
strong coupling between lattice and electrons !

Fe2+ (3d6)





strong covalency effects
in eg and d-xz orbitals
CO
Fe3+ (3d5)
VM Fe2.5+ (3d5.5)
majority-spin fully occupied
very localized states at lower energy than Fe2+
minority-spin states
d-xz fully occupied (localized)
empty
short bond in y
short bond in z (missing O)
d-z2 partly occupied
FM Fe-Fe; distances in z ??
Difference densities Dr=rcryst-ratsup

CO phase
VM phase
Fe2+: d-xz
Fe3+: d-x2
O1 and O3: polarized
toward Fe3+
Fe: d-z2 Fe-Fe interaction
O: symmetric
dxz spin density (rup-rdn) of CO phase




Fe3+: no contribution
Fe2+: dxz
weak p-bond with O
tilting of O3 p-orbital
Mössbauer spectroscopy:

Isomer shift: d = a (r0Sample – r0Reference); a=-0.291 au3mm s-1


proportional to the electron density r at the nucleus
Magnetic Hyperfine fields: Btot=Bcontact + Borb + Bdip

Bcontact = 8p/3 mB [rup(0) – rdn(0)]
…
spin-density at the nucleus
…
orbital-moment
… spin-moment
S(r) is reciprocal of the relativistic mass enhancement
nuclear quadrupole interaction
1  2V (0)
e
 ( x) xi x j dx 
ij  Q



2 ij xi x j
2h



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


  eQ / h
electric field gradient (EFG)
1
 ij  Vij  d ij 2V
3
 2V (0)
Vij 
xi x j
Electric field gradients (EFG)

EFG
traceless tensor
1
 ij  Vij  d ij 2V
3
 2V (0)
Vij 
xi x j
with Vxx  V yy  Vzz  0
Vaa Vab Vac
Vba Vbb Vbc 
Vca Vcb Vcc
V xx 0 0
0 V yy 0
0 0 Vzz
similarity
transformation
traceless
|Vzz|
 |Vyy|  |Vxx|
„EFG“: Vzz
principal component

/ Vxx /  / V yy /
/ Vzz /
asymmetry parameter
…0-1
directions of Vxx, Vyy, Vzz
cubic: no EFG; hex, tetragonal (axial symm.): only Vzz
theoretical EFG calculations
EFG is tensor of second derivatives of VC at the nucleus:
 2V (0)
Vij 
xi x j
Vzz  
Vc (r )  
r ( r )Y20
1
V  3
r
r3
p
zz
1
V  3
r
p
d
xy
r  r
dr VLM (r )YLM (rˆ)
LM
x
 p y )  pz
Vzz  V20 ( r  0)
Vxx   1 V20  V22
2

 d x 2  y 2  1 (d xz  d yz )  d z 2
2
Cartesian LM-repr.
V yy   1 V20  V22
2
dr  Vzzp  Vzzd
12 ( p
d
d
zz
r (r)

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 d = a (r0Sample – r0Ref.)
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.
 a is proportional to the change of nuclear radii during the transition,
(in principle a “known nuclear constant” ). However, a meaningful a
depends on details of the calculations (we use a=-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+
Fe2.5+
+1.84
+1.62
0.28
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 = 8p/3 mB [rup(0) – rdn(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
2s,p
3s,p
4s,p
total
- 1.4 T
- 113.7 T opposite sign; exchange effect with 3s wf.
+ 60.7 T same sign as 3d moment, strong overlap of 3d and 3s,p wf.
+ 16.3 T often even smaller, but can be large due to ligand
- 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 r (electronic +
nuclear) in the system.
Vc (r )  
r (r )
r  r
dr  VLM (r )YLM (rˆ)
LM
• The EFG is a tensor of second derivatives of VC at the nucleus:
 2V (0)
Vij 
;
xi x j
Vzz  
r (r )Y20
r
3
dr
• 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 r. 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)
theoretical EFG calculations

r (r )Y20
The charge density r in the integral Vzz  
r3
be decomposed in various ways for analysis:

dr
can
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
We write the charge density and the potential inside the atomic spheres in
an lattice-harmonics expansion
r (r )   r LM (r )YLM (rˆ)
V (r )   vLM (r )YLM (rˆ)
LM
LM
spatialdecomposition :
Vzz  
r (r )Y20
r3

d r 
3
Vzz 
r
LM
r3
r 20 (r )

sphere
r
3

d r
3
LM
sphere

(r )YLM Y20
int erstital
r (r )Y20
r3
d 3r
dr  interstitial
orbitaldecomposition :
r 20 (r )  

 Y drˆ 
nk * nk
lm l m 20
k , n.l ,l , m , m
Vzz  Vzzpp  Vzzdd  ..... interstitial
p  p; d  d ; ( s  d ) contr.
theoretical EFG calculations
Vzz  Vzzpp  Vzzdd  ..... interstitial
V
V
pp
zz
dd
zz
1
 3
r
1
 3
r
p
1 2 ( p
d
d
x
 p y )  pz

1

d

(d xz  d yz )  d z 2
2
2
xy
x y
2

• 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.
Nuclear Quadrupole moment Q of
56Fe
Compare theoretical and experimental EFGs
1
D Q  eQV zz
2
(
previous value)
FeSi
Fe2O3
FeS2
FeCl2
YFe2
Fe4N
FeNi
FeZr3
Fe2P
FeBr2
(exp.)
Q
FeF2
testing various DFT approximations: GGA
works surprisingly well in FeF2:
LSDA
• 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
LSDA:
GGA:
exp:
in FeF2:
6.2
16.8
16.5
GGA
EFGs in fluoroaluminates
10 different phases of known structures from CaF2-AlF3,
BaF2-AlF3 binary systems and CaF2-BaF2-AlF3 ternary system
Isolated octahedra
Isolated chains of
octahedra linked by
corners
a-BaAlF5
a-CaAlF5, b-CaAlF5,
b-BaAlF5, g-BaAlF5
Rings formed by four
octahedra sharing
corners
a-BaCaAlF7
Ca2AlF7, Ba3AlF9-Ib,
b-Ba3AlF9
Ba3Al2F12
Q and Q calculations using XRD data
AlF3
Q,exp = 0,803 Q,cal
R2 = 0,38
a-CaAlF
-16 |V | with R2 = 0,77
Q = 4,712.10performed
Attributions
with respect to the proportionality
between |Vzz|
zz
b-CaAlF
1,0
AlF
and Q for the multi-site
compounds Ca
a-BaAlF
5
5
1,8e+6
2
AlF3
5
a-CaAlF5
1,6e+6
b-BaAlF5
b-CaAlF5
g-BaAlF5
Ca2AlF7
Ba3AlF9-Ib
b-BaAlF5
b-Ba3AlF9
g-BaAlF5
a-BaCaAlF7
Ba3Al2F12
Experimental Q
Experimental Q (Hz)
1,2e+6
Ba3Al2F12
0,8
a-BaAlF5
1,4e+6
Ba3AlF9-Ib
b-Ba3AlF9
1,0e+6
7
a-BaCaAlF7
Régression
8,0e+5
Regression
Q,mes. =Q, cal.
0,6
0,4
6,0e+5
4,0e+5
0,2
2,0e+5
0,0
0,0
0,0
1,0e+21
2,0e+21
3,0e+21
-2
Calculated Vzz (V.m )
0,0
0,2
0,4
0,6
0,8
Calculated Q
Important discrepancies when structures are used which
were determined from X-ray powder diffraction data
1,0
Q and Q after structure optimization
1,8e+6
1,0
Q =
Vzz
R2 = 0,993
5,85.10-16
1,6e+6
0,8
1,4e+6
1,2e+6
Experimental Q
Experimental Q (Hz)
Q, exp = 0,972 Q,cal
R2 = 0,983
1,0e+6
AlF3
a-CaAlF5
8,0e+5
b-CaAlF5
Ca2AlF7
6,0e+5
0,6
AlF3
a-CaAlF5
b-CaAlF5
Ca2AlF7
a-BaAlF5
0,4
a-BaAlF5
b-BaAlF5
b-BaAlF5
g-BaAlF5
g-BaAlF5
4,0e+5
Ba3Al2F12
Ba3Al2F12
Ba3AlF9-Ib
Ba3AlF9-Ib
0,2
b-Ba3AlF9
b-Ba3AlF9
2,0e+5
a-BaCaAlF7
a-BaCaAlF7
Regression
Q,exp. =Q, cal.
Regression
0,0
0,0
0,0
5,0e+20
1,0e+21
1,5e+21
2,0e+21
2,5e+21
3,0e+21
0,0
0,2
-2
Calculated Vzz (V.m )
0,4
0,6
0,8
Calculated Q
Very fine agreement between experimental and calculated values
M.Body, et al., J.Phys.Chem. A 2007, 111, 11873
(Univ. LeMans)
1,0
EFG (1021 V/m2) in YBa2Cu3O7



















Site
Y
theory
exp.
Ba
theory
exp.
Cu(1) theory
exp.
Cu(2) theory
exp.
O(1) theory
exp.
O(2) theory
exp.
O(3) theory
exp.
O(4) theory
exp.
Vxx
-0.9
-8.7
8.4
-5.2
7.4
2.6
6.2
-5.7
6.1
12.3
10.5
-7.5
6.3
-4.7
4.0
Vyy
2.9
-1.0
0.3
6.6
7.5
2.4
6.2
17.9
17.3
-7.5
6.3
12.5
10.2
-7.1
7.6
Vzz
-2.0
9.7
8.7
-1.5
0.1
-5.0
12.3
-12.2
12.1
-4.8
4.1
-5.0
3.9
11.8
11.6

0.4
0.8
0.9
0.6
1.0
0.0
0.0
0.4
0.3
0.2
0.2
0.2
0.2
0.2
0.3
standard LDA calculations give
good EFGs for all sites except Cu(2)
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
O(1)
O(4)
z
O(2),O(3)
y
Asymmetry count
x
1
Dn p  p z  ( p x  p y )
2
EFG (p-contribution)
EF
Cu1-d
1
p
Vzz  Dn p  3  p
r
difference density Dr
EFG is proportional to the
asymmetric charge distribution
around a given nucleus
O1-py
partly occupied
Cu(2) and O(4) EFG as function of r


EFG is determined by the non-spherical charge density inside
sphere
r (r )Y20
r (r )   r LM (r )YLM
Vzz  
dr   r 20 (r ) / r dr
3
r
LM
Cu(2)
r

O(4)
r
final EFG
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 “hybridfunctionals” (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 GGAcalculations, 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
AMF
exp.
FLL
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 !!)
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)
LDA (NM)
-8.1
AMF-LDA+U (NM)
-4.6
AMF-LDA+U*(NM) -8.0
AMF-LDA+U (AF)
-4.5
AMF-LDA+U*(AF)
-8.0
DFT-LDA+U(AF)
-4.8
FLL-LDA+U (AF)
-8.3
FLL-LDA+U*(AF)
-8.0
Experiment
11.8
Vzz -Cu(2)
-3.7
-7.3
-7.2
-13.3
-13.3
-12.0
-12.3
-13.3
9.0
Cu(2) too small, Cu(1) ok
Cu(2) still too small, Cu(1) wrong
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
EFG analysis in YBa2Cu3O6:
Vzzp 
1
r3
12 ( p  p )  p 
x
p
y
z
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
5.8
large charge in z 
p-p (dn) -12.2
-12.2
5.7
6.3
large negative EFG
d-d (up)
7.1
8.8
-7.4
5.5
d-d (dn)
7.1
8.8
-7.4
-31.1
Semicore 2.1
2.0
-0.1
1.7
large charge in xy 
Total
-8.1
-4.8
-3.7
-12.0
large positive EFG
Exp.
11.8
9.0
Partial charges and anisotropy counts Dn 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
0.028
4px+py
0.041
0.042
0.131
0.140
Dnp
-0.083
-0.083
0.039
0.042
2
2
dx -y
1.776
1.782
1.433
1.228
Cu(1) anisotropy increased in
2
dz
1.474
1.398
1.757
1.784
DFT and AMF schems: wrong!
Cu(2) has O in xy plane
daver
1.802
1.818
1.815
1.841
Dnd
0.257
0.324
-0.294
-0.529
Cu(2) dx2-y2 depopulated: ok
Cu(1) has O only in z
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.
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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
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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
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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
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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
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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 chargeordered and valence-mixed modification of YBaFe2O5; Phys. Rev. B 79 (2009),
115123.