Overview: Tuesday Density-Functional-Theory Lecture...
Transcript of Overview: Tuesday Density-Functional-Theory Lecture...
8/10/2009
1
Density-Functional-TheoryLecture IIDr. Boris KieferPhysics Department
New Mexico State University
Overview: TuesdayReview:• Periodic table. • Hartree-Fock: Exchange Interaction.• Density-Functional-Theory (DFT)• Density-Functional-Theory (DFT).• Exchange-correlation energy.
DFT: ins and outs:• Emax, kmax.• Pseudopotentials vs. all-electron calculations. • Metals and insulators.
DFT lectures, 08/08+08/09/2009, UNM 2
• Convergence tests. • Bulk/surface/molecules.• Examples.
Periodic Table
3DFT lectures, 08/08+08/09/2009, UNM
Adiabatic Decoupling
Decouple electronic and nuclear degrees of freedom: Nuclei: classic treatment.
If T=0 K Kinetic energy of the nuclei is zero. Note: Nuclei are not eliminated from the
bl U d U
Nuclei: classic treatment. Electrons: quantum mechanics.
DFT lectures, 08/08+08/09/2009, UNM 4
problem, Unuc-nuc and Unuc-el.Static problem.
If T>0 K Nuclei are no longer fixed. Lattice vibrations; thermodynamics.
8/10/2009
2
rrrnrVm iiieffr
el
,
22
2
Kohn and Sham EquationsKohn and Sham (1965)
rnE
rdrr
rne
rVrnrV
XC
eff
2
,
Coulomb interaction.
External potential.
DFT lectures, 08/08+08/09/2009, UNM 5
rn
rnEXC
Exchange-correlation.
Can be generalized to include magnetism:
nnn
What is the Exchange-Correlation Potential, EXC?
Thermodynamics: ground state == lowest energy state
Pauli exclusion principle Exchange energy.
Repulsion of opposite spins Correlation energy.
6DFT lectures, 08/08+08/09/2009, UNM
Exchange Correlation Functionals
Active research field to develop better EXC functionals:• LDA : XCXC EE LDA :
Ceperley and Alder (1981)…
• GGA :
XCXC EE
,XCXC EE
DFT lectures, 08/08+08/09/2009, UNM 7
PW91 (1991), PBE (1996), revPBE (1996), rPBE (1999)…
Exchange Correlation Functionals:Which one to choose?
Depends on system and variable of interest:Depends on system and variable of interest:Iron: LDA gives incorrect ground state (hcp rather
than bcc); GGA correct ground state.
DFT lectures, 08/08+08/09/2009, UNM 8
SiO2: LDA gives better agreement with experiment athigh pressures then GGA.
8/10/2009
3
Suitable Electronic Wave FunctionsConstant potential
Solutions to Schrödinger equation with constant
Steep potential
DFT lectures, 08/08+08/09/2009, UNM 9http://gauss.sdsmt.edu/~bhemmelm/quantu1.gif
Solutions to Schrödinger equation with constant potential:
Planewaves: ψ~exp(i k r)Works best for “flat” potentials.
Brillouin zonesCrystal: Periodic Structure in 3-d space
Planewaves: ψ~exp(ikr) Fourier transform.
1st Brillouin zone: range of independent values.
Definition: aab 322
;i bnk
Face-centered cubic
cellVb1 2
1,2,3 cyclic
Same as in xrd.10DFT lectures, 08/08+08/09/2009, UNM
;ii
i bN
k
Energy CutoffPlanewaves: ψ~exp(ikr)Energy:
22
kkE 2
2k
mkE
e
Define cutoff energy: Emax
2max
2
max 2k
mE
e
DFT lectures, 08/08+08/09/2009, UNM 11
Note: minmaxmax
2 kEk
Increase Emax to resolve finer details of the charge density.
Pseudopotentials“Problem”: Steep potential close to nuclei
plane waves are not well suited.
Idea: Most material/chemical properties rely onvalence electrons.
DFT lectures, 08/08+08/09/2009, UNM 12
Pseudopotential.
Eliminate core electrons
8/10/2009
4
Generation of Pseudopotentials
DFT lectures, 08/08+08/09/2009, UNM 13http://upload.wikimedia.org/wikipedia/en/thumb/7/7e/SketchPseudopotenials.png/300px-SketchPseudopotenials.png
Matching Radius: rc
Oxygen: an example
Flat
DFT lectures, 08/08+08/09/2009, UNM 14
http://th.physik.uni-frankfurt.de/~engel/ncpp/fig4.jpg
Steep Coulomb potential
Carbon
Smaller rc: B tt t f bilit Better transferability. higher energy cutoff.
DFT lectures, 08/08+08/09/2009, UNM 15Pickett, 1989
AlternativeSeparate space in different regions:
Interstitial regions plane waves as before.Regions around the nucleus:
use atomic-like states.
DFT lectures, 08/08+08/09/2009, UNM 16
All-electron methods.(structural optimization difficult to
implement, i.e. WIEN2k)
8/10/2009
5
Computational ApproachesPlane waves
Treatment of Space Close to NucleiInterstitial Space: Plane Waves
Pseudo Potential Method All Electron Method
Plane waves Atomic states
DFT lectures, 08/08+08/09/2009, UNM 17
Pseudo‐Potential Method
VASP Siesta
PWScfAbinit
DACAPOGPAW
CASTEP
All Electron Method
Wien2K
FleurLAPW
Computational Procedure
DFT lectures, 08/08+08/09/2009, UNM 18
Payne et al. (1992)
Types of MaterialsDensity of States
DFT lectures, 08/08+08/09/2009, UNM 19http://www.physics.udel.edu/~watson/scen103/semi1.gif
Metal: States available just above the highest occupied state
Insulator: NO states available just above the highest occupied state
MetalsFermi-Dirac Distribution Function
TkBWidth
DFT lectures, 08/08+08/09/2009, UNM 20
http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg
Metal: Occupation is a step-function.“Softens” as temperature increases.
8/10/2009
6
Metals: Consequences - IProblem:
At T=0 K (low temperatures)Rapid change of occupation from 1 to zero.
DFT lectures, 08/08+08/09/2009, UNM 21
During optimization: unoccupied wave functionsbecome occupied and vice versa Oscillations and instabilities.
Metals: Consequences-IINeed to know charge density (wave functions) in the (1st) Brillouin zone very well
Expectation that we need a dense k-point mesh.
DFT lectures, 08/08+08/09/2009, UNM 22
Convergence TestCarefully check variation of
E=E(Emax,k-point mesh)
Two steps: 1) Fix Emax and increase k-point mesh size. 2) Fix k-point mesh size and vary Emax.
Changes between subsequent calculations should be less than 1 meV/atom
DFT lectures, 08/08+08/09/2009, UNM 23
should be less than 1 meV/atom. (criterion may vary for specific applications)
-12.400 -12.340
1010
10//3
00
1010
10//4
00
1010
10//5
00
1010
10//6
00
1010
10//7
00
888/
600
1010
10//6
00
1212
12//6
00
1414
14//6
00
1616
16//6
00
Convergence Test: RheniumRhenium (Re):• Convergence:hcp-Re: 600 eV; 12 12 8
-12.420
-12.415
-12.410
-12.405
Ene
rgy
(eV
/ato
m)
-12.360
-12.355
-12.350
-12.345
hcp-Re fcc-Re
12x12x8.fcc-Re: 600 eV; 10x10x10.• E(hcp) < E(fcc)hcp-Re more stable as observed
DFT lectures, 08/08+08/09/2009, UNM 24
-12.42512
128/
/300
1212
8//4
00
1212
8//5
00
1212
8//6
00
1212
8//7
00
1010
6//6
00
1212
8//6
00
1414
8//6
00
1616
10//6
00
2020
12//6
00
2424
16//6
00
Run parameters
-12.365
stable, as observed experimentally.(Not shown: Pt: 600 eV; 14x14x14).
8/10/2009
7
InsulatorsOccupied and unoccupied states are separatedby an energy gap of finite width.
We do not expect instabilities and oscillationsdue to changing occupation numbers.
Less dense k-point meshes than for metals.
Even though not shown here experience shows
DFT lectures, 08/08+08/09/2009, UNM 25
Even though not shown here experience shows that this expectation is supported.
es
Anhydrous (Na2C2O4) metal
Example: Computed Density of States
troni
c D
ensi
ty o
f Sta
te
Hydrous (HCOONa)
Na-formate (HCOONa)
metal
insulator
DFT lectures, 08/08+08/09/2009, UNM 26
Ele
ct
-25 -20 -15 -10 -5 0 5 10 15E-EF (eV)
Na formate (HCOONa)
insulator
Temperature: Extended System Approach (Anderson, 1980)
Idea: 0th law of th d i B iS thermodynamics: Bring system in contact with heat bath.
Thermostat: <T> = constant.
Bath
System
Coupling
Choose coupling such that the proper ensemble is modeled (Nose, 1984a, 1984b).
But other options exist: Lattice-dynamics.
27DFT lectures, 08/08+08/09/2009, UNM
Lattice Dynamics
DFT lectures, 08/08+08/09/2009, UNM 28
8/10/2009
8
Bulk-Structures-IBulk-calculations:• Periodic good match
with plane waves.
Hypothetical:
p
• Solid solutions (alloying):Vary composition.Pd8Zn8 Pd9Zn7…
Di d
DFT lectures, 08/08+08/09/2009, UNM 29
ypCubic-Pd8Zn8Cubic-Pd50Zn50
• Disorder: fixed composition.Exchange Pd Zn.
Bulk-Structures-IIGraphite:• sp2 – hybridization.• intra-plane sigma bonds.
i t l d W l• inter-plane van der Waals.Origin: fluctuating dipoles.
But:• theory here: time independent van der
DFT lectures, 08/08+08/09/2009, UNM 30
independent van der Waals compounds not necessarily well described.(New functionals).
Surfaces-I• 2-d periodic.• Symmetry broken in 3rd
dimension. V l (10 20 A)• Vacuum layer (10-20 A).
• No electrons in vacuum. But: we use planewaves…Need higher energy cutoff to
make charge density zero in vacuum
DFT lectures, 08/08+08/09/2009, UNM 31
fcc-Pd(111)-slab.
vacuum.Alternative: do not use planewaves.But more localized functions: wavelets, finite-elements…
Surfaces-II• slab thick enough such that surface relaxation can be accounted for. 4 layers sufficient, relax top two l (t t )layers (test convergence). • asymmetric slab: Artificial induced (electrical) dipole moment. Long ranged two options: 1) make vacuum thick enough 15 –
DFT lectures, 08/08+08/09/2009, UNM 32
fcc-Pd(111)-slab.
1) make vacuum thick enough, 15 20 A.
2) Compensate by additional dipole. (VASP: middle of the cell).
8/10/2009
9
Surfaces-III• adsorption.
Example: CO, MeOH, EtOH,… Many molecules are polar
fcc-Pd(111)-slab
• Same problem as with the asymmetric slab, same options to solve the problem.
y p(∆ EN ≠ 0)
DFT lectures, 08/08+08/09/2009, UNM 33
fcc Pd(111) slab.(top-view)
solve the problem.
Molecules• Electrons are localized. • Same difficulty as for surfaces.
Note: Adsorption energies are energy differencesAt least partial compensation.
Alternatives to plane waves are: W l t d fi it l t
DFT lectures, 08/08+08/09/2009, UNM 34
Wavelets and finite elements.
Quantum Mechanics + Molecular Mechanics/dynamics
Hybrid methods: QM//MM/MD• “Local” quantum mechanics.
“N l l” l i l h i /
Advantage: • Focus on active portion.
L l d t il• “Non-local” classical mechanics/molecular dynamics.
• Large scale: details are “less” important.
Disadvantage:• Boundary: QM + MM?• “Cutting” through bonds.
http://bio.phys.uniroma1.it/goase.gif
Classical Quantum Mechanics
?35DFT lectures, 08/08+08/09/2009, UNM
Some Applications of DFT
Earth’s and Planetary Sciences:• Electrons in planetary interiors.• Melts at high pressure.
Environmental Geosciences:• Structure of (H3O)+Fe(SO4)2
Student presentations
DFT lectures, 08/08+08/09/2009, UNM 36
• Andrew: Pt1-xRex solid solutions.• Eric: Pd on gamma-Al2O3.• Levi: Pd on alpha-Al2O3.• Sam: non-Pt based catalysts.
8/10/2009
10
Planetary Interiors(Stixrude et al.,1998)
3/25 207.0407.01176.0 SS rZrPPlasma model
Explanation:Coulomb attraction important.
3/1
43
S
SS
r Gellman and Brueckner (1949)Hubbard (1984)
DFT lectures, 08/08+08/09/2009, UNM 37
pElectrons are not a plasma.Predicted density is too HIGH.Electronic screening.Observations:
Charge density too high.
Melts at High Pressure(Mookherjee et al., 2008)
Background:• Constitution of the Earth’s interior.• Abundance and distribution of hydrogen. • Hydrogen affects: density viscosity elasticity electrical conductivity
1 GPa ~ 104 atm
Hydrogen affects: density, viscosity, elasticity, electrical conductivity, …
T=3000 KT=3000 K
12 * MgSiO3 + 8 * H2O = 84 atoms. 0 – 120 GPa, 3000 – 6000 K.
DFT lectures, 08/08+08/09/2009, UNM 38
Pressure dependent H2O speciation:
Low pressure: OH-, H2O, Mg-OH2.
High pressure: also Si-O-H2O-O-Si
• Neutrally buoyant? Hydrous melt lenses.Seismic observations(Song et al., 2004).
Implications
T=3000 K
T=6000 K
( g , )• Conduction dominated by
hydrogen diffusion. • P=14 GPa, T=1800 K:
σ~18 S/m. • Conduction should be high
enough to be detectable,
T=3000 K
T 3000 K
DFT lectures, 08/08+08/09/2009, UNM 39
through electromagnetic sounding.
Environment Geochemistry
Berkeley Pit; Butte, MontanaMining operation: 1955 - . Copper, silver, gold. Toxic metals and
formation of SO42‐.
2007: 1249 Superfund-sites in the USA
DFT lectures, 08/08+08/09/2009, UNM 40
Objectives: • Restoration of mining areas.• Environmental hazards.• Superfund sites.
• Water management.• Ecology.
8/10/2009
11
Crystallography of Complex Structured Sulfate Minerals - (H3O)+Fe(SO4)2
Objective:
(Majzlan and Kiefer, in prep.)
Problem:• Hydrogen positions cannot be determined reliably structure only partially solved.
Id H ti f
Crystal Structure =Theory + Neutrons + X-rays?
DFT lectures, 08/08+08/09/2009, UNM 41
Idea: Heavy cations from experiment and hydrogen positions from theory.
Neutron scattering
90120150180
eg)
Ab-Initio Structure Refinement(H3O)+Fe(SO4)2
-180-150-120-90-60-30
0306090
Azim
utha
l ang
le (d
e
4.54.03.53.02.52.01.51.00.50.0
Time (ps)I iti l St t
DFT lectures, 08/08+08/09/2009, UNM 42
Time (ps)
Majzlan and Kiefer (in prep.)
Initial Structure:Heavy ions neutron scattering; H planar group
After ~1‐1.5 ps rotations of half the H3O+ ions.
Structure EvaluationFinal structure: Ab-initio MD
Activation energy:~4.1 meV/atom
Degenerate State
(300 K == 25 meV)
• Groundstate of (H3O)+Fe(SO4)2: superposition of two states.• Dynamical Disorder
DFT lectures, 08/08+08/09/2009, UNM 43
atommeVE /07.0
• Dynamical Disorder. • May explain difficulty of hydrogen refinements.
Summary• Classical Potentials: Pro’s and Con’s. • Density-Functional-Theory (DFT):
Universal, EXC. Pseudopotentials. p Planewaves. Predictive power. Complementary to experiment.
• Finite temperature ab-initio Molecular Dynamics.
Theory addresses different classes of problems:
DFT lectures, 08/08+08/09/2009, UNM 44
Theory addresses different classes of problems:• Determination of significant fact.• Matching of facts with theory.• Articulation of theory. (Categorization after: T. Kuhn, 1962)
8/10/2009
12
Some References1) Frenkel, D., and B. Smit; Understanding Molecular Simulation. Academic Press (2002).2) Allen, M. P., and Tildesley; D. J.; Computer Simulation of Liquids; Oxford University Press (2002).3) Haile, J. M.; Molecular Dynamics Simulation; John Wiley & Sons (1997).4) Leach, A. R.; Molecular Modelling (2nd ed.); Pearson Education Limited (2001).5) Thijssen, J. M.; Computational Physics; Cambridge University Press (2000).6) Nosé, S. (1984a) A unified formulation of the constant temperature molecular dynamics method. J. Chem.Phys.,
81: 511 51981: 511-519.7) Nosé, S. (1984b) A molecular dynamics method for simulation in a canoical ensemble. Mol. Phys., 52: 255-268.8) Anderson, H. C. (1980) Molecular dynamics at constant pressure and/or temperature. J. Chem. Phys., 72: 1384-
2893.9) Parrinello, M., and Rahman, A. (1980) Crystal structure and pair-potentials: A molecular dynamcics study.Phys.
Rev. Lett., 45: 1196-1199.10) Parrinello, M., and Rahman, A. (1981) Polymorphic transitions in single crystals: A new molecular dynamics
method. J. Appl. Phys., 52: 7182-7190.11) Wentzcovitch, R. M. (1991) Invariant molecular-dynamics approach to structural phase transitions. Phys. Rev. B,
44: 2358-2361. 12) Martin, R. M. Electronic Structure. Cambridge University Press (2004).13) Ziman, J. M. Principles of the Theory of Solids. Cambridge University Press (1964).14) Fulde, P. Electron Correlation in Molecules and Solids. Springer, Solid State Sciences (1995). 15) Ashcroft, N. W. and Mermin, N. D. Solid State Physics, 2nd ed., Brooks Cole (1976).16) Kittel, C. Introduction to Solid State Physics, 8th ed., Wiley (2004).17) Pickett, W. E. (1989) Pseudopotential methods in condensed matter applications. Computer Phys. Reports 9, 115
-197. 18) Payne, M. P. (1992) Iterative minization techniques for ab-initio total-energy calculations: molecular dynamics and
conjugate gradient. Rev. Mod. Phys. 64, 1045 -1097.
45DFT lectures, 08/08+08/09/2009, UNM
Other LinksUnits and fundamental constants:http://physics.nist.gov/cuu/Constants/Table/allascii.txt
DFT lectures, 08/08+08/09/2009, UNM 46