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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.

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• Convergence tests. • Bulk/surface/molecules.• Examples.

Periodic Table

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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.

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problem, Unuc-nuc and Unuc-el.Static problem.

If T>0 K Nuclei are no longer fixed. Lattice vibrations; thermodynamics.

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2

rrrnrVm iiieffr

el

,

22

2

Kohn and Sham EquationsKohn and Sham (1965)

rnE

rdrr

rne

rVrnrV

XC

eff

2

,

Coulomb interaction.

External potential.

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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.

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Exchange Correlation Functionals

Active research field to develop better EXC functionals:• LDA : XCXC EE LDA :

Ceperley and Alder (1981)…

• GGA :

XCXC EE

,XCXC EE

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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.

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SiO2: LDA gives better agreement with experiment athigh pressures then GGA.

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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

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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.

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Pseudopotential.

Eliminate core electrons

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Generation of Pseudopotentials

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Matching Radius: rc

Oxygen: an example

Flat

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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.

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All-electron methods.(structural optimization difficult to

implement, i.e. WIEN2k)

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Computational ApproachesPlane waves

Treatment of Space Close to NucleiInterstitial Space: Plane Waves

Pseudo Potential Method All Electron Method

Plane waves Atomic states

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Pseudo‐Potential Method

VASP Siesta

PWScfAbinit

DACAPOGPAW

CASTEP

All Electron Method

Wien2K

FleurLAPW

Computational Procedure

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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

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http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg

Metal: Occupation is a step-function.“Softens” as temperature increases.

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Metals: Consequences - IProblem:

At T=0 K (low temperatures)Rapid change of occupation from 1 to zero.

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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.

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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

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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

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-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).

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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

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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

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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.

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Lattice Dynamics

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Bulk-Structures-IBulk-calculations:• Periodic good match

with plane waves.

Hypothetical:

p

• Solid solutions (alloying):Vary composition.Pd8Zn8 Pd9Zn7…

Di d

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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

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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

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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 –

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fcc-Pd(111)-slab.

1) make vacuum thick enough, 15 20 A.

2) Compensate by additional dipole. (VASP: middle of the cell).

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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)

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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

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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

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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

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• Andrew: Pt1-xRex solid solutions.• Eric: Pd on gamma-Al2O3.• Levi: Pd on alpha-Al2O3.• Sam: non-Pt based catalysts.

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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)

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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.

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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

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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

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Objectives: • Restoration of mining areas.• Environmental hazards.• Superfund sites.

• Water management.• Ecology.

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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?

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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

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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

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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:

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Theory addresses different classes of problems:• Determination of significant fact.• Matching of facts with theory.• Articulation of theory. (Categorization after: T. Kuhn, 1962)

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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.

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Other LinksUnits and fundamental constants:http://physics.nist.gov/cuu/Constants/Table/allascii.txt

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