Density Functional Theory The Basis of Most Modern Calculations
Transcript of Density Functional Theory The Basis of Most Modern Calculations
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Density Functional Theory The Basis of Most Modern Calculations
Richard M. Martin – UIUC
Lecture at Summer SchoolHands-on introduction to Electronic Structure
Materials Computation CenterUniversity of Illinois – June, 2005
Hohenberg-Kohn; Kohn-Sham – 1965Defined a new approach to the
many-body interacting electron problem
Reference: Electronic Structure: Basic Theory and Practical Methods,
Richard M. Martin (Cambridge University Press, 2004)R. Martin - Hands-on Introduction to Electronic Structure - DFT - 6/2005 1
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electrons in an external potentialInteracting
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The basis of most modern calculationsDensity Functional Theory (DFT)
• Hohenberg-Kohn (1964)
• All properties of the many-body system are determined by the ground state density n0(r)
• Each property is a functional of the ground state density n0(r) which is written as f [n0]
• A functional f [n0] maps a function to a result: n0(r) → fR. Martin - Hands-on Introduction to Electronic Structure - DFT - 6/2005 4
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The Hohenberg-Kohn Theorems
n0(r) → Vext(r) (except for constant)
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The Hohenberg-Kohn Theorems
Minimizing E[n] for a given Vext(r) → n0(r) and EIn principle, one can find all other properties and they are functionals of n0(r).
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The Hohenberg-Kohn Theorems - Proof
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The Hohenberg-Kohn Theorems - Continued
• Generalization by Levy and Lieb– Recast as a two step process
• Consider all many-body wavefunctions Ψ with the same density• First, minimize for a given density n• Next, minimize n to find density with lowest energy n0
• What is accomplished by the Hohenberg-Kohn theorems?
• Existence proofs
• A Nobel prize for this???• The genius is the next step –
to realize that this provides a new way to approach the many-body problem
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The Kohn-Sham Ansatz
• Kohn-Sham (1965) – Replace original many-body problem with an independent electron problem – that can be solved!
• The ground state density is required to be the same as the exact density
• Only the ground state density and energy are required to be the same as in the original many-body system
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The Kohn-Sham Ansatz II
• From Hohenberg-Kohn the ground state energy is a functional of the density E0[n], minimum at n = n0
• From Kohn-Sham
Exchange-CorrelationFunctional – Exact theorybut unknown functional!
Equations for independentparticles - soluble
• The new paradigm – find useful, approximate functionalsR. Martin - Hands-on Introduction to Electronic Structure - DFT - 6/2005 10
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The Kohn-Sham Ansatz III
• Approximations to the functional Exc[n]• Requires information on the many-body system of
interacting electrons• Local Density Approximation - LDA
• Assume the functional is the same as a model problem –the homogeneous electron gas
• Exc has been calculated as a function of densityusing quantum Monte Carlo methods (Ceperley & Alder)
• Gradient approximations - GGA• Various theoretical improvements for electron density
that is varies in space R. Martin - Hands-on Introduction to Electronic Structure - DFT - 6/2005 11
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What is Exc[n] ?
• Exchange and correlation → around each electron, other electrons tend to be excluded – “x-c hole”
• Excis the interaction of the electron with the “hole” –spherical average – attractive – Exc[n] < 0.
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nx(r, r′)→
→ →
→
50
25
−0.2 −0.1 0.10(r′ − r)/o0
LD
Exact
nx(r, r′)
2.0
3.0
1.0
Exact
LD
−1.0 −0.5 0.0 0.5(r′ − r)/o0
rr = 0.09o0
r = 0.09o0
r = 0.4o0
r = 0.4o0
r′ − r r′ − r
+
+
0.05
0.10
0.15
Exact
0.0 0.5 1.0r′′
r′′ nx3.0 (r, r′′)
LD
(b)
(a)
Exactr′′ nx
5.0 (r, r′′)
r r′′
0.0 0.1 0.2 0.3 0.4
1.0
0.5
LD
r′′/on
Exchange hole in Ne atomFig. 7.2 Gunnarsson, et. al. [348]
Very non-spherical
Spherical average very closeto the hole in a homogeneouselectron gas!
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Exchange-correlation (x-c) hole in silicon
• Calculated by Monte Carlo methods
1
(b)(a)
−0.1
−0.075−0.05−0.025
00.950.850.750.65
0.55
−0.1
0.75
1
0.5
Exchange Correlation
Fig. 7.3 - Hood, et. al. [349]
Hole is reasonably well localized near the electronSupports a local approximation
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Exchange-correlation (x-c) hole in silicon
• Calculated by Monte Carlo methods
Exchange-correlation hole – spherical averageBond Center Interstitial position Comparison to scale
ρ xc(
r, R
)
(c )
0.0 2.0 4.0 6.0 8.0
R (a.u.)
VMCLDA
VMCLDA
−0.060
−0.040
−0.020
0.000
4.0 6.0 8.0
VMCLDA
0.0 2.0−0.060
−0.040
−0.020
0.000(a )
ρ xc(
r, R
)
10.04.0 6.0 8.0
VMCLDA
0.0 2.0
R (a.u.)R (a.u.)
(b )
−0.0040
−0.0030
−0.0020
−0.0010
0.0000
0.0010
ρ xc(
r, R
)
x-c hole close to that in the homogeneous gas in the most relevant regions of spaceSupports local density approximation ! Fig. 7.4 - Hood, et. al. [349]
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The Kohn-Sham Equations
• Assuming a form for Exc[n]• Minimizing energy (with constraints) → Kohn-Sham Eqs.
Constraint – requiredExclusion principle forindependent particles
Eigenvalues are approximationto the energies to add or subtract electrons –electron bandsMore later
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Solving Kohn-Sham Equations
• Structure, types of atoms
• Guess for input
• Solve KS Eqs.
• New Density and Potential
• Self-consistent?• Output:
– Total energy, force, stress, ...– Eigenvalues
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Example of Results – Test Case
• Hydrogen molecules - using the LSDA(from O. Gunnarsson)
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Calculations on Materials Molecules, Clusters, Solids, ….
• Basic problem - many electrons in the presence of the nuclei
• Core states – strongly bound to nuclei – atomic-like• Valence states – change in the material – determine
the bonding, electronic and optical properties, magnetism, …..
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The Three Basic Methods for Modern Electronic Structure Calculations
• Localized orbitals– The intuitive appeal of atomic-like states– Simplest interpretation in tight-binding form– Gaussian basis widely used in chemistry– Numerical orbitals used in SIESTA
• Augmented methods– “Best of both worlds” – also most demanding– Requires matching inside and outside functions– Most general form – (L)APW
• Plane waves– The simplicity of Fourier Expansions– The speed of Fast Fourier Transforms– Requires smooth pseudopotentials
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Plane Waves
• Kohn-ShamEquations in a crystal
• The most general approach
• Kohn-ShamEquations in a crystal
• The problem is the atoms! High Fourier components!R. Martin - Hands-on Introduction to Electronic Structure - DFT - 6/2005 20
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Plane Waves• (L)APW method
• Augmentation: represent the wave function inside each sphere in spherical harmonics– “Best of both worlds”– But requires matching inside and outside functions– Most general form – can approach arbitrarily precision
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Plane Waves• Pseudopotential Method – replace each potential
solid 2
Pseudopotential atom 1
• Generate Pseudopotential in atom (spherical) – use in solid• Pseudopotential can be constructed to be weak
– Can be chosen to be smooth– Solve Kohn-Sham equations in solid directly in Fourier space
1 2
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Charge Density of Si – Experiment- LAPW calculations with LDA, GGA
• Electron density difference from sum of atoms– Experimental density from electron scattering– Calculations with two different functionals
• J. M. Zuo, P. Blaha, and K. Schwarz, J. Phys. Cond. Mat. 9, 7541 (1997).
– Very similar results with pseudopotentials• O. H. Nielsen and R. M. Martin (1995)
Exp LDA GGA
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Comparisons – LAPW – PAW -- Pseudopotentials (VASP code)
• a – lattice constant; B – bulk modulus; m – magnetization
• aHolzwarth , et al.; bKresse & Joubert; cCho & Scheffler; dStizrude, et al.
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Phase Transitions under PressureSilicon is a Metal for P > 110 GPa
• Demonstration that pseudopotentials are an accurate “ab initio” method for calculations of materials
• Results are close to experiment!– M. T. Yin and M. L. Cohen, Phys. Rev. B 26, 5668 (1982).– R. Biswas, R. M. Martin, R. J. Needs and O. H. Nielsen, Phys. Rev. B 30, 3210 (1982).
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The Car-Parrinello Advance
• Car-Parrinello Method – 1985– Simultaneous solution of Kohn-Sham equations for electrons
and Newton’s equations for nuclei– Iterative update of wavefunctions - instead of diagonalization– FFTs instead of matrix operations – N lnN instead of N2 or N3
– Trace over occupied subspace to get total quantities (energy, forces, density, …) instead of eigenfunction calculations
– Feasible due to simplicity of the plane wave pseudopotentialmethod
• A revolution in the power of the methods– Relaxation of positions of nuclei to find structures– Simulations of solids and liquids with nuclei moving thermally– Reactions, . . .
• Stimulated further developments - VASP, ABINIT, SIESTA, . . .R. Martin - Hands-on Introduction to Electronic Structure - DFT - 6/2005 26
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Simulation of Liquid Carbon
• Solid Line: Car-Parrinello plane wave pseudopotentialmethod (Galli, et al, 1989-90)
• Dashed Line: TB potential of Xu, et al (1992)
2 4 600.0
0.5
1.0
1.5
2.0
2.5
r (a.u.)
Rad
ial d
ensi
ty d
istri
butio
n g(
r)
“snapshot of liquid”
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Example of Thermal Simulation
• Phase diagram of carbon• Full Density Functional “Car-Parrinello” simulation • G. Galli, et al (1989); M. Grumbach, et al. (1994)
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Nitrogen under pressure – Recent discoveries
W. D. Mattson, S. Chiesa,R. M. Martin, PRL, 2004.
Squeezed&
Cooled•P > 100 Gpa and 0K•Network solid•Predicted > 15 years ago (DFT)•Found experimentally in 2000•New Prediction of Metallic N
•Hot Molecular Liquid•58 Gpa 7600 K•Nitrogen Molecules Disassociate and Reform
• Used SIESTA code for MD simulation• Sample structures tested using ABINIT
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What about eigenvalues?• The only quantities that are supposed to be correct in the
Kohn-Sham approach are the density, energy, forces, ….• These are integrated quantities
– Density n(r ) = Σi |Ψi(r )|2– Energy Etot = Σi εi + F[n] – Force FI = - dEtot / dRI where RI = position of nucleus I
• What about the individual Ψi(r ) and εi ?– In a non-interacting system, εi are the energies to add and subtract
“Kohn-Sham-ons” – non-interacting “electrons”– In the real interacting many-electron system, energies to add and
subtract electrons are well-defined only at the Fermi energy
• The Kohn-Sham Ψi(r ) and εi are approximate functions- a starting point for meaningful many-body calculations
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Electron Bands
• Understood since the 1920’s - independent electron theories predict that electrons form bands of allowed eigenvalues, withforbidden gaps • Established by experimentally for states near the Fermi energy
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Extra added electronsgo in bottom of conduction band
Missing electrons(holes) go in top of
valence band
Empty Bands
Silicon
Gap
Filled Bands
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Bands and the “Band Gap Problem”• Excitations are NOT well-predicted by
the “standard” LDA, GGA forms of DFTExample of Germanium
Ge is a metal in LDA!
L Γ χ W K Γ
−10Ge
−5
0
5
10
Ene
rgy
(eV
)
L Γ ∆ χΛ
Γ1
−10
L2′
L1Ge χ
1
−5
0
5
10
L2′
L3
L1
L3′
Γ15
Γ2'
Γ25′χ
4
χ1
Ene
rgy
(eV
)
Many-body theorywithExperimental points
LDA bands
Many-body bands
M. Rohlfing, et alR. Martin - Hands-on Introduction to Electronic Structure - DFT - 6/2005 32
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The “Band Gap Problem”• Excitations are NOT well-predicted by
the “standard” LDA, GGA forms of DFTThe “Band Gap Problem”
Orbital dependent DFT is more complicated but gives improvements -treat exchange better, e.g,“Exact Exchange”
M. Staedele et al, PRL 79, 2089 (1997)
Ge is a metal in LDA!
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Failures!• All approximate functionals fail at some point!• Most difficult cases
– Mott Insulators – often predicted to be metals – Metal-insulator Transitions– Strongly correlated magnetic systems– Transiton metal oxides – Hi-Tc materials – . . .
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Conclusions I• Density functional theory is by far the most widely
applied “ab intio” method used in for “real materials” in physics, chemistry, materials science
• Approximate forms have proved to be very successful
• BUT there are failures• No one knows a feasible approximation valid for
all problems – especially for cases with strong electron-electron correlations
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Conclusions II• Exciting arenas for theoretical predictions
– Working together with Experiments– Realistic simulations under real conditions– Molecules and clusters in solvents, . . .– Catalysis in real situations– Nanoscience and Nanotechnology– Biological problems
• Beware to understand what you are doing– Limitations of present DFT functionals– Care to use codes properly
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