Transcript es2002 1573

Density Functional Theory
Richard M. Martin
University of Illinois
Cu
d orbitals
Electron density in La2CuO4 - difference from
sum of atom densities - J. M. Zuo (UIUC)
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Outline
• DFT is an approach to Interacting Many-Body Problems
• Hohenberg-Kohn Theorems & Levy-Lieb Construction
• Kohn-Sham Ansatz allows in principle exact solution for
ground state of many-body system using independent
particle methods
• Classes of functionals: LDA, GGA, OEP, ….
• Examples of Results
• Locality Principles and linear scaling
• Electric polarization in crystals - deep issues that bring
out stimulating questions about DFT, and the differences
between the Hohenberg-Kohn and Kohn-Sham
approaches
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Questions for you
• Why were “orbitals” mentioned on the introductory slide
and not simply “density”
• Can you tell whether La2CuO4 is an insulator or a metal
just by looking at the density?
If so, what aspects of the density?
• Is Kohn-Sham theory the same as Density Functional
Theory?
• If not, what is the difference? What did Kohn-Sham add?
What did they subtract?
• Do locality principles in independent particle methods
carry over to the real many-body world?
• Is the electric polarization of a ferroelectric an intrinsic
ground state property? Is it determined by the density?
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Assumes
non-degenerate
ground state
H-K Functional
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Wavefunctions with density n( r )
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What have we gained so far?
• Apparently Nothing!
• The only result is that the density
determines the potential
• We are still left with the original many-body
problem
• But the proofs suggest(ed) the next step
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Kohn-Sham Ansatz
• If you don’t like the answer,
change the question
• Replace the original interacting-particle
problem with a different problem more
easily solved
• Kohn-Sham auxiliary system:
non-interacting ”electrons” assumed to
have the same density as the
interacting system
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Auxiliary
System
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• Replace interacting problem with
auxiliary non-interacting problem
• Each term in figure is uniquely related
to each other term!
• The ansatz has been shown to be fulfilled
in several simple cases – but not in general
• We will proceed assuming the ansatz is justiified
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Negative energy:
electron – positive hole
Kinetic energy:
positive
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Exchange-correlation
hole in
homogeneous
electron gas
• Exchange dominates
at high density
(small rs)
• Correlation dominates
at low density
(large rs)
Gori-Giorgi, Sacchetti and Bachelet,
PRB 61, 7353 (2000).
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Exchange hole in Ne atom
• Spherical average close to LDA!
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Gunnarsson, et al,
PRB 20, 3136 (79).
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Exchange hole in Si Crystal
• Variational
Monte
Carlo
Hood, et al,
PRB 57, 8972(98).
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Examples of Results
• Hydrogen molecules - using the LSDA
(from O. Gunnarsson)
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Examples of Results
• Phase transformations of Si, Ge
• from Yin and Cohen (1982)
Needs and Mujica (1995)
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Graphite vs Diamond
• A very severe test
• Fahy, Louie, Cohen calculated energy along a path
connecting the phases
• Most important - energy of graphite and diamond
essentially the same!
~ 0. 3 eV/atom barrier
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Less compressible than Diamond
• Bulk Modulus B (Gpa)
Exp
C
444
Os
462
Th (LDA)
467
Cynn, et al,
444
PRL March 14 (2002)
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Slater
average
exchange
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Phonons - LDA and GGA
• Calculated by response function
Baroni, et al,
method RMP 73, 515 (2000).
LDA
GGA
Exp
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The “Band Gap Problem”
• Often said that the eigenvalues have no
meaning – just Lagrange multipliers
• Energy to add or subtract an electron in the
non-interacting system - not an excitation
energy of the interacting system
• Naïve use of the eigenvalues as exciation
energies is the famous “band gap problem”
• To understand the effcets we first examine
the potential
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Exchange potential in atoms
• 2-electron systems
• LDA Vxc is too shallow
Almbladh and Pedroza, PR A 29, 2322 (84).
Density
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The “Band Gap Problem”
• Excitations are NOT well-predicted by the
“standard” LDA, GGA forms of DFT
The “Band Gap Problem”
Orbital dependent DFT is
more complicated but
gives improvements treat exchange better, e.g,
“Exact Exchange”
Ge is a
metal
in LDA!
M. Staedele et al, PRL 79, 2089 (1997)
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Status of “Band Gap Problem”
• It should be possible to calculate all
excitation energies from the Kohn-Sham
approach
• But not clear how close Kohn-Sham
eigenvalues should be to true excitation
energies
• Not clear how much of the “band gap
problem” is due to approximate functionals
• Size of derivative discontinuity?
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Locality and Linear Scaling
• DFT provides a fundamental basis for
“nearsightedness” (W. Kohn) -- if properties
in a region are determined only by densities
in a neighborhood -- so that an “Order N”
method must be possible
• Used, e.g., by W. Yang in his divide and
conquer method
• Orbital picture in Kohn-Sham method
provides the concrete methods
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Linear Scaling ‘Order-N’ Methods
• Computational complexity ~ N = number of
atoms (Current methods scale as N2 or N3)
•
•
•
•
•
•
“Divide and Conquer”
Green’s Function
Fermi Operator Expansion
Density matrix “purification”
Generalized Wannier Functions
Spectral “Telescoping”
(Review by S. Goedecker in Rev Mod Phys)
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Example of Our work
Prediction of Shapes of Giant Fullerenes
S. Itoh, P. Ordejon, D. A. Drabold and R. M. Martin, Phys Rev B 53, 2132 (1996).
See also C. Xu and G. Scuceria, Chem. Phys. Lett. 262, 219 (1996).
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Simulations of DNA with the SIESTA code
• Machado, Ordejon, Artacho, Sanchez-Portal, Soler
• Self-Consistent Local Orbital O(N) Code
• Relaxation - ~15-60 min/step (~ 1 day with diagonalization)
Iso-density surfaces
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Conclusions - I
• DFT is a general approach to interacting many-body
problems
Kohn-Sham approach makes it feasible
• Ground state properties are predicted with remarkable
success by LDA and GGAs.
Structures, phonons (~5%), ….
• Excitations are NOT well-predicted by the LDA, GGA
approximations
The “Band Gap Problem”
Orbital dependant functionals increase
the gaps - agree better with experiment
“Derivative discontinuity” natural in orbital
functionals
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Conclusions - II
• Locality inherent for properties of a region that depend
only on the density in a neighborhood
Forces, stress, ..
“Order N” linear scaling method should be possible
• Density matrix shows the locality in the quantum system
Several feasible methods for insulators
• Carries over to interacting many-body system
• Some propreties are not local in real space
Fermi surface of a metal, etc.
But states near Fermi energy have universal behavior
that should make linear scaling possible
• When is the functional an extremely non-local functional
of the density? A polarized insulator, where the KohnDensity
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Sham theory must
beFunctional
fundamentally
revised