Electronic structure theory of organicinorganic interfaces

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Electronic structure theory of organic/inorganic
interfaces
Leeor Kronik Dept. of Materials and
Interfaces Weizmann Institute of
Science Rehovoth, Israel
Minerva School on Unique Molecular Effects in
Electronic Materials and Devices Safed,
Israel, 2007
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Electronic structure from first principles

• Ideal first principles computation brute
  force solution of the Schrödinger equation.
• , where
  and

kinetic
ion-e
e-e
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Electronic structure from first principles?

• "The underlying physical laws necessary for the
  mathematical theory of a large part of physics
  and the whole of chemistry are thus completely
  known and the difficulty is only that the exact
  application of these laws leads to equations much
  too complicated to be soluble."
• P.A.M. Dirac, Proc. Royal Soc. Lond. Ser. A 123,
  714 (1929)

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The Hohenberg-Kohn theorem
density as the basic variable! (Hohenberg Kohn,
Phys. Rev., 1964)
Many-body wave function
Electron Density
ion-e
Plausibility argument
of electrons
Atomic positions
Atomic numbers
Electron Density
(Koch Holthausen, A Chemists guide to DFT,
2001)
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The Kohn-Sham equation
Exact mapping to a single particle problem! (Kohn
Sham, Phys. Rev., 1965)
Exc is a unique, but complicated and unknown
functional of the charge density.
In principle Exact In practice approximate
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The Hohenberg-Kohn Variational principle
Rayleigh-Ritz variational principle
Levys constrained search
Hohenberg-Kohn variational principle
Fn is universal !!
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Exact mapping to an equivalent non-interacting
electron system
Is there an equivalent non-interacting (Vee0)
electron system that yields the same density as
the true one? If so, what external potential, VKS
, yields the true density?
Is there
Such that
Is exactly equal to the electron density of the
real system?
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Finding VKS
Model system
Real System
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The Kohn-Sham equation
Exact mapping to a single particle problem! (Kohn
Sham, Phys. Rev., 1965)
Exc is a unique, but complicated and unknown
functional of the charge density.
In principle Exact In practice approximate
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How are we doing in practice?
Principle and practice are the same in principle
but not in practice
Ground rules
- Relating formal and practical difficulties -
No technical issues, only fundamental ones -
Only discuss well-established functionals -
Physical insight over mathematical rigor
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The local density approximation (LDA)
At each point in space, xc energy per particle is
given by its value for a homogeneous electron
gas
Yin Cohen, Phys. Rev. Lett., 1980 Structure
of Phases Lattice constants Transition
pressure Transition volumes
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The generalized gradient approximation (GGA)
- Add info from immediate vicinity via gradient
term - Not a gradient expansion!
Some quantitative issues with LDA that are
mitigated with GGA - Atomization enrgies - 0.3
eV versus 1 eV (Becke, J. Chem. Phys., 1993) -
Overbinding (e.g., Na Kronik et al., J. Chem.
Phys., 2001) - Over emphasis of packed
structures (e.g., Fe Zhu et al., Phys. Rev.,
1992)
but NOT a qualitative improvement!
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Qualitative limitations of semi-local functionals
(1) The self-interaction error!
- In Hartree term, repulsion from all electron
density - Imperfect cancellation by xc term gt
spurious repulsion of an electron from itself
!! - Old (Fermi and Amaldi, 1934) but painful
problem.
Important consequence - Incorrect asymptotic
potential - anions may be poorly described -
image potential poorly described
DFT
Bally Sastry, J. Phys. Chem. A, 1997
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What do the Kohn-Sham eigenvalues and orbitals
mean?
(very) strictly speaking nothing.
But then how do you explain this?
Kronik et al., Nature Materials 1, 49 (2002).
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A more subtle view of Kohn-Sham eigenvalues and
orbitals
Kohn-Sham eigenvalues have a well defined
physical meaning as zeroth-order excitation
energies in an effective series expansion
(Goerling, 1996)
Kohn-Sham orbitals are density optimal
one-electron orbitals (Baerends and Gritsenko,
1997)
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Self-interaction example gas phase valence
electronic structure of PTCDA
?
Dori et al., Phys. Rev. B, 2006.
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Qualitative failure of semi-local functionals
(2) absence of derivative discontinuity!
Real World Stretch the Li-H bond -gt Li
H LDA/GGA world Stretch the Li-H bond -gt
Li0.25 H-0.25
What happened??
Real World Ionization potential ? electron
affinity. ILi gt AH LDA/GGA world There
should be a discontinuity in the chemical
potential but its not fully expressed. xc
energy continuous in density!
Perdew et al., Phys. Rev. Lett., 1982.
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Consequence 1 Shifted spectra(example alkyl
chains on Si)
Ultimate functional highest eigenvalue at
ionization potential. LDA/GGA highest eigenvalue
between I and A.
Segev et al., Phys. Rev. B, 2006.
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Consequence 2 charge-transfer catastrophe
LDA/GGA make large errors in linear and huge
errors in the non-linear response in the
direction of the chain
Gisbergen et al., Phys. Rev. Lett., 1999.
Exact exchange calculations on a model hydrogen
chain emphasize role of derivative discontinuity!
Kümmel, Kronik, and Perdew, Phys. Rev. Lett.,
2004
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Exact exchange within DFT
but Orbital dependence. Density dependence
implicit.
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The KLI approximation
The optimized effective potential

KLI
Where
Accurate for ground state properties!
Krieger, Li, and Iafrate, 1992
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Hybrid functionals (1) the adiabatic connection
Assume one gradually turns off the
electron-interaction but keeps the ground state
density constant
Define
Adiabatic connection theorem
Harris and Jones, Phys. Rev. B, 1974.
Practical example the N2 molecule
Ernzerhof et al., Int. J. Quant. Chem., 1997.
Exact exchange limit!
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Hybrid functionals (2) practical forms
1-parameter hybrids
3-parameter hybrids
Becke, J. Chem. Phys., 1993 (two different papers)
Most popular form B3LYP, an odd variant of
Beckes original parameter fit
Strange fits of passion have I known
(Perdew citing Wordsworh, 2005)
Hybrid functionals because of minimization with
respect to orbitals,not density, of Fock exact
exchange part!
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Hybrid functionals pros and cons
Pros - Often highly accurate for organic
molecules, incl. atomization and
bindingenergies - mitigate (but dont solve!)
self-interaction (e.g., PTCDA spectrum yes
H2 dissociation no) Cons - Not very good
for metals. Not easy to apply to extended
systems - Still fail badly for transition
states, charge transfer complexes - Does not
enjoy Kohn-Sham theoretical protection
Very practical issue Functional chosen must be
equally accurate for both sides of the interface!
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The van der Waals quandary
Dispersion not described properly in any of the
above functionals! Example atop-parallel
benzene dimer (Dion et al., Phys. Rev. Lett.,
2004)
Caution sometimes (e.g., Kr2) fortuitously
reasonable due to error cancellation!
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The gap or not the gap is that the question?
The Kohn-Sham gap severely underestimates the
real gap

• Myth a failure of LDA/GGA that is fixed by
  hybrids

Reality An inherent limitation of ground state
theory
Perdew and Levy, Phys. Rev. Lett, 1983 Sham and
Schlüter, Phys. Rev. Lett, 1983
derivative discontinuity!
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Instructive gap comparisons
I-A 5.0 eV LDA gap 1.5 eV B3LYP gap 2.6 eV
Optical gap 2.75 eV
Exciton energy
Dori et al., Phys. Rev. B, 2006.
Is the Kohn-Sham gap the optical gap? Is the
hybrid gap the Kohn-Sham gap?
Barone et al., Nano Lett., 2005.
B3LYP overestimates as much as LDA
underestimates Agreement requires less exact
exchange
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Concluding Remarks
DFT is a subtle method. It is not based on an
arbitrarily accurate, controlled
approximation User friendliness of code does not
imply license to use as a black box -
Nevertheless - Judicious choice of functional
based on physical insights can yield highly
accurate, predictive results and its not
always the functionals fault!!
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Safed Summer School on Density Functional Theory

• September 2-6, 2007
• www.weizmann.ac.il/conferences/DFT
• Partial list of speakers Roi Baer (Jerusalem),
  Roberto Car (Princeton), Martin Head-Gordon
  (Berkeley), Eberhard Gross (Berlin), Stephan
  Kuemmel (Bayreuth), Steven Louie (Berekeley), Jan
  Martin (Weizmann), Daniel Neuhauser (UCLA),
  Gotthard Seifert (Dresden)

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THANKS
Stephan Kümmel
Kümmel and Kronik, Rev. Mod. Phys., in press