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1
spacem
mmContinuum Equations,
Rate Equationsand Finite Element
Modeling
Predictive modeling and simulations must
address all time and space scalesdensity-
functional th
ElectronicStructure
ab initioMolecularDynamics
μm
nm
ab initiokinetic Monte Carlo
Modelingtheory(and beyond)Multi-Scale Modeling from First
Principles
(Introduction to Electronic Structure Theory)
time
f s p s n s μ s m s s hours years
TheoryOne long MD run is not (always) useful; after a few hops
the memory about the past is lost. Many MD runs with different
starting conditions give a better impression about possibly
relevant processes.
spacem
mmContinuum Equations,
Rate Equationsand Finite Element
Modeling
Predictive modeling and simulations must
address all time and space scalesdensity-
functional th
ElectronicStructure
ab initioMolecularDynamics
μm
nm… +Friday
ab initiokinetic Monte Carlo
Modelingtheory(and beyond)
time
f s p s n s μ s m s s hours years
Theory
Now! – “the base”
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Modeling Materials and Bio-Molecular Propertiesand Functions:
The Many-Body Schrödinger Equation
Wi hWith: 1,
??? We know the ope-rators and the inter-actions We canactions.
We can
write them down.
No open question here!
Born-Oppenheimer Approximation
h l i f h “ l i il i ”
({rk})
Where Φν are solutions of the “electronic Hamiltonian”:
frequently (commonly) applied approximations:
({rk})({rk}) =
q y ( y) pp pp• neglect non-adiabatic coupling (terms of order
m/MI )• keep onlyΛ0
the dynamics of electrons and nuclei decouple
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Some Limits of the Born-Oppenheimer Approximation
It d t t f l t d d i f iIt does not account for correlated
dynamics of ionsand electrons. For example:
- polaron-induced superconductivity- dynamical Jahn-Teller
effect- some phenomena of diffusion in solids- non-adiabaticity in
molecule-surface scattering
- etc.
Wave-Function Theories
Restrict the study to a selected subclass of functions Φ
(Hartree and
functions Φ (Hartree and Hartree-Fock theory);
or Quantum Monte Carlo.
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The Hohenberg-Kohn Theorem (1964)n(r) = n[Φ]
= i
N
The set of particle densities n(r) belonging to non-degenerate
ground states of the N-electron problem.
The set of non-degenerate ground state wave functions Φ of
arbitrary N-electron Hamiltonians.
The dashed arrow is not possible
Comparison of Wave-Function and Density-Functional theory
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Comparison of Wave-Function and Density-Functional theory
Summary of Hohenberg-Kohn Density-Functional Theory (DFT) --
1964
-- The many-body Hamiltonian determines everything.(standard
quantum mechanics)
-- There is a one-to-one correspondence between the ground-state
wave function and the many-body Hamiltonian [or the nuclear (or
ionic) potential, υ(r)].(standard quantum mechanics)( q )
-- There is a one-to-one correspondence between the ground-state
electron-density and the ground-state wave function. (Hohenberg and
Kohn)
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The Kohn-Sham Ansatz-- Kohn-Sham (1965) – Replace the original
many-
body problem with an independent electron problem that can be
solved!
-- Only the ground state density and the ground state energy are
required to be the same as in the original many-body problem.
M b th t Exc[ ] f ti l t b itt-- Maybe the exact Exc[n]
functional cannot be written as a closed mathematical expression.
Maybe there is a detour similar to that taken for Ts[n]? The
challenge is to find useful approximate xc functionals.
Ts , EHartree , and Exc are all universal functionals in n(r),
i.e., they are independent of the special system studied. (general
theory: see the work by Levy and Lieb)
neglecting
is the localCeperley and Alder (1980)
Exc-LDA = ∫ ϵxc-jellium(n) n(r) d3rϵxc-jellium(n)
n
is the local-densityapproximation
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The Exchange-Correlation Hole
Hatree Comparison of Hartree, Hartree-Fock,
anddensity-functional theories for jellium
Hatree
Hartree-Fock
DFT (exact)
nxc(r, r’)Exc[n] =
For non-jellium systems and the LDA (or the GGA) the shape of
nxc(r, r’) is incorrect. However, only its spherical average
enters:
Exchange-Correlation Hole in Silicon
R. Q. Hood, M. Y. Chou, A. J. Williamson, G. Rajagopal, and R.
J. Needs, PRB 57, 8972 (1998)
The spherically averaged exchange-correlation hole in
variational Monte Carlo (VMC) and DFT-LDA with (a) one electron
fixed at the bond center, (b) one electron fixed at the tetrahedral
interstitial site, and (c) plots (a) and (b) superimposed with the
same scale.
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Most-Cited Papers in APS Journals
11 papers published in APS journals since 1893 with >1000
citations in APS journals (~5 times as many references in all
science journals)references in all science journals)
From Physics Today, June, 2005
Certainties about Density Functional Theory
1. DFT in principle: It is exact; a universal Exc[n] functional
exists.[ ]
2. DFT in practice: It is probably not possible to write down
Exc[n] as a closed mathematical expression. We need
approximations.
The success of DFT proves that “simple” approximationsThe
success of DFT proves that simple approximations to the
exchange-correlation functional can provide good results – if one
knows what one is doing.
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Some remarks about excited statesSome remarks about excited
states
Bulk and Surface Defects in Semiconductors; Example: The P
Vacancy at InP (110)
Ph. Ebert et al., Phys. Rev. Lett. 84 (2000)
l l d l i
InI InII [110]
1a’’ level of neutral P vacancy atInP(110)
calculated electronic structure[001]
“... the systematic error for the
[110]Calculated geometry
calculated energies of the chargetransfer levels is too large to
identify the symmetry of the vacancy on the position of the defect
level only.”
Use Kohn-Sham Density-Functional Theory
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Kohn-Sham eigenvalues are not excitation energies.
However, they may serve as a starting point.
The ionization energy is:
What About Kohn-Sham Eigenvalues?
n(r) = ∑ fi |φi (r)|2 ∞
i = 1
Here we assume that the positions of the atoms don’t change upon
ionization or that they change with some delay (Franck Condon
(Well defined for the highestoccupied state. Otherwise, thisonly
holds approximately.)
fi are occupation numbersi 1
ionization, or that they change with some delay (Franck-Condon
principle). Using the mean-value theorem of integral algebra
gives:
This is the Slater-Janak “transition state”. It is the DFT
analog of Koopmans’ theorem.
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Kohn-Sham Electron Bands
Kohn-Sham band gap:The measured (optical) band gap is something
else:
l dditi
ΔKS = єCB - єVB of the N-particle system
Silicon conductionband(empty states)
valence
removal addition
} the KS gap Δ Egap= ΔKS + Δxcband
(filled states)
L Γ X W
We don’t know how to determine Δxc within DFT
There is a discontinuity in Vxc at integer values of occupation
numbers.
Δ Δxc
What is measured and what is calculated?
Energies of direct or inverse photoemission are given by
total-
Photoemission Inverse Photoemission
p g yenergy differences or, equi-valently, by the Green
function.
This can be interpreted as a propagator: For t > t’ it
describes a process in which an extra electron is added to the
system at time t’process in which an extra electron is added to the
system at time t .A Fourier analysis yields the energy spectrum
(for j = (N+1) ... ∞)Likewise, for t’ > t the Green function
describes the propagation of an extra hole between t and t’,
yielding the energies
(for j = 1 ... N )
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Insert a complete set of eigenstates between the field operators
and Fourier transform to the frequency axis. This gives:
The poles of the Green function correspond directly to the
quasiparticle energies These are the “true band structure”. The q p
ghighest occupied state equals the exact ionization potential and
the lowest unoccupied state the exact electron affinity.
Many-body perturbation theory provides a convenient way to
analyze both defect states and charge-transition levels.
Many-Body Perturbation Theory; The Self-Energy
Occupied and unoccupied states and their energies are obtained
from the quasiparticle equation
The self-energy Σxc incorporates all contributions from exchange
and correlation processes. In contrast to the xc-potential of DFT,
it is nonlocal, energy-dependent and has a finite imaginary part.
We
l Σ i h G W i ievaluate Σxc in the G0W0 approximation:
W0 is the screened Coulomb interaction, and G0 is the zeroth
order Green function, i.e. that calculated from Kohn-Sham wave
functions
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Exploiting the formal similarity between the self-energy and the
Kohn-Sham equations by evaluating the quasiparticle energies within
first-order perturbation theory gives
This treatment (first-order perturbation theory) is justified if
the eigenvalues of the underlying Kohn-Sham system are already
sufficiently close to the expected quasiparticle band structure.
Using Σxc = i G0W0 (i.e. refraining from a self-consistent
evaluation of G and W) implies that the quasi-particle
spectrumevaluation of G and W) implies that the quasi-particle
spectrum depends on the input Green’s function, G0.
Still, the approach provides a good account of the discontinuity
as well as other shortcomings of the employed xc functional.
Exploiting the formal similarity between the self-energy and the
Kohn-Sham equations by evaluating the quasiparticle energies within
first-order perturbation theory gives
This treatment (first-order perturbation theory) is justified if
the eigenvalues of the underlying Kohn-Sham system are already
sufficiently close to the expected quasiparticle band structure.
Using Σxc = i G0W0 (i.e. refraining from a self-consistent
evaluation of G and W) implies that the quasi-particle
spectrumevaluation of G and W) implies that the quasi-particle
spectrum might depend on the input Green’s function, G0.
Still, the approach provides a good account of the discontinuity
as well as other shortcomings of the employed xc functional.
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Band Gaps of II-VI Semiconductors and of GaN
P. Rinke, A. Qteish, J. Neugebauer, C. Frey-
ldt d M Ssoldt, and M.S.,
Combining GW calcula-tions with exact-exchange
density-functional theory: an analysis of valence-band
photoemission for compound semiconduc-tors.
New J. Phys. 7, 126 (2005).
−0
This is a total-energy difference and well defined i hi DFT ? N
f h k
How to evaluate charge-transition levels? Example: The (+/0)
level of VP at InP(110)
CB
0+ within DFT. ... ? Not true for the known xc functionals. As
different charge states occur, the Δxc
term is needed. -- We rewrite the equation as
lattice relaxation energyof the neutral charge state
electron affinity level of thepositive charge state
VB
of the neutral charge state positive charge state
Charge-transition levels for the P vacancy at InP(110) in eV
LDA 0.47 0.54G0W0 0.82 1.09experiment 0.75±0.1 M. Hedström, et
al.
PRL 97 (2006).
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Summary-- Interacting electrons determine the properties and
function of real materials and bio molecules.-- Approximate xc
functionals have been very successful,
but for highly correlated situations and for excited statesh
blthere are problems.
Important arenas for future theoretical work:-- Correlated
systems, e.g. f-electron chemistry-- Thermodynamic phase
transitions, e.g. melting.-- Surfaces, interfaces, nanostructures –
in realistic environments.-- Modeling the kinetics, e.g. of
catalysts or crystal growth
(self-assembly and self-organization)( y g )-- Molecules and
clusters in solvents, electrochemistry,
fuel cells, external fields, transport.-- Biological problemsThe
challenges:-- Find practical ways to correct the xc
approximation.-- Develop methods for bridging the length and time
scales.