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”

2

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

3

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.

4

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

5

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)

6

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

7

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.

8

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.

9

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

10

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.

11

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 )

12

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

13

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.

14

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).

15

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.