Excited States and Spectroscopy

Tutorial 7

Dorothea Golze

Aalto University

Content

1. Photoemission Spectroscopy

2. DFT-based methods

3. GW methods

1

Photoemission Spectroscopy

This tutorial is about

. calculation of electron removal and addition energies

2

Spectroscopies: Important distinction

3

Our example in the tutorial

4

Our example in the tutorial

4

Experimental photoemission spectrum152

CH,=CH2

CH,-CH3

25 20 15 10

Phot

oem

issi

on sp

ectru

m

Energy [eV]

C2H4

5

DFT-based methods

DFT eigenvalues

• direct connection of ionization

potentials (IP) to KS-DFT or HF

orbital energies εKS/HFn

• ionization potentials: IPn = −εKS/HFn

• deviation from experiment for valence

states ≈ 3 eV

Figure 1: Energy levels H2O

6

Ionization potentials from DFT and HF eigenvalues

Exercise 1 and 2

• Calculate electron removal energies from KS-DFT and Hartree-Fock

eigenvalues for C2H4 and optionally H2O.

• Compare the results to experiment.

7

The DFT-∆SCF approach

Ionization potentials

IP = Etot(N − 1)− Etot(N)

• more accurate than eigenvalues

• constraints needed for levels other than HOMO and LUMO

• several conceptual problems, e.g., for periodic systems

8

Ionization potentials and affinities from DFT-∆SCF

Exercise 3

Calculate the ionization potential for the HOMO from the ∆SCF

approach

• IPHOMO = Etot(N − 1)− Etot(N)

Optional : Calculate the electron affinity (A) for the LUMO with ∆SCF

• ALUMO = Etot(N)− Etot(N + 1)

Compare the IPs to the experiment and the DFT and HF eigenvalues.

9

GW methods

Quasiparticle energies from GW

Basic idea

• in analogy to DFT: replacement of XC potential by self-energy

• self-energy

Σ(r, r′, ω) =i

2π

∫dω′G(r, r′, ω + ω′)W (r, r′, ω′) eiω

′η

• G : Green’s function; W : screened Coulomb interaction

10

The G0W0 approach

Single-shot perturbation: G0W0

• correction to KS-DFT orbital energies εKSn

εG0W0n = εKSn + 〈ψKS

n |Σ(εG0W0n )− vxc |ψKS

n 〉• self-energy

Σ(r, r′, ω) =i

2π

∫dω′G0(r, r′, ω + ω′)W0(r, r′, ω′) eiω

′η

Procedure:

1. run DFT calculation

2. calculate G0 and W0 from DFT orbital energies εKSn and MOs {ψKSn }

3. calculate self-energy from G0 and W0

4. solve quasi-particle equation

11

The G0W0 approach

Single-shot perturbation: G0W0

• correction to KS-DFT orbital energies εKSn

εG0W0n = εKSn + 〈ψKS

n |Σ(εG0W0n )− vxc |ψKS

n 〉• self-energy

Σ(r, r′, ω) =i

2π

∫dω′G0(r, r′, ω + ω′)W0(r, r′, ω′) eiω

′η

Procedure:

1. run DFT calculation

2. calculate G0 and W0 from DFT orbital energies εKSn and MOs {ψKSn }

3. calculate self-energy from G0 and W0

4. solve quasi-particle equation

11

The G0W0 approach

Non-interacting KS Green’s function G0

G0(r, r′, ω) =∑m

ψKSm (r)ψKS

m (r′)

ω − εKSm − iη sgn(εF − εKSm )

Screened Coulomb interaction W0

W0(r, r′, ω) =

∫dr′′ε−1(r, r′′, ω)v(r′′, r′),

with

ε(r, r′, ω) = δ(r, r′)−∫

dr′′v(r, r′′)P0(r′′, r′, ω)

ε ... dielectric function

P0 ... polarizability

12

The G0W0 approach

Non-interacting KS Green’s function G0

G0(r, r′, ω) =∑m

ψKSm (r)ψKS

m (r′)

ω − εKSm − iη sgn(εF − εKSm )

Screened Coulomb interaction W0

W0(r, r′, ω) =

∫dr′′ε−1(r, r′′, ω)v(r′′, r′),

with

ε(r, r′, ω) = δ(r, r′)−∫

dr′′v(r, r′′)P0(r′′, r′, ω)

ε ... dielectric function

P0 ... polarizability

12

G0W0 output with FHI-aims

State index

Occupation number

Figure 2: Output for the hydrogen molecule

13

Ionization potentials from G0W0

Exercise 4.1

Calculate IPs for HOMO, HOMO-1, HOMO-2 and HOMO-3 levels with

the G0W0 approach starting from PBE. Compare the results to

experiment and the previous calculations

14

G0W0 basis set convergence

Green’s function G0

G0(r, r′, ω) =all states∑

m

ψKSm (r)ψKS

m (r′)

ω − εKSm − iη sgn(εF − εKSm )

Polarizability P0

P0(r, r′, ω) =occ∑i

virt∑a

ψKSa (r′)ψKS

i (r′)ψKSi (r)ψKS

a (r)

×{

1

ω − (εKSa − εKSi ) + iη+

1

−ω − (εKSa − εKSi ) + iη

}.

Slow convergence!!!

15

G0W0 basis set convergence

Exercise 4.2

Test the basis set convergence of G0W0 using basis sets of increasing

size.

16

G0W0 starting point dependence

17

G0W0 starting point dependence

• “Screening”: eigenvalue difference in polarizability:

P0(r, r′, ω) =occ∑i

virt∑a

ψKSa (r′)ψKS

i (r′)ψKSi (r)ψKS

a (r)

×

{1

ω − (εKSa − εKSi ) + iη+

1

−ω − (εKSa − εKSi ) + iη

}.

• “Self-interaction”: directly from DFT eigenvalues:

εG0W0n = εKSn + 〈ψKS

n |Σ(εG0W0n )− vxc |ψKS

n 〉

18

G0W0 starting point dependence

• “Screening”: eigenvalue difference in polarizability:

P0(r, r′, ω) =occ∑i

virt∑a

ψKSa (r′)ψKS

i (r′)ψKSi (r)ψKS

a (r)

×

{1

ω − (εKSa − εKSi ) + iη+

1

−ω − (εKSa − εKSi ) + iη

}.

• “Self-interaction”: directly from DFT eigenvalues:

εG0W0n = εKSn + 〈ψKS

n |Σ(εG0W0n )− vxc |ψKS

n 〉

18

G0W0 starting point dependence

Exercise 4.3 and 4.4

Test the starting point dependence of G0W0 with different functionals

for the underlying DFT calculations. Visualize the G0W0 spectra and

compare to the experimental photoemission spectrum of ethylene.

19

Self-interaction error for naphthalene

20

Self-interaction error for naphthalene

20

Self-interaction error for naphthalene

Exercise 6

Compare the energetic ordering of the DFT orbitals and the G0W0

quasiparticle energies for naphthalene. Assess the self-interaction error.

21

Self-consistent GW - Procedure

• solve Dyson equation: G = G0 + G0ΣG

22

Self-consistent GW - Spectral function

Spectral function

A(ω) = − 1

π

∫dr lim

r′→rImG (r, r′, ω) sgn(ω − εF )

• fully interacting G from self-consistent GW calculation

• poles of G correspond to excitations

• peaks in A are the excitation energies

23

Self-consistent GW - Spectrum

Figure 3: Spectral function for benzene

24

Self-consistent GW

Exercise 5

Perform self-consistent GW for ethylene

• Compute the spectral functions and extract the quasiparticle

energies

• Test the starting point dependence of self-consistent GW

25

GW total energy

Galitskii-Migdal equation

EGM = −i∫

dω

2πTr {[ω + h0]G (ω)}+ Eion,

h0 ... single particle term

. V. Galitskii and A. Migdal, Sov. Phys. JETP 7, 96 (1958)

. G from self-consistent GW

26

GW total energies

Exercise 7

Calculate the binding energy curve of H2 in self-consistent GW and

compare to full-CI.

27

Happy computing!

27