VASP:BSE

MennoBokdam

UniversityofVienna,FacultyofPhysicsandCenterforComputaAonalMaterialsScience,

Vienna,Austria

PhotoabsorpAon

ConducAonband

Valenceband

~! = ✏f � ✏i

energyandmomentumconservaAon

p = ~k

‘Likelinessorintensity’ofthetransiAonisgivenbytheoscillatorstrength:

| < f |V0|i > |2

✏i, ~ki✏f , ~kf

~!, p

Excitons

ConducAonband

Valenceband

Intheexcitedstate,quasiparAclescanformwithfinitelifeAmescalledexcitons.

Thetotalenergyoftheexcitedsystemislowered

bytheelectron-holeCoulombinteracAonp = ~(kf � ki)

✏f � ✏i � Eeh

BetheSalpeterEquaAon(BSE)H (r1, ..., rN ) = E (r1, ..., rN )

✓�1

2�+ Vext(r) + VH(r)

◆ nk(r) +

Z⌃(r, r0, Enk) nk(r

0)dr0 = Enk nk(r)

⌃ = iGW AtthispointwehavequasiparAcleenergiesandwavefuncAonsincludingmanybodye-einteracAonsintheGWapproximaAon.

WiththeBSEweincludee-hinteracAonsintotheelectronicdescripAon.OTennecessarytoimprovethecalculatedopAcalspectraw.r.t.experiment.

ItintroduceshigherorderinteracAondiagramsandimprovestheelectronicdescripAonsystemaAcallyontopofGW.

P = PIQP + PIQP(2� �W )P

�n =elecX

c

holeX

v

An

cv�c(r)�

⇤v(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (Ec(k)�Ev(k))�cc0�vv0�kk0�W

cvkc0v0k0+2�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤ck(r)�c0k0(r)

"�1(r, r0)

|r� r0| �vk(r0)�⇤

v0k0(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤ck(r)�vk(r)

1

|r� r0|�c0k0(r0)�⇤

v0k0(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + 2�cvk

c0v0k0

1

E.E.Salpeter,H.A.Bethe,Phys.Rev.84,1232(1951),L.J.Sham,T.M.Rice,PhysRev.144,708(1966)

TheBethe-SalpeterequaAonissolvedforthePolarizaAonpropagatorinthefrequencydomaingivenbyaDyson-likeequaAon:

IndependentquasiparAcles(IQP)

ProductbasisofoccupiedandunoccupiedorbitalstoexpressthequasiparAcle(excitonic)wavefuncAon:

BSEconAnued

P = PIQP + PIQP(2� �W )P

�n =elecX

c

holeX

v

An

cv�c(r)�

⇤v(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (Ec(k)�Ev(k))�cc0�vv0�kk0�W

cvkc0v0k0+2�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤ck(r)�c0k0(r)

"�1(r, r0)

|r� r0| �vk(r0)�⇤

v0k0(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤ck(r)�vk(r)

1

|r� r0|�c0k0(r0)�⇤

v0k0(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + 2�cvk

c0v0k0

1

(Tamm-DancoffapproximaAon)

Instantaneousscreening: (staAcscreening) (dynamicaleffectsareexcluded)

Remarks

W (! ! 0)

PhysicallyintuiAvepictureofinteracAnge-hpairs.

!+IP = ✏c � ✏v � 0

!�IP = ✏v � ✏c 0

OverviewofBSEtheory:G.Onidaetal.,Rev.Mod.Phys.74,601(2002)

BSEEVP

S.Albrechtetal.PRL80,4510(1998)|G.Onidaetal.,Rev.Mod.Phys.74,601(2002)|F.Fuchsetal.,Phys.Rev.B78,085103(2008)

InteracAonwitheachotherbytwoterms:a

i

a a a

i

i

b

j

j

b

j

b

j

b

i

(A) (B) (D)(C)

+!

+!

+!

!!

+! +!+!

!!

P = PIQP + PIQP(2� �W )P

�n =elecX

c

holeX

v

An

cv�c(r)�

⇤v(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (Ec(k)�Ev(k))�cc0�vv0�kk0�W

cvkc0v0k0+2�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤ck(r)�c0k0(r)

"�1(r, r0)

|r� r0| �vk(r0)�⇤

v0k0(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤ck(r)�vk(r)

1

|r� r0|�c0k0(r0)�⇤

v0k0(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + 2�cvk

c0v0k0

1

Direct

P = PIQP + PIQP(� �W )P

�n =holeX

v

elecX

c

An

cv�v(r)�

⇤c(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (Ec(k)�Ev(k))�cc0�vv0�kk0�W

cvkc0v0k0+�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤v(r)�c(r

0)"�1(r, r0)

|r� r0| �v(r)�⇤c(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤v(r)�c(r

0)1

|r� r0|�v(r)�⇤c(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + �

cvkc0v0k0

1

+

P = PIQP + PIQP(2� �W )P

�n =elecX

c

holeX

v

An

cv�c(r)�

⇤v(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (✏ck�✏vk)�cc0�vv0�kk0�W

cvkc0v0k0+2�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤ck(r)�c0k0(r)

"�1(r, r0)

|r� r0| �vk(r0)�⇤

v0k0(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤ck(r)�vk(r)

1

|r� r0|�c0k0(r0)�⇤

v0k0(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + 2�cvk

c0v0k0

1

Problemcanbeformulatedasaneigenvalueproblem(EVP):

a

i

a a a

i

i

b

j

j

b

j

b

j

b

i

(A) (B) (D)(C)

+!

+!

+!

!!

+! +!+!

!!

P = PIQP + PIQP(2� �W )P

�n =elecX

c

holeX

v

An

cv�c(r)�

⇤v(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (Ec(k)�Ev(k))�cc0�vv0�kk0�W

cvkc0v0k0+2�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤ck(r)�c0k0(r)

"�1(r, r0)

|r� r0| �vk(r0)�⇤

v0k0(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤ck(r)�vk(r)

1

|r� r0|�c0k0(r0)�⇤

v0k0(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + 2�cvk

c0v0k0

1

Exchange

ScageringandannihilaAon/creaAonprocesses

BSEEVP

X

c0v0k0

pwkwk0H

cvkc0v0k0

pwk0A

n

c0v0k0 = (En�(✏ck�✏vk))pwkA

n

cvk

"i(!) = 1� 2e2

hc✏0⌦

X

n

|Pc,v,k

(fck�fvk)h c|pi| vi✏ck�✏vk

wkAn

cvk|2

! � En + i�

|An

k|2 =

X

c,v

An⇤c,v,kA

n

c,v,k

↵(!) = !

vuut�<"(!) +q<"(!)2 + ="(!)2

2

⌦

(2⇡)3

Z

⌦BZ

f (k)dk �!X

k2⌦BZ

wkf (k)

2

DiscreAzaAonon(some)gridgivesthegeneralizedeigenvalueproblemtosolve:

X

c0v0k0

pwkwk0H

cvkc0v0k0

pwk0A

n

c0v0k0 = (En�(✏ck�✏vk))pwkA

n

cvk

"i(!) = 1� 2e2

hc✏0⌦

X

n

|Pc,v,k

(fck�fvk)h c|pi| vi✏ck�✏vk

wkAn

cvk|2

! � En + i�

|An

k|2 =

X

c,v

An⇤c,v,kA

n

c,v,k

↵(!) = !

vuut�<"(!) +q<"(!)2 + ="(!)2

2

2

P = PIQP + PIQP(2� �W )P

�n =elecX

c

holeX

v

An

cv�c(r)�

⇤v(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (Ec(k)�Ev(k))�cc0�vv0�kk0�W

cvkc0v0k0+2�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤ck(r)�c0k0(r)

"�1(r, r0)

|r� r0| �vk(r0)�⇤

v0k0(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤ck(r)�vk(r)

1

|r� r0|�c0k0(r0)�⇤

v0k0(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + 2�cvk

c0v0k0

1

S.Albrechtetal.PRL80,4510(1998)|G.Onidaetal.,Rev.Mod.Phys.74,601(2002)|F.Fuchsetal.,Phys.Rev.B78,085103(2008)

Buildmatrixbasedon ,diagonaliseandobtain .{En, Ancvk}{EIQP , DFT , ✏RPA(r, r

0,!)}

TypicalBSEcalculaAon

1.  PreformagroundstateDFTorHybridcalculaAon

2.  Increasethenumberofunoccupiedorbitals

3.  PreformaGWcalculaAon(keepingorbitalsfixed)andcalculatethe

quasiparAcleenergiesandscreenedCoulombkernel.

(KeeptheWxxx.tmpandWFULLxxx.tmpfiles)

4.  PreformBSEcalculaAon(dielectricfuncAonwillbewrigeninvasprun.xml)

PHYSICAL REVIEW B 92, 045209 (2015)

Beyond the Tamm-Dancoff approximation for extended systems using exact diagonalization

Tobias Sander, Emanuele Maggio, and Georg KresseUniversity of Vienna, Faculty of Physics and Center for Computational Materials Science, Sensengasse 8/12,

A-1090 Vienna, Austria(Received 27 November 2014; revised manuscript received 22 June 2015; published 20 July 2015)

Linear optical properties can be accurately calculated using the Bethe-Salpeter equation. After introducing asuitable product basis for the electron-hole pairs, the Bethe-Salpeter equation is usually recast into a complex non-Hermitian eigenvalue problem that is difficult to solve using standard eigenvalue solvers. In solid-state physics, itis therefore common practice to neglect the problematic coupling between the positive- and negative-frequencybranches, reducing the problem to a Hermitian eigenvalue problem [Tamm-Dancoff approximation (TDA)]. Weuse time-inversion symmetry to recast the full problem into a quadratic Hermitian eigenvalue problem, whichcan be solved routinely using standard eigenvalue solvers even at a finite wave vector q. This allows us to accessthe importance of the coupling between the positive- and negative-frequency branch for prototypical solids. As astarting point for the Bethe-Salpeter calculations, we use self-consistent Green’s-function methods (GW), makingthe present scheme entirely ab initio. We calculate the optical spectra of carbon (C), silicon (Si), lithium fluoride(LiF), and the cyclic dimer Li2F2 and discuss why the differences between the TDA and the full solution are tiny.However, at finite momentum transfer q, significant differences between the TDA and our exact treatment arefound. The origin of these differences is explained.

DOI: 10.1103/PhysRevB.92.045209 PACS number(s): 71.35.!y, 78.20.!e, 78.40.!q

I. INTRODUCTION

The study of optical properties of condensed matter andmolecular systems is a field of growing interest, not leastbecause of the emerging importance of renewable energiesand the requirement to accurately predict the optical propertiesof novel composite materials and nanostructures. Time-dependent density functional theory (TDDFT) has certainlybeen the most widely used approach to date, although itis not without problems. In TDDFT, an effective two-pointDyson-like equation relates the density response functionof the noninteracting Kohn-Sham system !0(r,r",t ! t ") tothe (linear) density response function of the interactingsystem !TD(r,r",t ! t "): !TD = !0 + !0(v + fxc)!TD [1]. The“interaction” terms are described by the Coulomb kernelv and the exchange-correlation kernel fxc. Unfortunately,though, the interaction kernel does not allow for a systematicimprovable expansion of the microscopic particle-particleinteraction as would be the case, e.g., for Green’s-functionmethods. Furthermore, or rather resultantly, only few two-point kernels, fxc(r,r",t ! t "), yield a satisfactory descriptionof excitonic effects [2,3]. Among them, the most successfulapproximate kernels are the nanoquanta kernel [4–6], thebootstrap kernel of Sharma [7], and kernels based on thejellium with a gap [8]. The nanoquanta kernel requires oneto explicitly calculate the two-electron four-orbital integrals,making it almost as expensive as the methods discussed below,whereas the latter two are yet not satisfactorily derived fromfirst principles and fail to describe bound excitons accurately[7].

Alternative descriptions rely on the so-called Bethe-Salpeter equation (BSE). After some manipulation, theconventional Bethe-Salpeter equation—known from nucleartheory—can be cast into a Dyson-like equation,

P = P0 + P0IP,

where P (1,2,3,4) is the four-point time-ordered polarizationpropagator and I denotes the interaction kernel [9], and we usethe common notation for space and time points 1 = (r1,t1).This equation resembles the response equation for !TD fromTDDFT, where P can be regarded to be a generalized lineardensity matrix response function to a nonlocal perturbation(cf. Eq. (63) in Ref. [10]).

Obviously, manipulation of such four-point quantities ismuch more involved than the simpler TDDFT two-pointquantities. In practice, the polarization propagator P (1,2,3,4)is expressed in a suitable two-orbital basis made up of allrelevant combinations of electron and hole pairs. Furthermore,the electron-hole interaction kernel I is approximated bythe Coulomb kernel v and a static (or, more correctly,instantaneous) screened interaction W . This static approx-imation is commonly applied to simplify the calculations.Inclusion of frequency-dependent kernels is possible and,e.g., important for the description of double excitations,but computationally much more demanding [11,12]. Also,it has been shown that quasiparticle (QP) renormalizationeffects cancel against dynamical effects in the interac-tion kernel [11]. Hence, neglecting dynamical effects asdone throughout this work is expected to yield accurateresults.

The excitation energies are determined by calculating theresolvent of the polarization propagator. This usually requiresthe diagonalization of a large matrix, where the matrixdimension equals the number of occupied states times thenumber of unoccupied states. Formally, the solution of thisequation is then entirely equivalent to solving the so-calledCasida equation for time-dependent DFT and time-dependentHartree-Fock [13]. For hybrid functionals, the only differenceis that in Casida’s equation, the screened interaction Wbetween electrons and holes is replaced by the Coulomb kernelv “screened” by the mixing parameter ". The mixing parameter" determines how much of the nonlocal exchange is included(in most cases, " = 1/4).

1098-0121/2015/92(4)/045209(14) 045209-1 ©2015 American Physical Society

AlldetailsabouttheBSEImplementaAoninVASP5.4.1:

INCARTagsandlinks

ALGO=BSE TurnonBSEcalculaAonANTIRES=0/1/2 0:Tamm-DancoffapproximaAon

1:exactatomega=0 2:BeyondTD

NBANDSO=n Numberofoccupiedbandsincluded(INTEGER)NBANDSV=m Numberofunoccupiedbandsincluded(INTEGER)OMEGAMAX=x.x MaximalfrequencyincludedinBSEbasis(REAL[eV])BSEBethe-SalpetercalculaAonsintheVaspmanual

OMEGAMAXBandgap

E

k

Truncateyoure-hproductbasisorfaceMEMORYshortage!

ExampleSiliconAbsorpAonspectrumisgivenbytheImaginarypartofthedielectricfuncAoninthelongwavelengthlimit.

✏(!)

Example:ExcitoninMAPbI3

Bandgapatdifferentlevelsoftheory:

�opt �EXPopt Exb EEXP

xb �GW �DFT �wsDFT Vol.

(eV) (eV) (meV) (meV) (eV) (eV) (eV) (A3)

MAPbI3 1.63 1.52-1.67 45 6-55 1.67 0.77 1.69 251.60

M.Bokdametal.,ScienAficReports,6,28618,(2016)

Exb = EGW � EBSE�opt = �GW � Exb

WhichquasiparAcleenergiesshouldbeused?

TheshiTintheonsetofabsorpAonistherelevantquanAty.

BSEcalculaAonsbecomemoreexpensiveforsmallgapsemiconductorswithhighdielectricconstants.Thee-hinteracAonisheavilyscreenedandtheextentoftheexcitonwavefuncAonislarge.->Largesupercells/densek-pointmeshesarerequired.

ExampleMAPbX3

-1

0

1

2

3

4

1.67

-1

0

1

2

3

4

2.34

M R Z

-1

0

1

2

3

4

3.07

0

1

2

0

1

2

3

R

0

1

2

3

MAPbI3

MAPbBr3

MAPbCl3

Exb

= 45 meV

Exb

= 71 meV

Exb

= 106 meV

P = PIQP + PIQP(2� �W )P

�n =elecX

c

holeX

v

An

cv�c(r)�

⇤v(r0)

⌦

(2⇡)3

X

c0v0

Z

⌦BZ

Hcv

c0v0(k,k0)An

c0v0(k0)dk0 = E

nA

n

cv(k)

Hcv

c0v0(k,k0) = (Ec(k)�Ev(k))�cc0�vv0�kk0�W

cvkc0v0k0+2�

cvkc0v0k0

Wcvkc0v0k0 =

Z Z�⇤ck(r)�c0k0(r)

"�1(r, r0)

|r� r0| �vk(r0)�⇤

v0k0(r0)drdr0

�cvkc0v0k0 =

Z Z�⇤ck(r)�vk(r)

1

|r� r0|�c0k0(r0)�⇤

v0k0(r0)drdr0

Hcvkc0v0k0 = �W

cvkc0v0k0 + 2�cvk

c0v0k0

1

EventhoughbandgapnotatR,excitoniscenteredatR:

M.Bokdametal,ScienAficReports,6,28618,(2016)

ModelBSEcalculaAons:Excitons

TruncateBSEbasis:•  TDA•  2conducAon/valencebands•  q=k-k’=0

3.

1. StartfromDFTorbitals+SCISSORtogetGW0gap

ModelScreenedCoulomb:

isreplacedbylocalmodeldielectricresponsefuncAon.

2.

ConvergeExbwithincreasinggridandlinearlyextrapolateExbtoinfinitek-points.4.

M.Bokdametal,ScienAficReports,6,28618,(2016)

0 1 2 3 4 5 6 72!|k+G|

0

0.2

0.4

0.6

0.8

1

"-1

(k+

G)

s6-GWmodel"

#

-1

ModelBSEcalculaAons:Excitons

M.Bokdametal,ScienAficReports,6,28618,(2016)

0 0.005 0.01 0.015 0.02

NK

-1

-250

-200

-150

-100

-50

0

50

100

150

200E

-!G

W (

me

V)

0 0.0001 0.0002-60-50-40-30-20-10

010

E-!

GW

(m

eV

)

Exb

4x4x4

6x6x6

8x8x8

20x20x20

16x16x16

18x18x18

Extrapolatetoinfinitek-pintgridsF.Fuchsetal.Phys.Rev.B78,085103(2008)

Example:FASnI3

INCARTagsandlinksALGO=BSE TurnonBSEcalculaAonANTIRES=0/1/2 0:Tamm-DancoffapproximaAon

1:exactatomega=0 2:BeyondTD

NBANDSO=n Numberofoccupiedbandsincluded(INTEGER)NBANDSV=m Numberofunoccupiedbandsincluded(INTEGER)OMEGAMAX=x.x MaximalfrequencyincludedinBSEbasis](REAL[eV])AlternaAvely,theCassidaEquaAonorModelBSE:ALGO=TDHF TurnonBSEcalculaAonLHFCALC=.TRUE. TurnonmodelscreeningAEXX=0.3;HFSCREEN=0.2 (Typicalvalues,formodelBSEtheseparameters

needtobedeterminedbyfivngtoaprecedingRPA calculaAon.)

BSEBethe-SalpetercalculaAonsintheVaspmanual

TheEnd

Thankyou!