Electronic excitation in semiconductor nanoparticles: A ... · To describe electron-hole screening without using ... (BSE), equation-of-motion methods (EOM-CC, EOM-GF), CIS, ... Renormalizes

Electronic excitation in semiconductor nanoparticles: A real-space quasiparticle

perspective

Department of Chemistry

Syracuse University

Funding: NSF, ACS-PRF, Syracuse University

Objective: To describe electron-hole screening without using unoccupied states

Motivation:Calculation of unoccupied states are expensive. Judicious elimination of these states can lead to faster algorithm(e.g. WEST method by Galli et al.)

Strategy:Treating electron correlation in real-space representation by using explicitly correlated operators

Chemical applications:The developed method was used for calculations of optical gap and exciton binding energies

Overview of this talk

Charge-neutral excitation energy Matrix equation for excitation energies

ω

=

* *

A B x 1 0 xB A y 0 -1 y

Can be obtained using linear-response (LR-TDDFT), MBPT (BSE), equation-of-motion methods (EOM-CC, EOM-GF), CIS, ADC,…

Electron-hole interaction kernel (Keh)eh

, ,( )ia jb ij ab a i ia jbA Kδ δ= − + eh, ,ia jb ia jbB K=

Two important considerations:[1] Choice of 1-particles basis functions[2] Choice for treating e-e correlation

Effective single-particle Hamiltonian

22

eff ext eff2h v v

m−

= + +∇

0 eff ( )N

iH h i=∑

0H H W= +

ee eff ( )N

iW V v i= −∑

Non-interaction system Interacting system

Can be:eff HF MBPT ps modelKS ,, , , , v v v vv v∈ …

00 0

00

0 0 00 0

| 0 | 0| |a a

i n i

n n

EH

HE E

E

ω

=

=

⟩ ⟩

Φ ⟩ = Φ ⟩

−

0 0

0 0

| | 0| |n n n

n n

H EH E

EEω

Ψ ⟩ ⟩Ψ ⟩ =

=Ψ ⟩−

=

( , )N

i jW w i j

<

=∑

0 00 (?) (?) (?)n nω ω= + + +Goal of today’s talk:

Definitions: Correlation operator

0 0| | 0G=Ψ ⟩ ⟩ | |na

n iGΨ ⟩ Φ= ⟩

0, 0, ( , )N

n ni j

g i jG<

=∑ 2 212

1

/(1, 2)g

kr dN

kk

g b e=

−=∑

00 | 1⟨ Ψ ⟩ = 1| nai⟨Φ Ψ =⟩

Intermediate normalization condition

Electron-electron correlated is treated by operators that are local in real-space representation

0, 0,| | ' ( ) ( ')n nG G δ⟨ ⟩ = −x x x x x

G is a two-body operator and is represented by a linear combination of Gaussian-type geminal functions


00 | 1⟨ Ψ ⟩ = 1| nai⟨Φ Ψ =⟩



0 0| | 0G=Ψ ⟩ ⟩ | |na

n iGΨ ⟩ Φ= ⟩

00 | 1⟨ Ψ ⟩ = 1| nai⟨Φ Ψ =⟩



0, 0,| | ' ( ) ( ')n nG G δ⟨ ⟩ = −x x x x x


0 0| | 0G=Ψ ⟩ ⟩ | |na

n iGΨ ⟩ Φ= ⟩

0, 0, ( , )N

n ni j

g i jG<

=∑ 2 212

1

/(1, 2)g

kr dN

kk

g b e=

−=∑

00 | 1⟨ Ψ ⟩ = 1| nai⟨Φ Ψ =⟩



0, 0,| | ' ( ) ( ')n nG G δ⟨ ⟩ = −x x x x x

G is a two-body operator and is represented by a linear combination of Gaussian-type geminal functions

See also: geminal correlator (Rassolov et al.), NEO-XCHF (Hammes-Schiffer et al.), geminalMCSCF (Varganov & Martinez), trans-correlated Hamiltonian, Jastrow functions in VMC…

Connection to Configuration Interaction (CI)

G

0 00010

| | | || |k

k k k kk

ck

G G G∞

=

∞

=

= =Φ Φ ⟩⟨Φ Φ ⟩ ⟨Φ Φ ⟩ Φ ⟩∑ ∑

CIN

kk

kcΨ = Φ∑CIConfiguration interaction (CI):

Explicitly correlated wave function:

: finite number of independly optimizable coefficientskc

The explicitly correlated wave function is an infinite-order CI expansion with constrained CI coefficients

0G GΨ = Φ

(must be a functional of G)

Electron-hole interaction kernel

0

0 0 00

0 0 0

| |0 |

||

| 0 | |an n

an i

n

n

i

H W

W

E

WH WE

ω ω

⟨Φ Ψ ⟩⟨ Ψ ⟩

= + ⟨Φ

= +=

Ψ

+

⟩ − ⟨ Ψ ⟩

The excitation energies for the interacting and non-interacting system are related by the W operator


0

0 0 00

0 0 0

| |0 | |

| | 0 | |

an i n

an n i n

H W

W

EH W

WEω ω

⟨Φ Ψ ⟩⟨ Ψ ⟩

= += +

= + ⟨Φ Ψ ⟩ − ⟨ Ψ ⟩

00 0 0| | 0 | | 0a a

n n i n iWG WGω ω ⟨Φ ⟩ − ⟨+ Φ= ⟩


Expressing in terms of non-interacting states using (G)


0

0 0 00

0 0 0

| |0 | |

| | 0 | |

an i n

an n i n

H W

W

EH W

WEω ω

⟨Φ Ψ ⟩⟨ Ψ ⟩

= += +

= + ⟨Φ Ψ ⟩ − ⟨ Ψ ⟩

00 0 0| | 0 | | 0a a

n n i n iWG WGω ω ⟨Φ ⟩ − ⟨+ Φ= ⟩

0 † †0 0 00 | | 0 00 | |n n ni a aWG i WGω ω ⟨ ⟩ − ⟨= + ⟩


Expressing in terms of non-interacting states using (G)

Expressing in term of vacuum expectation value

Can be simplified using diagrammatic techniques

Contribution from the linked terms

Subset #1: All linked diagrams (all vertices are connected) Subset #2: All unlinked diagrams

† †0 | | 0nWG a ii a⟨ ⟩ Only fully contracted terms have non-zero contribution

to this term (Wick’s theorem)

The set of all resulting Hugenholtz diagrams, can be factored into sets of linked and unlinked diagrams

† † † †00 | | 0 | | 0 |0 | 0nn Lni a a i aWG ai WG i WG⟨ ⟩ = ⟨ ⟩ ⟨+ ⟩

†0 | | 0 (normal ord0 ered)ia⟨ ⟩ =

Because (algebraically): † †0 | | 10i a a i⟨ ⟩ =

Bayne, Chakraborty, JCTC, ASAP (2018)

Contribution from the linked terms

Subset #1: All linked diagrams (all vertices are connected) Subset #2: All unlinked diagrams

† †0 | | 0nWG a ii a⟨ ⟩ Only fully contracted terms have non-zero contribution

to this term (Wick’s theorem)

The set of all resulting Hugenholtz diagrams, can be factored into sets of linked and unlinked diagrams

Unlinked diagrams in excited state are exactly canceled by the ground state contributions

† †† † 0 | 0 | | 0 | | | 00 0n n Lni a a i aW i G aW WG iG ⟨⟨ ⟩ ⟨ ⟩ = ⟩−

(Important point used in the next slide)Bayne, Chakraborty, JCTC, ASAP (2018)

Elimination of unlinked diagrams

[ ]00 0

† †0 0 | | 0 | |0 | 0 |0 0 | 0 0|n n nnn i a WG a i WW GG WGω ω = + ⟨ ⟩ − + ⟩ ⟨ ⟨⟨ −⟩ ⟩

† † † †0 | | 0 | | 0 0 | 0 | 0n n Lni a aWG i WG WGi a ia⟨ ⟩ − ⟨ ⟩ = ⟨ ⟩

0 † †0 0 00 | | 0 | | 00 ( )n n n nLWG ii a a W G Gω ω= + + −⟨ ⟩ ⟨ ⟩

Adding zero to the expression…

Only linked terms contribute in the following expression

Expression for the excitation energy

Depends on particle-hole states

Depends only on occupied states

Generalized Hugenhotlz vertices

2 vefrom rtex 2-bo y (d )Ω

( , ) ( , , ) ( , , ,( ( , ), ) )N N N N N

i j i j i j i j i jn n n n

k ln i jWG w i j g i i j k i j k lj θ θ θ

< < < < < < <

= = + + ∑ ∑ ∑ ∑ ∑

2 3 4nWG = +Ω +Ω Ω

Product of two two-body operators generates 2, 3, and 4-body operators

Contributing diagrams to the excitation energy


Key result: Only connected diagrams contribute to the exciting energy and electron-hole interaction kernel



(effective 1-body)

e h,U U =

ehK =

(effective 2-body)


Contribution from different treatment of e-e correlation for ground and excited state wave function

0 19 20 210 | )( | 0n DW G G D D⟨ ⟩ + +=−

0 0| | 0G=Ψ ⟩ ⟩ | |na

n iGΨ ⟩ Φ= ⟩

Impacts excitation energy

Does not impact electron-hole interaction kernel

Is zero if Gn = G0


Effective 1-body (quasi) electron and hole operators

Renormalizes quasiparticle energy levels due to e-e correlation

Depends only on excited-state correlation operator Gn

Impacts excitation energy

Does not impact electron-hole interaction kernel

e h,U U =


It is an 2-particle operator that simultaneous operate on both (quasi) electron and hole states

The loops represent renormalization of 3- and 4-body operators as effective 2-body operators

All diagrams contribute to the electron-hole interaction kernel

Depends only on excited-state correlation operator Gn

ehK =

eh0 0nG K= → =

(eh screening is a consequence of ee correlation)

Interpretation of the closed-loops diagrams

ehK =

Closed-loops represent summation over occupied-state

They represent effective 2-body operators generated from a 4-body operator by treating the additional coordinates at mean-field level

Interpretation of the closed-loops diagrams

ehK =

Closed-loops represent summation over occupied-state

They represent effective 2-body operators generated from a 4-body operator by treating the additional coordinates at mean-field level

( , ) ( , , ) ( , , ,( ( , ), ) )N N N N N

i j i j i j i j i jn n n n

k ln i jWG w i j g i i j k i j k lj θ θ θ

< < < < < < <

= = + + ∑ ∑ ∑ ∑ ∑

o, cc

1 (3) (4) (3) (| (1, 2,3, 4) | 4)4! i j n

i ji j Aχ χ θ χ χ

∈

= ⟨ ⟩∑

Just-in-time (JIT) source code generation† †

1 2 1 20 | ... | 0pqrs

pqrs X p q sX YrI h Y⟨ … ⟩= ∑ Numerically zero mo integrals are

eliminated

Non-unique values are mapped to unique terms

All non-zero & unique mo integeralsare assigned a unique id

The unique id of the mo integrals are used to consolidate terms in the reduction step

Many-to-one map

Key point: Incorporating molecular integrals in the reduction step

Computer assisted Wick’s contraction

The strings of second quantized operators were evaluated using generalized Wick’s theorem

Strategy#1: All contractions are performed computationally

Strategy#2: Contractions are performed diagrammatically and the implementation is done computationally

Just-in-time (JIT) source code generation

Disadvantages: Source code is generated every time the mo integrals are updated (new system and change of basis)

Can be impractical for large codes that have long compilation time

Advantages:

The generated source code is optimized for the specific system

Can reduce the overall memory footprint

Chemical application using first-order diagrams


Application to chemical systems( )eh 12(1, 2) (1, 1 )2)(IK w g P= −

System Energy difference in eV

Ne 0.06

H2O 0.03

0 0[this work] [EOM-CCSD]n nω ω−

0( , ) ( , )eh e,h

approximations :0

nII III II III

G GK U

= = =

System Energy difference in eV

Cd6Se6 0.04 (Ref. 1 & 3)

Cd20Se19 -0.04 (Ref. 2 & 3)

0 0[this work] [GW/BSE]n nω ω−

Excitation energies in small molecules and clusters

Results from this work show reasonable agreement with many-body methods that use unoccupied states

1 Noguchi, Sugino, Nagaoka, Ishii, Ohno, JCP, 137, 024306 (2012)2 Wang, Zunger, PRB, 53, 9579 (1996)3 Bayne, Chakraborty, to be submitted (this work) Bayne, Chakraborty, JCTC, ASAP (2018)

1st exciton level

Quasiparticleenergy gap

Optical absorption (OA) gap

Exciton binding (EBE) gapConduction band

Valence band

Exciton binding energies in CdSe clusters

Exciton binding energies in CdSe quantum dots0

binding 0 0n nE ω ω= −

Exciton binding energies in CdSe quantum dots0

binding 0 0n nE ω ω= −

experiment

This work

theory

Summary

It was shown that the electron-hole interaction kernel (eh-kernel) can be expressed without using unoccupied states.

The derivation was performed using a two-body correlation operator which is local in real-space representation.

Using diagrammatic techniques, it was shown that the eh-kernel can be expressed only in terms of linked-diagrams.

The derived expression provides a route to make additional approximations to the eh-kernel

The 1st order approximation of eh-kernel was used for calculating electron-hole binding energies and excitation energies in atoms, molecules, clusters, and quantum dots.

Title here

Deformation potential: Which basis?0

defη η= +h h v

Both deformed and reference Hamiltonian use identical basis functions

Space-filling basis functions

Examples: plane-waves, real-space grid, distributed Guassian functions, Harmonic osc. basis, particle-in-box basis

Atom-centered basis functions

Deformed and reference Hamiltonian use different basis functions

We transform into the eigenbasis of the reference Hamiltonian

Transformation to ref. eigenbasis-I

0 0 0 0 0= λF C S C Step #1: Get quantities from the converged SCF calculation on reference

structure

Step #2: Perform symmetric or orthogonal transformation such that the S matrix is diagonal in that basis

0† 0

0† 0 0

=

=

X S X IX F X F

Step #3: Find the U matrix that diagonalizes the transformed Fock matrix

0 0 † 0 0 0[ ]≡ =F F λU U

The U0 matrix is the matrix needed to transform operators in the eigenbasisof the reference structure

(single tilde transformation)

(double tilde transformation)

Transformation to ref. eigenbasis-II

η η η η η= λF C S C Step #4: Get quantities from the converged SCF calculation on the deformed

structure

0 † 0[ ]η η=F U F U

Step #6: Transform the Fock in the eigenbasis of the reference Hamiltonian

0†

0†

η

η η

=

=

X S X IX F X F

Step #5: Perform orthogonal transformation

Step #7: Calculate the deformation potential0

defη η= −V F F

Info#: Contributing diagrams to the excitation energy

0 ) ( , ) ( , , ) ( , , ,( , ) ( , )( )n n n n

N N N N N

i j i j i j i j i j k lW G G w i j g i j i j i j k i j k lθ θ θ

< < < < < < <

− = = + +

∑ ∑ ∑ ∑ ∑

0 0) ( ,( ( , ) ( , ))n n

N N

i j i jG G g i j g g i ji j

< <

= =− −∑ ∑

1 2

1 2 3

1 2 3 4

3

0 1 2 2 12 1 2

1 2 3 3 1 2 3

1 2 3 4 1

!

1

4!

4 2 3 41

( (1

ˆ( 1)

ˆ( 1

10 | ) | 0 | ) |21 | |

1 )

3!

| |4!

k

k

N

ni

Np

ki

Np

i

i i k

i i i kk

i

W G G i P i

i i P i i

i i i P i i

i i

i i

i ii

θ

θ

θ

=

=

⟨ ⟩ = ⟨ ⟩

+ ⟨

−

⟩

⟩

−

−⟨

−

+

∑∑

∑

∑

∑

Info#: Determination of geminal parameters

For quantum dots, G was obtained from parabolic QD

model harme h ehH T T V V= + + +

For small molecules and clusters, G was obtained varaitionally

†model†mi | |

| |n e h

G

e h

e h e h

GG G

G Hχ χ χ χχ χ χ χ

⟨ ⟩⟨ ⟩ modelG

modelnG G

†

†

0 | | 00 | | 0

minG

GG

G HG

⟨ ⟩⟨ ⟩ 0G

0 nG G=

0nG G=

If we are ready to admit unoccupied states

Phys. Rev. A 89, 032515 (2014)

Challenge: Avoiding 3, 4, 5, 6-particle integrals

Step 1: Project the correlation function in a finite basis

†0 0 0 0

'||

M

k kk kkk

GE G HG H G′ ′⟨Φ Φ ⟩ == ∑

Step 2: Write the energy expression term of diagrams 1 2 36

M

kkE D D D= + …++∑

Step 3: Obtain a renormalized 2-body operator by performing infinite-order summation over a subset of diagrams

112 12 12

1 10 11 36

) )( (

limM M

M kk kk

g r r g r

E D D D D

−

→∞…+ …+= + + +∑ ∑

Partial infinite-order diagrammatic summation

PCTH-PIOS

lim

lim

M

k kM

k k

M

k kM

M

M

k k

D DE

D D

→∞

→∞

+ = +

∑ ∑

∑ ∑

PCTH

M

kk

k

M

k

DE

D=∑

∑

XCHF

lim

lim

M

kkM

kM

M

k

DE

D

→∞

→∞

=∑

∑

All diagrams are added to finite order

All diagrams are added to infinite order

Some diagrams are added to infinite order

42

, , ,a b c d

∞

∑

Info#: Summation over intermediate particle-hole states

Phys. Rev. A 89, 032515 (2014)

=

Finite and independent

1c2c

10c9c8c7c6c5c4c3c

Finite number of terms

FCIΨInfinite and independent

exactΨ

∞

1c2c

10c9c8c7c6c5c4c3c

Infinite number of terms

0 | | iG⟨Φ Φ ⟩

Comparison of the three methods

∞

1c2c

10c9c8c7c6c5c4c3c

Infinite number of terms

Infinite and dependent

ic ic icXCHFΨ

Unconstrained optimization over

infinite domain

Unconstrained optimization over

finite domain

Constrained optimized over infinite domain

1. Elward, Hoja, Chakraborty Phys. Rev. A. 86, 062504 (2012)

Varganov and Martinez, JCP, 132, 054103 (2010) ; Xu and Jordan, JPCA, 114, 1365 (2010)

• FCI - multi-determinant minimization.

XCHF - single determinant minimization.

• XCHF gives lowerenergy than FCI for all the basis sets shown.

• XCHF converges much faster with respect to size

1-particle basis

3 Key Points

See also: Geminal augmented MCSCF for H2, QMC calculations on H2O

Ground state energy of helium atom

e ee h hhext ext

e hinteraction termhole subsystemelectronic subs

eh

ystem

H T V V T V V V= + + + + + +

0 ehH H V= +•Configuration interaction (CI): Zunger, Efros, Sundholm, Wang, Bester, Rabani, Franceschetti, Califano, Bittner, Hawrylak,…

•Many-body perturbation theory (MBPT): Baer, Neuhauser, Galli,...

•Quantum Monte Carlo method (QMC): Hybertsen, Shumway,…

•GW+BSE: Louie,Chelikowsky, Galli, Rohlfing , Rubio, …

•All-electron TDDFT/DFT: Prezhdo, Tretiak, Kilina, Akimov, Ullrich, Li,….

Coulomb attraction term is responsible for electron-hole coupling

This talk:Explicitly correlated Hartree-Fock

Electron-hole Hamiltonian

Documents

Electronic excitation in semiconductor nanoparticles: A ... · To describe electron-hole screening without using ... (BSE), equation-of-motion methods (EOM-CC, EOM-GF), CIS, ... Renormalizes