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
Definitions: Correlation operator
00 | 1⟨ Ψ ⟩ = 1| nai⟨Φ Ψ =⟩
Intermediate normalization condition
Definitions: Correlation operator
0 0| | 0G=Ψ ⟩ ⟩ | |na
n iGΨ ⟩ Φ= ⟩
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
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
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
Electron-hole interaction kernel
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= ⟩
The excitation energies for the interacting and non-interacting system are related by the W operator
Expressing in terms of non-interacting states using (G)
Electron-hole interaction kernel
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 ⟨ ⟩ − ⟨= + ⟩
The excitation energies for the interacting and non-interacting system are related by the W operator
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
Contributing diagrams to the excitation energy
Key result: Only connected diagrams contribute to the exciting energy and electron-hole interaction kernel
Bayne, Chakraborty, JCTC, ASAP (2018)
Contributing diagrams to the excitation energy
(effective 1-body)
e h,U U =
ehK =
(effective 2-body)
Contributing diagrams to the excitation energy
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
Contributing diagrams to the excitation energy
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 =
Contributing diagrams to the excitation energy
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
Bayne, Chakraborty, JCTC, ASAP (2018)
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