basics and state of the art of quantum-chemistry methods ......basics and state of the art of...
TRANSCRIPT
Basics and State of the Art of Quantum-ChemistryMethods for Molecules, Clusters and Materials
张颖 (Igor Ying Zhang)
复旦大学化学系Department of Chemistry, Fudan University
2018-08-02, 北京Hands-on Workshop Density-functional Theory and Beyond
Electronic Structure Theory
Materials science and engineering: Properties of solids
Solve many-electron Schrödinger Equation
Hψ(r1, ..., rN ) =
[−
12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
]ψ(r1, ..., rN ) = Eψ(r1, ..., rN )
3N-dimensional problem.Find a good approximation!
Igor (FDU) QCM 2018-08-02 2 / 46
Different Ways to Approach the Exact Solutions
Wave-function Theory (WFT) Density-functional Theory (DFT)
A. E. Mattsson, and J. M. Wills, IJQC, 116, 834 (2016).G. H. Booth, et. al., Nature, 493, 365 (2013).
Improvable accuracy & potentially richer information
Igor (FDU) QCM 2018-08-02 3 / 46
Different Ways to Approach the Exact Solutions
Wave-function Theory (WFT)
Density-functional Theory (DFT)
A. E. Mattsson, and J. M. Wills, IJQC, 116, 834 (2016).G. H. Booth, et. al., Nature, 493, 365 (2013).
Improvable accuracy & potentially richer informationIgor (FDU) QCM 2018-08-02 3 / 46
▶ The 2nd-order Møller-Plessetperturbation theory
▶ The simplest wave function-basedmethod
▶ The computational cost is O(N5)with system size N
MP2
▶ Coupled-cluster approach withsingle, double, and perturbativetriple excitations
▶ “Gold standard” in quantumchemistry
▶ The computational cost is O(N7)with system size N
CCSD(T)
MP2 and CCSD(T) are the mostpopular wave function-basedquantum-chemistry methods
Igor (FDU) QCM 2018-08-02 4 / 46
▶ The 2nd-order Møller-Plessetperturbation theory
▶ The simplest wave function-basedmethod
▶ The computational cost is O(N5)with system size N
MP2
▶ Coupled-cluster approach withsingle, double, and perturbativetriple excitations
▶ “Gold standard” in quantumchemistry
▶ The computational cost is O(N7)with system size N
CCSD(T)
MP2 and CCSD(T) are the mostpopular wave function-basedquantum-chemistry methods
Igor (FDU) QCM 2018-08-02 4 / 46
Performance of CCSD(T) and MP2
F. Neese, et. al. Acc. of Chem. Research, 42, 641 (2009).J. Zheng, et. al. J. Chem. Theory Comput., 3, 569 (2007).
▶ CCSD(T) shows an overwhelmingperformance, serving the ultimatebenchmark for the development ofother methods.
State of the Art▶ Large-scale parallel CCSD(T)
implementations → >= 1,000s nodes.▶ Reduced-scaling CCSD(T) algorithms:
O(N7) → O(N?)▶ Coupled-electron pair approximations
(CEPA) and others▶ Molecules → Solids
▶ MP2 surpasses DFAs in manyimportant properties, including weakinteraction, charge-transfer-drivenproperties and others.
State of the Art▶ Large-scale parallel MP2
implementations → >= 1,000s nodes.▶ Reduced-scaling MP2 algorithms:
O(N5) → O(N3), and even O(N)▶ Double hybrid density functional
approximations▶ Molecules → Solids
Igor (FDU) QCM 2018-08-02 5 / 46
Performance of CCSD(T) and MP2
F. Neese, et. al. Acc. of Chem. Research, 42, 641 (2009).J. Zheng, et. al. J. Chem. Theory Comput., 3, 569 (2007).
▶ CCSD(T) shows an overwhelmingperformance, serving the ultimatebenchmark for the development ofother methods.
State of the Art▶ Large-scale parallel CCSD(T)
implementations → >= 1,000s nodes.▶ Reduced-scaling CCSD(T) algorithms:
O(N7) → O(N?)▶ Coupled-electron pair approximations
(CEPA) and others▶ Molecules → Solids
▶ MP2 surpasses DFAs in manyimportant properties, including weakinteraction, charge-transfer-drivenproperties and others.
State of the Art▶ Large-scale parallel MP2
implementations → >= 1,000s nodes.▶ Reduced-scaling MP2 algorithms:
O(N5) → O(N3), and even O(N)▶ Double hybrid density functional
approximations▶ Molecules → Solids
Igor (FDU) QCM 2018-08-02 5 / 46
Basics of MP2 and CCSD(T)Elementary configuration space concepts
References:▶ Szabo, Attila and Ostlund, Neil S. Modern Quantum Chemistry, publisher
McGraw-Hill, New York, 1996. BOOK▶ Rodney J. Bartlett and Monika Musial, Coupled-cluster theory in quantum
chemisty, Rev. Modern Phys. 79:291:352, 2007▶ So Hirata, Thermodynamic limit and size-consistent design, Theor. Chem.
Acc. 129:727-746, 2011
Igor (FDU) QCM 2018-08-02 6 / 46
Definition of the Correlation Energy in Quantum Chemistry
Ecorr = Eexact − EHFMost accepted definition of correlationenergy (Löwdin)
Hatree-Fock (HF) approximation:
▶ The common ground of allquantum-chemistry methods.
▶ A mean-field approximation (orindependent particle) model
▶ A single slater determinant|Φ0⟩ = |ϕ1ϕ2...ϕn⟩
▶ A set of single-particle HForbitals {ϕi} (occupied +virtual)
|Φ0⟩ = 1√N !
∣∣∣∣∣∣∣∣ϕ1(x1) ϕ1(x2) · · · ϕ1(xN )ϕ2(x1) ϕ2(x2) · · · ϕ2(xN )
......
...ϕN (x1) ϕN (x2) · · · ϕN (xN )
∣∣∣∣∣∣∣∣∂⟨Φ0∣∣H∣∣Φ0
⟩∂ϕi
= 0;⟨Φ0∣∣H∣∣Φ0
⟩= EHF
(−1
2 ∇2 + Vext + VCoul + K)
|ϕi⟩ =εi |ϕi⟩
F |ϕi⟩ =εi |ϕi⟩
Configuration Space concepts:
▶ HF ground state: |Φ0⟩▶ nth-order excitations: |Φa···
i··· ⟩
▶ T3 |Φ0⟩ =∑
ijkabc Cabcijk
∣∣∣Φabcijk
⟩▶ Ground state + excited states
→ Configuration space for theElectron Correlation problem
Φ0 Φai Φab
ij Φabcijk
T1 T2 T3
Ψ0 = Φ0 +∑
iaCa
i Φai +∑
ijabCab
ij Φabij +
∑ijkabc
Cabcijk Φabc
ijk + · · ·
Igor (FDU) QCM 2018-08-02 7 / 46
Definition of the Correlation Energy in Quantum Chemistry
Ecorr = Eexact − EHFMost accepted definition of correlationenergy (Löwdin)
Hatree-Fock (HF) approximation:
▶ The common ground of allquantum-chemistry methods.
▶ A mean-field approximation (orindependent particle) model
▶ A single slater determinant|Φ0⟩ = |ϕ1ϕ2...ϕn⟩
▶ A set of single-particle HForbitals {ϕi} (occupied +virtual)
|Φ0⟩ = 1√N !
∣∣∣∣∣∣∣∣ϕ1(x1) ϕ1(x2) · · · ϕ1(xN )ϕ2(x1) ϕ2(x2) · · · ϕ2(xN )
......
...ϕN (x1) ϕN (x2) · · · ϕN (xN )
∣∣∣∣∣∣∣∣∂⟨Φ0∣∣H∣∣Φ0
⟩∂ϕi
= 0;⟨Φ0∣∣H∣∣Φ0
⟩= EHF
(−1
2 ∇2 + Vext + VCoul + K)
|ϕi⟩ =εi |ϕi⟩
F |ϕi⟩ =εi |ϕi⟩
Configuration Space concepts:
▶ HF ground state: |Φ0⟩▶ nth-order excitations: |Φa···
i··· ⟩
▶ T3 |Φ0⟩ =∑
ijkabc Cabcijk
∣∣∣Φabcijk
⟩▶ Ground state + excited states
→ Configuration space for theElectron Correlation problem
Φ0 Φai Φab
ij Φabcijk
T1 T2 T3
Ψ0 = Φ0 +∑
iaCa
i Φai +∑
ijabCab
ij Φabij +
∑ijkabc
Cabcijk Φabc
ijk + · · ·
Igor (FDU) QCM 2018-08-02 7 / 46
Configuration Interaction (CI) Expansion
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Intermediatenormalization: ⟨Ψ0|Φ0⟩ = C0 = 1; ⟨Ψ0|Ψ0⟩ = 1; ⟨Ψ0|Ψ0⟩ ≍ N
Convention:
Due to the electron indistinguishability, the summations in the wavefunction should only go through the combination of i < j, a <b for Cab
ij and i < j < k, a < b < c for Cabcijk , and so forth.
For simplicity, we do not explicitly write this constrain down in thistalk.
CI energy:(H − EHF
)|Ψ0⟩ = (Eexact − EHF ) |Ψ0⟩
⟨Φ0∣∣(H − EHF
)∣∣Ψ0⟩
= Ecorr ⟨Φ0|Ψ0⟩
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Igor (FDU) QCM 2018-08-02 8 / 46
Configuration Interaction (CI) Expansion
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Intermediatenormalization: ⟨Ψ0|Φ0⟩ = C0 = 1; ⟨Ψ0|Ψ0⟩ = 1; ⟨Ψ0|Ψ0⟩ ≍ N
Convention:
Due to the electron indistinguishability, the summations in the wavefunction should only go through the combination of i < j, a <b for Cab
ij and i < j < k, a < b < c for Cabcijk , and so forth.
For simplicity, we do not explicitly write this constrain down in thistalk.
CI energy:(H − EHF
)|Ψ0⟩ = (Eexact − EHF ) |Ψ0⟩
⟨Φ0∣∣(H − EHF
)∣∣Ψ0⟩
= Ecorr ⟨Φ0|Ψ0⟩
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Igor (FDU) QCM 2018-08-02 8 / 46
Configuration Interaction (CI) Expansion
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Intermediatenormalization: ⟨Ψ0|Φ0⟩ = C0 = 1; ⟨Ψ0|Ψ0⟩ = 1; ⟨Ψ0|Ψ0⟩ ≍ N
Convention:
Due to the electron indistinguishability, the summations in the wavefunction should only go through the combination of i < j, a <b for Cab
ij and i < j < k, a < b < c for Cabcijk , and so forth.
For simplicity, we do not explicitly write this constrain down in thistalk.
CI energy:
(H − EHF
)|Ψ0⟩ = (Eexact − EHF ) |Ψ0⟩
⟨Φ0∣∣(H − EHF
)∣∣Ψ0⟩
= Ecorr ⟨Φ0|Ψ0⟩
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Igor (FDU) QCM 2018-08-02 8 / 46
Configuration Interaction (CI) Expansion
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Intermediatenormalization: ⟨Ψ0|Φ0⟩ = C0 = 1; ⟨Ψ0|Ψ0⟩ = 1; ⟨Ψ0|Ψ0⟩ ≍ N
Convention:
Due to the electron indistinguishability, the summations in the wavefunction should only go through the combination of i < j, a <b for Cab
ij and i < j < k, a < b < c for Cabcijk , and so forth.
For simplicity, we do not explicitly write this constrain down in thistalk.
CI energy:
(H − EHF
)|Ψ0⟩ = (Eexact − EHF ) |Ψ0⟩
⟨Φ0∣∣(H − EHF
)∣∣Ψ0⟩
= Ecorr ⟨Φ0|Ψ0⟩
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩Igor (FDU) QCM 2018-08-02 8 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩Ecorr =
∑ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩Ecorr =
∑ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩
Ecorr =∑
ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩Ecorr =
∑ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩Ecorr =
∑ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩Ecorr =
∑ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩Ecorr =
∑ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
Nesbet’s and Brillouin’s Theorems
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑
I
CI
⟨Φ0∣∣H∣∣ΦI
⟩
Hamiltonian: H = −12
N∑i
∇2i −
N∑i
v(ri) +N∑
i<j
1rij
Correlationenergy:
Ecorr =∑
ia
Cai
⟨Φ0∣∣H∣∣Φa
i
⟩+∑ijab
Cabij
⟨Φ0∣∣H∣∣Φab
ij
⟩Ecorr =
∑ia
Cai
⟨ϕi
∣∣F ∣∣ϕa
⟩+∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Brillouin’sTheorem:
⟨ϕi
∣∣F ∣∣ϕa
⟩∗=⟨ϕa
∣∣F ∣∣ϕi
⟩= εi ⟨ϕi|ϕa⟩ = 0
For HF orbitals, the matrix elements of the Hamil-tonian with single excited configurations are zero!
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Nesbet’sTheorem:
If we would know the precise values of the double excitationcoefficients, we would know the EXACT correlation energy
BUT all coefficients of excited configura-tions are indirectly depends on each others!
The central task in quantum chemistry is to find bet-ter approximations to the doubly excited coefficients CD
Igor (FDU) QCM 2018-08-02 9 / 46
MP2: First-order perturbation treatment of Cabij
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Correlationenergy:
Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
Coefficientsby MP2: Cab
ij = −⟨ϕaϕb||ϕiϕj⟩
εa + εb − εi − εj→∣∣ΨMP2
⟩EMP2
c = −∑ijab
|⟨ϕiϕj ||ϕaϕb⟩|2
εa + εb − εi − εj
Igor (FDU) QCM 2018-08-02 10 / 46
Configuration Interaction (CI) Equation
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Intermediatenormalization: ⟨ΨI |Φ0⟩ = CI ; ⟨Ψ0|Φ0⟩ = C0 = 1
Secular Equation:⟨
ΦI
∣∣H − EHF
∣∣Ψ0⟩
= Ecorr ⟨ΦI |Ψ0⟩
∑J
⟨ΦI
∣∣H − EHF
∣∣ΦJ
⟩CJ = Ecorr
∑J
CJ ⟨ΦI |ΦJ ⟩
Igor (FDU) QCM 2018-08-02 11 / 46
Configuration Interaction (CI) Equation
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Intermediatenormalization: ⟨ΨI |Φ0⟩ = CI ; ⟨Ψ0|Φ0⟩ = C0 = 1
Secular Equation:
⟨ΦI
∣∣H − EHF
∣∣Ψ0⟩
= Ecorr ⟨ΦI |Ψ0⟩
∑J
⟨ΦI
∣∣H − EHF
∣∣ΦJ
⟩CJ = Ecorr
∑J
CJ ⟨ΦI |ΦJ ⟩
Igor (FDU) QCM 2018-08-02 11 / 46
Truncated Configuration Interaction with DoubleExcitations (CID)
Secular Equation:∑klcd
⟨Φab
ij
∣∣H − EHF
∣∣Φcdkl
⟩Ccd
kl = Ecorr
∑klcd
Ccdkl
⟨Φab
ij |Φcdkl
⟩(
0 B∗
B D
)(1
C
)= Ecorr
(1
C
)
Bijab =⟨
Φabij
∣∣H∣∣Φ0⟩
Dijab,klbc =⟨
Φabij
∣∣H − EHF
∣∣Φbckl
⟩
CI Coefficients: C = −(D − 1Ecorr)−1B
CI Correlation: Ecorr = −B∗(D − 1Ecorr)−1B
Igor (FDU) QCM 2018-08-02 12 / 46
Truncated Configuration Interaction with DoubleExcitations (CID)
Secular Equation:∑klcd
⟨Φab
ij
∣∣H − EHF
∣∣Φcdkl
⟩Ccd
kl = Ecorr
∑klcd
Ccdkl
⟨Φab
ij |Φcdkl
⟩(
0 B∗
B D
)(1
C
)= Ecorr
(1
C
)
Bijab =⟨
Φabij
∣∣H∣∣Φ0⟩
Dijab,klbc =⟨
Φabij
∣∣H − EHF
∣∣Φbckl
⟩
CI Coefficients: C = −(D − 1Ecorr)−1B
CI Correlation: Ecorr = −B∗(D − 1Ecorr)−1B
Igor (FDU) QCM 2018-08-02 12 / 46
Size ExtensivityIf we treat a supersystem consisting of non-interacting subsystems (A · · · B),we should obtain the sum of the individual subsystem energies.
Ecorr[A · · · B] = Ecorr[A] + Ecorr[B]
Equivalent definition is that the correlation energy should be asympotitcallypropotional to the number of electrons N
Ecorr ≍ N
▶ In chemistry one is interested in the relative energies of molecules ofdifferent size, making this property a prerequisite for the development ofelectronic-structure methods.
▶ In condensed matter systems that contain an infinite number ofelectrons, the approximation that is not size extensive doesn’t work atall.
Igor (FDU) QCM 2018-08-02 13 / 46
Analysis of Size Extensivity in H2 · · · H2
For a single minimal basis H2 molecule, we found that the CID matrix is ofthe form:
H =(
0 VV ∆
)with
∆ =⟨ΦD
∣∣H − EHF
∣∣ΨD
⟩V =
⟨Φ0∣∣H − EHF
∣∣ΨD
⟩ σ∗
σGround state of theminimal basis H2system.With the lowest eigenvalue:
E0 = 12
(∆ −
√∆2 + 4V 2
)
It is easy to show that for N noninteracting H2 molecules CID gives:
E0 = 12
(∆ −
√∆2 + 4NV 2
)≍ N1/2
Igor (FDU) QCM 2018-08-02 14 / 46
Analysis of Size Extensivity in H2 · · · H2
For a single minimal basis H2 molecule, we found that the CID matrix is ofthe form:
H =(
0 VV ∆
)with
∆ =⟨ΦD
∣∣H − EHF
∣∣ΨD
⟩V =
⟨Φ0∣∣H − EHF
∣∣ΨD
⟩ σ∗
σGround state of theminimal basis H2system.With the lowest eigenvalue:
E0 = 12
(∆ −
√∆2 + 4V 2
)It is easy to show that for N noninteracting H2 molecules CID gives:
E0 = 12
(∆ −
√∆2 + 4NV 2
)≍ N1/2
Igor (FDU) QCM 2018-08-02 14 / 46
CID in Iterative Solution
CI Coefficients: C = −(D − 1Ecorr)−1B
CI Correlation: Ecorr = −B∗(D − 1Ecorr)−1B
Large-dimentional eigenvalue problem → iterative solution (Davidson algorithm){E(0)
corr = EMP2c ;C(0) → ΨMP2
}→ E(1)
corr = − EMP2c + ∆EMP3
c
1 + ⟨ΨMP2|ΨMP2⟩
{EMPn
c ≍ N ;⟨ΨMP2|ΨMP2⟩ ≍ N
}→ E(1)
corr ≍ N0
Igor (FDU) QCM 2018-08-02 15 / 46
CID in Iterative Solution
CI Coefficients: C = −(D − 1Ecorr)−1B
CI Correlation: Ecorr = −B∗(D − 1Ecorr)−1B
Large-dimentional eigenvalue problem → iterative solution (Davidson algorithm){E(0)
corr = EMP2c ;C(0) → ΨMP2
}→ E(1)
corr = − EMP2c + ∆EMP3
c
1 + ⟨ΨMP2|ΨMP2⟩
{EMPn
c ≍ N ;⟨ΨMP2|ΨMP2⟩ ≍ N
}→ E(1)
corr ≍ N0
Igor (FDU) QCM 2018-08-02 15 / 46
Brueckner-Goldstone Theorem
EMP2c = ▼ ▲ ▲ ▼ + ▼ ▲
ErMP2c = ▼ ▲ ▲ ▼ + ▼ ▲ ×S
= −−▼ ▲▲ ▼
▼ ▲▲ ▼▼ ▲▲ ▼▼ ▲▲ ▼
renormalizedterm in CID
unlinked 4th-excited terms
linked 2nd-excited terms
Many-body Corr. Unlinked Corr.No size extensive
Linked Corr.Size extensive
Renormalized diagrams in the truncated CI correlation scale as N2 or higher with theelectron number N , which, however, can be exactly eliminated by the disconnected (orunlinked) higher-order excited diagrams (except for exclusion principle violating terms).
Igor (FDU) QCM 2018-08-02 16 / 46
Brueckner-Goldstone Theorem
EMP2c = ▼ ▲ ▲ ▼ + ▼ ▲
ErMP2c = ▼ ▲ ▲ ▼ + ▼ ▲ ×S
= −−▼ ▲▲ ▼
▼ ▲▲ ▼▼ ▲▲ ▼▼ ▲▲ ▼
renormalizedterm in CID
unlinked 4th-excited terms
linked 2nd-excited terms
Many-body Corr. Unlinked Corr.No size extensive
Linked Corr.Size extensive
Renormalized diagrams in the truncated CI correlation scale as N2 or higher with theelectron number N , which, however, can be exactly eliminated by the disconnected (orunlinked) higher-order excited diagrams (except for exclusion principle violating terms).
Igor (FDU) QCM 2018-08-02 16 / 46
Brueckner-Goldstone Theorem
EMP2c = ▼ ▲ ▲ ▼ + ▼ ▲
ErMP2c = ▼ ▲ ▲ ▼ + ▼ ▲ ×S
= −−▼ ▲▲ ▼
▼ ▲▲ ▼▼ ▲▲ ▼▼ ▲▲ ▼
renormalizedterm in CID
unlinked 4th-excited terms
linked 2nd-excited terms
Many-body Corr. Unlinked Corr.No size extensive
Linked Corr.Size extensive
Renormalized diagrams in the truncated CI correlation scale as N2 or higher with theelectron number N , which, however, can be exactly eliminated by the disconnected (orunlinked) higher-order excited diagrams (except for exclusion principle violating terms).
Igor (FDU) QCM 2018-08-02 16 / 46
Linked and Unlinked Clusters for Correlations
Many-body perturbation theory:
▶ Linked diagrams are size extensive, but unlinked ones not (withoutproof; look in the references[1,2])
▶ Rayleigh-Schrödinger (RS) perturbation theory includes both linkedand unlinked terms, and thus is not size extensive.
▶ Excluding unlinked diagrams in the perturbation series, RSPT →Møller-Plesset (MP) perturbation theory.
▶ MPn methods (inlcuding MP2) are therefore size extensive.
1) J. Goldstone, Proc. Royal Soc. London A 1957, 239:12172) R.J. Bartlett, Ann. Rev. Phys. Chem. 1981, 32:359
Configuration interaction approach:
▶ CI approaches renormalize the linked diagrams, meanwhile includeunlinked diagrams, both of which are not size extensive
▶ In Full CI solution, the renormalized low-order diagrams cancel outexactly with some unlinked diagrams in higher-order.
▶ To be specific, CID is not size extensive, because it containsrenormalized second-order diagrams, however the unlinked 4th-orderdiagrams are completely excluded.
Igor (FDU) QCM 2018-08-02 17 / 46
Linked and Unlinked Clusters for Correlations
Many-body perturbation theory:
▶ Linked diagrams are size extensive, but unlinked ones not (withoutproof; look in the references[1,2])
▶ Rayleigh-Schrödinger (RS) perturbation theory includes both linkedand unlinked terms, and thus is not size extensive.
▶ Excluding unlinked diagrams in the perturbation series, RSPT →Møller-Plesset (MP) perturbation theory.
▶ MPn methods (inlcuding MP2) are therefore size extensive.
1) J. Goldstone, Proc. Royal Soc. London A 1957, 239:12172) R.J. Bartlett, Ann. Rev. Phys. Chem. 1981, 32:359
Configuration interaction approach:
▶ CI approaches renormalize the linked diagrams, meanwhile includeunlinked diagrams, both of which are not size extensive
▶ In Full CI solution, the renormalized low-order diagrams cancel outexactly with some unlinked diagrams in higher-order.
▶ To be specific, CID is not size extensive, because it containsrenormalized second-order diagrams, however the unlinked 4th-orderdiagrams are completely excluded.
Igor (FDU) QCM 2018-08-02 17 / 46
Linked and Unlinked Clusters for Correlations
CI wave function:
Ψ0 = Φ0 +∑
ia
Cai Φa
i +∑ijab
Cabij Φab
ij +∑
ijkabc
Cabcijk Φabc
ijk + · · ·
= Φ0 +∑
I
CIΦI , where I = S(1), D(2), T (3), · · ·
Electron correlation can be seperated clusters by clusters:
▶ N -electrons correlation clusters, which covers all the correlations in N electrons:1) one-electron correlation cluster: Ψ(1) =
∑iaCa
i Φai
2) two-electron correlation cluster: Ψ(2) =∑
ijabCab
ij Φabij
▶ Linked and unlinked clusters, denoted by CJ with J = S(1), D(2), T (3):
1) Cabcijk = Cabc
ijk + Cai C
bcjk − Ca
j Cbcik + · · · = Cabc
ijk + Aabcijk
[Ca
i Cbj Cc
k
3! + · · ·]
,
where Aabcijk is an anti-symmetric operator to ensure the anti-symmetry property in
the CI expansion.
Igor (FDU) QCM 2018-08-02 18 / 46
Coupled-cluster Approach with Singles and Doubles(CCSD)
Cai =Ca
i
Cabij =Cab
ij + Cai Cb
j − Cbi Ca
j = Cabij + Aab
ij
(Ca
i Cbj
2!
)Cabc
ijk =Cabcijk + Ca
i Cbcjk − Ca
j Cbcik + · · · = Cabc
ijk + Aabcijk
[Ca
i Cbj Cc
k
3! + · · ·]
Cabcdijkl =Cabcd
ijkl + Aabcdijkl
[Ca
i Cbj Cc
kCdl
4! +Ca
i Cbj Ccd
kl
(2!)22! +Ca
i Cbcdjkl
(3!)2 +Cab
ij Ccdkl
(2!)3(2!)22!
]
Seperate linked and unlinked clusters up to 4th excitations:
▶ Terms in blue: linked clusters, but renormalized partially▶ Terms in red: unlinked clusters
CCSDwave function:
ΨCCSD0 =
(1 + T1 + T2 +
(T1 + T2
)2
2!+
(T1 + T2
)3
3!+ · · ·
)Φ0
= eT1+T2 Φ0
Igor (FDU) QCM 2018-08-02 19 / 46
Coupled-cluster Approach with Singles and Doubles(CCSD)
Cai =Ca
i
Cabij =Cab
ij + Cai Cb
j − Cbi Ca
j = Cabij + Aab
ij
(Ca
i Cbj
2!
)Cabc
ijk =Cabcijk + Ca
i Cbcjk − Ca
j Cbcik + · · · = Cabc
ijk + Aabcijk
[Ca
i Cbj Cc
k
3! + · · ·]
Cabcdijkl =Cabcd
ijkl + Aabcdijkl
[Ca
i Cbj Cc
kCdl
4! +Ca
i Cbj Ccd
kl
(2!)22! +Ca
i Cbcdjkl
(3!)2 +Cab
ij Ccdkl
(2!)3(2!)22!
]
Seperate linked and unlinked clusters up to 4th excitations:
▶ Terms in blue: linked clusters, but renormalized partially▶ Terms in red: unlinked clusters
CCSDwave function:
ΨCCSD0 =
(1 + T1 + T2 +
(T1 + T2
)2
2!+
(T1 + T2
)3
3!+ · · ·
)Φ0
= eT1+T2 Φ0
Igor (FDU) QCM 2018-08-02 19 / 46
Coupled-cluster Approaches
CC wave function:ΨCCSD
0 =(
1 + T +T 2
2!+T 3
3!+ · · ·
)Φ0 = eT Φ0
with T = T1 + T2 + T3 + · · ·
▶ The full CC with T = T1 + T2 + T3 + · · · is identical to FCI▶ CCSD = T = T1 + T2 (scales as O[N6])▶ CCSDT = T = T1 + T2 + T3 (scales as O[N8])▶ · · ·▶ CC approaches are size extensive
Igor (FDU) QCM 2018-08-02 20 / 46
CCSD(T): “Gold Standard”In recent years it has become possible for small molecules to pursue very ac-curate calculations in CCSDT and CCSDTQ. However, in most of the cases,CCSDT is too expensive, as it scales as O(N8). On the other hand it is clearthat one has to go beyond CCSD if high accuracy (“chemical accuracy” of1-3 kcal/mol) should be reached.
This compromise is the “gold standard” CCSD(T) model, which takes a per-turbative correction to the converged doubly excited coefficients of CCSDcalculations. This triple-excitation correction features an O(N7) effort andreads
∆E(T ) = −∑
ijkabc
tabcijk
(tabcijk + tabc
ijk
)(εa + εb + εc − εi − εj − εk)
tabcijk = − P (ijk)P (abc)
∑d tab
ij ⟨bc||dk⟩ −∑
l tilab ⟨lc||jk⟩
εa + εb + εc − εi − εj − εk
tabcijk = − P (ijk)P (abc) ti
a ⟨bc||jk⟩εa + εb + εc − εi − εj − εk
Igor (FDU) QCM 2018-08-02 21 / 46
Convergence of CC EnergiesDeviation from full-CI (CO molecules, cc-pVDZ basis, frozen core) in mEhfor CI and CC models with various excitation levels:
CI CCSD 30.804 12.120
SDT 21.718 1.011SD(T) – 1.470SDTQ 1.775 0.061
SDTQP 0.559 0.008SDTQPH 0.035 0.002
For a given excitation level, the CC models are about one order of magni-tude more accurate than CI models (which becomes even more significantfor larger molecules)!
Gauss, J.; Lecture notes for ”Coupled Cluster Theory“, Workshop, Mariapfarr, Austra, 2004.Neese, F.; Lecture notes for ”Density-functional theory and beyonds“, Workshop, Trieste, Italy, 2013.
Igor (FDU) QCM 2018-08-02 22 / 46
Summary: Basics of MP2 and CCSD(T)
• Configuration Space: {Φ0, Φa
i , Φabij , Φabc
ijk · · ·}
• CI Expansion: Ψ0 = Φ0 +∑ia
Cai Φa
i +∑ijab
Cabij Φab
ij + · · ·
• Correlation energy: Ecorr =∑ijab
Cabij ⟨ϕiϕj ||ϕaϕb⟩
• Size extensivity: E[A · · ·B] = E[A] + E[B], E ≍ N
Both MP2 and CCSD(T) are size extensive,which is the theoretical guarantee of using these
methods to large and even extended systemsIgor (FDU) QCM 2018-08-02 23 / 46
Challenges and State-of-the-art of MP2 and CCSD(T)
• Error in MP2 Correlation
• Slow Basis-set Convergence
• High Computational Cost
• Molecules → Solids
Igor (FDU) QCM 2018-08-02 24 / 46
Nonempirical Improvement over MP2
Renormalized Second-order Perturbation Theory (rPT2):
Xinguo Ren, Patrick Rinke, Gustavo E. Scuseria, and Matthias Scheffler. Phys. Rev. B, 2013, 88:035102
▲ ▼
▲
▲ ▼
X
X
▲
▲ ▼
+
+
+
▲ ▼
▲
▲ ▼
X
X
X
▲ ▲
▲
▲ ▼
▲ ▼
+ · · · (= rSE)
+ · · · (= SOSEX)
+ · · · (= RPA)
Igor (FDU) QCM 2018-08-02 25 / 46
Empirical Improvement over MP2
Hybrid (Becke):
EHxc [ρ] = ELDA
xc + a(EHF
x − ELDAx
)+ b∆EGGA
x + c∆EGGAc
Double hybrid (XYG3):
EDHxc [ρ] = ELDA
x + a(EHF
x − ELDAx
)+ b∆EGGA
x + cEGGAc + (1 − c)EP T 2
c
Three parameters {a, b, c} were optimized against 223 molecules in G3/99 set
XYG3 {a = 0.8033; b = 0.2107; c = 0.6789}
Ying Zhang , Xin Xu, and William A. Goddard III. Proc. Natl. Acad. Sci. USA, 2009, 106:4963-4968
Igor (FDU) QCM 2018-08-02 26 / 46
Performance of XYG3
Basis set: 6-311+G(3df,2p)
Igor (FDU) QCM 2018-08-02 27 / 46
Slow Basis-set Convergence
The basis-set convergence in theHartree-Fock theory that is not toohard. However, the slow convergenceof CI expansion has presented a forb-biding barrier to high-accuracy calcu-lations.
The explicit correlation expansionmodels in configuration space re-quire an accurate representation ofthe two-electron density with a cuspwhen two electrons are getting closeto each other.
Igor (FDU) QCM 2018-08-02 28 / 46
State-of-the-Art of Basis-set Issue
Basis-set Extrapolations:
Ecorrlmax
= Ecorr∞ + A/(lmax + 1)3 + O
[(lmax + 1)−4]
• Gaussian-type basis sets, including cc-pVnZ, aug-cc-pVnZ,def-nZVP, and so forth
• Plane-wave basis sets provide an intrinsically and systemati-cally improvable solution of the electronic correlation by increasingthe momentum cutoff parameters.
• Numeric atom-centered orbital (NAO) basis sets: NAO-VCC-nZ is a series of NAO basis sets with valence-correlation con-sistency (VCC). NAO-VCC-nZ is now the default choice in FHI-aims for advanced correlation methods, including MP2, CCSD(T),RPA, GW, and so forth.
T.H. Dunning 1989 J. Chem. Phys. 90:1007
J. J. Shepherd et. al. 2012 Phys. Rev. B 86:035111
I. Y. Zhang et. al. 2013 New. J. Phys. 15:123033
Igor (FDU) QCM 2018-08-02 29 / 46
State-of-the-Art of Basis-set IssueExplictly correlated methods: R12/F12:
1) ”Correlation Consistent Basis Sets and Explicitly CorrelatedWavefunctions in a Numerical Atom-Centered Framework“, TH4Poster, FHI Fachbeirat 20162) Ten-no Research Group Home-page
References:
▶ ”Explicitly Correlated Electrons inMolecules”, Chem. Rev. (2012) 112:4
▶ ”Explicitly Correlated R12/F12 Methods forElectronic Structure”, Chem. Rev. (2012)1:75
▶ “Explicitly Correlated Electronic StructureTheory from R12/F12 ansätze”, WIREsComput. Mol. Sci. (2012) 2:114
Igor (FDU) QCM 2018-08-02 30 / 46
State-of-the-Art of Basis-set IssueExplictly correlated methods: R12/F12:
1) ”Correlation Consistent Basis Sets and Explicitly CorrelatedWavefunctions in a Numerical Atom-Centered Framework“, TH4Poster, FHI Fachbeirat 20162) Ten-no Research Group Home-page
References:
▶ ”Explicitly Correlated Electrons inMolecules”, Chem. Rev. (2012) 112:4
▶ ”Explicitly Correlated R12/F12 Methods forElectronic Structure”, Chem. Rev. (2012)1:75
▶ “Explicitly Correlated Electronic StructureTheory from R12/F12 ansätze”, WIREsComput. Mol. Sci. (2012) 2:114
Igor (FDU) QCM 2018-08-02 30 / 46
High Computational Cost
O(N7)
O(N5)
O(N4)
O(N4) O(O2V 2)
O(O4)
N = O + V
(H2O)n; D-ζ Basis Set
Resolution of Identity (RI)
vpqrs = ⟨ϕpϕq|ϕrϕs⟩ ≈
∑µ
MµprMµ
qs
O(N4) O(N3)
RI-MP2• M. Feyereisen, et. al. Chem. Phys. Lett. 1993, 208:359• D. E. Bernholdt, et. al. Chem. Phys. Lett. 1996, 250:477• T. Najajima, et. al. Chem. Phys. Lett. 2006, 427:225• F. Aquilante et. al. J. Chem. Phys. 2007, 127:114107• X. Ren, et. al. New J. Phys. 2012, 14:053020
RI-CCSD and/or RI-CCSD(T):
• A.P. Rendell, et. al. J. Chem. Phys. 1994, 101:400• T.B. Pedersen, et. al. J. Chem. Phys. 2004, 120:8887• M. Pitonak, et. al. Chem. Comm. 2011, 76:713• A.E. DePrince, et. al. JCTC 2013, 9:2687• A.E. DePrince, et. al. JCTC 2013, 9:2687
Local Resolution of Identity(RI-LVL)
⟨ϕpϕq|ϕrϕs⟩ ≈∑µ′ν′
Mµ′pr Vµ′ν′ Mν′
qs
µ′, ν′ ∼ O(N0) O(N2)
• A.C. Ihrig, et. al. New J. Phys. 2015, 17:093020• S.V. Levchenko, et. al. Comput. Phys. Comm. 2015,192:60
Tensor HyperContraction(THC) Scheme
⟨ϕpϕq|ϕrϕs⟩ ≈ XPp XP
r V P QXQq XQ
s
O(N2)
• S. Schumacher, et. al. JCTC 2015, 11:3052• R.M. Rarrish, et. al. J. Chem. Phys. 2014, 140:181102
Reduced-scaling Algorithms
3 Nearsightedness of the correlation
3 Sparsity in real space
7 Delocalization of molecular or-bitals (MO)
MO =⇒{
Atom-centered orbitalsLocalized MOs
References:• M. Schtz, et. al. J. Chem. Phys. 1999, 111:5691• P. Pinski, et. al. J. Chem. Phys. 2015, 144:024109• M. Häser, et. al. J. Chem. Phys. 1992, 96:489• S.A. Maurer, et. al. J. Chem. Phys. 2014, 140:224112
Igor (FDU) QCM 2018-08-02 31 / 46
High Computational Cost
O(N7)
O(N5)
O(N4)
O(N4) O(O2V 2)
O(O4)
N = O + V
(H2O)n; D-ζ Basis Set
Resolution of Identity (RI)
vpqrs = ⟨ϕpϕq|ϕrϕs⟩ ≈
∑µ
MµprMµ
qs
O(N4) O(N3)
RI-MP2• M. Feyereisen, et. al. Chem. Phys. Lett. 1993, 208:359• D. E. Bernholdt, et. al. Chem. Phys. Lett. 1996, 250:477• T. Najajima, et. al. Chem. Phys. Lett. 2006, 427:225• F. Aquilante et. al. J. Chem. Phys. 2007, 127:114107• X. Ren, et. al. New J. Phys. 2012, 14:053020
RI-CCSD and/or RI-CCSD(T):
• A.P. Rendell, et. al. J. Chem. Phys. 1994, 101:400• T.B. Pedersen, et. al. J. Chem. Phys. 2004, 120:8887• M. Pitonak, et. al. Chem. Comm. 2011, 76:713• A.E. DePrince, et. al. JCTC 2013, 9:2687• A.E. DePrince, et. al. JCTC 2013, 9:2687
Local Resolution of Identity(RI-LVL)
⟨ϕpϕq|ϕrϕs⟩ ≈∑µ′ν′
Mµ′pr Vµ′ν′ Mν′
qs
µ′, ν′ ∼ O(N0) O(N2)
• A.C. Ihrig, et. al. New J. Phys. 2015, 17:093020• S.V. Levchenko, et. al. Comput. Phys. Comm. 2015,192:60
Tensor HyperContraction(THC) Scheme
⟨ϕpϕq|ϕrϕs⟩ ≈ XPp XP
r V P QXQq XQ
s
O(N2)
• S. Schumacher, et. al. JCTC 2015, 11:3052• R.M. Rarrish, et. al. J. Chem. Phys. 2014, 140:181102
Reduced-scaling Algorithms
3 Nearsightedness of the correlation
3 Sparsity in real space
7 Delocalization of molecular or-bitals (MO)
MO =⇒{
Atom-centered orbitalsLocalized MOs
References:• M. Schtz, et. al. J. Chem. Phys. 1999, 111:5691• P. Pinski, et. al. J. Chem. Phys. 2015, 144:024109• M. Häser, et. al. J. Chem. Phys. 1992, 96:489• S.A. Maurer, et. al. J. Chem. Phys. 2014, 140:224112
Igor (FDU) QCM 2018-08-02 31 / 46
High Computational Cost
O(N7)
O(N5)
O(N4)
O(N4) O(O2V 2)
O(O4)
N = O + V
(H2O)n; D-ζ Basis Set
Resolution of Identity (RI)
vpqrs = ⟨ϕpϕq|ϕrϕs⟩ ≈
∑µ
MµprMµ
qs
O(N4) O(N3)
RI-MP2• M. Feyereisen, et. al. Chem. Phys. Lett. 1993, 208:359• D. E. Bernholdt, et. al. Chem. Phys. Lett. 1996, 250:477• T. Najajima, et. al. Chem. Phys. Lett. 2006, 427:225• F. Aquilante et. al. J. Chem. Phys. 2007, 127:114107• X. Ren, et. al. New J. Phys. 2012, 14:053020
RI-CCSD and/or RI-CCSD(T):
• A.P. Rendell, et. al. J. Chem. Phys. 1994, 101:400• T.B. Pedersen, et. al. J. Chem. Phys. 2004, 120:8887• M. Pitonak, et. al. Chem. Comm. 2011, 76:713• A.E. DePrince, et. al. JCTC 2013, 9:2687• A.E. DePrince, et. al. JCTC 2013, 9:2687
Local Resolution of Identity(RI-LVL)
⟨ϕpϕq|ϕrϕs⟩ ≈∑µ′ν′
Mµ′pr Vµ′ν′ Mν′
qs
µ′, ν′ ∼ O(N0) O(N2)
• A.C. Ihrig, et. al. New J. Phys. 2015, 17:093020• S.V. Levchenko, et. al. Comput. Phys. Comm. 2015,192:60
Tensor HyperContraction(THC) Scheme
⟨ϕpϕq|ϕrϕs⟩ ≈ XPp XP
r V P QXQq XQ
s
O(N2)
• S. Schumacher, et. al. JCTC 2015, 11:3052• R.M. Rarrish, et. al. J. Chem. Phys. 2014, 140:181102
Reduced-scaling Algorithms
3 Nearsightedness of the correlation
3 Sparsity in real space
7 Delocalization of molecular or-bitals (MO)
MO =⇒{
Atom-centered orbitalsLocalized MOs
References:• M. Schtz, et. al. J. Chem. Phys. 1999, 111:5691• P. Pinski, et. al. J. Chem. Phys. 2015, 144:024109• M. Häser, et. al. J. Chem. Phys. 1992, 96:489• S.A. Maurer, et. al. J. Chem. Phys. 2014, 140:224112
Igor (FDU) QCM 2018-08-02 31 / 46
High Computational Cost
O(N7)
O(N5)
O(N4)
O(N4) O(O2V 2)
O(O4)
N = O + V
(H2O)n; D-ζ Basis Set
Resolution of Identity (RI)
vpqrs = ⟨ϕpϕq|ϕrϕs⟩ ≈
∑µ
MµprMµ
qs
O(N4) O(N3)
RI-MP2• M. Feyereisen, et. al. Chem. Phys. Lett. 1993, 208:359• D. E. Bernholdt, et. al. Chem. Phys. Lett. 1996, 250:477• T. Najajima, et. al. Chem. Phys. Lett. 2006, 427:225• F. Aquilante et. al. J. Chem. Phys. 2007, 127:114107• X. Ren, et. al. New J. Phys. 2012, 14:053020
RI-CCSD and/or RI-CCSD(T):
• A.P. Rendell, et. al. J. Chem. Phys. 1994, 101:400• T.B. Pedersen, et. al. J. Chem. Phys. 2004, 120:8887• M. Pitonak, et. al. Chem. Comm. 2011, 76:713• A.E. DePrince, et. al. JCTC 2013, 9:2687• A.E. DePrince, et. al. JCTC 2013, 9:2687
Local Resolution of Identity(RI-LVL)
⟨ϕpϕq|ϕrϕs⟩ ≈∑µ′ν′
Mµ′pr Vµ′ν′ Mν′
qs
µ′, ν′ ∼ O(N0) O(N2)
• A.C. Ihrig, et. al. New J. Phys. 2015, 17:093020• S.V. Levchenko, et. al. Comput. Phys. Comm. 2015,192:60
Tensor HyperContraction(THC) Scheme
⟨ϕpϕq|ϕrϕs⟩ ≈ XPp XP
r V P QXQq XQ
s
O(N2)
• S. Schumacher, et. al. JCTC 2015, 11:3052• R.M. Rarrish, et. al. J. Chem. Phys. 2014, 140:181102
Reduced-scaling Algorithms
3 Nearsightedness of the correlation
3 Sparsity in real space
7 Delocalization of molecular or-bitals (MO)
MO =⇒{
Atom-centered orbitalsLocalized MOs
References:• M. Schtz, et. al. J. Chem. Phys. 1999, 111:5691• P. Pinski, et. al. J. Chem. Phys. 2015, 144:024109• M. Häser, et. al. J. Chem. Phys. 1992, 96:489• S.A. Maurer, et. al. J. Chem. Phys. 2014, 140:224112
Igor (FDU) QCM 2018-08-02 31 / 46
Finite Clusters → Periodic Systems
CRYSCOR: [1]▶ Methods: Local MP2▶ Basis set: Gaussian-type orbitals▶ Core states: Considered explicitly▶ K-grid: Gamma-centered k-mesh
CP2K [2]▶ Methods: Canonical MP2▶ Basis set: Gaussian & plane waves▶ Core states: Pseudo potentials▶ K-grid: Gamma-only
VASP [3,4]▶ Methods: Canonical MP2 and CCSD(T)▶ Basis set: Plane waves▶ Core states: Pseudo potentials▶ K-grid: Gamma-center k-mesh
[1] C. Pisani, et al., J. Comput. Chem. 29, 2113 (2008).
[2] M. Del Ben, et al. J. Chem. Theory Comput. 8, 4177(2012).
[3] A. Grüneis, et al. J. Chem. Phys. 133, 074107 (2010).
[4] G. H. Booth, et al. Nature, 493, 365 (2013).
Igor (FDU) QCM 2018-08-02 32 / 46
MP2 and CCSD(T) in FHI-aims
3(released) Canonical MP2 for both Molecules and Solids
3(released) Canonical CCSD(T) for Molecules
3(in testing) Reduced-scaling MP2 for both Molecules and Solids
3(in testing) Canonical CCSD for Solids
ø (in developing) Canonical CCSD(T) for Solids
Features:▶ Resolution-of-Identity, RI-MP2 and RI-CCSD(T)▶ Gamma-centered multiple k-points▶ Large-scale parallelization▶ The reduced-scaling Laplace-Transform MP2 ∼ O(N<3)
Igor (FDU) QCM 2018-08-02 33 / 46
Initialize a bunch of tasks {δk,k, q′}Flowchart of Peri-odic MP2 in FHI-aims
Communicate the requiredmatrices for each task
Generate L,R, and calculate EP T 2c
for a given batch {δk,k, q′}
Finish
update the nextbatch of tasks
{δk,k, q′}
Traverse{δk,k, q′}?
Preparerestartfile?
Store EP T 2c
and {δk,k, q′}
δk1, k1, q′1 δk2, k2, q′
2 · · · δkn, kn, q′n
yes
no
yes
no
{k, k′, q, q′}
V, c, ϵ, C
N2o ∗ N2
v ∗ Na
3 Parallelization w.r.t k-grid and MO
3 Scalapack/BLACS
3 85% parallel efficiency in com-modity supercomputers (up to 1000scores)
Igor (FDU) QCM 2018-08-02 34 / 46
Basis-set convergence of the periodic MP2 methodDiamond Si
E (eV) a0 (Å) B0 (GPa) E (eV) a0 (Å) B0 (GPa)N2Z 7.65 3.56 451 4.62 5.44 98N3Z 7.81 3.55 454 4.92 5.41 101N4Z 7.96 3.54 454 5.07 5.41 101CBS[34] 8.06 3.54 454 5.17 5.41 102VASP 2010[1] 7.97 3.55 450 5.05 5.42 100(PW) 2013[2] 8.04
MgO AlPE (eV) a0 (Å) B0 (GPa) E (eV) a0 (Å) B0 (GPa)
N2Z 5.11 4.23 160 4.06 5.48 92N3Z 5.39 4.23 162 4.41 5.45 94N4Z 5.47 4.24 164 4.55 5.45 95CBS[34] 5.53 4.24 165 4.66 5.45 95VASP 2010[1] 5.35 4.23 153 4.32 5.46 93(PW) 2013[2] 4.63
[1] A. Grüneis, et al. J. Chem. Phys. 133, 074107 (2010).
[2] G. H. Booth, et al. Nature, 493, 365 (2013).
Igor (FDU) QCM 2018-08-02 35 / 46
Accuracy: Small Molecules v.s. Extended Systems
Errors (unit in meV) in 10 cohesive energies and 148 atomization energies in G2BLYP B3LYP PBE PBE0 SCAN XYG3
C -545 -423 165 65 15 -32Si -720 -558 -110 -120 50 126
SiC -710 -527 -80 -100 10 46BN -308 -225 172 62 122 86BP -460 -319 150 110 200 196AlN -510 -384 -150 -230 -40 33AlP -722 -533 -242 -232 -72 27LiF -170 -128 -130 -240 -40 107LiCl -370 -259 -220 -220 -70 28MgO -480 -383 -210 -290 50 -23MAE 500 374 163 167 67 70G2
MAE 319 129 751 221 247 73
J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).J.W. Sun, A. Ruzsinszky, and J.P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).
I.Y. Zhang and X. Xu, ChemPhysChem 13, 1486 (2012).
Igor (FDU) QCM 2018-08-02 36 / 46
Accuracy: Small Molecules v.s. Extended Systems
Errors (unit in meV) in 10 cohesive energies and 148 atomization energies in G2BLYP B3LYP PBE PBE0 SCAN XYG3
C -545 -423 165 65 15 -32Si -720 -558 -110 -120 50 126
SiC -710 -527 -80 -100 10 46BN -308 -225 172 62 122 86BP -460 -319 150 110 200 196AlN -510 -384 -150 -230 -40 33AlP -722 -533 -242 -232 -72 27LiF -170 -128 -130 -240 -40 107LiCl -370 -259 -220 -220 -70 28MgO -480 -383 -210 -290 50 -23MAE 500 374 163 167 67 70G2
MAE 319 129 751 221 247 73
J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).J.W. Sun, A. Ruzsinszky, and J.P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).
I.Y. Zhang and X. Xu, ChemPhysChem 13, 1486 (2012).
Igor (FDU) QCM 2018-08-02 36 / 46
Accuracy: Small Molecules v.s. Extended Systems
Errors (unit in meV) in 10 cohesive energies and 148 atomization energies in G2BLYP B3LYP PBE PBE0 SCAN XYG3
C -545 -423 165 65 15 -32Si -720 -558 -110 -120 50 126
SiC -710 -527 -80 -100 10 46BN -308 -225 172 62 122 86BP -460 -319 150 110 200 196AlN -510 -384 -150 -230 -40 33AlP -722 -533 -242 -232 -72 27LiF -170 -128 -130 -240 -40 107LiCl -370 -259 -220 -220 -70 28MgO -480 -383 -210 -290 50 -23MAE 500 374 163 167 67 70G2
MAE 319 129 751 221 247 73
J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).J.W. Sun, A. Ruzsinszky, and J.P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).
I.Y. Zhang and X. Xu, ChemPhysChem 13, 1486 (2012).
Igor (FDU) QCM 2018-08-02 36 / 46
Reduced-scaling MP2: Laplace Transformation + RI-LVL
EMP2 = −2occ∑i,j
virt∑a,b
basis∑s,u,t,v
Csi C
uj Cu
aCvb (su|tv)
2
ϵa + ϵb − ϵi − ϵj
Xqsu =
occ∑i
Csi C
ui eϵitq Y q
su =virt∑a
CsaC
ua e−ϵatq
ELTMP2 = −
Nq∑q
wq
basis∑s,u,t,v
(su∣∣tv)q [2 (su|tv) − (sv|tu)]2
1x
=∫ ∞
0e−xtdt ≈
Nq∑q
wqe−xtq
[1] PY. Ayala, and GE. Scuseria, J. Chem. Phys. 110, 3660 (1999).[2] M. Häser, Theor Chim Acta 87, 147 (1993)
by Arvid C. Ihrig
Igor (FDU) QCM 2018-08-02 37 / 46
Reduced-scaling MP2: Laplace Transformation + RI-LVL
EMP2 = −2occ∑i,j
virt∑a,b
basis∑s,u,t,v
Csi C
uj Cu
aCvb (su|tv)
2
ϵa + ϵb − ϵi − ϵj
Xqsu =
occ∑i
Csi C
ui eϵitq Y q
su =virt∑a
CsaC
ua e−ϵatq
ELTMP2 = −
Nq∑q
wq
basis∑s,u,t,v
(su∣∣tv)q [2 (su|tv) − (sv|tu)]2
1x
=∫ ∞
0e−xtdt ≈
Nq∑q
wqe−xtq
[1] PY. Ayala, and GE. Scuseria, J. Chem. Phys. 110, 3660 (1999).[2] M. Häser, Theor Chim Acta 87, 147 (1993)
by Arvid C. Ihrig
Igor (FDU) QCM 2018-08-02 37 / 46
Reduced-scaling MP2: Laplace Transformation + RI-LVL
EMP2 = −2occ∑i,j
virt∑a,b
basis∑s,u,t,v
Csi C
uj Cu
aCvb (su|tv)
2
ϵa + ϵb − ϵi − ϵj
Xqsu =
occ∑i
Csi C
ui eϵitq Y q
su =virt∑a
CsaC
ua e−ϵatq
ELTMP2 = −
Nq∑q
wq
basis∑s,u,t,v
(su∣∣tv)q [2 (su|tv) − (sv|tu)]2
1x
=∫ ∞
0e−xtdt ≈
Nq∑q
wqe−xtq
[1] PY. Ayala, and GE. Scuseria, J. Chem. Phys. 110, 3660 (1999).[2] M. Häser, Theor Chim Acta 87, 147 (1993)
by Arvid C. Ihrig
Igor (FDU) QCM 2018-08-02 37 / 46
Reduced-scaling MP2: Laplace Transformation + RI-LVL
EMP2 = −2occ∑i,j
virt∑a,b
basis∑s,u,t,v
Csi C
uj Cu
aCvb (su|tv)
2
ϵa + ϵb − ϵi − ϵj
Xqsu =
occ∑i
Csi C
ui eϵitq Y q
su =virt∑a
CsaC
ua e−ϵatq
ELTMP2 = −
Nq∑q
wq
basis∑s,u,t,v
(su∣∣tv)q [2 (su|tv) − (sv|tu)]2
1x
=∫ ∞
0e−xtdt ≈
Nq∑q
wqe−xtq
[1] PY. Ayala, and GE. Scuseria, J. Chem. Phys. 110, 3660 (1999).[2] M. Häser, Theor Chim Acta 87, 147 (1993)
by Arvid C. Ihrig
Igor (FDU) QCM 2018-08-02 37 / 46
Performance & Scaling
Scaling Analysis
(H2O)10 – (H2O)150NAO-VCC-2Z basis sets
Igor (FDU) QCM 2018-08-02 38 / 46
Computational Scaling of the LT-MP2 with RI-LVL
0 20 40 60 80 100 120 140 160
number of water molecules
0
1
2
3
4
5
6
7
wal
ltim
ein
hour
s
canonical MP2 ∼ N4.93
LT-MP2 ∼ N2.31
canonical MP2 + RI coeffs ∼ N3.89
LT-MP2 + RI coeffs ∼ N2.23
MP2 correlationenergy errors
consistently below1 meV/atom
Igor (FDU) QCM 2018-08-02 39 / 46
H2O on TiO2 (110)
Periodic Implementation
H2O on a TiO2 (110) surface
O,H – NAO-VCC-2Z basis setTi – tier1 basis set
Γ-point calculations
Igor (FDU) QCM 2018-08-02 40 / 46
H2O on TiO2 (110) - System Size Convergence
0 20 40 60 80 100 120
number of atoms in surface unit cell
0.98
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
abso
rpti
on e
nerg
y in e
V
z-layers
x-axis (2 layers)
y-axis (2 layers)
3 layers
3 unit cellsin x-axis 4 unit cells
in y-axis
(4X4X4,384 atoms)
(3X4X3,216 atoms)
Igor (FDU) QCM 2018-08-02 41 / 46
H2O on TiO2 (110) - System Size Convergence
0 20 40 60 80 100 120
number of atoms in surface unit cell
0.98
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
abso
rpti
on e
nerg
y in e
V
z-layers
x-axis (2 layers)
y-axis (2 layers)
3 layers
3 unit cellsin x-axis 4 unit cells
in y-axis
(4X4X4,384 atoms)
(3X4X3,216 atoms)
Igor (FDU) QCM 2018-08-02 41 / 46
RI-CCSD(T) in FHI-aims: Accuracy
Igor (FDU) QCM 2018-08-02 42 / 46
RI-CCSD in FHI-aims: Parallelization Efficiency
Igor (FDU) QCM 2018-08-02 43 / 46
RI-CCSD in FHI-aims: Cluster v.s. Periodic
Igor (FDU) QCM 2018-08-02 44 / 46
Advanced first-principle methods for ma-terials science and engineering
http://th.fhi-berlin.mpg.de/site/index.php?n=Groups
▶ Canonical RI-MP2: Towards accurate MP2 calculations for solids
▶ Reduced-scaling MP2: Laplace-transformation + local RI
▶ RI-CCSD(T): Large-scale parallel implementations for bothmolecules and solids
Igor (FDU) QCM 2018-08-02 45 / 46
Acknowledgement
Prof. Matthias Scheffler
Advanced first-principle methods for ma-terials science and engineering
http://th.fhi-berlin.mpg.de/site/index.php?n=Groups
Arvid C. IhrigReduced-scaling MP2
Dr. Tonghao ShenRI-CCSD(T)
Igor (FDU) QCM 2018-08-02 46 / 46