hartree, hartree-fock and post-hf methods
TRANSCRIPT
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Hartree, Hartree-Fock and post-HF methods
Nicolas Onofrio School of Materials Engineering DLR 428 Purdue University [email protected]
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MSE697 fall 2015
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Let’s consider a multi electron WF
• We want to solve the Schrödinger equation
The curse of dimensionality
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(x1, x2, x3, . . . xN )
H = E
E = h | H | i
E =
Z ⇤(x1, x2, x3, . . . xN )H ⇤(x1, x2, x3, . . . xN )d3Nx
Hydrogen: 1e: 1003 = 106 op Silicon: 14e: 1003x14 = 1084 op SC: ~PFLOPS = 1015 op/sH: 106/1015 ~ 1nsSi: 1084/1015 ~1069 s ~ 1062 years!!! Marcoscale ~ 1023 electrons...
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Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Helium: Hartree approximation
• Let’s define the WF as a product of orbitals
• We want to solve the Schrödinger equation
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+2e
-e
-e
r1
r2
R
(r1, r2) = '1(r1)'2(r2)
H = � ~22m
r21 �
~22m
r22 �
2e2
|R� r1| �2e2
|R� r2| +e2
|r1 � r2|
H = E
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• We replace the WF by the Hartree product in the Schrödinger equation
• We multiply and integrate
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Helium: Hartree approximation
� ~22m
r21 �
~22m
r22 �
2e2
|R� r1|� 2e2
|R� r2|+
e2
|r1 � r2|
�'1(r1)'2(r2) = E'1(r1)'2(r2)
⇥Z
'⇤2(r2)dr2
2
6664� ~22m
r21 �
~22m
Z'2(r2)
⇤r22'2(r2)dr2
| {z }C1
� 2e2
|R� r1|� 2e2
Z'2(r2)⇤'2(r2)
|R� r2|dr2
| {z }C2
+e2Z
'2(r2)⇤'2(r2)
|r1 � r2|dr2
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7775'1(r1) = E'1(r1)
= E'1(r1)C1 and C2 are constants and do not act on '1(r1)
E0 = E � C1 � C2� ~22m
r21 �
2e2
|R� r1|+ e2
Z'2(r2)⇤'2(r2)
|r1 � r2|dr2
�'1(r1) = E0'1(r1)
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Remark (1)
• The starting point was:
• We end-up with equations of the form:
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Helium: Hartree approximation
H(r1, r2) (r1, r2) = E�(r1, r2)dimension: n3D (+spin..) = 2x3 = 6 (8 with spin)
f1(r1)'1(r1) = E0'1(r1)
f2(r2)'2(r2) = E00'2(r2)
dimension: n3D (+spin..) = 1x3 = 3 (4 with spin)
single-electron equations!but no free lunch…
{
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Remark (2)
• The operator depends on the function we are looking for the solutions…
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Helium: Hartree approximation
SCF: self-consistent fieldf1(r1,'2)
iterative procedure
See for example in ORCA:
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Remark (3)
• Electron-electron interaction
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Helium: Hartree approximation
Mean-field approximation!
+2e
-e
-e
r1
r2
R
e2Z
'2(r2) ⇤ '2(r2)
|r1 � r2|dr2 ⇥ '1(r1)
⇢2 ⇠ |'2|2
average density of electron 2 interacting
with electron 1
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Helium: Hartree approximation• Remark (4)
• Probability density:
• Considering the Hartree product
• What is dP1 for the Hartree product?
• What is the probability dP12 of finding electron 1 in dr1 and electron 2 in dr2?
dP1 =
Z| (r1, r1, . . . , rN )|2dr2dr3 . . . drN
Probability of finding electron 1 in dr1
(r1, r2, . . . , rN ) = '1(r1)'2(r2) . . .'N (rN )
ques
tion
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Remark (4)
• Probability of finding electron 1 in dr1
• Probability of finding electron 1 in dr1 and electron 2 in dr2
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Helium: Hartree approximation
dP1 = |'1(r1)|2Z
|'2(r2)|2dr2Z
|'3(r3)|2dr3 . . .Z
|'N (rN )|2drN
dP1 = |'1(r1)|2
dP12 =
Z| (r1, r2, . . . , rN )|2dr3 . . . drN
dP12 = |'1(r1)|2|'2(r2)|2 = dP1dP2
Electrons are uncorrelated + do not respect Pauli!(remember oxygen singlet/triplet)
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Hartree product: generalization
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Hartree product: generalization
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Hartree product: generalization
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Energy
• Hamiltonian
• What is the energy for Helium considering the Hartree WF?
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Helium: Hartree approximation
E = h |H | i =Z ⇤H dr
H = � ~22m
r21 �
~22m
r22 �
2e2
|R� r1| �2e2
|R� r2| +e2
|r1 � r2|
h1(r1) = � ~22m
r21 �
2e2
|R� r1|
simplifications:
h2(r2) . . . g12(r1, r2) . . .
ques
tion
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Helium: Hartree approximation
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Helium: Hartree approximation
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Quantum character of the WF• Identical particle (indistinguishable)
• All electrons in the universe have the same charge, mass, etc.
• Can’t measure the position of an electron with infinite precision (Heisenberg)
⇒ Symmetry in the WF
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Particles WF Spin Example
Fermions AS 1/2 integer electrons, protons, etc.
Bosons S integer phonons
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Anti-symmetric WF
• Back to Hartree WF
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Quantum character of the WF
(r1, r2) = � (r2, r1) (r, r) = 0{
Pauli exclusion!
(r1, r2) = '1(r1)'2(r2)
Pauli AS
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
The Slater determinant• Antisymmetric WF
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• Can’t distinguish between electrons • Antisymmetric (swap 2 particle change total sign) • Same spin and position ⇒ P = 0
ques
tion
• Demonstrate the antisymmetry for 2 electrons
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Overview of the lectures• Hartree-Fock
• Energy & equations
• Application to H2
• Energy & Wave function
• Simulations with ORCA
• HF limitations
• Post Hartree-Fock methods
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Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Antisymmetric WF: Slater determinant
Slater determinant
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• Can’t distinguish between electrons • Antisymmetric (swap 2 particle change total sign) • Same spin and position ⇒ P = 0
SD characteristics
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Hartree-Fock energy
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Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Helium
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Hartree-Fock energy
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Helium
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Hartree-Fock energy
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Exchange integral
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sum (x1, x2, . . . xN ) =
1pN !
N !X
1
(�1)P⇧N1 '1(xi)
Hartree-Fock energy
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CoulombExchange
Hartree-Fock energy
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Hartree vs. Hartree-Fock
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Hartree-Fock equations
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Lagrange multiplier
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Form
F (x, y, z,�) = f(x, y, z)� �(g(x, y, z)� k)
Solve
F
x
= 0
F
y
= 0
F
z
= 0
F
�
= 0
Back to f. . .
max/min of
f(x,y,z)
subject to the constraint
g(x,y,z)=k
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Hartree-Fock equations
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Hartree-Fock equations
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Hartree-Fock equations
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 33
Hartree vs. Hartree-Fock
• Mean field approximation • Spin correlation: exchange K • SCF
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Simulations with ORCA
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ques
tion
• Perform PES H2 dissociation at HF and DFT levels
https://nanohub.org/tools/orcatool/
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 35
Problem: H2 minimal basis
• Are those all real spin states?
ques
tion
• Find the HF energies of all the configurations • Are these configurations actual spin states?
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Spin operators (Extra)
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ques
tion
• Demonstrate S and S2 = 0 for GS configuration
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Spin operators (Extra)
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Spin operators (Extra)
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For some details about spin projection
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
ORCA tool
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Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
ORCA tool: overview• Tasks: SP, relaxation, PES, etc.
• Coordinates: cartesian and internal
• Spin/Charge state
• Methods: HF, DFT, post HF
• Basis sets
• Options
• Constraints
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Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
SCF, Relaxation, PES, etc.
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N loops task
SCF(electronic structure) 1 min E = <Ψ|H|Ψ>
Ionic relaxation(geometry optimization) 2 min F = -∇E
min E
Potential energy surface (PES) 2 N-Constraint min E
Relaxed potential energy surface (PES) 3
N-constraint min F min E
Ionic + cell relaxation 3min Stress min F min E
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
ORCA tool: overview• Tasks: SP, relaxation, PES, etc.
• Coordinates: cartesian and internal
• Spin/Charge state
• Methods: HF, DFT, post HF
• Basis sets
• Options
• Constraints
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Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Cartesian vs. internal (or Z-matrix)
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O
H H
O 0.0 0.0 0.0 H x y 0.0 H -x y 0.0
O(1)
H(2) H(3)
O(1) 0 0 0 0.0 0.0 0.0 H(2) 1 0 0 0.9 0.0 0.0 H(3) 1 2 0 0.9 109.5 0.0
Cartesian coordinates Internal coordinates (or Z-matrix)
O(2)H(1)
O(3)H(4)
H(1) 0 0 0 0.0 0.0 0.0 O(2) 1 0 0 0.9 0.0 0.0 O(3) 1 2 0 0.8 120.0 0.0 H(4) 3 2 1 0.9 120.0 180.0
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 44
ORCA tool: Potential energy surface H2
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 45
ORCA tool: Potential energy surface H2
Edis
a0
λ
E(x) = �Edise(� x�a0
a0⇤� ) ⇥✓1 +
x� a0
a0�
◆+ E0
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Electronic correlation
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‘exact’
HF DFT
MP2
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Correlation energy
• HF fails at dissociation, bad for transition state and open shell
• Two type of electronic correlation: dynamical << static
The electronic correlation
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Ecorr = Eexact � EHF
What approximations have we made?
‘exact’
HFEcorr
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Remark (3)
• Electron-electron interaction
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Dynamical correlation
Mean-field approximation!
+2e
-e
-e
r1
r2
R
e2Z
'2(r2) ⇤ '2(r2)
|r1 � r2|dr2 ⇥ '1(r1)
⇢2 ⇠ |'2|2average density of electron 2 interacting
with electron 1
‘exact’
HFmostly
dynamical corr.
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
The static correlation
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HF wave function (SD) fails at dissociation
For some details about spin projection
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
• Let’s develop the these WF
Intuitive approach
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a b
u ⇠ (a� b)
g ⇠ (a+ b)
|gg| |uu|
|uu| ⇠ |aa|+ |bb|� |ab|� |ba| = I � C
|gg| ⇠ |aa|+ |bb|+ |ab|+ |ba| = I + C
CI ⇠ |gg|+ c|uu|What would be a good value for c at the dissociation limit?
• if c = 1: pure ionic • if c = -1: pure covalent
CI ⇠ |aa|+ |bb| CI ⇠ |ab|+ |ba|
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Configuration interaction
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Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 52
Configuration interaction
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 53
Configuration interaction
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 54
Configuration interaction
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 55
Configuration interaction
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 56
Configuration interaction: H2
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 57
Configuration interaction: H2
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Configuration interaction: H2
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 59
Configuration interaction: H2
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Complete active space: CASSCF• CAS(n,m)
• n: number of electrons
• m: number of orbitals
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CAS(3,3)
CAS(2,2)
HF
Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation
Other methods• Perturbation theory (Moller-Plesset or MP2,MP4,…)
• Coupled clusters (CCSD,CCSDT,…)
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