Hartree, Hartree-Fock and post-HF methods

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

mailto:[email protected]


• Let’s consider a multi electron WF

• We want to solve the Schrödinger equation

The curse of dimensionality

2

(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...

100

100

100


Helium: Hartree approximation

• Let’s define the WF as a product of orbitals

• We want to solve the Schrödinger equation

3

+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


• We replace the WF by the Hartree product in the Schrödinger equation

• We multiply and integrate

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� ~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

3

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)


• Remark (1)

• The starting point was:

• We end-up with equations of the form:

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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…

{


• Remark (2)

• The operator depends on the function we are looking for the solutions…

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SCF: self-consistent fieldf1(r1,'2)

iterative procedure

See for example in ORCA:


• Remark (3)

• Electron-electron interaction

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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

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 8

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


• 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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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)


Hartree product: generalization






• Energy

• Hamiltonian

• What is the energy for Helium considering the Hartree WF?

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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






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


• 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


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


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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• 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


Hartree-Fock energy

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• Helium

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Hartree-Fock energy


• Helium

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Hartree-Fock energy


Exchange integral


sum (x1, x2, . . . xN ) =

1pN !

N !X

1

(�1)P⇧N1 '1(xi)

Hartree-Fock energy


CoulombExchange

Hartree-Fock energy


Hartree vs. Hartree-Fock


Hartree-Fock equations


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








Hartree vs. Hartree-Fock

• Mean field approximation • Spin correlation: exchange K • SCF


Simulations with ORCA

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ques

tion

• Perform PES H2 dissociation at HF and DFT levels

https://nanohub.org/tools/orcatool/

https://nanohub.org/tools/orcatool/


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?


Spin operators (Extra)

36

ques

tion

• Demonstrate S and S2 = 0 for GS configuration





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For some details about spin projection

https://en.wikipedia.org/wiki/Spin_%28physics%29#Spin_projection_quantum_number_and_multiplicity


ORCA tool

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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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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


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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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


ORCA tool: Potential energy surface H2


ORCA tool: Potential energy surface H2

Edis

a0

λ

E(x) = �Edise(� x�a0

a0⇤� ) ⇥✓1 +

x� a0

a0�

◆+ E0


Electronic correlation

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‘exact’

HF DFT

MP2


• 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


• Remark (3)

• Electron-electron interaction

48

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.


The static correlation

49

HF wave function (SD) fails at dissociation

For some details about spin projection

https://en.wikipedia.org/wiki/Spin_%28physics%29#Spin_projection_quantum_number_and_multiplicity


• 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|


Configuration interaction

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Configuration interaction: H2








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


Other methods• Perturbation theory (Moller-Plesset or MP2,MP4,…)

• Coupled clusters (CCSD,CCSDT,…)

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Documents

Hartree, Hartree-Fock and post-HF methods