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Home , Electronic correlation

Hartree, Hartree-Fock and post-HF methods

MSE697 fall 2015

Nicolas Onofrio School of Materials Engineering DLR 428 Purdue University [email protected]

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 1 The curse of dimensionality

• Let’s consider a multi WF

(x1,x2,x3,...xN )

• We want to solve the Schrödinger equation

Hˆ =E E = Hˆ h | | i 3N E = ⇤(x1,x2,x3,...xN )Hˆ ⇤(x1,x2,x3,...xN )d x 100 Z Hydrogen: 1e: 1003 = 106 op Silicon: 14e: 1003x14 = 1084 op 100 SC: ~PFLOPS = 1015 op/s H: 106/1015 ~ 1ns Si: 1084/1015 ~1069 s ~ 1062 years!!! Marcoscale ~ 1023 ... 100

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 2 Helium: Hartree approximation

-e

r2

R +2e -e r1

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

(r1,r2)='1(r1)'2(r2) • We want to solve the Schrödinger equation =E H ~2 ~2 2e2 2e2 e2 = 2 2 + H 2mr1 2mr2 R r R r r r | 1| | 2| | 1 2| Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 3 Helium: Hartree approximation

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

~2 ~2 2e2 2e2 e2 2 2 + ' (r )' (r )=E' (r )' (r ) 2mr1 2mr2 R r R r r r 1 1 2 2 1 1 2 2  | 1| | 2| | 1 2|

• We multiply and integrate '⇤(r )dr ⇥ 2 2 2 Z 2 2 2 2 ~ 2 ~ 2 2e 2 '2(r2)⇤'2(r2) 2 '2(r2)⇤'2(r2) 3 1 '2(r2)⇤ 2'2(r2)dr2 2e dr2 +e dr2 '1(r1)=E'1(r1) 2mr 2m r R r1 R r2 r1 r2 6 Z | | Z | | Z | | 7 6 C 7 6 1 C2 7 4 5 | {z } | {z } = E'1(r1) C1 and C2 are constants and do not act on '1(r1) E0 = E C1 C2 2 2 ~ 2 2e 2 '2(r2)⇤'2(r2) + e dr ' (r )=E0' (r ) 2mr1 R r r r 2 1 1 1 1  | 1| Z | 1 2| Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 4 Helium: Hartree approximation

• Remark (1)

• The starting point was: (r ,r ) (r ,r )=E(r ,r ) H 1 2 1 2 1 2 dimension: n3D (+spin..) = 2x3 = 6 (8 with spin)

• We end-up with equations of the form:

fˆ1(r1)'1(r1)=E0'1(r1)

{fˆ (r )' (r )=E00' (r ) 2 2 2 2 2 2 dimension: n3D (+spin..) = 1x3 = 3 (4 with spin) single-electron equations!

but no free lunch…

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 5 Helium: Hartree approximation

• Remark (2)

• The operator depends on the function we are ˆ looking for the solutions… f1(r1,'2) SCF: self-consistent field iterative procedure

See for example in ORCA:

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 6 Helium: Hartree approximation

' (r ) ' (r ) • Remark (3) e2 2 2 ⇤ 2 2 dr ' (r ) r r 2 ⇥ 1 1 Z | 1 2| • Electron-electron interaction Mean-field approximation!

⇢ ' 2 -e 2 ⇠| 2|

r2

R +2e average density of electron 2 interacting -e with electron 1 r1

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 7 Helium: Hartree approximation

• Remark (4)

• Probability density: dP = (r ,r ,...,r ) 2dr dr ...dr 1 | 1 1 N | 2 3 N Z Probability of finding electron 1 in dr1 • Considering the Hartree product (r1,r2,...,rN )='1(r1)'2(r2) ...'N (rN )

• What is dP1 for the Hartree product?

• What is the probability dP12 of finding electron 1 in

question dr1 and electron 2 in dr2?

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

• Remark (4)

• Probability of finding electron 1 in dr1 dP = ' (r ) 2 ' (r ) 2dr ' (r ) 2dr ... ' (r ) 2dr 1 | 1 1 | | 2 2 | 2 | 3 3 | 3 | N N | N Z Z Z dP = ' (r ) 2 1 | 1 1 | • Probability of finding electron 1 in dr1 and electron 2 in dr2

dP = (r ,r ,...,r ) 2dr ...dr 12 | 1 2 N | 3 N dP =Z ' (r ) 2 ' (r ) 2 = dP dP 12 | 1 1 | | 2 2 | 1 2 Electrons are uncorrelated + do not respect Pauli! (remember oxygen singlet/triplet)

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 9 Hartree product: generalization

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 10 Hartree product: generalization

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 11 Hartree product: generalization

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 12 Helium: Hartree approximation

• Energy

E = = ⇤ dr h |H| i H Z • Hamiltonian ~2 ~2 2e2 2e2 e2 = 2 2 + H 2mr1 2mr2 R r R r r r | 1| | 2| | 1 2| • What is the energy for Helium considering the Hartree WF? question simplifications: ~2 2e2 hˆ (r )= 2 1 1 2mr1 R r | 1| hˆ2(r2) ... gˆ12(r1,r2) ...

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 13 Helium: Hartree approximation

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 14 Helium: Hartree approximation

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 15 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

Particles WF Spin Example

electrons, Fermions AS 1/2 integer protons, etc.

Bosons S integer phonons

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 16 Quantum character of the WF

• Anti-symmetric WF (r ,r )= (r ,r ) 1 2 2 1 { (r, r)=0 Pauli exclusion!

• Back to Hartree WF

(r1,r2)='1(r1)'2(r2)

Pauli AS

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 17 The

• Antisymmetric WF

• Can’t distinguish between electrons • Antisymmetric (swap 2 particle change total sign) • Same spin and position ⇒ P = 0

• Demonstrate the antisymmetry for 2 electrons question

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 18 Overview of the lectures

• Hartree-Fock

• Energy & equations

• Application to H2

• Energy &

• Simulations with ORCA

• HF limitations

• Post Hartree-Fock methods

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 19 Slater determinant

• Antisymmetric WF: Slater determinant

SD characteristics

• Can’t distinguish between electrons • Antisymmetric (swap 2 particle change total sign) • Same spin and position ⇒ P = 0

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 20 Hartree-Fock energy

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 21 Hartree-Fock energy

• Helium

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 22 Hartree-Fock energy

• Helium

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 23 Exchange integral

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 24 Hartree-Fock energy

N! 1 P N (x1,x2,...xN )= ( 1) ⇧1 '1(xi) sum p N! 1 X

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 25 Hartree-Fock energy

Coulomb Exchange

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 26 Hartree vs. Hartree-Fock

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 27 Hartree-Fock equations

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 28 Lagrange multiplier

max/min of f(x,y,z) subject to the constraint g(x,y,z)=k

Form F (x, y, z, )=f(x, y, z) (g(x, y, z) k) Solve Fx =0 Fy =0 Fz =0 F =0 Back to f. . .

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 29 Hartree-Fock equations

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 30 Hartree-Fock equations

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 31 Hartree-Fock equations

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 32 Hartree vs. Hartree-Fock

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

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 33 Simulations with ORCA

• Perform PES H2 dissociation at HF and DFT levels question

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

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 34 Problem: H2 minimal basis

• Find the HF energies of all the configurations • Are these configurations actual spin states? • Are those all real spin states? question

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 35 Spin operators (Extra)

• Demonstrate S and S2 = 0 for GS configuration question

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 36 Spin operators (Extra)

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 37 Spin operators (Extra)

For some details about spin projection

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 38 ORCA tool

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 39 ORCA tool: overview

• Tasks: SP, relaxation, PES, etc.

• Coordinates: cartesian and internal

• Spin/Charge state

• Methods: HF, DFT, post HF

• Basis sets

• Options

• Constraints

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 40 SCF, Relaxation, PES, etc.

N loops task

SCF 1 min E = <Ψ|H|Ψ> ()

Ionic relaxation min F = -∇E 2 (geometry optimization) min E

N-Constraint Potential energy surface (PES) 2 min E

N-constraint Relaxed potential energy surface 3 min F (PES) min E min Stress Ionic + cell relaxation 3 min F min E

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 41 ORCA tool: overview

• Tasks: SP, relaxation, PES, etc.

• Coordinates: cartesian and internal

• Spin/Charge state

• Methods: HF, DFT, post HF

• Basis sets

• Options

• Constraints

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 42 Cartesian vs. internal (or Z-matrix)

O O(1) H(1) O(2) O(3) H H H(2) H(3) H(4)

O 0.0 0.0 0.0 O(1) 0 0 0 0.0 0.0 0.0 H(1) 0 0 0 0.0 0.0 0.0 H x y 0.0 H(2) 1 0 0 0.9 0.0 0.0 O(2) 1 0 0 0.9 0.0 0.0 H -x y 0.0 H(3) 1 2 0 0.9 109.5 0.0 O(3) 1 2 0 0.8 120.0 0.0 H(4) 3 2 1 0.9 120.0 180.0 Cartesian Internal coordinates (or Z-matrix) coordinates

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 43 ORCA tool: Potential energy surface H2

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 44 ORCA tool: Potential energy surface H2

x a 0 x a ( a ) 0 E(x)= E e 0⇤ 1+ + E dis ⇥ a 0 ✓ 0 ◆

Edis

λ

a0

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 45 Electronic correlation

HF DFT

MP2 ‘exact’

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 46 The electronic correlation

HF Ecorr

‘exact’

• Correlation energy E = E E corr exact HF • HF fails at dissociation, bad for transition state and open shell

• Two type of electronic correlation: dynamical << static

What approximations have we made?

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 47 Dynamical correlation

' (r ) ' (r ) • Remark (3) e2 2 2 ⇤ 2 2 dr ' (r ) r r 2 ⇥ 1 1 Z | 1 2| • Electron-electron interaction Mean-field approximation!

average density 2 ⇢2 '2 of electron 2 interacting -e ⇠| | with electron 1

r2 HF R +2e mostly -e dynamical r1 corr. ‘exact’

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 48 The static correlation

For some details about spin projection

HF wave function (SD) fails at dissociation

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 49 Intuitive approach u (a b) ⇠

a b gg¯ uu¯ | | | | g (a + b) ⇠ • Let’s develop the these WF gg¯ aa¯ + b¯b + a¯b + ba¯ = + | | ⇠ | | | | | | | | I C uu¯ aa¯ + b¯b a¯b ba¯ = | | ⇠ | | | | | | | | I C gg¯ + c uu¯ CI ⇠ | | | | What would be a good value for c at the dissociation limit? ¯ • if c = 1: pure ionic CI aa¯ + bb ⇠ | | | | • if c = -1: pure covalent a¯b + ba¯ CI ⇠ | | | | Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 50 Configuration interaction

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 51 Configuration interaction

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: H2

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 56 Configuration interaction: H2

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 57 Configuration interaction: H2

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 58 Configuration interaction: H2

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 59 Complete active space: CASSCF

• CAS(n,m)

• n: number of electrons

• m: number of orbitals

HF CAS(3,3)

CAS(2,2)

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 60 Other methods

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

• Coupled clusters (CCSD,CCSDT,…)

Nicolas Onofrio - Atomistic View of Materials: Modeling & Simulation 61