Hartree, Hartree-Fock and post-hf methods

Transcription

1 Hartree, Hartree-Fock and post-hf methods MSE697 fall 2015 Nicolas Onofrio School of Materials Engineering DLR 428 Purdue University 1

2 The curse of dimensionality Let s consider a multi electron WF (x 1,x 2,x 3,...x N ) We want to solve the Schrödinger equation Ĥ =E E = h Z Ĥ i E = (x 1,x 2,x 3,...x N )Ĥ (x 1,x 2,x 3,...x N )d 3N x 100 Hydrogen: 1e: = 10 6 op Silicon: 14e: 100 3x14 = op SC: ~PFLOPS = op/s H: 10 6 /10 15 ~ 1ns Si: /10 15 ~10 69 s ~ years!!! Marcoscale ~ electrons... 2

3 Helium: Hartree approximation -e r2 R +2e r1 -e Let s define the WF as a product of orbitals (r 1,r 2 )=' 1 (r 1 )' 2 (r 2 ) We want to solve the Schrödinger equation H =E H = ~2 ~ 2 2e 2 2e 2 2m r2 1 2m r2 2 R r 1 R r 2 + e 2 r 1 r 2 3

4 Helium: Hartree approximation apple We replace the WF by the Hartree product in the Schrödinger equation ~ 2 2m r2 1 ~ 2 2m r2 2 2e 2 R r 1 2e 2 R r 2 + e 2 r 1 r 2 We multiply and integrate ~ 2 2m r2 1 ~ 2 Z ' 2 (r 2 ) r 2 2m 2' 2 (r 2 )dr 2 {z } C 1 2e 2 R r 1 Z Z 2e 2 '2 (r 2 ) ' 2 (r 2 ) dr 2 R r 2 {z } C 2 ' 1 (r 1 )' 2 (r 2 )=E' 1 (r 1 )' 2 (r 2 ) ' 2(r 2 )dr 2 +e 2 Z '2 (r 2 ) ' 2 (r 2 ) r 1 r 2 C1 and C2 are constants and do not act on ' 1 (r 1 ) apple ~ 2 2m r2 1 2e 2 R r 1 + e2 E 0 = E C 1 C 2 3 dr ' 1(r 1 ) = E' 1 (r 1 ) Z '2 (r 2 ) ' 2 (r 2 ) dr 2 ' 1 (r 1 )=E 0 ' 1 (r 1 ) r 1 r 2 4

5 Helium: Hartree approximation Remark (1) The starting point was: H(r 1,r 2 ) (r 1,r 2 )=E (r 1,r 2 ) dimension: n3d (+spin..) = 2x3 = 6 (8 with spin) We end-up with equations of the form: { ˆf 1 (r 1 )' 1 (r 1 )=E 0 ' 1 (r 1 ) ˆf 2 (r 2 )' 2 (r 2 )=E 00 ' 2 (r 2 ) dimension: n3d (+spin..) = 1x3 = 3 (4 with spin) single-electron equations! but no free lunch 5

6 Helium: Hartree approximation Remark (2) The operator depends on the function we are ˆf looking for the solutions 1 (r 1,' 2 ) SCF: self-consistent field iterative procedure See for example in ORCA: 6

7 Helium: Hartree approximation Remark (3) e 2 Z '2 (r 2 ) ' 2 (r 2 ) r 1 r 2 dr 2 ' 1 (r 1 ) Electron-electron interaction Mean-field approximation! -e 2 ' 2 2 r2 R +2e r1 -e average density of electron 2 interacting with electron 1 7

8 Helium: Hartree approximation Remark (4) Probability density: Z dp 1 = (r 1,r 1,...,r N ) 2 dr 2 dr 3...dr N Probability of finding electron 1 in dr1 Considering the Hartree product (r 1,r 2,...,r N )=' 1 (r 1 )' 2 (r 2 )...' N (r N ) question What is dp 1 for the Hartree product? What is the probability dp 12 of finding electron 1 in dr 1 and electron 2 in dr 2? 8

9 Helium: Hartree approximation Remark (4) Probability of finding electron 1 in dr1 dp 1 = ' 1 (r 1 ) 2 Z ' 2 (r 2 ) 2 dr 2 Z ' 3 (r 3 ) 2 dr 3... Z ' N (r N ) 2 dr N dp 1 = ' 1 (r 1 ) 2 Probability of finding electron 1 in dr1 and electron 2 in dr2 Z dp 12 = (r 1,r 2,...,r N ) 2 dr 3...dr N dp 12 = ' 1 (r 1 ) 2 ' 2 (r 2 ) 2 = dp 1 dp 2 Electrons are uncorrelated + do not respect Pauli! (remember oxygen singlet/triplet) 9

10 Hartree product: generalization 10

11 Hartree product: generalization 11

12 Hartree product: generalization 12

13 Helium: Hartree approximation Energy E = h H i = Z H dr question Hamiltonian H = ~2 2m r2 1 What is the energy for Helium considering the Hartree WF? ~ 2 2m r2 2 2e 2 R r 1 simplifications: ĥ 1 (r 1 )= ~2 2e 2 2m r2 1 R r 1 ĥ 2 (r 2 )... ĝ 12 (r 1,r 2 )... 2e 2 R r 2 + e 2 r 1 r 2 13

14 Helium: Hartree approximation 14

15 Helium: Hartree approximation 15

16 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 Fermions AS 1/2 integer electrons, protons, etc. Bosons S integer phonons 16

17 Quantum character of the WF Anti-symmetric WF { (r 1,r 2 )= (r 2,r 1 ) (r, r) =0 Pauli exclusion! Back to Hartree WF (r 1,r 2 )=' 1 (r 1 )' 2 (r 2 ) Pauli AS 17

18 The Slater determinant Antisymmetric WF Can t distinguish between electrons Antisymmetric (swap 2 particle change total sign) Same spin and position P = 0 question Demonstrate the antisymmetry for 2 electrons 18

19 Overview of the lectures Hartree-Fock Energy & equations Application to H2 Energy & Wave function Simulations with ORCA HF limitations Post Hartree-Fock methods 19

20 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 20

21 Hartree-Fock energy 21

22 Hartree-Fock energy Helium 22

23 Hartree-Fock energy Helium 23

24 Exchange integral 24

25 Hartree-Fock energy (x 1,x 2,...x N )= p 1 XN! ( 1) P N 1 ' 1 (x i ) N! sum 1 25

26 Hartree-Fock energy Coulomb Exchange 26

27 Hartree vs. Hartree-Fock 27

28 Hartree-Fock equations 28

29 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 F x =0 F y =0 F z =0 F =0 Back to f... 29

30 Hartree-Fock equations 30

31 Hartree-Fock equations 31

32 Hartree-Fock equations 32

33 Hartree vs. Hartree-Fock Mean field approximation Spin correlation: exchange K SCF 33

34 Simulations with ORCA question Perform PES H2 dissociation at HF and DFT levels 34

35 Problem: H 2 minimal basis question Find the HF energies of all the configurations Are these configurations actual spin states? Are those all real spin states? 35

36 Spin operators (Extra) question Demonstrate S and S 2 = 0 for GS configuration 36

37 Spin operators (Extra) 37

38 Spin operators (Extra) For some details about spin projection 38

39 ORCA tool 39

40 ORCA tool: overview Tasks: SP, relaxation, PES, etc. Coordinates: cartesian and internal Spin/Charge state Methods: HF, DFT, post HF Basis sets Options Constraints 40

41 SCF, Relaxation, PES, etc. 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 Relaxed potential energy surface (PES) Ionic + cell relaxation 3 3 N-Constraint min E N-constraint min F min E min Stress min F min E 41

42 ORCA tool: overview Tasks: SP, relaxation, PES, etc. Coordinates: cartesian and internal Spin/Charge state Methods: HF, DFT, post HF Basis sets Options Constraints 42

43 Cartesian vs. internal (or Z-matrix) O O(1) H(1) O(2) O(3) H H H(2) H(3) H(4) O H x y 0.0 H -x y 0.0 O(1) H(2) H(3) H(1) O(2) O(3) H(4) Cartesian coordinates Internal coordinates (or Z-matrix) 43

44 ORCA tool: Potential energy surface H2 44

45 ORCA tool: Potential energy surface H2 E(x) = E dis e ( x a 0 a 0 ) 1+ x a 0 a 0 + E 0 Edis λ a0 45

46 Electronic correlation HF DFT MP2 exact 46

47 The electronic correlation HF Ecorr exact Correlation energy E corr = E exact E HF HF fails at dissociation, bad for transition state and open shell Two type of electronic correlation: dynamical << static What approximations have we made? 47

48 Dynamical correlation Remark (3) e 2 Z '2 (r 2 ) ' 2 (r 2 ) r 1 r 2 dr 2 ' 1 (r 1 ) Electron-electron interaction Mean-field approximation! average density of electron 2 interacting with electron 1 -e 2 ' 2 2 r2 R +2e HF r1 -e mostly dynamical corr. exact 48

49 The static correlation For some details about spin projection HF wave function (SD) fails at dissociation 49

50 Intuitive approach u (a b) a b g (a + b) gḡ uū Let s develop the these WF gḡ aā + b b + a b + bā = I + C uū aā + b b a b bā = I C CI gḡ + c uū What would be a good value for c at the dissociation limit? if c = 1: pure ionic if c = -1: pure covalent CI aā + b b CI a b + bā 50

51 Configuration interaction 51

52 Configuration interaction 52

53 Configuration interaction 53

54 Configuration interaction 54

55 Configuration interaction 55

56 Configuration interaction: H2 56

57 Configuration interaction: H2 57

58 Configuration interaction: H2 58

59 Configuration interaction: H2 59

60 Complete active space: CASSCF CAS(n,m) n: number of electrons m: number of orbitals HF CAS(3,3) CAS(2,2) 60

61 Other methods Perturbation theory (Moller-Plesset or MP2,MP4, ) Coupled clusters (CCSD,CCSDT, ) 61