Benoît BRAÏDA* Label chimie théorique d Ile de France Valence Bond theory. Laboratoire de Chimie Théorique Sorbonne Université - CNRS

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1 Benoît BRAÏDA* Label chimie théorique d Ile de France 2019 Valence Bond theory Laboratoire de Chimie Théorique Sorbonne Université - CNRS * benoit.braida@sorbonne-universite.fr

2 VB lectures Part. 0 - Purpose and istory Part. 1 - Basics of VB theory and VB formalism Part. 2 - Ab initio VB Part. 3 - Application to 1,3-dipoles

3 VB references Book : Reviews articles :

4 Part 0. Motivation and history

5 euristic models Lewis model : - Lewis (1916) : electron pairing - Langmuir (1919) : octet rule

6 euristic models Lewis model : - Lewis (1916) : electron pairing - Langmuir (1919) : octet rule Mesomery / resonance : - Arndt, Robinson, Ingold (1924) : mesomery

7 euristic models Arrow-pushing language : describe the rearrangement of electrons during a reaction (mechanisms)

8 euristic models Chemist s models: ave shaped chemist s mind Now form the chemist s basic language Allow to organize and rationalize a incredibly large quantity of chemical facts

9 Quantum Chemistry Accurate quantum theory based calculations can provide : E 1 - accurate barriers E ; E 2 - details about reaction mechanisms

10 Quantum Chemistry Accurate quantum theory based calculations can provide : Nouvelle F Cl F + Cl F + Cl - complete exploration of the PES and reaction dynamics

11 Quantum Chemistry... but it does not (directly) provide : - human type comprehension of the computation outcome - general laws and trends over a family of compounds/reactions - description in terms of chemists local picture + + E = 22 kcal.mol -1 low barrier, easy? have to do the computation...

12 Chemists «schizophrenia» Concepts and heuristic models based on a localized vision : Quantitative theory based on a delocalized particles vision : C Lewis model, arrow-pushing language, VSEPR, hybridization,... - Localized electron pairs - Chemical bond concept - delocalized particles (e -, n + ) - indistinguishable and allinteracting (no chemical bond)

13 Chemists «schizophrenia» Chemical Question + Ψ rnx Quantum Calculations etc.

14 Chemists «schizophrenia» Chemical Question + Ψ rnx Quantum Calculations etc. «I am very glad the computer understands this. But I would like to understand it too» (Eugene Wigner)

15 Chemists «schizophrenia» Interpretative model Chemical Question + Ψ rnx Quantum Calculations etc. «I am very glad the computer understands this. But I would like to understand it too» (Eugene Wigner)

16 Chemists «schizophrenia» Chemical Question + ΨVB rnx Directly interpretative Quantum Calculations Structure weights Resonance energies etc. «I am very glad the computer understands this. But I would like to understand it too» (Eugene Wigner)

17 Birth and origins: G.N. Lewis L. Pauling VB: a quantum dressing of Lewis model

18 ~ s: Rise and glory L. Pauling VB dominated the mental map of chemistry

19 ~ : The M-VB rivalry Successes of M theory vs. VB «failures»

20 ~ : The downfall Sir John A. Pople M programs are developed, VB had nothing

21 ~ : small but active community New models, methods, programs, applications

22 2017- : awakening of the sleeping beauty? All elements for a Valence Bond revival are ready

23 : Two Nobel prices Rudolph A. Marcus ET theory Ariel Warshel EVB Developments of VB theory

24 Part 1. Basics of VB theory

25 eitler-london Notations : Dihydrogen molecule 2 : a a b b ab = (1, s 1 ) (2, s 2 ) a(1) α(s 1 ) b(1) β(s 1 ) a(2) α(s 2 ) b(2) β(s 2 ) 1 = (x 1, y 1,z 1 ) : spatial coordinates for electron 1 s 1 : spin coordinate for electron 1

26 eitler-london Notations : Dihydrogen molecule 2 : a b b bb = (1, s 1 ) (2, s 2 ) b(1) α(s 1 ) b(1) β(s 1 ) b(2) α(s 2 ) b(2) β(s 2 ) 1 = (x 1, y 1,z 1 ) : spatial coordinates for electron 1 s 1 : spin coordinate for electron 1

27 Stop me at any time!

28 eitler-london Dihydrogen molecule 2 : a b eitler-london (1927) : Ψ L = a b b a + - Electrons in atomic orbitals - Shared electron pair (covalent bond) basis of VB theory

29 eitler-london Dihydrogen molecule 2 : a b und-mulliken (1927) : eitler-london (1927) : σ u a b σ g a + b Ψ L = a b b a + - Electrons in atomic orbitals Ψ M = σ g σ g basis of M theory (F wave function) - Shared electron pair (covalent bond) basis of VB theory

30 eitler-london Ψ S = ab + 2(1+ S 2 ab ) ba [a(1)b(2) + a(2) b(1)] [α(s 1 ) β(s 2 ) α(s 2 ) β(s 1 )], antisymetric Singlet Ψ T = ab 2(1 S 2 ab ) ba [a(1)b(2) a(2) b(1)] [α(s 1 ) β(s 2 ) + α(s 2 ) β(s 1 )], symetric Triplet (MS=0) Note that : Ψ ab = [a(1)b(2) a(2) b(1)] [α(s 1 ) α(s 2 ) ] => MS = +1 T Ψ ab = [a(1)b(2) a(2) b(1)] [β(s 1 ) β(s 2 ) ] => MS = 1 T... all triplets have the same energy as Ĥ is spin-independent

31 eitler-london Exchanging two columns of a determinants change sign: Ψ S ab + ba = ab a b Ψ T ab ba = ab + a b

32 eitler-london E (kcal/mole) Quasi-Classical (QC) state : Ψ QC = a b R E QC = él ab ab ab ab = h aa + h bb + J ab E QC ~ cte along the dissociation curve no spin exchange no bonding -100 Ψ exact

33 eitler-london E (kcal/mole) eitler-london (F) wf : Ψ L = ab + ba E L = (h aa + h bb + J! #" ## $ + 2h ab abs ab + K! #" ## $ ) ab E QC <0 R verlap (distance) dependant Ψ L Ψ exact Physical origin of the chemical bond : spin exchange between As

34 eitler-london E(T ) = 1 (1 S 2 ab ) (h + h + J 2h S K ) aa bb ab ab ab ab E QC E(R ) >0 S T gap 2De E(S) = 1 (1+ S 2 ab ) (h aa + h bb + J ab + 2h ab S ab + K ab ) E QC E(R ) <0

35 The VB wave function ow to improve upon the L wave function? Linus Pauling (1931) : Covalent + ionic supperposition: Ψ VB : quantum dressing of Lewis model

36 The VB wave function E (kcal/mole) ow to improve upon the L wave function? Linus Pauling (1931) : Ψ VB = c 1 ( ab + ba ) + c2( aa + bb ) R Covalent + ionic superposition Ψ L Ψ VB Ψ VB : quantum dressing of Lewis model -100 Ψ «exact»

37 VB vs. M Difference between eitler-london and artree-fock? a b σ u = σ g = 1 (a b) = 2(1+ S) 1 (a + b) = 2(1+ S) Ψ F = σ g σ g

38 VB vs. M Difference between eitler-london and artree-fock? a b σ u = σ g = 1 (a b) = 2(1+ S) 1 (a + b) = 2(1+ S) Ψ F = σ g σ g = ab + b a + aa + bb

39 VB vs. M Difference between eitler-london and artree-fock? a b σ u = σ g = 1 (a b) = 2(1+ S) 1 (a + b) = 2(1+ S) Ψ F = σ g σ g = ab + b a + aa + bb % covalent 50% ionic

40 VB vs. M E (kcal/mole) Difference between eitler-london and artree-fock? 50% covalent 50% ionic a b σ u = σ g = 1 (a b) = 2(1+ S) 1 (a + b) = 2(1+ S) R Ψ F = σ g σ g ΨF = ab + b a + aa + bb % covalent ΨL 50% covalent 50% ionic -100 Ψ exact

41 VB vs. M E (kcal/mole) Difference between eitler-london and artree-fock? 50% covalent 50% ionic wrong dissociation limit R ΨL ΨF 100% covalent ptimal 80% covalent 20% ionic -100 Ψ exact

42 Writing VB wave functions Extension to the general case : Lewis : C - We want to construct a VB w.f. which corresponds to Lewis picture - Which orbitals to use?

43 Writing VB wave functions Extension to the general case / 1) general localized orbital: Lewis : C - We want to construct a VB w.f. which corresponds to Lewis picture - Which orbitals to use? Atomic rbitals ybrid rbitals 2p 2s Unitary transformation Four equivalent directional sp 3 orbitals c i

44 Writing VB wave functions Extension to the general case / 1) general localized orbital: Lewis : C - We want to construct a VB w.f. which corresponds to Lewis picture - Which orbitals to use? L : h 2 h 1 c 2 c 1 c 4 c h 3 4 h 3 Ψ L = (c 1 h 1 + h 1 c 1 )(c 2 h 2 + h 2 c 2 )(c 3 h 3 + h 3 c 3 )(c 4 h 4 + h 4 c 4 ) Electrons occupy localized orbitals (atomics, hybrids,...) A bond = two singlet-coupled electrons in two orbitals (+minor ionics)

45 Writing VB wave functions Extension to the general case / 2) active electrons/orbitals: Not all electrons are treated at the VB level : inactive/active separation

46 Writing VB wave functions Extension to the general case / 2) active electrons/orbitals: Not all electrons are treated at the VB level : inactive/active separation

47 Writing VB wave functions Extension to the general case / 2) active electrons/orbitals: Not all electrons are treated at the VB level : inactive/active separation Example : SN2 Transition state : a 4-e/3-orbital VB system a b c Φ 1 = σ 1 σ 2...σ 9 (ab + ba)cc - an active space of electrons/orbitals treated at the VB level - the rest (called inactive or «spectators») at the M level

48 Writing VB wave functions Extension to the general case / 3) multi-structure: p C N C N pc pn 1 2 Ψ(1 2) = C 1 (Ψ 1 ) + C 2 (Ψ 2 )

49 Writing VB wave functions Extension to the general case / 3) multi-structure: p C N C N pc pn 1 2 Ψ(1 2) = C 1 (Ψ 1 ) + C 2 (Ψ 2 ) = C 1 p N p N (p p C + p C p ) + C 2 p p (p C p N + p N p C ) VB wave function : two resonating components, each one corresponding to one of the 2 structures

50 Writing VB wave functions Exercise 1: Allyl cation We want to study the electronic structure of the allyl cation. 1. ow many active electron and orbitals do we have to consider? 2. Write a complete basis of Lewis structure for this problem. 3. Write the mathematical expression of the corresponding VB structures

51 Writing VB wave functions Exercise 1: Allyl cation 1. ow many active electron and orbitals do we have to consider? p a p b p c 2 " electrons in 3 active (") orbitals

52 Writing VB wave functions Exercise 1: Allyl cation p a p b p c 2. Write a complete basis of Lewis structure for this problem. 3. Write the mathematical expression of the corresponding VB structures

53 Writing VB wave functions Exercise 1: Allyl cation p a p b p c 2. Write a complete basis of Lewis structure for this problem. 3. Write the mathematical expression of the corresponding VB structures

54 Writing VB wave functions Exercise 1: Allyl cation p a p b p c 2. Write a complete basis of Lewis structure for this problem. 3. Write the mathematical expression of the corresponding VB struct.

55 Writing VB wave functions Exercise 1: Allyl cation p a p b p c 2. Write a complete basis of Lewis structure for this problem. 3. Write the mathematical expression of the corresponding VB struct. p a p b + p b p a

56 Writing VB wave functions Exercise 1: Allyl cation p a p b p c 2. Write a complete basis of Lewis structure for this problem. 3. Write the mathematical expression of the corresponding VB struct. p a p b + p b p a p b p c + p c p b p a p c + p c p a p a p a p c p c p b p b

57 Writing VB wave functions Exercise 2: SN2 reaction X: + (A Y) X A Y (X A) + :Y We want to study the SN2 reaction using VB theory. 1. ow many active electron and orbitals do we have to consider? 2. Write a complete basis of Lewis structure for this problem. 3. Write the mathematical expression of the corresponding VB structures 4. What structures describe the reactant electronic structure? The product electronic structure? 5. What will be the major structure(s) at the transition state geometry, for the SN2 reaction on the carbon? n the silicium?

58 Writing VB wave functions Exercise 2: SN2 reaction X: + (A Y) X A Y (X A) + :Y It is a 4e in 3 orbitals problem:

59 Writing VB wave functions Exercise 2: SN2 reaction X: + (A Y) X A Y (X A) + :Y It is a 4e in 3 orbitals problem: X A Y X A Y X A Y 1 3 2

60 Writing VB wave functions Exercise 2: SN2 reaction X: + (A Y) X A Y (X A) + :Y It is a 4e in 3 orbitals problem: X A Y X A Y X A Y X A Y X A Y X A Y 6 4 5

61 Writing VB wave functions Exercise 2: SN2 reaction X: + (A Y) X A Y (X A) + :Y X A Y X A Y X A Y 6 It is a 4e in 3 orbitals problem: xx(ay + ay) aa(xy + xy) yy(xa + ax) X A Y X A Y X A Y 4 5 xxaa xxyy aayy

62 Writing VB wave functions Exercise 2: SN2 reaction X: + (A Y) X A Y (X A) + :Y X A Y X A Y X A Y xx(ay + ay) aa(xy + xy) yy(xa + ax) X A Y X A Y X A Y 4 5 xxaa xxyy aayy

63 Writing VB wave functions Exercise 2: SN2 reaction X: + (A Y) X A Y (X A) + :Y X A Y X A Y X A Y xx(ay + ay) aa(xy + xy) yy(xa + ax) X A Y X A Y X A Y 4 5 xxaa xxyy aayy

64 Writing VB wave functions Exercise 2: SN2 reaction

65 Writing VB wave functions Exercise 3 : lone pairs of water Are the lone pairs of water best described by a " and a sigma lone pairs, of two equivalent rabbit-ears lone pairs?

66 Writing VB wave functions Exercise 3 : lone pairs of water

67 VB vs. M Exercise 3 (conclusion) : n + λp p n - λp n So why do we get two different images? one if more handy than the other depending on what you want to look for!

68 VB vs. M Exercise 3 (extra) : The «rabbit-ear» image is a faithful representation of the electron density in the isolated molecule : max ρ n-p max ρ n+p sp 3

69 VB vs. M Exercise 3 (extra) : The M image more readily account the two different IP in the water molecule : p n IP2 IP1 Canonical Ms allows to account to what will happen to a molecule when a perturbation is applied (Koopman s theorem, F theory)

70 VB vs. M Exercise 3 (extra) : Can VB predict two IPs for the water molecule? Ψ VB1 = Ψ VB2 Ψ VB2 = + Ψ VB1 = + lasy reasoning (very common!) : IP Ψ VB

71 VB vs. M Exercise 3 (extra) : Correct reasoning : Ψ VB2 = Ψ VB2 + + Ψ VB1 IP 1 IP 2 Ψ VB1 = + + Ψ VB

72 Part 2. Ab initio Valence Bond

73 VB methods including correlation The VBSCF* method : Basically a MCSCF method with nonorthogonal orbitals : Ψ VBSCF = K C K Φ K with : N Φ K (1,..., N ) = Â{ ϕ (1) Θ i K } : VB structures i=1 {ϕ i } : set of non-orthogonal localized orbitals expanded onto a set of basis functions χ m Θ K spin function { } : ϕ i (1) = d i m χ m (1) m *van Lenthe; Balint-Kurti, J. Chem. Phys. 1983, 78, 5699

74 VB methods including correlation The VBSCF* method : Basically a MCSCF method with nonorthogonal orbitals : Ψ VBSCF = K C K Φ K with : N Φ K (1,..., N ) = Â{ ϕ (1) Θ i K } : VB structures i=1 {ϕ i } : set of non-orthogonal localized orbitals expanded onto a set of basis functions χ m Θ K spin function { } : ϕ i (1) = d i m χ m (1) m All w.f. parameters : structure coef. {C K } and orb. coef. {d i m} are optimized simultaneously minimizing Ψ VBSCF Ĥ Ψ VBSCF *van Lenthe; Balint-Kurti, J. Chem. Phys. 1983, 78, 5699

75 VB methods including correlation The VBSCF method : Ψ VBSCF = C 1 Φ cov + C 2 Φ ion1 + C 2 Φ ion2

76 VB methods including correlation The VBSCF method : F : 50% 25% 25% L : 100% 0% 0% VBSCF : 80% 10% 10% The VBSCF method ensures a correct balance beween covalent and ionic configurations («left-right» static correlation)

77 VB methods including correlation The VBSCF method : Accuracy : E (kcal.mol -1 ) RF 37 VBSCF +15 Too ionic Why??? Exact +39 Some important physical ingredient is missing...

78 VB methods including correlation The VBSCF method : What the VBSCF method does : Same set of orbitals for all VB structures : optimized for a mean situation

79 VB methods including correlation The VBSCF method : What the VBSCF method does : Same set of orbitals for all VB structures : optimized for a mean situation A better wave function would be : Each structure has its own specific set of orbitals optimized for each situation

80 VB methods including correlation The BVB* method : BVB uses : same number of structures as VBSCF, but different orbitals for the different structures : K {ϕ Ψ BVB = C K Φ K i } K with: N Φ BVB K (1,..., N ) = Â{ ϕ K (1) Θ i K } : VB structures i=1 {ϕ K } : set of non-orthogonal (del)localized orbitals for the structure K i expanded onto a set of basis functions { χ m } : ϕ K (1) = d i,k χ i m m (1 m Θ K spin function All w.f. parameters : structure coef. {C K } and orb. coef. sets {d i m} K are optimized simultaneously minimizing Ψ BVB Ĥ Ψ BVB * iberty, P. C. ; umbel, S. ; Byrman, C. P. ; van Lenthe J.. J. Chem. Phys. 1994, 101, 5969

81 VB methods including correlation The VBSCF method : Ψ VBSCF = C 1 Φ cov + C 2 Φ ion1 + C 2 Φ ion2 Same orbital set for the different structures

82 VB methods including correlation The BVB method : {ϕ Ψ BVB = C 1 Φ cov i } {ϕ cov + C 2 Φ ion1 i } {ϕ ion1 + C 2 Φ ion2 i } ion2 Different orbitals for the different structures

83 VB methods including correlation The BVB method : Accuracy : E (kcal.mol -1 ) RF 37 VBSCF 15 L-BVB 28.2 SD-BVB 33.6 Exact +39 Basis set incompleteness

84 VB methods including correlation The VBSCF method The BVB method The VBCI method

85 VB computations in practice What do you get out of the calculation?

86 VB computations in practice (good) numbers...

87 VB computations in practice...but also insight!

88 zone and other 1,3 dipoles Let s start with a question freshman chemistry student are asked for: What is the electronic structure of zone?

89 zone and other 1,3 dipoles Let s start with a question freshman chemistry student are asked for: What is the electronic structure of zone? Reference chemistry textbooks (Klein): Same description in other reference organic chemistry textbooks (Vollhardt, Carey-Sundberg, ) and in wikipedia

90 zone and other 1,3 dipoles Contrasted reactivity of 1,3 dipoles: zone: cleaves multiple CC bonds very strong oxydent (E =+2,07V) addition of on terminal atoms: very exothermic Sulfur dioxide: unreactive toward mult. CC bonds weak oxydent (E =+0,50V) addition of on terminal atoms: slightly exothermic Do they really have the same electronic structure?

91 VB description of 1,3 dipoles The " system is active (and σ inactive) p 1 p 2 p 3 this is a 4e in 3 orb. case A complete basis of features 6 VB structures: D Z 1 Z 2 All kept in the calculations Z 3 Z 4 MI

92 VB description of 1,3 dipoles The " system is active (and σ inactive) p 1 p 2 p 3 this is a 4e in 3 orb. case A complete basis of features 6 VB structures: Major D Z 1 Z 2 Minor Z 3 Z 4 MI Neglected in the discussion

93 VB description of 1,3 dipoles The " system is active (and σ inactive) p 1 p 2 p 3 this is a 4e in 3 orb. case A complete basis of features 6 VB structures: Major D Z 1 Z 2 ψ BVB = c D Φ D + c Z1 Φ Z1 + c Z2 Φ Z Z 3 Z 4 MI

94 VB description of 1,3 dipoles Quantifying with the VB weights: ψ BVB = c D Φ D + c Z1 Φ Z1 + c Z2 Φ Z with: Φ i the VB structures Chirgwin-Coulson weights Norbek-Gallup ( inverse ) weights CC with: M KL the overlaps between structures K and L

95 VB description of 1,3 dipoles Quantifying with the VB weights (in %): (BVB/cc-pVTZ) S S S A B C Major 44.4% 56.1% 61.0% A B C 24.4% 17.7% 19.5% A B C 24.4% 26.2% 19.5%

96 VB description of 1,3 dipoles Quantifying with the VB weights (in %): (BVB/cc-pVTZ) S S S S S S A B C 15.9 (15.0) 22.7 (25.7) 34.6 (37.7) A A B B C C } 35.3 (38.4) 30.1 (35.7) 28.5 (29.0) Major 35.3 (38.4) 31.6 (28.7) 28.5 (29.0)

97 VB description of 1,3 dipoles S S S S B 50% diradical A B C A B C A C B. Braida, S. E. Galembeck, P. C. iberty JCTC, 2017

98 VB computations in practice Exemple 2 : Resonance energy / «diabatic states» It is possible to compute a VB w.f. which does not correspond to a real quantum state : a single structure of a subset of structures

99 VB computations in practice Exemple 2 : Resonance energy / «diabatic states» It is possible to compute a VB w.f. which does not correspond to a real quantum state : a single structure of a subset of structures computation of Resonance Energies (R.E.) : C N C N 1 2 Ψ(1 2) = C 1 (Ψ 1 ) + C 2 (Ψ 2 )

100 VB computations in practice Exemple 2 : Resonance energy / «diabatic states» It is possible to compute a VB w.f. which does not correspond to a real quantum state : a single structure of a subset of structures computation of Resonance Energies (R.E.) : 1) ptimize Ψ(1 2) C N C N 2) ptimize Ψ 1 separately 3) R.E. = E(Ψ 1 ) E(Ψ (1 2) ) 1 2 Ψ(1 2) = C 1 (Ψ 1 ) + C 2 (Ψ 2 )

101 VB computations in practice Exemple 2 : Resonance energy / «diabatic states» It is possible to compute a VB w.f. which does not correspond to a real quantum state : a single structure of a subset of structures computation of Resonance Energies (R.E.) : 1) ptimize Ψ(1 2) C N C N 2) ptimize Ψ 1 separately 3) R.E. = E(Ψ 1 ) E(Ψ (1 2) ) 1 2 Ψ(1 2) = C 1 (Ψ 1 ) + C 2 (Ψ 2 ) ~30 kcal/mol! => no free rotation

102 VB computations in practice Chemical insight / «diabatic states» : It is possible to compute a VB w.f. which does not correspond to a real quantum state : a single structure of a subset of structures Valence Bond diagrams (Shaik and Pross) for reactivity : Part 4. lecture

103 VB theory VB theory : provides a wave-function ansatz which enables to compute high level wf that are quantum dressing of Lewis model ; retrieves fundamental chemical concepts, such as : resonance/mesomery, hybridization, arrow-pushing language, and provides a theoretical support for them ; incorporates interpretative tools which are both directly connected to quantum mechanics and to the local vision of chemists (VB weights, resonance energies, VB diagrams)

104 To go further...