1. Valence Bond Theory * Why doing VB? * Basic principles * Ab initio methods * Qualitative VB theory

2. The Valence Bond model electron pairs are in local bonds one bond = an interaction between two singly occupied atomic orbitals Each bond is mainly covalent, but has some minor ionic character The Valence bond wave function is the Quantum Mechanical translation of theLewis structure

3. Systems that cannot be described by a single Lewis structure: Molecules displaying electron conjugation The VB wave function  (1  2) for the ground state is a combination of two VB structures  1 and  2 :  (1  2) = C 1 (  1 ) + C 2 (  2 ) Estimation of C 1, C 2, resonance energy…

4. Why Doing Valence Bond and not only MO? Some vital concepts - hybridization - resonance - VSEPR - Lewis structures - Arrow-pushing language - IR frequencies of C-H bonds: sp 3 -H, sp 2 -H, sp-H Application to reactivity (the VBSCD model) Standard teaching: VB has failed The only correct theory is MO

5. Writing VB functions Exemple: the covalent H 2 bond Working hypothesis: the electrons remain in atomic orbitals At equilibrium distance, 2 possible déterminants :  a   b  b   a  Correct wave function (for the covalent bond) :  a   b  b   a  +  HL 

6. Writing VB functions beyond the two-electron/two center case Exemple: the π-system of butadiene  cov 

7. -20 -40 -60 -80 -100 E (kcal/mole) R H-H  a   b  Dissociation curve of H 2

8. -20 -40 -60 -80 -100 E (kcal/mole) R H-H  exact  a   b  b   a  +  HL  Dissociation curve of H 2  HL

9. Physical origin of the bond: spin exchange between AOs -20 -40 -60 -80 -100 E (kcal/mole) R H-H  exact  a   b  b   a  +  HL  Dissociation curve of H 2  HL

10. Comparaison with MO description (Hartree-Fock)  HF  =  g  g   =  a  b   b  a   a  a   b  b   H—HH – H + + H + H –  g  g   = 50% covalent 50% ionic  HL = 100% covalent Simple MO description Simple VB description

11. Comparaison with MO description (Hartree-Fock)  HF  =  g  g   =  a  b   b  a   a  a   b  b   H—HH – H + + H + H –  g  g   = 50% covalent 50% ionic  HL = 100% covalent 72-79% covalent 21-28% ionic Simple MO description Simple VB description Exact description

12. In the VB framework  exact  =   a  b   b  a     a  a   b  b   H—HH – H + + H + H –,  and the orbitals are optimized simultaneously: VBSCF method (Balint-Kurti, van Lenthe)

13. Exact description In the VB framework  exact  =   a  b   b  a     a  a   b  b    g  g   =  a  b   b  a     a  a   b  b   H—HH – H + + H + H – In the MO framework  u  u   =  a  b   b  a     a  a   b  b   C 1  g  g   C 2  u  u   =  exact

14. The Generalized Valence Bond Method (GVB)  a  b H a  a  b  GVB = +  b  a H b  a  a  b  b  b  a H a H b

15. The Generalized Valence Bond Method (GVB)  a  b H a  a  b  GVB = +  b  a H b  a  a  b  b  b  a H a H b  GVB =       a  b   b  a     a  a   b  b   H—HH – H + + H + H –  GVB is formally covalent, but physically covalent-ionic optimized

16. The Generalized Valence Bond Method (GVB)  a  b  GVB = +  b  a « GVB pair » Overlapping distorted AOs

17. The Generalized Valence Bond Method (GVB)  a  b  GVB = +  b  a « GVB pair » Overlapping distorted AOs  GVB = Four GVB pairs Generalization:

18. Hartree-Fock, GVB, CASSCF … and correlation energy CASSCF (1764 conf) SCF 80% Non-dynamical correlation energy GVB

19. Ab initio methods Test case: the bonding energy of F–F Nature of the wave function for F 2 : Hartree-Fock: too much ionic VBSCF: GVB: equivalent to VBSCF F—F F – F + F + F – Optimized covalent vs ionic coefficients

20. Accuracy of the various methods Test case: the dissociation of F 2 F–F F + F EE Calculation of ∆E for F-F=1.43Å, 6-31G(d) basis: Hartree-Fock: - 37 kcal/mol (repulsive!) Reason: too much ionic GVB, VBSCF Full configuration interaction (6-31G(d) basis) 30-33 kcal/mol Only 15.7 kcal/mol Reason: we miss dynamic correlation. What does this physically mean?

21. What is wrong with GVB and VBSCF? GVB/VBSCF: a closer examination F—F F – F + F + F – The coefficients and orbitals are optimized, but… - The same set of AOs is used for all VB structures: optimized for a mean neutral situation

22. What is wrong with GVB and VBSCF? GVB/VBSCF: a closer examination F—F F – F + F + F – The coefficients and orbitals are optimized, but… - The same set of AOs is used for all VB structures: optimized for a mean neutral situation A better wave function:

23. The « Breathing-Orbital » VB method (BOVB) Provides optimized covalent-ionic coefficients (like GVB) Different orbitals for different VB structures - Orbitals for F—F will be the same as VBSCF - Orbitals for ionic structures will be much improved F—FF – F + F + F – One expects - A better description of ionic structures - A better bonding energy

24. IterationDe(kcal)F–F F + F –  F – F + Classical VB - 4.60.8130.187 GVB,VBSCF 15.70.7680.232 BOVB 124.60.7310.269 227.90.7120.288 328.40.7090.291 428.50.7100.290 528.60.7070.293 Full CI 30-33 Test case: the dissociation of F 2 F–F F + F EE Calculation of ∆E for F-F=1.43Å, 6-31G(d) basis:

25. Improvements of the BOVB method Improvement of the ionic VB structures - basic level: The « active » orbital is split. This brings radial electron correlation - improved level (« split-level » or S)

26. Improvement of the interactions between spectator orbitals - local atomic orbitals - bonding and antibonding combinations Spectator orbitals can be: Slightly better (« Delocalized » level or D)

27. The three levels of the BOVB method Basic: L-BOVB - Active orbitals are split in ionics - Spectator orbitals are delocalized in all structures SL-BOVB All orbitals are localized, ionics are closed-shell All orbitals are localized, but active orbitals in ionics are split SD-BOVB

28. Performances of the various BOVB levels MethodReq (Å) De(kcal/mol) 6-31+G* basis set GVB1.50614.0 CASSCF1.49516.4 L-BOVB1.48527.9 SL-BOVB1.47331.4 SD-BOVB1.44933.9 Dunning-Huzinaga DZP basis set SD-BOVB1.44331.6 Estimated full CI 1.44±0.005 28-31 Experimental1.41238.3 Test case: the dissociation of F 2 F–F F + F EE

29. Electron correlation in BOVB Non-dynamic correlation (GVB, CASSCF) - Non dynamic correlation gives the correct ionic/covalent ratio

30. Electron correlation in BOVB Non-dynamic correlation (GVB, CASSCF) - Non dynamic correlation gives the correct ionic/covalent ratio Dynamic correlation - All the rest. This is what is missing in GVB. - BOVB brings that part of dynamic correlation that varies in the reaction

31. What is an accurate description of two-electron bonding? Spin exchange between two atomic orbitals - Electrons are on different atoms and they exchange their positions

32. What is an accurate description of two-electron bonding? Spin exchange between two atomic orbitals - Electrons are on different atoms and they exchange their positions Charge fluctuation - Sometimes both electrons are on the same atom. - There is some charge fluctuation. All orbitals instantaneously rearrange in size and shape to follow the charge fluctuation (orbitals « breathe »). This is differential dynamic correlation

33. A Qualitative valence bond theory Effective hamiltonian: H eff = (h (1) + h (2) + h (3) +….) Parameters: , S,  Similar model in the MO framework:

34. A Qualitative valence bond theory Overlap between determinants: Generate permutations… - between identical spins - only one side

35. A Qualitative valence bond theory Hamiltonian matrix elements: Choice of an origin of energies:

36. A Qualitative valence bond theory The two-electron bond: ab ab ba ba  +  VB  a,b = pure AO => purely covalent bond a,b = GVB pair => fully correlated bond Reminder: In MO theory: D e = 2  /(1+S)  VB   2ßS 1+ S 2  D e (2-e)

37. A Qualitative valence bond theory The triplet Pauli repulsion: ab ab ba ba  – T T  Reminder: MO theory: The same!  T   -2ßS 1- S 2

38. Triplet Pauli repulsion:why  MO =   VB ? The MO picture:  MO =  g  u   =  a+b)(a-b)   aa   ab   ba   bb    VB   a b  The VB picture: Whenever the MO and VB wave functions of an electronic state are equivalent, the VB energy can be estimated using qualitative MO theory

39. The three-electron bond Example: the helium cation dimer He 2 + The MO picture: The VB picture:  MO =  g  g  u   a  a  b   a  b  b    VB  a  a  b  a  b  b  Other cases where  MO =   VB ? Interaction energy: D e = E(He  He) + - E(He) - E(He + ) = ß(1-3S) 1- S 2

40. Elementary interaction energies in qualitative VB vs MO theories Triplet repulsion (A  B) = -2ßS 1- S 2 4-e repulsion (A  B) = VB MO One-electron bond (A  B) = Two-electron bond (A  B) = Three-electron bond (A  B) = -2ßS 1- S 2 -4ßS 1- S 2 -4ßS 1- S 2 ß(1-3S) 1- S 2 ß(1-3S) 1- S 2 2ßS 1+ S 2 2ß 1+ S ß ß

41. Thumb rule: VB energy of a single determinant Energy of a determinant with n (neighboring  -2nßS -2ßS/(1-S 2 )triplet repulsion spin-alternated determinant 4-e repulsion 3-e repulsion bond… single electron bond… bond -4ßS/(1-S 2 ) 0 -2ßS/(1-S 2 ) -ßS/(1-S 2 ) (VB and MO) (VB only) 1- S 2

42. The « failures » of valence bond theory 1) Dioxygen molecule: O=O or  O–O  ? O=O, singlet state  O–O , diradical displaying two 3-e bonds, triplet state

43. The « failures » of valence bond theory 1) Dioxygen molecule: O=O or  O–O  ? E(S) = E(T) E(S) - E(T) = Qualitative VB predicts O 2 to be a triplet diradical Pauling, 1931 (!)

44. 2) The two ionization potentials of CH 4 The MO model: Two valence MO energies => two IPs The VB model: Four equivalent local bonds => only one IP ? The « failures » of valence bond theory

45. 2) The two ionization potentials of CH 4 The « failures » of valence bond theory -E -  -  -  -  -E -  -  -  -  -E -  -  -  -  -E = 0 Qualitative VB predicts CH 4 to have two IPs

46. Conclusions VB and MO: two complementary theories. - lower level: MO too much ionic, VB not enough - elaborate level: MO  VB - There are nothing such as VB failures VB specific concepts: - Lewis structures, arrow pushing language, - transferable local bonds, hybridization, resonance energy Applications to chemical reactivity Valence bond state crossing diagrams (next lecture)

47. Some reading… Valence Bond Theory. Its History, Fundamentals, and Applications. A Primer. S. Shaik and P.C. Hiberty, Reviews in Computational Chemistry 20, 1-100 (2004) A conversation on VB vs MO Theory: A Never-Ending Rivalry? R. Hoffmann, S. Shaik and P.C. Hiberty, Acc. Chem. Res. 36, 750-756 (2003)