1. 1 Electronic correlation VB method and polyelectronic functions IC DFT The charge or spin interaction between 2 electrons is sensitive to the real relative position of the electrons that is not described using an average distribution. A large part of the correlation is then not available at the HF level. One has to use polyelectronic functions (VB method), post Hartree-Fock methods (CI) or estimation of the correlation contribution, DFT.

2. 2 Electronic correlation A of the correlation refers to HF: it is the “missing energy” for SCF convergence: E corr = E – E SCF E corr < 0 ( variational principle) E corr ~ -(N-1) eV

3. 3 Importance of correlation effects on Energy RHF in blue, Exact in red. Energy (kcal/mol)

4. 4 Importance of correlation effects on reactions − Bond cleavage: H 2 O → HO + H : ~ 130 kcal/mol − same number of electron pairs : H 2 O + H + → H 3 O + : ~ 7 kcal/mol − intermediate case: 2 BH 3 → B 2 H : ~ 70 kcal/mol − weak interactions (H-bonds, van der Waals) H 2 O - H 2 O: ~ 3 kcal/mol Ne−Ne: ~ 0.5 (kcal/mol however De is only ~ 0.3 kcal/mol) − ions are strongly correlated systems F - : 246.5 kcal/mol Al 3+ : 253.66 kcal/mol

5. 5 Importance of correlation effects on distances RHF in blue, Exact in red. Distances pm

6. 6 Relative errors in % Relative errors in ppm Weak contribution: 0.1 Ả, 5%

7. 7 Fermi hole – Coulomb hole Each electron should be surrounded by an “empty volume” excluding the presence of another electron – of the same spin (Fermi hole) –of opposite spins (Coulomb hole) In HF, the Probability of finding an electron with the same spin at the same place is 0; that of finding an electron with  spin is not! P(↑↑) 2 =dV 1 dV 2 /2 [a 2 (r 1 )b 2 (r 2 )+a 2 (r 2 )b 2 (r 1 )-2a(r 1 )b(r 2 )a(r 2 )b(r 1 )] P(↑↑) 2 = 0 for r 1 = r 2 P(↑↓) 2 =dV 1 dV 2 /2 [a 2 (r 1 )b 2 (r 2 )+a 2 (r 2 )b 2 (r 1 )] P(↑↓²) 2  0 for r 1 = r 2 HF account for the Fermi hole not the Coulomb hole! This term vanishes for ↓

8. 8 Correlation Left-Right The probability of finding 1 electron in the left region and one in the right is 1. In HF it is only ½; Taking into account correlation reduces the probability of finding the electrons together; it decreases the weight of the ionic contribution.

9. 9 types of correlation How electrons avoid each other? Left-right radial (or In-out) angular

10. 10 Valence Bond

11. 11 Heitler-London 1927 Walter Heinrich Heitler German 1904 –1981 Fritz Wolfgang London German 1900–1954          Electrons are indiscernible: If      is a valid solution      also is. The polyelectronic function is therefore: To satisfy Pauli principle this symmetric expression is associated with an antisymmetric spin function: this represents a singlet state! Each atomic orbital is occupied by one electron: for a bond, this represents a covalent bonding. 

12. 12 The total function has to be antisymmetric; for a triplet state since the spin is symmetric, the spatial function has to be antisymmetric.          Associated with one of the 3 spin functions  The triplet state 

13. 13 The resonance, ionic functions Other polyelectronic functions :                             According to Pauli principle, these functions necessarily correspond to the singlet state:  H 1 - -H 2 + H 1 + -H 2 - H 1 - -H 2 + ↔ H 1 + -H 2 - symmetric antisymmetric

14. 14 H + + H - E = 2 √(1-  )3/p = -0.472 a.u. with  =0.31 H 2 dissociation The cleavage is whether homolytic, H 2 → H + H or heterolytic: H 2 → H + + H - H + H E = -1 a.u;

15. 15 MO behavior of H 2 dissociation Energy A-B distance The cleavage is whether homolytic, H 2 → H + H or heterolytic: H 2 → H + + H -  g 2 = (  A +  B ) 2 = [(  A (1)  A (2) +  B (1)  B (2)] + [(  A (1)  B (2) +  B (1)  A (2)] 50% ionic + 50% covalent The MO description fails to describe correctly the dissociation!  u 2 = (  A -  B ) 2 = [(  A (1)  A (2) +  B (1)  B (2)] - [(  A (1)  B (2) +  B (1)  A (2)] 50% ionic - 50% covalent

16. 16 H + + H - E = 2 √(1-  )3/p = -0.472 a.u. with  =0.31 H 2 dissociation The cleavage is whether homolytic, H 2 → H + H or heterolytic: H 2 → H + + H - H + H E = -1 a.u; The MO approach, -0.769 a.u. *

17. 17 E= -0.769 a.u. What are the solutions? 1.Uncouple ionic and covalent functions; this is the VB approach. 2.Let interact the  g 2 and  u 2 states: these are of the same symmetry and interact. This is the IC approach.  g 2 and  u 2 E= -0.472 a.u. E= -1 a.u. ionic covalent

18. 18 The Valence-Bond method It consists in describing electronic states of a molecule from AOs by eigenfunctions of S 2, S z and symmetry operators. There behavior for dissociation is then correct. These functions are polyelectronic. To satisfy the Pauli principle, functions are determinants or linear combinations of determinants build from spinorbitals.

19. 19 Covalent function for electron pairs a (1) b(1) IabI = a (2) b(2)  = IabI + IbaI  = IabI - IabI Ordered on electrons Ordered on spins or  = [a(1).b(2) +a(2). b(1)].[  (1).  (2) -  (2).  (1)]  = [a(1).  (1)b (2)  (2) - b(1).  (1). a(2).  (2)] + [b(1).  (1) a(2)  (2) - a(1).  (1).b (2).  (2)] This is the Heitler-London expression  = IabI + IbaI

20. 20 Triplet states  = IabI + IbaI  = IabI - IabI Ordered on electrons Ordered on spins or  = [a(1).b(2) -a(2). b(1)].[  (1).  (2) +  (2).  (1)]  = [a(1).  (1)b (2)  (2) - b(1).  (1). a(2).  (2)] - [b(1).  (1) a(2)  (2) - a(1).  (1).b (2).  (2)]  = IabI - IbaI + -  = IabI

21. 21 2 electrons – 2 orbitals J ij -K ij K ij J ij │ ab │ │ ab │- │ ba │ │aa│- │ bb │J ij +K ij │ aa │ │ bb │ Ground state First excited States Diexcited States +│ bb │+ │ aa │+ │ ab │+ │ ba │ │ ab │+ │ ba │-

22. 22 Valence Bond Make the list of all the resonance structures. (a complete treatment necessitates considering all of them). To each one is associated a VB expression (and a Lewis structure). Rule 1: electrons belonging to a pair have opposite spins Rule 2: bonds are represented by covalent singlet functions Rule 3: The total wavefunction is antisymmetric relative to exchange of spins (symmetric relative to exchange of electrons). The VB functions interact to give linear combinations (coefficients and energies are obtained through a variational principle and a secular determinant) The square of the coefficients are the weights of the VB structure

23. 23 + + + + - - - - Resonance structures for the  electrons of benzene: Neutral species have larger weights than charged structures, Kekulé more than Dewar. Kekulé It is easy to make approximation, This structure has small weight Kekulé Dewar

24. 24 Example of Butadiene a b c d C=C-C=C A determinant: IabcdI ab is a bond:IabcdI+IbacdI = IabcdI-IabcdI cd is a bond:IabcdI+IbacdI + IbadcI+IabdcI or IabcdI-IabcdI - IabcdI+IabcdI

25. 25 excited Butadiene (zwiterion) a b c d C-C=C-C  C-C=C-C A determinant: IaabcI (2 electrons in a, none in d) aa is a pair: IaabcI cd is a bond: IaabcI+IaacbI = IaabcI-IaabcI Resonance: IaabcI+IaacbI + IbcddI+IcbddI = IaabcI-IaabcI + IbcddI-IbcddI +- + -

26. 26 Calculation of matrix elements Q = H = H mono + H bi = + = h aa +h bb = J ab Q = = h aa + h bb + J ab K = = + = 2h ab S bb = K ab K = = 2S ab h ab + K ab

27. 27 Calculation of matrix elements General Method: 1/n! Remove 1/n! and retain only one permutation Keep only permutations involving the same spin. 1/24 = = - - + = 1 - S 2 - S 2 + S 4 = 1 -2S 2 +S 4 Using this rule, it is simpler to order the determinants on spins rather than on electrons

28. 28 covalent states singlet AB distance

29. 29 H 2 singlet states: the symmetric ionic and covalent functions mix to generate the ground state and the diexcited state; the antisymmetric covalent state does not mix (Brillouin theorem) 2K ab │ ab ││ ab│ triplets │ ab │-│ ab │ │aa│- │ bb │Antisymmetric Ground state, mixed character more covalent than ionic First excited States Diexcited States, mixed character More covalent than ionic │ aa │+ │ bb │Sym │ab│+ │ ba │Sym 2K ab /(1+S 2 ) 2S ab h ab +J ab

30. 30 Energies of pure VB structures IabI + IbaI Singlet (covalent) Sym h aa +h bb +J ab +2S ab h ab +K ab 1+S ab 2 IaaI + IbbI Singlet (ionic) h aa +h bb +J ab +2S ab h ab -K ab 1+S ab 2 IabI, IabI, IabI - IbaI triplet h aa +h bb +J ab -2S ab h ab -K ab 1-S ab 2 IaaI - IbbI Singlet (ionic) Antisym h aa +h bb -J ab -2S ab h ab +K ab 1-S ab 2

31. 31 mixing of the singlet states Brillouin theorem 1934 Two functions are symmetric and one is antisymmetric: the ionic, antisymmetric state does not mix. It is essentially a non-bonding state corresponding to  g  u (monoexcitation) h aa +h bb +J ab +2S ab h ab -K ab 1+S ab 2 Léon Nicolas Brillouin 1889 – 1969) was a French physicist. His father, Marcel Brillouin, grand- father, Éleuthère Mascart, and great-grand-father, Charles Briot, were physicists as well. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory.

32. 32 Interaction term between symmetric singlets │ aa │+ │ bb │Sym Ground state, mixed character more covalent (80%) than ionic │ab│+ │ ba │Sym 2K ab /(1+S 2 ) │ aa │+ │ bb │Sym 1/√(1+S 2 ){│ aa │+ bb │} │H│ 1/√(1+S 2 ){│ ab │+ │ ba │} = {(h aa +h bb )S+2h ab + + } / (1+S 2 ) = {(h aa +h bb )S+2h ab + (aa│ab)+(bbba) } / (1+S 2 )

33. 33 Sign of K Earlier we have define K>0 from a single determinant; it was a consequence of the Pauli principle Here we have the determinant with permutation; This changes the sign: K<0. The covalent energy is the lowest. IabI + IbaI Singlet (covalent) Sym h aa +h bb +J ab +2S ab h ab +K ab 1+S ab 2 IaaI + IbbI Singlet (ionic) h aa +h bb +J ab +2S ab h ab -K ab 1+S ab 2

34. 34

35. 35 Resonance & covalence, Both structures contribute to the bonding, equally for MOs or when K is neglected: (1+S) (aa+bb)+(1-S) (ab+ba) There is 1 electron in each orbital so that the density is the same: The ionic VB structure: H +  H - The covalent: H-H In the middle plane, the amplitude is not zero for the two VB structures. When they combine equally, they double (bonding state) or vanish (antibonding state).

36. 36 Resonance & covalence, electron pair & AF diradical The ionic VB structure: H +  H - (aa+bb) matches better the description of electron pairs, the two electrons being located at the same place The covalent: H - H (ab+ba) corresponds better to an antiferromagnetic state with one electron on each side and a diradical.

37. 37 Interaction term between symmetric singlets within Hückel approximation │ aa │+ │ bb │Sym {(h aa +h bb )S+2h ab + (aa│ab)+(bbba) } / (1+S 2 ) becomes 2h ab } and the two states are degenerate → forming 50% ionic 50% covalent states The bonding state is therefore the sum: this is 2 E  g the energy of the  g 2 state! h aa +h bb E = 2h ab 2 E  g 2h ab 2 E  u │ab│+ │ ba │Sym

38. 38 Interaction term between symmetric singlets Hückel approximation with spin 2 E  g = 2(h aa +h ab )/(1+S) 2 E  u = 2(h aa -h ab )/(1-S) (h aa +h bb )S+2h ab - E 1+S 2 (h aa +h bb ) +2h ab S - E S 1+S 2 1+S 2 (h aa +h bb ) +2h ab S - E S 1+S 2 1+S 2 (h aa +h bb )S+2h ab - E 1+S 2 = 0 with S, the average energy is above the energy of pure VB structures

39. 39 Interaction term between symmetric singlets Hückel approximation with spin │ aa │+ │ bb │ 2 h bb +2h ab S 1+S 2 E = 2 E  g = 2(h aa +h ab )/(1+S) │ab│+ │ ba │ 2 E  u = 2(h aa -h ab )/(1-S) h aa = h bb E = 2h aa -2h ab S 1-S 2 E = Mean energy Atomic energy Pure VB structures

40. 40 Generalized Valence Bond, GVB It starts by an orthogonalization William A. Goddard III,

41. 41 Interaction Configuration The OM calculated at the HF level are eigenfunctions of H°+  that is not H°+ . We can form linear combinations of the determinants using variational theory. This is the IC. If the basis set is “complete”, a full IC should lead to experimental results (excepting relativistic effects, Born-Oppenheimer…)

42. 42 Interaction Configuration OM/IC: In general the OM are those calculated in an initial HF calculation. Usually they are those for the ground state. MCSCF: The OM are optimized simultaneously with the IC (each one adapted to the state).

43. 43 Interaction Configuration: mono, di, tri, tetra excitations… Monexcitation: promotion of i to k Diexcitation promotion of i and j to k and l kjikji lkjilkji I > kiki k l I j

44. 44 Branching diagram How many configurations for a spin state? 41 7/21 317 5/216 21520 3/21414 113928 1/212514 0125 12345678 Number of open shells S Each number is The sum of the two previous ones. See circles!

45. 45 Matrix elements within a state and an excited state: Slater rules =  [(ppIqr)-(pqIpr)] -  [(ttIqr)-(tqItr)] pqpq p q All the matrix elements are bielectronic terms (since H-H HF concerns the bielectronic repulsion). Many of them are equal to zero. For a single excitation, the terms are the off- diagonal terms of the Fock-matrix, which are 0 for HF eigenfunctions. There is no mixing between the GS and the mono-excited states. Brillouin theorem (already seen using symmetry). = (ppIqr)-(pqIpr) - (ttIqr)-(tqItr) pqpq

46. 46 Mind the spatial symmetry Only determinants with the same symmetry interact. To perform an IC: Generate all the configurations corresponding to an electronic state (solutions of S z and S 2 ) and eliminate those that are of different symmetry.

47. 47 Full symmetry, truncated symmetry Since it is expensive, instead of making a full IC, one restricts the “space of configuration” to a small space. Note that VB includes more physics in truncations. excitations% of excitations single0.6 % ( Brillouin) diexcitations94 % tri0.8 % ( similar to Brillouin) tetra4.4 % All the others0.17 % Example of H 2 O

48. 48 For other properties (Dipole moment) the monoexcitations count! Energy  (D) SCF-112.788-0.108 SCF+di-112.016-.068 SCF+mono+di-113.018+0.030 Exp.+.044 :C≡O: or :C=O ¨ ¨ C δ- O δ+

49. 49 perturbations Moller-Plesset MP2 perturbation for di-excitations MP3 perturbation up to tri-excitations MP4 perturbation up to tetra-excitations Coherence in size Correlation should depend on N Ecorr~ -(N-1) eV Considering mono and diexcitations (SDIC) on gets a size dependence in N 1/2. This requires correction as proposed by Davidson.

50. 50 IC: increasing the space of configuration Exact energy levels 2x2 3x3 4x4

51. 51 Which linear combination of  g 2 and  u 2 is 80% covalent and 20% ionic?  g 2 and  u 2 are 50% covalent and 50% ionic:  g 2 = 1/√2 (aa+bb) + 1/√2 (ab+ba)  u 2 = 1/√2 (aa+bb) - 1/√2 (ab+ba)  =  g 2 +  u 2 =(  √  (aa+bb)+(  √  (ab+ba)  2  = 0.2  2  = 0.8  2  = 0.4  2  = 1.6  = 0.9467  = -0.3162  The diexcitation allows flexibility between covalency and ionicity. The square of the coefficents is the weight

52. 52 VB vs. IC Both methods are equivalent provided that The same basis set are used there is no simplification (truncation of the # of VB structures or limited IC) They both take into account correlation and allow mixing covalent and ionic contributions in variable amounts. Advantage of VB: Close to Lewis structures and chemical language Easier to visualize and then easier for making approximation Advantage of IC: Many efficient softwares Less thinking

53. 53

54. 54 Density Functional Theory What is a functional? A function of another function: In mathematics, a functional is traditionally a map from a vector space to the field underlying the vector space, which is usually the real numbers. In other words, it is a function that takes a vector as its argument or input and returns a scalar. Its use goes back to the calculus of variations where one searches for a function which minimizes a certain functional. E = E[  (r)] E(  ) = T(  ) + V N-e (  ) + V e-e (  )

55. 55 Thomas-Fermi model (1927): The kinetic energy for an electron gas may be represented as a functional of the density. It is postulated that electrons are uniformely distributed in space. We fill out a sphere of momentum space up to the Fermi value, 4/3  p Fermi 3. Equating #of electrons in coordinate space to that in phase space gives: n(r) = 8  /(3h 3 ) p Fermi 3 and T(n)=c ∫ n(r) 5/3 dr T is a functional of n(r). Llewellen Hilleth Thomas 1903-1992 Enrico Fermi 1901-1954 Italian nobel 1938

56. 56 DFT Two Hohenberg and Kohn theorems : Walter Kohn 1923, Austrian-born American nobel 1998 Pierre C Hohenberg Kohn 1923, german-born american The existence of a unique functional. The variational principle.

57. 57 First theorem: on existence The first H-K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density. It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to only 3 spatial coordinates, through the use of functional of the electron density. This theorem can be extended to the time-dependent domain to develop time- dependent density functional theory (TDDFT), which can be used to describe excited states. The external potential, and hence the total energy, is a unique functional of the electron density.

58. 58 First theorem on Existence : demonstration The external potential, and hence the total energy, is a unique functional of the electron density. The proof of the first theorem is remarkably simple and proceeds by reductio ad absurdum. Let there be two different external potentials, V 1 and V 2, that give rise to the same density . The associated Hamiltonians,H 1 and H 2, will therefore have different ground state wavefunctions,  1 and  2, that each yield . E 1 = + = E 2 + ∫  r  V 1  r  V 2  r))dr E 2 = E 1 + ∫  r  V 2  r  V 1  r))dr E 1 + E 2 < E 1 + E 2 Therefore ∫  r  V 1  r  V 2  r))dr=0 and  V 1  r  V 2  r) The electronic energy of a system is function of a single electronic density only.

59. 59 Second theorem: Variational principle The second H-K theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional : If  (r) is the exact density, E[  (r)] is minimum and we search for  by minimizing E[  (r)] with ∫  (r)dr = N  (r) is a priori unknown  As for HF, the bielectronic terms should depend on two densities  i (r) and  j (r) : the approximation  2e (r i,r j ) =  i (r i )  j (r j ) assumes no coupling.

60. 60 Kohn-Sham equations 3 equations in their canonical form: Lu Jeu Sham San Diego Born in Hong-Kong member of the National Academy of Sciences and of the Academia Sinica of the Republic of China

61. 61 Kohn-Sham equations Equation 1: This reintroduces orbitals: the density is defined from the square of the amplitudes. This is needed to calculate the kinetic energy.

62. 62 Kohn-Sham equations Equation 2: The intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. Using an effective potential, one has a one-body expression.

63. 63 Kohn-Sham equations Equation 3: the writing of an effective single-particle potential E eff (  ) = T eff+ V eff = T + V mono + Rep Bi V eff = V mono + Rep Bi + T – T eff V eff (r) = V(r) + ∫e 2  (r’)/(r-r’) dr’ + V XC [  (r)] as in HF Aslo a mono electronic expression Unknown except for free electron gas

64. 64 Exchange correlation functionals V XC [  (r)] This term is not known except for free electron gas: LDA E XC [  (r)] = ∫  (r) E XC (r) dr E XC [  (r)] =  V XC [  (r)]/  (r) = E XC (  ) +  (r)  E XC (  )/  E XC (  ) = E X (  ) + E C (  ) = -3/4(3/  ) 1/3  (r) + E C (  ) determined from Monte-Carlo and approximated by analytic expressions

65. 65 Exchange correlation functionals SCF-X  The introduction of an approximate term for the exchange part of the potential is known as the X  method. V X   ()  -6  (3/4   ) 1/3   is the local density of spin up electrons and  is a variable parameter. with a similar expression for  ↓

66. 66 Exchange correlation functionals V XC [  (r)] E XC = E XC (  ) LDA or LSDA (spin polarization) E XC = E XC (  ) GGA or GGSDA - Perdew-Wang - PBE: J. P. Perdew, K. Burke, and M. Ernzerhof E XC = E XC (  ) metaGGA

67. 67 Hybrid methods: B3-LYP (Becke, three-parameters, Lee-Yang-Parr) Axel D. Becke german 1953 Incorporating a portion of exact exchange from HF theory with exchange and correlation from other sources : a 0 =0.20 a x =0.72 a C =0.80 List of hybrid methods: B1B95 B1LYP MPW1PW91 B97 B98 B971 B972 PBE1PBE O3LYP BH&H BH&HLYP BMK

68. 68 Weitao Yang Duke university USA Born 1961 in Chaozhou, China got his undergraduate degree at the University of Peking Chengteh Lee received his Ph.D. from Carolina in 1987 for his work on DFT and is now a senior scientist at the supercomputer company Cray, Inc. Robert G. Parr Chicago 1921

69. 69 DFT Advantages : much less expensive than IC or VB. adapted to solides, metal-metal bonds. Disadvantages: less reliable than IC or VB. One can not compare results using different functionals*. In a strict sense, semi-empical,not ab- initio since an approximate (fitted) term is introduced in the hamiltonian. * The variational priciples applies within a given functional and not to compare them. The only test for validity is comparison with experiment, not a global energy minimum!

70. 70 DFT good for IPs IP for Au (eV)Without fWith f functions SCF7.44 SCF+MP28.008.91 B3LYP9.08 Experiment9.22 Electron affinity H Exp.SCFICB3LYP Same basis set 0.735-0.5280.3820.635

71. 71 DFT good for Bond Energies Exp.HFLDAGGA B2B2 3.10.93.93.2 C2C2 6.30.87.36.0 N2N2 9.95.711.610.3 O2O2 5.21.37.66.1 F2F2 1.7-1.43.42.2 Bond energies (eV); dissociation are better than in HF

72. 72 DFT good for distances Exp.HFLDAGGA B2B2 1.591.531.601.62 C2C2 1.240.87.36.0 N2N2 1.101.061.091.10 O2O2 1.211.151.201.22 F2F2 1.411.321.381.41 Distances (Å)

73. 73 DFT polarisabilities (H 2 O) Exp.HFLDA  0.7280.7870.721  xx 9.267.839.40  xx 10.019.1010.15  xx 9.628.369.75 Distances (Å)

74. 74 DFT dipole moments (D) Exp.HFLDAGGA CO-0.110.33-0.17-0.15 CS1.981.262.112.01 LiH5.835.555.655.74 HF1.821.981.861.80 Distances (Å)

75. 75 Jacobs’scale, increasing progress according Perdew Paradise = exactitude StepsMethodExample 5 th stepFully non local- 4 th step Hybrid Meta GGAB1B95 Hybrid GGAB3LYP 3 rd stepMeta GGABB95 2 nd stepGGABLYP 1 st stepLDASPWL Earth = Hartree-Fock Theory