Master Course for Theoretical Chemistry and Computational Modelling (TCCM)

Using Valence Bond Theory to Model (Bio)Chemical Reactivity

Fernanda Duarte Department of Chemistry, Oxford

September 7th 2016

1 General Outline of Lecture

• Motivation & History of VB

• Basic Concepts

• ab initio VB theory

• Multiscale VB simulations

• Empirical VB theory Motivation

Concepts and heuristic models Quantitative theory (Localised view ) (Delocalised view) H = E H X C Y

H H

Lewis model Curly-Arrows VSEPR Hybridisation

... Delocalised particles Localised electron pairs indistinguishable and interacting Chemical bond concept 3 (no chemical bond) Historical Timeline

ROOTS OF VB THEORY 3 Valence bond theory in Pauling’s view is6 a Aquantum BRIEF STORY chemical OF VALENCE version BOND THEORY of Lewis’s theory of

valence Pauling-Wheland's resonating picture HL- Wave function Covalent-ionic superpositionHuckel's delocalized π-MO picturePauling's three-electron in a bond, A–B bond Interaction Between Neutral Atoms and Homopolar Binding K1 K2 AB AB AB • HH HH O •• O ••• a b Rumer’s A B O D1 D2 D3 Heitler & London canonical structures 2 1 2 Ψ = 3c1(K1 + K2) + c2(D1 + D2 + D3) c1 > c2 σu 5 6 Scheme 1.2 σg 1916 1927 1929 antiaromaticity,1931 and its articulation1933 by organic chemists1950 in the 1950s 1970s À G.N. Lewis Slater 4 will constitute a major cause for the acceptance of MO theory and the rejection of VB theory (4). Determinant The description of benzene in terms of a superposition (resonance) of two The Atom and The Molecule Scheme 1.1 Pauling (1939) Kekule´ structures appeared for the first time in the work of Slater, as a case belonging to a class of species in which each atomWheland possesses (1955) more neighbors than electrons it can share, much like in metals (21).VB TwoTextbooks years later, Pauling and Wheland (37) applied HLVB theory to benzene. They developed a less As demonstrated by Heisenberg, the mixing of [wn(1)wm(2)] and [wn(2)wm(1)] led cumbersome computational approach, compared with Hu¨ ckel’s previous to a new energy term that caused aHLVB splitting treatment, betweenL. using Pauling the the five two canonical wave structures functions in 6, and approximated the matrix elements between the structures by retaining only close neighbor CA and CB. He called this term ‘‘resonance’’The Nature using of the a Chemical classical Bond analogy of two resonance interactions. Their approach allowed them to extend the treatment oscillators that, by virtue of possessingto naphthalene the same and frequency, to a great variety form of aother resonating species. Thus, in the HLVB A Chemist’ssituation Guide with to Valence characteristic Bond Theory, by Sason exchange Shaikapproach, and energy. Philippe benzene C. Hiberty is described as a ‘‘resonance hybrid’’ of the two4 Kekule´ structures and the three Dewar structures; the latter had already appeared In modern terms, the bonding inbefore H2 incan Ingold’s be idea accounted of mesomerism, for which by itself the is rooted wave in Lewis’s concept function drawn in 1, in Scheme 1.1.of This electronic wave tautomerism function (6). In is his a book, superposition published for the of first time in 1944, Wheland explains the resonance hybrid with the biological analogy of two covalent situations in which, in themule first = donkey form + ( horsea) one (38). electron The pictorial has a representation spin-up of the wave (a spin), while the other has spin-downfunction, (b spin), the link and to Kekul vicee´’s versa oscillation in the hypothesis, second and to Ingold’s mesomerism, which were known to chemists, made the HLVB representation form (b). Thus, the bonding in H2veryarises popular due among to practicing the quantum chemists. mechanical ‘‘resonance’’ interaction between the twoWith patterns these two of seemingly spin arrangement different treatments that of benzene, are the chemical required in order to form a singletcommunity electron was pair. faced with This two ‘‘resonance alternative descriptions energy’’ of one of its molecular icons, and this began the VB MO rivalry that seems to accompany chemistry accounted for 75% of the totalto bonding the Twenty-first of Century the molecule,À (5). This rivalry and involved thereby most of the prominent projected that the wave function in chemists1, which of various is referred periods (e.g., to henceforth Mulliken, Hu¨ ckel, as J. the Mayer, Robinson, Lapworth, Ingold, Sidgwick, Lucas, Bartlett, Dewar, Longuet-Higgins, HL-wave function, can describe the chemicalCoulson, Roberts, bonding Winstein, in a Brown). satisfactory A detailed manner. and interesting account of This ‘‘resonance origin’’ of the bondingthe nature was of this a rivalry remarkable and the major feat players of can the be found new in the treatment of Brush (3,4). Interestingly, back in the 1930s, Slater (22) and van Vleck and quantum theory, since until then it was not obvious how two neutral species could be at all bonded. In the winter of 1928, London extended the HL-wave function and drew the general principles of the covalent bonding in terms of the resonance interaction between the forms that allow interchange of the spin-paired electrons between the two atoms (10,12). In both treatments (9,12) the authors considered ionic structures for homopolar bonds, but discarded their mixing as being too small. In London’s paper, there is also a consideration of ionic (so-called polar) bonding. In essence, the HL theory was a quantum mechanical version of Lewis’s electron-pair theory. Thus, even though Heitler and London did their work independently and perhaps unaware of the Lewis model, the HL-wave function still precisely described the shared-pair bond of Lewis. In fact, in his letter to Lewis (8), and in his landmark paper (13), Pauling points out that the HL and London treatments are ‘entirely equivalent to G.N. Lewis’s successful theory of shared electron pair ...’’. Thus, although the final formulation of the 6 A BRIEF STORY OF VALENCE BOND THEORY

Huckel'sHistorical delocalized π-MO picture TimelinePauling-Wheland's resonating picture

6 A BRIEF STORY OFROOTS VALENCE OF BOND VB THEORY THEORY 3 K1 K2

Pauling-Wheland's resonating picture Covalent-ionic superpositionHuckel's delocalized π-MO picture HL- Wave functionLennard-Jones MO Theory Pauling's three-electron in a bond, A–B D1 D2 Dbond3 O2 MullikenΨ = c1(K 1& + K Hund2) + c2(D1 + D2 + D3) c1 > c2 K1 K2 AB AB5 AB 6 • HH HH Huckel Model O •• O Scheme 1.2 ••• a b Benzene Rumer’s antiaromaticity, and A its articulationB by organic chemists in the 1950s O1970s2D1 D2 D3 Heitler & London will constitute a major cause for thecanonical acceptance of structures MO theory and the rejectionÀ 1 2 Ψ = 3c1(K1 + K2) + c2(D1 + D2 + D3) of VB theory (4). c > c The description of benzene in terms of a superposition (resonance) of two 1 2 Kekule´ structuresσu appeared for the first time5 in the work of Slater, as a case 6 belonging to a class of species in which each atom possesses more neighbors Scheme 1.2 than electronsσ itg can share, much like in metals (21). Two years later, Pauling and Wheland (37) applied HLVB theory to benzene. They developed a less 1916 1927 1929 antiaromaticity,1931 and its articulation1933 by organic chemists1950 in the 1950s 1970s cumbersome computational approach, compared with Hu¨ ckel’s previous À G.N. Lewis SlaterHLVB 4 treatment, usingwill the constitute five canonical a major structures cause for in 6 the, and acceptance approximated of MO theory and the rejection the matrix elementsof between VB theory the structures (4). by retaining only close neighbor Determinant resonance interactions.Scheme TheirThe 1.1 description approach allowed of benzene them to in extend terms the of treatment a superpositionPauling (resonance)(1939) of two to naphthalene andKekul to a greate´ structures variety of appeared other species. for Thus, the first in the time HLVB in the work of Slater, as a case approach, benzene isbelonging described as to a a ‘‘resonance class of species hybrid’’ in of which the two each Kekul atomWhelande´ possesses (1955) more neighbors structures and the three Dewar structures; the latter had already appeared than electrons it can share, much like in metals (21). Two years later, Pauling before in Ingold’s idea of mesomerism, which itself is rooted in Lewis’s conceptVB Textbooks and Wheland (37) applied HLVB theory to benzene. They developed a less As demonstrated by Heisenberg,of electronic tautomerism the mixing (6). In ofhis book, [wn(1) publishedwm(2)] for the and first time [wn in(2) 1944,wm(1)] led Wheland explains thecumbersome resonance hybrid computational with the biological approach, analogy compared of with Hu¨ ckel’s previous to a new energy term thatmule caused = donkey + a horseHLVB splitting (38). treatment, The between pictorialL. using Pauling representation the the five two canonical of wave the structures wave functions in 6, and approximated function, the linkthe to Kekul matrixe´’s elements oscillation between hypothesis, the structures and to Ingold’s by retaining only close neighbor CA and CB. He called this term ‘‘resonance’’The Nature using of the a Chemical classical Bond analogy of two mesomerism, which wereresonance known interactions. to chemists, made Their the approach HLVB representation allowed them to extend the treatment oscillators that, by virtuevery of popular possessing amongto practicing naphthalene the chemists. same and frequency, to a great variety form of aother resonating species. Thus, in the HLVB With these two seemingly different treatments of benzene, the chemical A Chemist’ssituation Guide with to Valence characteristic Bond Theory,community by Sason exchange was Shaik facedapproach, and with energy. Philippe two alternative benzene C. Hiberty descriptions is described of one as a of ‘‘resonance its molecular hybrid’’ of the two5 Kekule´ icons, and this beganstructures the VB MO and rivalry the that three seems Dewar to accompany structures; chemistry the latter had already appeared In modern terms, theto the bonding Twenty-first in Centurybefore H2À (5).incan Ingold’s This berivalry idea accounted involved of mesomerism, most of for the which prominent by itself the is rooted wave in Lewis’s concept function drawn in 1, inchemists Scheme of various 1.1.of periods This electronic (e.g., wave tautomerism Mulliken, function Hu¨ ckel, (6). In is J. his Mayer, a book, superposition Robinson, published for the of first time in 1944, Lapworth, Ingold,Wheland Sidgwick, Lucas, explains Bartlett, the resonance Dewar, Longuet-Higgins, hybrid with the biological analogy of two covalent situations inCoulson, which, Roberts, in the Winstein,mule first = Brown). donkey form A+ (detailed horsea) one (38).and electron interesting The pictorial account has a of representation spin-up of the wave the nature of this rivalry and the major players can be found in the treatment of function, the link to Kekule´’s oscillation hypothesis, and to Ingold’s (a spin), while the otherBrush has (3,4). spin-down Interestingly, back (b spin), in the 1930s, and Slater vice (22) and versa van Vleck in the and second mesomerism, which were known to chemists, made the HLVB representation form (b). Thus, the bonding in H2veryarises popular due among to practicing the quantum chemists. mechanical ‘‘resonance’’ interaction between the twoWith patterns these two of seemingly spin arrangement different treatments that of benzene, are the chemical required in order to form a singletcommunity electron was pair. faced with This two ‘‘resonance alternative descriptions energy’’ of one of its molecular icons, and this began the VB MO rivalry that seems to accompany chemistry accounted for 75% of the totalto bonding the Twenty-first of Century the molecule,À (5). This rivalry and involved thereby most of the prominent projected that the wave function in chemists1, which of various is referred periods (e.g., to henceforth Mulliken, Hu¨ ckel, as J. the Mayer, Robinson, Lapworth, Ingold, Sidgwick, Lucas, Bartlett, Dewar, Longuet-Higgins, HL-wave function, can describe the chemicalCoulson, Roberts, bonding Winstein, in a Brown). satisfactory A detailed manner. and interesting account of This ‘‘resonance origin’’ of the bondingthe nature was of this a rivalry remarkable and the major feat players of can the be found new in the treatment of Brush (3,4). Interestingly, back in the 1930s, Slater (22) and van Vleck and quantum theory, since until then it was not obvious how two neutral species could be at all bonded. In the winter of 1928, London extended the HL-wave function and drew the general principles of the covalent bonding in terms of the resonance interaction between the forms that allow interchange of the spin-paired electrons between the two atoms (10,12). In both treatments (9,12) the authors considered ionic structures for homopolar bonds, but discarded their mixing as being too small. In London’s paper, there is also a consideration of ionic (so-called polar) bonding. In essence, the HL theory was a quantum mechanical version of Lewis’s electron-pair theory. Thus, even though Heitler and London did their work independently and perhaps unaware of the Lewis model, the HL-wave function still precisely described the shared-pair bond of Lewis. In fact, in his letter to Lewis (8), and in his landmark paper (13), Pauling points out that the HL and London treatments are ‘entirely equivalent to G.N. Lewis’s successful theory of shared electron pair ...’’. Thus, although the final formulation of the Historical Timeline

Mulliken, Nobel Prize

Shaik Goddard Hiberty Exp. verification GVB Landis Zhang Huckel rules Thrular Fukui Woodward & Hoffmann Vogh Frontier MO Theory

1950 1952 1960 1965 1970 1980 2000 O2 failure GAUSSIAN70 EVB XIAMEN-99 Dewar & Coulson Localized<—> delocalized

3rd Ed. Valence Pauling Book

J. A. Pople A. Warshel

A Chemist’s Guide to Valence Bond Theory, by Sason Shaik and Philippe C. Hiberty 6 Valence Bond Theory

• VB theory describes system as a collection of structures differing mainly in composition of valence electrons.

• Overall description is a linear combination of various VB structures formed by different distributions, arrangements and/or pairing of these electrons.

VB = Cii i X each i is a VB structure

7 Bridge Between MO and VB Theory

H H H + H

ΨVB = λ( χ a χb − χ a χb )+ µ( χ a χ a + χb χb ) λ > µ Heitler & London

ΦHL = χ a χb − χ a χb

1 " (1) (1) (2) (2) (2) (2) (1) (1)$ χ a χb = #χ a α χb β − χ a α χb β % 2

1 " (1) (1) (2) (2) (2) (2) (1) (1)$ χ a χb = #χ a β χb α − χ a β χb α % 2

At the equilibrium bond distance, the bonding is predominantly covalent (about 75%) As the bond is stretched, the weight of the ionic structures gradually decreases

8 Bridge Between MO and VB Theory

VB ΨVB = λ( χ a χb − χ a χb )+ µ( χ a χ a + χb χb ) λ > µ

MO * σ = χ a + χb σ = χ a − χb

Ψ = σ σ = χ χ − χ χ + χ χ + χ χ MO ( a b a b ) ( a a b b ) HL Ionic

The wave function is always half-covalent and half-ionic, irrespective of distances!!

9 Bridge Between MO and VB Theory

Ψ = σ σ = χ χ − χ χ + χ χ + χ χ MO ( a b a b ) ( a a b b ) HL Ionic

The wave function is always half-covalent and half-ionic, irrespective of distances!!

Ψ* = σ uσ u = −( χ a χ b − χ a χ b )+( χ a χ a + χ b χ b )

ΨMO−CI = c1 σσ − c2 σ uσ u

= (c1 + c2 )( χ a χ b − χ a χ b )+ (c1 − c2 )( χ a χ a + χ b χ b ) (c1 + c2 ) = λ; (c1 − c2 ) = µ HL Ionic

Both theories, when properly implemented, are correct and mutually transformable

10 ab initio Valence Bond Theory

VB = Cii i X Not all electrons are treated at the VB level

inactive/active separation

= A[ inactives actives ] VB { }{ } - an active space of electrons/orbitals treated at the VB level - the rest as MOs (spectator orbitals)

The active space chosen depending on the chemical problem

11 ab initio Valence Bond Theory

VB = Cii i X

Example: the π system of benzene 6 electrons, 6 centres

How many possible VB structures do we have?

12 ab initio Valence Bond Theory

Rumer’s rule for a covalent n-centre/n-electron system: 1. Put the orbitals around an imaginary circle 2. Crossing bonds are not allowed

13 ab initio Valence Bond Theory

Number of covalent structures for N-e systems (f)

(2S + 1)N! f N = S ( 1 N + S + 1)!( 1 N S)! 2 2 S: total spin ; N : number of electrons/centres

N 4 6 8 10 12

f 2 5 14 42 121

14 ab initio Valence Bond Theory

Rumer’s rule for N-e/N-c ionic structures

1.Choose a distribution of charges

2. Apply Rumer’s rules on the rest

3.Choose another distribution of charges...

4. Another one …

15 ab initio Valence Bond Theory

Number of covalent+ionic structures for N-e/m-c systems

(2S + 1) m +1 m +1 gN,m = S m +1 N + S +1 N S ✓ 2 ◆✓ 2 ◆ S: total spin; N : number of electrons; m=orbitals/centres

N 4 6 8 14 28 2.76 x 106 2.07 x 1014 f 20 175 1764

16 Describing Chemical SN2 Reactions

= A[ inactives actives ] VB { }{ }

H X C Y

H H

- - Cl + CH3 – Cl Cl – CH3 + Cl

17 Describing Chemical SN2 Reactions H X C Y

H H

- - Cl + CH3 – Cl Cl – CH3 + Cl

22 core electrons (1s, 2s and 2p of Cl, 1s of C) described by 11 doubly occupied MOs.

4 active valence electrons will occupy VB vectors localised on the corresponding fragments.

18 Describing Chemical SN2 Reactions

H X C Y

H H Depending on identity of nucleophile (Nu) and leaving group (LG) overall number of valence electrons can vary.

There will always be at least four active electrons (i.e. electrons involved in the bond breaking/forming)

1 4 4 4-e/3-orbital/centre VB system g4,3 = 0 4 3 2 ✓ ◆✓ ◆

19 VB Description of SN2 Reaction

6 VB structures

(1) X + H3C Y (4) X + + CH3 + Y

(2) X CH3 + Y (5) X + H3C + Y

(3) X + CH3 + Y (6) X + C + Y

20 VB Description of SN2 Reaction

6 VB structures

(1) X + H3C Y (4) X + + CH3 + Y

xx¯(cy¯ +¯cy) ccy¯ y¯ | | | | (2) X CH3 + Y (5) X + H3C + Y

yy¯(xc¯ + cx¯) xxc¯ c¯ | | | |

(3) X + CH3 + Y (6) X + C + Y xxy¯ y¯ cc¯(xy¯ +¯xy) | | | |

21 VB Description of SN2 Reaction

Qualitative Description using

VB State Correlation Diagrams (VBSCD)

H X C Y

H H

Sason S. Shaik J. Am. Chem. Soc., 1981, 103 , 692 Pross, A.; Shaik, S. S. Acc. Chem. Res. 1983, 16, 363 Shaik, S. and Shurki, A. Angew. Chem. Int. Ed. 1999, 38: 586

22 VB Description of SN2 Reaction H RS in the PS geometry X C Y

E H H X CH3 Y

deformation toward P

X + (H3C Y) Reactant = xx¯(cy¯ +¯cy) R | |

23 VB Description of SN2 Reaction H

PS in the RS geometry RS in the PS geometry X C Y H E CH H X + CH3 Y X 3 Y

X CH3 + Y X +H3C Y Reactant Product = xx¯(cy¯ +¯cy) = yy¯(xc¯ + cx¯) R | | P | |

24 VB Description of SN2 Reaction

PS in the RS geometry RS in the PS geometry

E CH X + CH3 Y X 3 Y

G = I⇤ A⇤C Y X

X CH3 + Y X +H3C Y Reactant Product = xx¯(cy¯ +¯cy) = yy¯(xc¯ + cx¯) R | | P | |

25 VB Description of SN2 Reaction

E CH X + CH3 Y X 3 Y

B: resonance energy

X CH3 + Y X +H3C Y a barrier formation is described as a result of avoided crossing between two state curves

26 VB Description of SN2 Reaction

E CH X + CH3 Y X 3 Y

excited state

B: resonance energy

ground state

X CH3 + Y X +H3C Y a barrier formation is described as a result of avoided crossing between two state curves

27 VB Description of SN2 Reaction

E R* P*

excited state

G: R→R* is an excited G diabatic state B: resonance energy of the TS due to VB mixing ΔE=fG

ground state

R P

G = fG B

28 VB Description of SN2 Reaction Principles

• Two-state (VBSCD) vs. multi-state diagrams (VBCMD) :

Barrier Stepwise

R and P mix to form the The intermediate has a different barrier and the TS for an electronic structure than R and P elementary process («internal catalysis»)

29 VB Description of SN2 Reaction

• General Applicability: Nucleophilic, electrophilic, radical, pericyclic reactions …

• Simple: could be applied «on the back of en envelope»

• Insightful: allows to create order among great families of reactions

• Qualitative reasonings: a few rules and elementary interactions - quantitive proof : by high level VB calculations

30 VB Description of SN2 Reaction

Quantitative Description

H X C Y

H H

31 VB Description of SN2 Reaction

• Overall wavefunction is simply a linear combination of the six structures:

VB = Cii each i is a VB structure i X

• Solving Schrödinger equation using variation principle gives following set

of linear equations: N ∑ (Hij − ESij )cj = 0 i =1,..., N j=1

32 VB Description of SN2 Reaction

• Overall wavefunction is simply a linear combination of the six structures:

VB = Cii each i is a VB structure i X

• Solving Schrödinger equation using variation principle gives following set

of linear equations: N ∑ (Hij − ESij )cj = 0 i =1,..., N j=1 coefficients of VB Hamiltonian matrix configurations overlap matrix

H ij = φi | H |φ j Sij = φi |φ j

33 VB Description of SN2 Reaction

(1) X + H3C Y

(2) X CH3 + Y

(3) X + CH3 + Y

(4) X + CH3 + Y

(5) X + H3C + Y

Shurki et al. Chem. Soc. Rev., 2015, 44, 1037 34 Softwares to Run ab initio VB

Program Capabilities Website Comments Reference

VBSCF, BOVB, VBCI, http:// Integrated into McWeeny and V2000 SCVB, CASVB www.scinetec.com GAMESS coworkers GVB

http://tc5.chem.uu.nl/ Integrated into TURTLE VBSCF ATMOL/turtle/ van Lenthe et al. GAMESS-UK turtle_main.html

VBSCF, BOVB, VBCI, VBPT2, http://ftcc.xmu.edu.cn/ Integrated into Wu and XMVB DFVB, VBPCM, xmvb/index.html GAMESS coworkers VBEFP, VBEFP/ PCM

35 ab initio Valence Bond Theory

Localised orbitals VBSCF

Different ways to introduce full correlation Breathing CI Orbitals VBCI PT2 BOVB VBCI

Adds on for Condense Phases: +PCM +SM +MM

VBPCM VBSM VB/MM

36 Wu et al. Chem. Rev., 2011, 111, 7557 How does one calculate VB wave functions with localized orbitals ? VBSCF Method

"VB =#Ci $ i where each Φi is a VB structure =i a The VBSCF0 method (Balint-Kurtik k where & eachvan Lenthe Φi is a )VB structure Example: the F moleculek 2 X ! • • + C • • • • C1 X X 2 X X + C2 X X C1 • F • • F • + C2 • F • F • + C2 • F • F • • • • • • • F•—•F F–F+ F+F– Coefficients C and orbitals optimized simultaneously (like MCSCF) • Orbitals and Ci coefficientsi of different structures are optimised simultaneously, All orbitals are optimized (active as well as spectator ones) leading to self-consistent field type wave function.

GVB, SCVB ~ equivalent to VBSCF • VBSCF basically multi-configuration SCF method with the added advantage of utilising chemically interpretable configurations.

• Analogous to MO-based CASSCF. However, VBSCF uses fewer structures and localised non-orthogonal orbitals. 37 Accuracy of the various methods

Computational errors of BD energies relative to the FCI method

What is VBSCF missing?

Wu et al. Chem. Rev., 2011, 111, 7557 38 How does one calculate VB wave functions with localized orbitals ?

"VB =#Ci $ i where each Φi is a VB structure i VBSCF Method The VBSCF method (Balint-Kurti & van Lenthe)

Example: the F2 molecule

! • • + C • • • • C1 X X 2 X X + C2 X X C1 • F • • F • + C2 • F • F • + C2 • F • F • • • • • • • F•—•F F–F+ F+F–

Coefficients Ci and orbitals optimized simultaneously (like MCSCF) 0All= orbitalsa arek optimizedk The same (active set as of well AOs as is spectatorused for allones VB) structures…. k X GVB, SCVBOptimised ~ equivalent for ato mean VBSCF neutral situation

39 How does one calculate VB wave functions with localized orbitals ?

"VB =#Ci $ i where each Φi is a VB structure i VBSCF Method The VBSCF method (Balint-Kurti & van Lenthe)

Example: the F2 molecule

! • • + C • • • • C1 X X 2 X X + C2 X X C1 • F • • F • + C2 • F • F • + C2 • F • F • • • • • • • F•—•F F–F+ F+F– Coefficients C and orbitals optimized simultaneously (like MCSCF) How does onei calculate VB wave functions with localized orbitals ? 0All= orbitalsa arek optimizedk The same (active set as of well AOs as is spectatorused for allones VB) structures…. k X Optimised for a mean neutral situation "VBGVB,=# SCVBCi $ i ~ equivalentwhere each to VBSCF Φi is a VB structure i The VBSCF method (Balint-Kurti & van Lenthe) Better Representation Example: the F2 molecule ! • • • • • • C • F + C2+ C ++ C FX X C1 X1 • •X• F • 2 •X F • XF • 2 • • F • • • • • • • F•—•F F–F+ F+F– 40

Coefficients Ci and orbitals optimized simultaneously (like MCSCF) All orbitals are optimized (active as well as spectator ones)

GVB, SCVB ~ equivalent to VBSCF Improvements on VBSCF Method

BOVB

BOVB removes average field restriction of VBSCF. The orbitals are variationally optimised with the freedom to be different for different VB structures.

- Orbitals for F•—•F will be the same as VBSCF - Orbitals for ionic structures will be much improved

41 Wu et al. Chem. Rev., 2011, 111, 7557 Improvements on VBSCF Method

Test case: the dissociation of F2 ΔE F–F F• + F•

Calculation of ∆E for F-F=1.43Å, 6-31G(d) basis:

Iteration De(kcal) F•–•F F+F– ↔ F–F+ Classical VB -4.6 0.813 0.187 GVB,VBSCF ~ 15 0.768 0.232 BOVB 1 24.6 0.731 0.269 2 27.9 0.712 0.288 3 28.4 0.709 0.291 4 28.5 0.710 0.290 5 28.6 0.707 0.293 Full CI 30-33

BOVB brings that part of dynamic correlation that varies in the process

42 Philippe C. Hiberty Improvements on VBSCF Method

VBCI

1. Start from VBSCF 0 = akk Xk 2. For each Φk define a set of strictly localised virtual orbitals

CI k k 3. Improve " by post-VBSCF configuration interaction: µ = Cµµ Xk k All µ are excitations that correspond to the same VB structure CI µ is a multi-determinant description of a unique VB structure.

BV CI CI CI 4. Do the configuration interaction: = Cµ µ µ X

43 Incorporation of Solvent Effects

VBPCM combined VB approach with PCM.

Interactions between solute charges and polarised electric field of solvent considered by embedding an interaction potential into the VB Hamiltonian:

VB VB (H0 +VR )Ψ = EΨ

VBSM uses SMX (currently X=6) to account for solvation.

solvent-solute electrostatic interactions are described by the generalised born (GB) approximation, with self-consistent solute charges.

environment is described as being homogenous

44 VB/MM Calculations

For meaningful description of environment, need to move to hybrid QM/MM description

How to deal with coupling between QM and MM parts?

• Simplest approach: QM and MM parts do not polarise each other, only mechanical embedding considered.

• MM point charges polarise QM region in process referred to as electrostatic embedding.

Shurki et al. J. Phys. Chem. B, 2010, 114, 2212 45 VB/MM Calculations

N ∑ (Hij − ESij )cj = 0 i =1,..., N j=1 coefficients of VB Hamiltonian matrix configurations overlap matrix

H ij = φi | H |φ j Sij = φi |φ j

Diagonal elements: energy of the respective VB configuration

0 int Hii = Hii + Hii

interaction energy of Φi configuration QM energy of configuration Φi with its surrounding

46 VB/MM Calculations

N ∑ (Hij − ESij )cj = 0 i =1,..., N j=1 coefficients of VB Hamiltonian matrix configurations overlap matrix

H ij = φi | H |φ j Sij = φi |φ j

Off-diagonal elements: resonance integral between VB configurations 1 H = H0 + (Hint + Hint)S0 ij ij 2 ii jj ij

Invariant to the environment

47 VB/MM Calculations

N (H "S )c =0 i =1,..., N ij ij i i=1 X 0 int Hii = Hii + Hii

1 H = H0 + (Hint + Hint)S0 ij ij 2 ii jj ij

Finally, the overall energy is given by

EVB/MM = " + E(MM )

classical interactions of the environment within itself

48 VB/MM Calculations

N (H "S )c =0 i =1,..., N ij ij i i=1 X 0 int Hii = Hii + Hii

1 H = H0 + (Hint + Hint)S0 ij ij 2 ii jj ij

Finally, the overall energy is given by

EVB/MM = " + E(MM )

classical interactions of the environment within itself

49