Using Valence Bond Theory to Model (Bio)Chemical...

September 7th 2016

1

Fernanda Duarte

Using Valence Bond Theory to Model

(Bio)Chemical Reactivity

Master Course for Theoretical Chemistry and Computational

Modelling (TCCM)

Department of Chemistry, Oxford

• Motivation & History of VB

• Basic Concepts

• ab initio VB theory

• Multiscale VB simulations

• Empirical VB theory

General Outline of Lecture

33

Motivation

Concepts and heuristic models (Localised view )

Quantitative theory (Delocalised view)

Lewis model

Curly-Arrows

VSEPR

Hybridisation

...

Localised electron pairs Chemical bond concept

Delocalised particles indistinguishable and interacting

(no chemical bond)

C Y

H

HH

X

H = E

4

Historical Timeline

L. Pauling The Nature of the Chemical Bond

1929 19331927G.N. Lewis

1916

Heitler & London

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

As demonstrated by Heisenberg, the mixing of [wn(1)wm(2)] and [wn(2)wm(1)] ledto a new energy term that caused a splitting between the two wave functionsCA and CB. He called this term ‘‘resonance’’ using a classical analogy of twooscillators that, by virtue of possessing the same frequency, form a resonatingsituation with characteristic exchange energy.

In modern terms, the bonding in H2 can be accounted for by the wavefunction drawn in 1, in Scheme 1.1. This wave function is a superposition oftwo covalent situations in which, in the first form (a) one electron has a spin-up(a spin), while the other has spin-down (b spin), and vice versa in the secondform (b). Thus, the bonding in H2 arises due to the quantum mechanical‘‘resonance’’ interaction between the two patterns of spin arrangement that arerequired in order to form a singlet electron pair. This ‘‘resonance energy’’accounted for !75% of the total bonding of the molecule, and therebyprojected that the wave function in 1, which is referred to henceforth as theHL-wave function, can describe the chemical bonding in a satisfactory manner.This ‘‘resonance origin’’ of the bonding was a remarkable feat of the newquantum theory, since until then it was not obvious how two neutral speciescould be at all bonded.

In the winter of 1928, London extended the HL-wave function and drew thegeneral principles of the covalent bonding in terms of the resonance interactionbetween the forms that allow interchange of the spin-paired electrons betweenthe two atoms (10,12). In both treatments (9,12) the authors considered ionicstructures 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 ofLewis’s electron-pair theory. Thus, even though Heitler and London did theirwork independently and perhaps unaware of the Lewis model, the HL-wavefunction still precisely described the shared-pair bond of Lewis. In fact, in hisletter to Lewis (8), and in his landmark paper (13), Pauling points out that theHL and London treatments are ‘entirely equivalent to G.N. Lewis’s successfultheory of shared electron pair . . .’’. Thus, although the final formulation of the

A BH H H H

A BA BOO •••

•••

B O2 Aa b

HL- Wave function Covalent-ionic superpositionin a bond, A–B

Pauling's three-electronbond

1 3

4

σg

σu

2

Scheme 1.1

ROOTS OF VB THEORY 3

Slater Determinant

1931

Rumer’s canonical structures

antiaromaticity, and its articulation by organic chemists in the 1950s!1970swill constitute a major cause for the acceptance of MO theory and the rejectionof VB theory (4).

The description of benzene in terms of a superposition (resonance) of twoKekule structures appeared for the first time in the work of Slater, as a casebelonging to a class of species in which each atom possesses more neighborsthan electrons it can share, much like in metals (21). Two years later, Paulingand Wheland (37) applied HLVB theory to benzene. They developed a lesscumbersome computational approach, compared with Huckel’s previousHLVB treatment, using the five canonical structures in 6, and approximatedthe matrix elements between the structures by retaining only close neighborresonance interactions. Their approach allowed them to extend the treatmentto naphthalene and to a great variety of other species. Thus, in the HLVBapproach, benzene is described as a ‘‘resonance hybrid’’ of the two Kekulestructures and the three Dewar structures; the latter had already appearedbefore in Ingold’s idea of mesomerism, which itself is rooted in Lewis’s conceptof electronic tautomerism (6). In his book, published for the first time in 1944,Wheland explains the resonance hybrid with the biological analogy ofmule= donkey+ horse (38). The pictorial representation of the wavefunction, the link to Kekule’s oscillation hypothesis, and to Ingold’smesomerism, which were known to chemists, made the HLVB representationvery popular among practicing chemists.

With these two seemingly different treatments of benzene, the chemicalcommunity was faced with two alternative descriptions of one of its molecularicons, and this began the VB!MO rivalry that seems to accompany chemistryto the Twenty-first Century (5). This rivalry involved most of the prominentchemists of various periods (e.g., Mulliken, Huckel, J. Mayer, Robinson,Lapworth, Ingold, Sidgwick, Lucas, Bartlett, Dewar, Longuet-Higgins,Coulson, Roberts, Winstein, Brown). A detailed and interesting account ofthe nature of this rivalry and the major players can be found in the treatment ofBrush (3,4). Interestingly, back in the 1930s, Slater (22) and van Vleck and

D1 D3

K2K1

D2

5 6

Huckel's delocalized π-MO picture

Ψ = c1(K1 + K2) + c2(D1 + D2 + D3)

c1 > c2

Pauling-Wheland's resonating picture

Scheme 1.2

6 A BRIEF STORY OF VALENCE BOND THEORY

1950

Pauling (1939)Wheland (1955)VB Textbooks

The Atom and The Molecule

Interaction Between Neutral Atoms and Homopolar Binding

Valence bond theory in Pauling’s view is a quantum chemical version of Lewis’s theory of valence

5

Historical Timeline

L. Pauling The Nature of the Chemical Bond

1929 19331927G.N. Lewis

1916

Heitler & London


As demonstrated by Heisenberg, the mixing of [wn(1)wm(2)] and [wn(2)wm(1)] ledto a new energy term that caused a splitting between the two wave functionsCA and CB. He called this term ‘‘resonance’’ using a classical analogy of twooscillators that, by virtue of possessing the same frequency, form a resonatingsituation with characteristic exchange energy.

In modern terms, the bonding in H2 can be accounted for by the wavefunction drawn in 1, in Scheme 1.1. This wave function is a superposition oftwo covalent situations in which, in the first form (a) one electron has a spin-up(a spin), while the other has spin-down (b spin), and vice versa in the secondform (b). Thus, the bonding in H2 arises due to the quantum mechanical‘‘resonance’’ interaction between the two patterns of spin arrangement that arerequired in order to form a singlet electron pair. This ‘‘resonance energy’’accounted for !75% of the total bonding of the molecule, and therebyprojected that the wave function in 1, which is referred to henceforth as theHL-wave function, can describe the chemical bonding in a satisfactory manner.This ‘‘resonance origin’’ of the bonding was a remarkable feat of the newquantum theory, since until then it was not obvious how two neutral speciescould be at all bonded.

In the winter of 1928, London extended the HL-wave function and drew thegeneral principles of the covalent bonding in terms of the resonance interactionbetween the forms that allow interchange of the spin-paired electrons betweenthe two atoms (10,12). In both treatments (9,12) the authors considered ionicstructures 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 ofLewis’s electron-pair theory. Thus, even though Heitler and London did theirwork independently and perhaps unaware of the Lewis model, the HL-wavefunction still precisely described the shared-pair bond of Lewis. In fact, in hisletter to Lewis (8), and in his landmark paper (13), Pauling points out that theHL and London treatments are ‘entirely equivalent to G.N. Lewis’s successfultheory of shared electron pair . . .’’. Thus, although the final formulation of the

A BH H H H

A BA BOO •••

•••

B O2 Aa b

HL- Wave function Covalent-ionic superpositionin a bond, A–B

Pauling's three-electronbond

1 3

4

σg

σu

2

Scheme 1.1

ROOTS OF VB THEORY 3

Slater Determinant

1931

Rumer’s canonical structures

MO Theory Mulliken & Hund

Lennard-Jones O2

Huckel ModelBenzene




D1 D3

K2K1

D2

5 6


Ψ = c1(K1 + K2) + c2(D1 + D2 + D3)

c1 > c2


Scheme 1.2





D1 D3

K2K1

D2

5 6


Ψ = c1(K1 + K2) + c2(D1 + D2 + D3)

c1 > c2


Scheme 1.2


1950

Pauling (1939)Wheland (1955)VB Textbooks

6

Historical Timeline

1952 1960 20001950

J. A. Pople A. Warshel

1970


Exp. verification Huckel rules

Valence

Dewar & Coulson Localized<—> delocalized

O2 failure

Mulliken, Nobel Prize

3rd Ed. Pauling Book

Fukui Frontier MO Theory

Woodward & Hoffmann

1965GAUSSIAN70

1980EVB

Goddard GVB

XIAMEN-99

Shaik Hiberty Landis Zhang Thrular Vogh

7

Valence Bond Theory

each is a VB structure �i

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

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

V B =X

i

Ci�i

8

Bridge Between MO and VB Theory

H + HH H

ΦHL = χaχb − χaχb

χaχb =12χa (1)α(1)χb(2)β(2)− χa (2)α(2)χb(1)β(1)"# $%

χaχb =12χa (1)β(1)χb(2)α(2)− χa (2)β(2)χb(1)α(1)"# $%

ΨVB = λ( χaχb − χaχb )+µ( χaχa + χbχb ) λ > µ

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

Heitler & London

9


ΨVB = λ( χaχb − χaχb )+µ( χaχa + χbχb ) λ > µ

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

σ * = χa − χbσ = χa + χb

ΨMO = σ σ = χaχb − χaχb( )+ χaχa + χbχb( )HL

VB

MO

Ionic

10


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

ΨMO = σ σ = χaχb − χaχb( )+ χaχa + χbχb( )HL Ionic

Ψ*= σ uσ u = − χaχb − χaχb( )+ χaχa + χbχb( )

ΨMO−CI = c1 σσ − c2 σ uσ u

= (c1 + c2 ) χaχb − χaχb( )+ (c1 − c2 ) χaχa + χbχb( ) (c1 + c2 ) = λ; (c1 − c2 ) = µ

HL Ionic

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

11

ab initio Valence Bond Theory

V B =X

i

Ci�i

Not all electrons are treated at the VB level

inactive/active separation

- an active space of electrons/orbitals treated at the VB level - the rest as MOs (spectator orbitals)

V B = A[{inactives}{actives}]

The active space chosen depending on the chemical problem

12


How many possible VB structures do we have?

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

V B =X

i

Ci�i

13


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

14


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

fNS =

(2S + 1)N !

( 12N + S + 1)!( 12N � S)!

N 4 6 8 10 12f 2 5 14 42 121

S: total spin ; N : number of electrons/centres

15


1.Choose a distribution of charges

2. Apply Rumer’s rules on the rest

3.Choose another distribution of charges...

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

4. Another one …

16


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

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

gN,mS =

(2S + 1)

m+ 1

✓m+ 1

N2 + S + 1

◆✓m+ 1N2 � S

◆

S: total spin; N : number of electrons; m=orbitals/centres

17

Describing Chemical SN2 Reactions

V B = A[{inactives}{actives}]

C Y

H

HH

X

Cl- + CH3 – Cl Cl – CH3 + Cl-

18


C Y

H

HH

X

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.

19


4-e/3-orbital/centre VB system g4,30 =1

4

✓43

◆✓42

◆

C Y

H

HH

X

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)

20

VB Description of SN2 Reaction

6 VB structures

(5)

(4)

X H3C Y+ +

Y+X + + CH3X H3C Y

(3)

(1)

(2)

+

X CH3 Y+

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

22

C Y

H

HH

X

Qualitative Description using

VB State Correlation Diagrams (VBSCD)


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

23

C Y

H

HH

X

R = |xx(cy + cy)|

deformation toward P

RS in the PS geometry


ReactantX (H3C Y)+

E X CH3 Y

24

C Y

H

HH

X

R = |xx(cy + cy)|


Reactant

RS in the PS geometry

P = |yy(xc+ cx)|Product

PS in the RS geometry

X CH3 Y+

X CH3 Y+

X H3C Y+

E X CH3 Y

25

RS in the PS geometry PS in the RS geometry

G = I⇤X �A⇤C � Y


R = |xx(cy + cy)|Reactant

P = |yy(xc+ cx)|Product

X CH3 Y+

X CH3 Y+

X H3C Y+

E X CH3 Y

26

a barrier formation is described as a result of avoided crossing between two state curves

B: resonance energy


X CH3 Y+

X CH3 Y+

X H3C Y+

E X CH3 Y

27

a barrier formation is described as a result of avoided crossing between two state curves

B: resonance energy


ground state

excited state

X CH3 Y+

X CH3 Y+

X H3C Y+

E X CH3 Y

28

B: resonance energy of the TS due to VB mixing


�G = fG�B

G

ΔE=fG

P

E

R

P*R*

ground state

excited state

G: R→R* is an excited diabatic state

R*

29


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

Barrier Stepwise

R and P mix to form the barrier and the TS for an

elementary process

The intermediate has a different electronic structure than R and P

(«internal catalysis»)

30

• 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


31

C Y

H

HH

X

Quantitative Description


32

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

• Solving Schrödinger equation using variation principle gives following set of linear equations:

Hij −ESij( )cj = 0j=1

N

∑ i =1,...,N


each is a VB structure �i V B =

X

i

Ci�i

33

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

• Solving Schrödinger equation using variation principle gives following set of linear equations:


N

∑ i =1,...,N


each is a VB structure �i V B =X

i

Ci�i

Hamiltonian matrixoverlap matrix

coefficients of VB configurations

Sij = φi |φ jHij = φi |H |φ j

34


X H3C Y(1)

(5)

(4)

(3)

(2)

+

X CH3 Y+

Y+X + CH3

X H3C Y+ +

Y+X + CH3

Shurki et al. Chem. Soc. Rev., 2015, 44, 1037

35

Softwares to Run ab initio VB

Program Capabilities Website Comments Reference

V2000

VBSCF, BOVB, VBCI,

SCVB, CASVB GVB

http://www.scinetec.com

Integrated into GAMESS

McWeeny and coworkers

TURTLE VBSCFhttp://tc5.chem.uu.nl/

ATMOL/turtle/turtle_main.html

Integrated into GAMESS-UK van Lenthe et al.

XMVB

VBSCF, BOVB, VBCI, VBPT2,

DFVB, VBPCM, VBEFP, VBEFP/

PCM

http://ftcc.xmu.edu.cn/xmvb/index.html

Integrated into GAMESS

Wu and coworkers

36


Adds on for Condense Phases:

VBPCM VBSM VB/MM

+PCM +SM +MM

VBCI BOVB

VBCI

VBSCF

Breathing OrbitalsCI

PT2

Different ways to introduce full correlation

Wu et al. Chem. Rev., 2011, 111, 7557

Localised orbitals

37

VBSCF Method

• Orbitals and Ci coefficients of different structures are optimised simultaneously, leading to self-consistent field type wave function.

• 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.

0 =X

k

ak�k where each Φi is a VB structure

GVB, SCVB ~ equivalent to VBSCF

F•—•F F–F+ F+F–

Coefficients Ci and orbitals optimized simultaneously (like MCSCF)

All orbitals are optimized (active as well as spectator ones)

How does one calculate VB wave functions with localized orbitals ?

!

"VB

= Ci

i

# $i where each Φi is a VB structure

The VBSCF method (Balint-Kurti & van Lenthe)

Example: the F2 molecule

F F•••

••

••

••••

••

• F F••

••

••

••••

••

F F

••

••

••••

••

C1 + C2 + C2•• ••••XX X XXC1 + C2 + C2X

38

Accuracy of the various methods

Computational errors of BD energies relative to the FCI method

Wu et al. Chem. Rev., 2011, 111, 7557

What is VBSCF missing?

39

VBSCF Method






!

"VB

= Ci

i




F F•••

••

••

••••

••

• F F••

••

••

••••

••

F F

••

••

••••

••

C1 + C2 + C2•• ••••

The same set of AOs is used for all VB structures…. 0 =X

k

ak�k

Optimised for a mean neutral situation

XX X XXC1 + C2 + C2X






!

"VB

= Ci

i




F F•••

••

••

••••

••

• F F••

••

••

••••

••

F F

••

••

••••

••

C1 + C2 + C2•• ••••

40

VBSCF Method






!

"VB

= Ci

i




F F•••

••

••

••••

••

• F F••

••

••

••••

••

F F

••

••

••••

••

C1 + C2 + C2•• ••••

The same set of AOs is used for all VB structures…. 0 =X

k

ak�k

Optimised for a mean neutral situation

XX XX XXC1 + C2 + C2

XX X XXC1 + C2 + C2X

Better Representation

41

Improvements on VBSCF Method

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

BOVB

- Orbitals for F•—•F will be the same as VBSCF

- Orbitals for ionic structures will be much improved

Wu et al. Chem. Rev., 2011, 111, 7557

42


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

Test case: the dissociation of F2

F–F F• + F• ΔE

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

BOVB brings that part of dynamic correlation that varies in the process Philippe C. Hiberty

43


1. Start from VBSCF

VBCI

0 =X

k

ak�k

3. Improve " by post-VBSCF configuration interaction: �CIµ =

X

k

Ckµ�

kµ

All are excitations that correspond to the same VB structure

�CIµ is a multi-determinant description of a unique VB structure.

�kµ

4. Do the configuration interaction:

2. For each Φk define a set of strictly localised virtual orbitals

BV CI =X

µ

CCIµ �CI

µ

44

Incorporation of Solvent Effects

(H0 +VR )ΨVB = EΨVB

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:

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.

environmentisdescribedasbeinghomogenous

45

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

46

VB/MM Calculations


N

∑ i =1,...,N




Diagonal elements: energy of the respective VB configuration

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

Hii = H0ii +Hint

ii

47

VB/MM Calculations


N

∑ i =1,...,N




Off-diagonal elements: resonance integral between VB configurations

Invariant to the environment

Hij = H0ij +

1

2(Hint

ii +Hintjj )S0

ij

48

VB/MM Calculations

i =1,...,N

Finally, the overall energy is given by

Hij = H0ij +

1

2(Hint

ii +Hintjj )S0

ij

Hii = H0ii +Hint

ii

classical interactions of the environment within itself

EVB/MM = "+ E (MM )

NX

i=1

(Hij � "Sij)ci = 0

49

VB/MM Calculations

i =1,...,N

Finally, the overall energy is given by

Hij = H0ij +

1

2(Hint

ii +Hintjj )S0

ij

Hii = H0ii +Hint

ii

classical interactions of the environment within itself

EVB/MM = "+ E (MM )

NX

i=1

(Hij � "Sij)ci = 0

Using Valence Bond Theory to Model (Bio)Chemical...

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