Guillaume Barbe (1979- )

Université de MontréalNovember 11th 2008

From Newton to WoodwardComplete Construction of the Diels-Alder Correlation Diagram

Sir Isaac Newton1642-1727

Robert B. Woodward1917-1979

Theoritical Model forConcerted Reactions

Outlines

• Construction of Schrödinger equation from classical mechanics and routine mathematics

• Hückel Model of molecular orbitals will provide a quantification of the energies and orbital coefficients for polyenes

• Quick excursion in the rational of the symmetry-allowed Diels-Alder Cycloaddition

Classical Mechanics

Sir Isaac Newton1642-1727

First LawEvery object in a state of uniform motion

tends to remain in that state of motion unless an external force is applied to it.

p(t) m (t) mdx(t)

d tconstan t v

Second LawThe relationship between an object's mass m, its acceleration a, and the

applied force F

F md x t

dxm a

d

dxU x

2

2

( )( )

Potential Energy

Em v P

mkin 2 2

2 2

E U x F x dxpo t

x

( ) ( )0

Kinetic Energy

Continuum

Plum-pudding (1904)

Joseph J. Thomson 1856-1940

1906

His sonGeorge Paget Thomson

1856-1940Nobel Prize Physics 1937

Disovery of the Particlelike property of Electron

7 of his students won the Nobel Prize

Planetary Model (1909)

Ernest Rutherford 1871-1937

By emitting radiation, the electron should lose energy and

collapse into the nucleous

Atom is not stable !

1908

Hydrogen Atom

Niels Bohr1885-1962

r

me

mn

m

r

e

r0e

2 2

2

v

Electron on a Stable Orbit

Hydrogen Atom Equilibrium

Em

U xm e

r

m e

rto te e r e

v v v2 2 2

2

2 2

2 2 2( )

Electric Force

Centrifugal Force

Total Energy of the Electron

Ee

rto t 2

2Continuum

1922

Hydrogen SpectraPhotoelectric effect (1905)

Albert Einstein1879-1955

hv hv Epho ton kin elec tron 0

h = Planck Constantphoton = frequency of the incident photonh0 = = Work function = Energy needed

to remove an electron

1921

Wave-Particle DualityPhoton Case

Albert Einstein1879-1955

E pc m c ( ) ( )2 2 2

Special Theory of Relativity

E m c 2

Limit CaseSpeed of object is low

m 0Photon

E pc hv c v

ph

xx t

c

'

v

v1

2

2

tt

cx

c

'

v

v

2

2

21

1921

Wave-Particle DualityElectron Case

Louis de Broglie1892-1987Destructive

r n2

Electron Wave-Particle Duality p

h

Angular momentum l rp rm v

r

phn

r

n

r

2

rm nv

1929

Ultraviolet CatastropheBeginning of quantum theory (1900)

Max Planck1858-1947

Black-body Radiation

d d Density of Energy

8

4

T

Rayleigh-Jeans Law

8

15

hc ehc

T

ehc

T

Planck Distribution

Planck SuggestionE nhvosc

1918

Bohr Model (1913)

Niels Bohr1885-1962

m

r

Z e

r0e

2 2

2

v

Electron on a Stable Orbit

Equilibrium

m Z e

re v 2 2

22

Electric Force

Centrifugal Force

Hydrogen Radius

Ee

rto t 2

2Continuum

rm nv Quantification

rn

Z e m e

2 2

2

EZ e m

nto t 2 4

2 22 QuantumMechanic

1922

Wave Equation

Lieou, C. K. C. Eur. J. Phys. 2007, 28, N17-N19

x t

dx d t 0

dx d t vWave Function

Classical Mechanic

Non-absortive and Non-dispersive medium

x t

1

v

x t t x

Continuously Differentiable

2

2 2

2

2

1 1

x x x x t tv v

2

2 2

2

2

1

x tv

Wave Equation

Stationary Wave

Lieou, C. K. C. Eur. J. Phys. 2007, 28, N17-N19

Wave Equation

Variable Separation ( , ) ( ) ( )x t X x T t

Stationary Wave Function

2

2 2

2

2

1

x tv

( , ) s in 2 sin co s co s sinx tx

v tx

v tx

v t

2

22

2

Trigonomeric Identity

( , ) co s sinx tx

v t

22

sin ax e a ix

Trigonomeric Identity

( , ) ( )x t x e iv t 2

Wave Function

Stationary Wave

Erwin Schrödinger1887-1961

Lieou, C. K. C. Eur. J. Phys. 2007, 28, N17-N19

Wave Equation Wave Function

2

2 2

2

2

1

x tv

Stationary Wave Equation

( , ) ( )x t x e iv t 2

2

22

2

2

( , ) ( )x te

xiv t

x x

1 22

2

2

2

2v v

( , )( )

x t ivx e iv t

t

2

2

220

( )( )

xx

x

2

2

2

0

( )( )

x px

x

1933

Schrödinger Equation

Erwin Schrödinger1887-1961

Stationary Wave Equation

2

2

2

0

( )( )

x px

x E

m v P

mkin 2 2

2 2

Kinetic Energy

EP

mV xto t

2

2( )

Total Energy

P m E V x2 2 ( )

2

2 2

20

( ) ( ( ))

( )x m E V x

xx

E xm

xV x x

( )

( )( ) ( )

2 2

22

x

Schrödinger Equation

1933

Free Electron

P = mv > 0P = mv < 0

x

P = momentumm = massv = speed (vector)

E xm

x

( )

( )

2 2

22

x

Schrödinger Equation

( , ) s inx t A ax

Stationary Wave Function

E xm

x

mA a x

( )

( )( )

2 2

2

22

2 2

x

EA a

m

A a h

m

2 2 2 2

22 8

Free Electron Wavefunction

( ) s inx A ax

Particle in a Box

P = mv > 0P = mv < 0

x

P = momentumm = massv = speed (vector)

L

V

0

2 Conditions

( )0 0 ( )L 0

Particle in a Box Wavefunction

( )x A e B eiax iax

A B

( ) s in ( )x A e e A i axiax iax 2

e e

iax

iax iax

2sin ( )

( ) s in ( )x A i aL 2 0

a 2

aL n n 1 2 3, , . . .

an

L

Particle in a Box

n = 1

n = 2

n = 3

LL/20

EA a h

m

2 2

28a

n

L

EA n h

m L

2 2

28

an

L

2

2 L

n

Example: -Carotene

nmXbondschemicalL 8.185.0 21

22 electrons

2

2

2

22

2

22

8

23

8

11

8

12

mL

h

mL

h

mL

hE

nmJs

mkgsm

h

cmL464

)10x626.6(23

)10x8.1)(10x109.9)(/10x998.2(8

23

834

293182

Absorption : blue-greenReflection : yellow-red = orange

E

n = 1

n = 11

n = 12

L0

11:

12:

nHOMO

nLUMO

Quantum Mechanics PostulatesPostulate 1

E xm

xV x x

( )

( )( ) ( )

2 2

22

x

Schrödinger Equation

1. Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system.

( , )x t single value

Quantum Mechanics PostulatesPostulate 2

E xm

xV x x

( )

( )( ) ( )

2 2

22

x

Schrödinger Equation

2. With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction.

E x H x ( ) ( )

Quantum Mechanics PostulatesPostulate 3

E xm

xV x x

( )

( )( ) ( )

2 2

22

x

Schrödinger Equation

3. Any operator Q associated with a physically measurable property q will be Hermitian.

Pi

x

EP

mkin 2

2

a b a bH H* ( ) ( ) *

Quantum Mechanics PostulatesPostulate 4

E xm

xV x x

( )

( )( ) ( )

2 2

22

x

Schrödinger Equation

4. The set of eigenfunctions of operator Q will form a complete set of linearly independent functions.

c c c ca a 1 1 2 2 3 3 . . .

E Ha a

Quantum Mechanics PostulatesPostulate 5

E xm

xV x x

( )

( )( ) ( )

2 2

22

x

Schrödinger Equation

5. For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction.

a b a bH H* ( ) ( ) *

Hermetian Operator

E H * ( )

Expectation Value

Quantum Mechanics

Erwin Schrödinger1887-1961

Paul Dirac1902-1984

« for the discovery of new productive forms of

atomic theory » 

The Nobel Prize in Physics

1933

Werner Heisenberg1901-1976

1932

« for the creation of quantum mechanics… »

Molecular Orbital Theory of Conjugated SystemsHückel Molecular Orbitals

C-2 C-3 C-4

C-5 C-6

Erich Hückel1896-1980

Secular Equations

E H Schrödinger Equation

c a a

Postulate 4

b aa

ac H E 0 c H Ea aa

0

Determination of ca and E

b a abH H* ( )

b a abS *

Overlap Integral

c H Ea b a b aa

* * 0

c H S Ea ab aba

0

Secular Equations

c H S Ea ab aba

0Secular Equations

c H c H c H E c S c S c S1 11 2 1 2 3 1 3 1 11 2 1 2 3 1 3 0

c H c H c H E c S c S c S1 2 1 2 2 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c H c H E c S c S c S1 3 1 2 3 2 3 3 3 1 3 1 2 3 2 3 3 3 0

We want to determine the value and sign of ca and E

Hückel TheoryPlanar/symmetric systems

c H c H c H E c S c S c S1 11 2 1 2 3 1 3 1 11 2 1 2 3 1 3 0

c H c H c H E c S c S c S1 2 1 2 2 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c H c H E c S c S c S1 3 1 2 3 2 3 3 3 1 3 1 2 3 2 3 3 3 0

Secular Equations

4 Approximations for planar and symmetrical polyenes

Erich Hückel1896-1980

Hückel TheoryPlanar/symmetric systems

c H c H c H E c S c S c S1 11 2 1 2 3 1 3 1 11 2 1 2 3 1 3 0

c H c H c H E c S c S c S1 2 1 2 2 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c H c H E c S c S c S1 3 1 2 3 2 3 3 3 1 3 1 2 3 2 3 3 3 0

Secular Equations

C-3

12

3

Mirror Plane of Molecule

x zy

Approximation 1

Hückel TheoryPlanar/symmetric systems C-3

12

3

Approximation 2 H Ha a a a * ( )

c H c H c H E c S c S c S1 11 2 1 2 3 1 3 1 11 2 1 2 3 1 3 0

c H c H c H E c S c S c S1 2 1 2 2 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c H c H E c S c S c S1 3 1 2 3 2 3 3 3 1 3 1 2 3 2 3 3 3 0

Secular Equations

= Coulomb Integral= Energy of bound electron

= Constant

Hückel TheoryPlanar/symmetric systems C-3

12

3

Approximation 2 H Ha a a a * ( )

c c H c H E c S c S c S1 2 1 2 3 1 3 1 11 2 1 2 3 1 3 0

c H c c H E c S c S c S1 2 1 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c H c E c S c S c S1 3 1 2 3 2 3 1 3 1 2 3 2 3 3 3 0

Secular Equations

= Coulomb Integral= Energy of bound electron

= Constant

Hückel TheoryPlanar/symmetric systems C-3

12

3

Approximation 3 H Ha b b a * ( ) a b 1 0:

a b 1 : constant

c c H c H E c S c S c S1 2 1 2 3 1 3 1 11 2 1 2 3 1 3 0

c H c c H E c S c S c S1 2 1 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c H c E c S c S c S1 3 1 2 3 2 3 1 3 1 2 3 2 3 3 3 0

Secular Equations

Hückel TheoryPlanar/symmetric systems C-3

12

3

Approximation 3 H Ha b b a * ( ) a b 1 0:

a b 1 : constant

c c H E c S c S c S1 2 1 2 1 11 2 1 2 3 1 3 0

c H c c H E c S c S c S1 2 1 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c E c S c S c S2 3 2 3 1 3 1 2 3 2 3 3 3 0

Secular Equations

Hückel TheoryPlanar/symmetric systems C-3

12

3

Approximation 4Kronecker Symbol

b a ab abS * Overlap Integral

a b ab : 0

a b ab : 1

c c H E c S c S c S1 2 1 2 1 11 2 1 2 3 1 3 0

c H c c H E c S c S c S1 2 1 2 3 2 3 1 2 1 2 2 2 3 2 3 0

c H c E c S c S c S2 3 2 3 1 3 1 2 3 2 3 3 3 0

Secular Equations

Hückel TheoryPlanar/symmetric systems C-3

12

3

Approximation 4

c c c E1 2 1 0

c c c c E1 2 3 2 0

c c c E2 3 3 0

Secular Equations

Kronecker Symbol

b a ab abS * Overlap Integral

a b ab : 0

a b ab : 1

Hückel TheoryPlanar/symmetric systems C-3

12

3

c c c E1 2 1 0

c c c c E1 2 3 2 0

c c c E2 3 3 0

Secular Equations

c E c1 2 0

c c E c1 2 3 0

c c E2 3 0

Secular Equations

E

Secular Determinant

0

E

E

0

0

Hückel TheoryPlanar/symmetric systems C-3

12

3

E

Secular Determinant

0

E

E

0

0

0 2,

E 2

E

E 2

Highest Energy

Lowest Energy3 Molecular Orbitals

is negative

Hückel TheoryPlanar/symmetric systems C-3

12

3

E 2

E

E 2

Highest Energy

Lowest Energy

E

Hückel TheoryPlanar/symmetric systems C-3

12

3

c E c1 2 0 c c E c1 2 3 0

c c E2 3 0

Secular Equations

E 2

E

E 2

Highest Energy

Lowest Energy

c c c1 2 3

1

20

1

2 ; ;

c c c12

22

32 1

Normalization

c c c1 2 3

1

2

1

2

1

2 ; ;

c c c1 2 3

1

2

1

2

1

2 ; ;

Hückel TheoryPlanar/symmetric systems C-3

12

3

E 2

E

E 2

Highest Energy

Lowest Energy

E 0.500

-0.707

0.500

0.707 -0.707

0.500 0.500

0.707

Secular EquationsExample: Butadiene

c H S Ea ab aba

0Secular Equations

c H c H c H c H E c S c S c S c S1 11 2 1 2 3 1 3 4 1 4 1 11 2 1 2 3 1 3 4 1 4 0

C-4

12

34

c H c H c H c H E c S c S c S c S1 2 1 2 2 2 3 2 3 4 2 4 1 2 1 2 2 2 3 2 3 4 2 4 0

c H c H c H c H E c S c S c S c S1 3 1 2 3 2 3 3 3 4 3 4 1 3 1 2 3 2 3 3 3 4 3 4 0

We want to determine the value and sign of ca and E

c H c H c H c H E c S c S c S c S1 4 1 2 4 2 3 4 3 4 4 4 1 4 1 2 4 2 3 4 3 4 4 4 0

Secular EquationsExample: Butadiene C-4

12

34

c c c E1 2 1 0

c c c c E1 2 3 2 0

c c c c E2 3 4 3 0

Secular Equations

c E c1 2 0

c c E c1 2 3 0

c c E c2 3 4 0

Secular Equations

c c c E3 4 4 0 c c E3 4 0

Secular EquationsExample: Butadiene C-4

12

34

c E c1 2 0

c c E c1 2 3 0

c c E c2 3 4 0

Secular Equations

c c E3 4 0

1 2 3 4

Symmetrical

c c1 4 c c2 3

Anti-Symmetrical

c c1 4 c c2 3

Secular EquationsExample: Butadiene C-4

12

34

c E c1 2 0

c c E c1 2 3 0

c c E c2 3 4 0

Secular Equations

c c E3 4 0

1 2 3 4

Symmetrical

c E c1 2 0

c c E c1 2 2 0

E 1 6 2.

E 0 6 2.

Secular EquationsExample: Butadiene C-4

12

34

c E c1 2 0

c c E c1 2 3 0

c c E c2 3 4 0

Secular Equations

c c E3 4 0

1 2 3 4

Anti-Symmetrical

c E c1 2 0

c c E c1 2 2 0

E 1 6 2.

E 0 6 2.

Secular EquationsExample: Butadiene C-4

12

34

Highest Energy

Lowest Energy

E

E 1 6 2.

E 0 6 2.

E 0 6 2.

E 1 6 2.

Secular EquationsExample: Butadiene C-4

12

34

Secular Equations

c c c c12

22

32

42 1

Normalization

c c c c1 2 3 40 3 7 1 0 6 0 0 0 6 0 0 0 3 7 1 . ; . ; . ; .

c E c1 2 0 c c E c1 2 3 0 c c E c2 3 4 0

c c E3 4 0

Highest Energy

Lowest EnergyE 1 6 2.E 0 6 2.E 0 6 2.E 1 6 2.

c c c c1 2 3 40 6 0 0 0 3 7 1 0 3 7 1 0 6 0 0 . ; . ; . ; .

c c c c1 2 3 40 6 0 0 0 3 7 1 0 3 7 1 0 6 0 0 . ; . ; . ; .

c c c c1 2 3 40 3 7 1 0 6 0 0 0 6 0 0 0 3 7 1 . ; . ; . ; .

Secular EquationsExample: Butadiene C-4

12

34

Highest Energy

Lowest Energy

E

0.371

-0.600 0.600

-0.371

0.600

-0.371 -0.371

0.600

0.600

0.371 -0.371

-0.600

0.371

0.600 0.600

0.371E 1 6 2.

E 0 6 2.

E 0 6 2.

E 1 6 2.

Sinusoidal Lobe Alternance

Ethene

Allyle

Butadiene

« Electron in a Box »

Diels-Alder Cycloaddition

Diels-Alder CycloadditionConservation of Orbital Symmetry

Robert B. Woodward1917-1979

Roald Hoffman1937-

Elias J. Corey1928-

?

1965 1981

1990

Diels-Alder CycloadditionSymmetry of Orbitals

plan symmetry

C2 rotation symmetry

plan symmetry

C2 rotation symmetry

Robert B. Woodward1917-1979

Roald Hoffman1937-

Ethylene

Butadiene

1965

1981

Diels-Alder CycloadditionSymmetry of Orbitals

E

SA

AS

SA

AS

Butadiene

SA

AS

Ethylene

, C2

Diels-Alder ReactionReaction Path: Plan Symmetry

Cyclobutene + Ethelyne Cyclohexene

Only a plan symmetry along the reaction path

Diels-Alder ReactionCorrelation Diagrams

LUMO dienophile

HOMO diene

A

A

S

S

A

S

A

S

A

S

A

S

Diels-Alder ReactionCorrelation Diagrams

LUMO dienophile

HOMO diene

A

A

S

S

A

S

A

S

A

S

A

S

[2+2] CycloadditionCorrelation Diagrams

Ethylene + Ethelyne Cyclobutane

Two plan symmetry along the reaction path

[2+2] CycloadditionCorrelation Diagrams

SA

SA

[2+2] CycloadditionCorrelation Diagrams

SA

SS

SA

SS

SA

AS

AA

AS

SA

AA

Diels-Alder CycloadditionFrontier Molecular Orbitals

E

Butadiene EthyleneFMO

Fukui Acc. Chem. Res. 1971, 4, 57.

HOMO

LUMO

HOMO

LUMO Kenichi Fukui1918-1988

1981

Spino et al. Angew. Chem., Int. Ed. 1998, 37, 3262.

Conclusions

• Schrödinger equation can be easily obtained from classical mechanics through routine mathematical procedures

• Application of Hückel Model to polyenes provides an approximate but reliable quantification of energies and orbital coefficients

• Conservation orbital symmetry and FMO are useful in predicting the course of concerted reactions