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OC-IV. Orbital Concepts and Their Applications in Organic Chemistry. Klaus Müller. Script ETH Zürich, Spring Semester 2009. Chapter 5. p -systems HMO and extended PMO method. Lecture assistants: Deborah Sophie Mathis HCI G214 – tel. 24489 [email protected]

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Oc iv

OC-IV

Orbital Concepts

and

Their Applications in Organic Chemistry

Klaus Müller

Script

ETH Zürich, Spring Semester 2009

Chapter 5

p-systems

HMO and extended PMO method

Lecture assistants:

Deborah Sophie MathisHCI G214 – tel. [email protected]

Alexey FedorovHCI G204 – tel. 34709 [email protected]


Oc iv

Sim

Sik

  • For planar unsaturated systems:

  • the p- and s-orbitals are orthogonal by symmetry

  • there are no (s,p)-orbital interactions

  • there are no orbital splitting effects between s- and p-orbitals

  • hence:the p-orbital system can be treated independently from the s-orbital systemhowever:

  • the orbital energies of the p-system are affected by the s-electron distribution

  • and vice-versa

Hik

interaction energy between

adjacent pp orbitals:bCC = b uniform interaction parameter for C-atoms in p-system

Hij

pp interaction energy

involving heteroatoms:in simplest approach bCX = b

Him

interaction energy between

non-adjacent pp orbitals:in simplest approach bC…C = 0

in refined approach:

bCX = kCXb ; typically

kCX < 1 (prop. SCX/SCC)

p

p

refined approach:

bim = kimb ;e.g.kim =

typically: b1,3 ~ 0.3 b ; b1,4 = 0

Hjj

pp orbital energy

for heteroatom :Hjj = a + hjb

Hii

pp orbital energies

Hkk

- characteristic for given atom- modulated by local s-electron density- modulated by p-electron density

  • - specified with reference to aC

  • modulation in units of b- hi numerical parameterhi > 0 : atom more el.neg. than C

  • hi < 0 : atom less el.neg. than C

in simplest HMO approach:Hii = Hkk = aC=aa : uniform energy parameter for pp-AO of C-atoms in p-systems

in refined HMO models:Hii = ai + hi bai: dependent on local topology

hi : numerical parameter (small)

b : all energy corrections in b units


Oc iv

p

S1,2 ~ 0.25

p-overlap integrals are relatively small

therefore, they are neglected in the eigenvalue problem

p

S1,3 ~ 0.08

the typical eigenvalue problem of the LCAO MO approach is:

H11 - e

H12 - eS12

H1n - eS1n

H21 – eS21

H22 - e

H2n – eS2n

= 0

. . .

Hn1 – eSn1

Hn2 – eSn2

Hnn – e

this simplifies in the ZOA to

H11 - e

H12

H1n

H21

H22 - e

H2n

= 0

. . .

Hn1

Hnn – e

Hn2

with a- and b-parameters of the HMO schemethis transforms into

a- e

b

0

b

a- e

0

= 0

. . .

0

a– e

0

dividing by the universal b parameter andsubstituting –x = (a – e) / b, results in

b

b

-x

0

1

b

e.g., for acrolein (above) in the standard (simple) approximation:

1

2

1

-x

0

3

4

a

aO

= 0

. . .

a

a

-x+1

0

0

1

aO = a + b

-x

1

1

0

0

0

-x

= 0

giving the x-polynomial:

1

-x

1

0

x4 – x3 – 3x2 + 2x + 1 = 0

the solutions x1, x2, …, xn ofthe polynomial in x provides the

eigenvalues (p-CMO energies)

viaei = a + xib

0

1

0

-x

with the solutions:

x1 = 1.88 : e1 = a + 1.88 b

x2 = 1.00 : e1 = a + b

back-substitution of xi into the above linear equations

provides the relative expansion coefficients for CMO yi

x3 = -0.35 : e1 = a - 0.35 b

x4 = -1.53 : e1 = a - 1.53 b


Oc iv

pCO

pCC

*

*

a - b

a - b

0.707

-0.707

a –0.618b

de = 1b

de = 0.618b

0.851

-0.526

symmetrical

orbital splitting

in ZOA

DE = 0

a

a

H = 1b

symmetrical

orbital splitting

in ZOA

DE = 1b

H = 1b

de = 1b

pCC

a + b

a + b

de = 0.618b

0.707

0.707

pCO

a +1.618b

a +2b

0.526

0.851

electron distribution in 2-center LCAO MO in the ZOA:

yp = c1f1 + c2f2

2

2

2

2

2

yp = c1f1 + 2c1c2f1f2 + c2f2

2

2

2

2

2

ypdv = c1f1dv + 2c1c2f1f2dv + c2f2 dv

1

0 (ZOA)

1

2

2

2

yp = c1 + c2 = 1

hence:

normalization condition in the ZOA

note:

there is no overlap population in the ZOA!

in its place, one has to resort to ‚bond orders‘ to discuss bonding or antibonding character

for example (for C=C and C=O):

bond order p12 = 2c1c2

*

*

p(pCC) = -1

p(pCO) = -0.895

for multi-orbital system:

occ

p(pCC) = 1

p(pCO) = 0.895

2

Nel = ∑ niyi dv

i

occ

= ∑ ∑ni qK

= ∑ qK

i

i

qK : partial p-AO population in yi at center K

i

K

K

qK : total p-electron population at center K

occ

i

pKM =∑ ni pKM

i

pKM : partial p-bond order in yi between centers K and M

pKM : total p-bond order between centers K and M

i


Oc iv

2-center C=X p-system with varying aX = a + hXb

0.707

-0.707

0.788

-0.615

0.851

-0.526

0.894

-0.447

0.924

-0.383

a - b

0.944

-0.331

de = 1.00b

de = 0.78b

de = 0.62b

de = 0.50b

de = 0.41b

de = 0.35b

a

DE = 0b

DE = 0.5b

DE = 1.0b

DE = 1.5b

DE = 2.0b

a + b

DE = 2.5b

0.707

0.707

a +2b

0.615

0.788

0.526

0.851

0.447

0.894

a +3b

..

0.383

0.924

..

0.331

0.944

Note:

The electrophilic character of the C=X p-system increases with increasing electronegativity of X, i.e. decreasing energy of the fX AO.

The increased electrophilicity manifests itself through

- the increased lowering of the p*-orbital of the C=X system

- the increased amplitude at the electrophilic C center in the p*-orbital

Thus, towards a given nucleophile with a relatively high-lying occupied orbital, e.g., the nN-dominated CMO of an amine or highest occupied MO (HOMO) of an enamine (see below), the possible coupling effect through intermolecular interaction of this HOMO with the p*-orbital of the C=X system increases with decreasing energy gap DEHOMO-p* and increasing p*-orbital amplitude at the C center of the C=X system.

Protonation (or complexation by a Lewis acid) of the O-atom in the s-plane of the C=O system results in a marked lowering the fO level and concomitant

increase of the p-electrophilicity of the C=O system.

The p-MO systems of the C=X units are useful orbital building blocks for the derivation of the p-orbital structures of more complex p-systems

using the extended perturbation MO (EPMO) method.


Oc iv

a -2b

a -2b

a - b

a - b

*

a

a

1

1

√2

√2

a + b

a + b

1

1

1

1

a +2b

a +2b

2

2

2

2

0.71

0.71

cp*p =0.52

cpp* =0.52

2.0

2.0

1

1

1

1

√2

√2

√2

√2

-

(fC1 + fC3)

(fC1 - fC3)

y2 = jC…C

-

DE = 0 bH = 2·1/√2 ~ 1.41b

+

-

jC…C =

jC…C =

Two approaches to the allyl system

A: formal union of C=C + C

*

y3 ~ pCC - 0.52 fC + 0.18 pCC

fC-induced mixing of p into p*:

pCC

*

DE = 1bH = 0.707b

de = 0.37b

c* = 0.52

*

y2 ~ fC - 0.52 pCC + 0.52 pCC

fC

note: exact cancellation

of orbital amplitude

DE = 1bH = 0.707b

de = 0.37b

pCC

c* = 0.52

note: build-up of amplitude of

equal absolute size at allylic center

fC-induced mixing of p* into p:

*

*

y1 ~ pCC + 0.52 fC + 0.18pCC

B: formal union of C1… C3 + Ccentral

+

a - 1.41 b

y3 = ( jC…C - fC2 )

symmetry-adapted

group

orbitals

fC2

note that fC2 interacts exclusively with jC…C

de = 1.41b

+

c* = 1.00

+

y1 = ( jC…C + fC2 )

a + 1.41 b


Oc iv

a -2b

a -2b

a - b

a - b

a

a

a + b

a + b

a +2b

a +2b

rel

ksolv, (allyl) = 15

rel

ksolv, (propyl) = 1

+

-

-

jC…C

jC…C

jC…C

jC…C

+

+

+

+

chemical associations with allyl orbital interaction schemes

pCC

pCC

pCC

*

*

*

symmetricsplitting in

2-center3-el

sytem in ZOA

repulsion in

2-center-4-el

sytem notcounted in ZOA

pCC

pCC

pCC

stabilization of anion

by allyl resonance

stabilization of cation

by allyl resonance

stabilization of radical

by allyl resonance

DEp ~ 2 ·0.4 b

DEp ~ 2 ·0.4 b

DEp ~ 2 ·0.4 b

in ZOA:

:B

C=C-assisted solvolysis

(45°C, H2O/EtOH):

C=C-assistedhomolytic bond cleavage:

C=C-promotedC-H acidity:

94.5

kcal/mol

82.3

kcal/mol

C-H acidity (DHº, gas):CH3CH2-H 420.1

CH2=CH-H 407.5

CH2=CH-CH2-H 390.8

(via SN2 not SN1 ?)

disrotatory process thermally

‘allowed’; stereochemistry

experimentally confirmed

at low temperature.

sCC

sCC

*

*

sCX

*

+

conrotatory process

thermally ‘forbidden’;

experimentally not

observed

SbF5, SO2ClF

-100ºC, by NMR

pC2

pC2

Experimentally, no cyclopropyl cation

intermediate can be observed; thus,

C-X solvolysis and ring opening may

occur in a synchronous fashion; for

transparent orbital analysis, the two

processes are treated sequentially.

ground state

correlates with

doubly excited state

nX

nX

+

solvolysis

of C-X

sCC

sCC

+

+

no inter-

action by

symmetry

sCX

disrotatory

ring opening

conrotatory

ring opening


Oc iv

..

a -2b

a -2b

pCC

pCC

*

*

a - b

a - b

a

a

a + b

a + b

a +2b

a +2b

0.71

0.71

cp*p = 0.52

cp*p = 0.71

2.0

2.0

..

enamine and enolether p-systems

*

de2

a - 1.19 b

fN-induced p*-mixing into p

reduces amplitude at Caand

augments amplitude at Cb

de2 = 0.19b

DE = 2.5 b

c* = 0.26

H = 0.707 b

y2 = pCC - 0.71 fN - 0.25 pCC

*

a + 0.5 b

de1

pCC

DE = 0.5 b

de1 = 0.50b

fN

a + 1.5 b

H = 0.707 b

c* = 0.71

de1

de2

*

y1 = fN + 0.71 pCC + 0.26 pCC

a + 2.19 b

Note: CMO’s approximated by EPMO method

are unnormalized to show mixing effects

de2

a - 1.16 b

*

de2 = 0.16b

DE = 3.0 b

c* = 0.22

H = 0.707 b

fN-induced p*-mixing into p

reduces amplitude at Caand

augments amplitude at Cb

y2 = pCC - 0.52 fO - 0.18 pCC

*

a + 0.63 b

pCC

de1

DE = 1.0 b

de1 = 0.37b

H = 0.707 b

c* = 0.52

fO

a + 2.0 b

de1

de2

a + 2.53 b

*

y1 = fO + 0.52 pCC + 0.22 pCC


Oc iv

*

pcc

the enol ether p-system

orbital interactions and mixing effects

0.707

c*pp* = 0.224

2.0

pCC mixes from belowinto pCC, thus enhancingthe antibonding characterwith fO

a – 1.16 b

*

y3 ≈ p* – 0.22 fO+ 0.08 p

a – b

a - b

Hfp* = 0.707 b

de2 = 0.16 b

c* = 0.22

DEfp* = 3.0 b

0.707

a

c*p*p = 0.518

2.0

DEpp* = 2 b

*

pCC mixes from above into pCC, thus enhancingthe bonding characterwith fO

a + 0.63 b

y2 ≈ p – 0.52 fO– 0.18 p*

pcc

a + b

Hfp= 0.707 b

de1 = 0.37 b

a + b

c* = 0.52

DEfp= 1.0 b

a + 2.0 b

fO

a + 2b

a + 2.53 b

y1 ≈ fO+ 0.52 p + 0.22 p*

polarization of y2 by admixture of p*

in a bonding mode to fO as p* admixes from above

polarization of y2

0.51

0.45

0.73

normalized amplitudes in y2

prior to polarization:

0.63

0.46

0.63

normalized amplitudes

in y2 after to polarization

HOMO-controlled electrophilic attack (by soft electrophile) occurs at Cbof enol ether.

Note that the large amplitude at Cb in the HOMO of the enol ether p-system arises

from polarization of the C=C double bond by the O-p lonepair, not from p-el.transfer!

(see next 2 slides)


Oc iv

*

pcc

..

the enol ether p-system

how much p-charge transfer from X into CC p-system?

generalized orbital interactions and mixing effects

assuming fX to lie below pCC-level

induced mixing effects

y3 ≈ p* – d* fX+ b* p

a - b

a – b

direct mixing effects

a

induced mixing effects

y2 ≈ p – c* fX– a* p*

pcc

a + b

direct mixing effects

a + b

fX

a + 2b

y1 ≈ fX+ c* p + d* p*

direct mixing effects

Net p-charge transfer arises only from the interaction of the

doubly occupied fX with the unoccupied p*CC orbital;

hence, net p-charge transfer can be estimated to be ≤ 2d*2 .

For a more quantitative estimate, the atomic p-charges from

the normalized p-orbitals y1 and y2 have to be considered:


Oc iv

a - b

*

pcc

a + b

a + hXb

p

p

p

p

qCC

qCC

qX

qX

2

2

2

2

2

2

2

2

N2

N1

N1

N2

induced mixing effects

y3 ≈ p* – d* fX+ b* p

direct mixing effects

a – b

induced mixing effects

a

2

y2 ≈ p – c* fX– a* p*

N2 = 1 + c*2 + a*2

pcc

a + b

direct mixing effects

fX

2

y1 ≈ fX+ c* p + d* p*

N1 = 1 + c*2 + d*2

direct mixing effects

(1) + (c*2)

total p-charge in fX unit:

=

2

2

(1 + 2c*2 + a*2)

(1 + c*2 + a*2 + c*2 + …) ≈

2

2

2

2

N1

N2

N1

N2

2

2

(1 + 2c*2 + a*2) -

N1

N2

p

2

dqX= - 2

net charge transfer from fX:

2

2

N1

N2

- 2d*2

(1 + 2c*2 + a*2) -

(1 + 2c*2 + a*2 + d*2)

2

(1 + 2c*2)

(1 + 2c*2 + a*2 + d*2)

(c*2 + d*2) + (1 + a*2)

=

total p-charge in CC-p-unit:

2

2

(1 + 2c*2 + a*2 + 2d*2)

(c*2 + d*2 + 1 + a*2 + c*2 + d*2 + …) ≈

2

2

2

2

N1

N2

N1

N2

2

2

(1 + 2c*2 + a*2 + 2d*2) -

N1

N2

p

2

dqCC = - 2 ≈

net charge transfer into CCp:

2

2

N1

N2

+ 2d*2

(1 + 2c*2 + a*2 + 2d*2) -

(1 + 2c*2 + a*2 + d*2)

2

(1 + 2c*2)

(1 + 2c*2 + a*2 + d*2)

for the specific example of the enol ether, net p-charge transfer is estimated to be

.

.

p

dq (X→CC)≤ 2 0.2182 / (1 + 2 0.5182) = 0.062; hence, not more than ca. 3%


Oc iv

a -2b

a -2b

-

-

a - b

a - b

-

a

a

1

1

1

1

√2

√2

√2

√2

a + b

a + b

exact solution: a - √2 b

exact solution: a +√2 b

1

1

1

1

a +2b

a +2b

2

2

2

2

pCO

*

comparison: allyl anion – carbanion a to C=O p-system

y3 = pCC - 0.52 fC + 0.18 pCC

*

0.71

c*pp* = 0.52

2.0

fC-induced mixing

of p into p*

pCC

*

DE = 1bH = 0.707b

de = 0.37b

c* = 0.52

..

0

fC

*

y2 = fC - 0.52 pCC + 0.52 pCC

DE = 1bH = 0.707b

de = 0.37b

fC-induced mixing

of p* into p

pCC

c* = 0.52

0.71

c*pp* = 0.52

2.0

*

y1 = pCO + 0.52 fC + 0.18 pCO

from exact HMO-solution of allyl system:

net p energy stabilization: ~ 2 · 0.4 b = 0.8 bnet p charge shift from fC to C=C: ~ - 0.5

Note: CMO’s approximated by EPMO method

are unnormalized to show mixing effects

net p energy stabilization: ~ 2 · 0.6 b = 1.2 b

net p chargeshift from fC to C=O: ~- 0.57

y3 = pCO - 0.70 fC + 0.17 pCO

*

0.53

0.85

a - 1.22 b

c*pp* = 0.70

2.24

fC-induced mixing

of pCO into pCO

-0.53

DE = 0.62bH = 0.85b

de = 0.60b

*

a – 0.62 b

c* = 0.70

..

fC

a + 0.44 b

*

y2 = fC - 0.30 pCO + 0.70 pCO

DE = 1.62bH = 0.53b

de = 0.16b

c* = 0.30

fC-induced mixing

of pCO into pCO

pCO

*

a + 1.62 b

0.85

c*pp* = 0.30

a + 1.78 b

2.24

0.53

0.85

*

y1 = pCO + 0.30 fC + 0.11 pCO

*

*

Note: the pCO orbital lies at a lower energy and has a larger amplitude at C than the pCC;

likewise, the energy pCO is lower and its amplitude at C is smaller compared to the pCC;

these combined factors result in a net downshift of the fCa to C=O to produce the CMO y2 with net bonding amplitudes (positive partial p bond order) between the two C atoms.


Oc iv

a -2b

a -2b

a - b

a - b

a

a

a + b

a + b

a +2b

a +2b

comparison: amide and ester p-systems

net p energy stabilization: ~ 2 · 0.3 b = 0.6 b

net p chargeshift from fN to CO:~- 0.13

the C-N torsion barrier disrupting N…C=O p conjugationis typically 18-20 kcal/mol

..

0.85

-0.53

y3 = pCO - 0.35 fN + 0.08 pCO

a – 0.92 b

*

de2

a - 0.62 b

0.53

pCo

*

c*pp* = 0.35

2.24

DE = 2.12 b

de = 0.30 b

fN-induced mixing

of pCO into pCO

H = 0.85 b

c* = 0.35

*

DE = 0.12 b

de = 0.47 b

H = 0.53 b

c* = 0.89

*

y2 = fN - 0.89 pCO + 0.35 pCO

de2

fN-induced mixing

of pCO into pCO

a + 1.33 b

de1

*

a + 1.5 b

pCO

0.85

fN

a + 1.62 b

c*pp* = 0.89

de1

2.24

a + 2.09 b

0.53

*

y1 = pCO + 0.89 fN + 0.34 pCO

0.85

Note: CMO’s approximated by EPMO method

are unnormalized to show mixing effects

..

fN-induced mixing

of pCO into pCO

net p energy stabilization: ~ 2 · 0.25 b = 0.5 b

net p chargeshift from fO to C=O:~- 0.11

*

0.53

c*pp* = 0.30

2.24

y3 = pCO - 0.30 fO + 0.07 pCO

*

0.85

-0.53

de2

a – 0.87 b

fN-induced mixing

of pCO into pCO

a - 0.62 b

pCo

*

*

de = 0.25 b

DE = 2.62 b

0.85

c* = 0.30

H = 0.85 b

c*pp* = 0.70

2.24

*

y2 = pCO - 0.70 fO - 0.27 pCO

DE = 0.38 b

de = 0.37 b

H = 0.53 b

c* = 0.70

a + 1.25 b

de1

pCO

a + 1.62 b

a + 2.0 b

fO

0.53

de1

0.85

de2

a + 2.62 b

*

y1 = fO + 0.70 pCO + 0.30 pCO


Oc iv

a -2b

a -2b

a - b

a - b

a

a

1

1

1

1

1

1

√2

√2

√2

√2

√2

√2

a + b

a + b

a +2b

a +2b

1,3-butadiene: from 2 conjugated ethylene p-systems

y4

induced

mixing

de2

a - 1.62 b

de1

y3

p1,CC

p2,CC

*

*

de1

a - 0.62 b

de2

*

pCC - pCC

DE = 2.0 b

de2= 0.12 b

induced

mixing

H = 0.5 b

c* = 0.24

DE = 0.0 b

de1= 0.50 b

induced

mixing

pCC - pCC

H = 0.5 b

c* = 1.00

de2

a + 0.62 b

de1

p2,CC

p1,CC

de1

de2

y2

a + 1.62 b

net p-energy stabilization: ~ 2 · 2 de2 = 0.47 b

induced

mixing

Note that the closed-shell (overlap) repulsion effect due to the pCC – pCC interaction is neglected in the ZOA; hence the net p energy stabilization is overestimated: the trans → cis torsional barrier is ca. 5 kcal/mol.

y1

PE spectrum of 1,3-butadiene: IP1 = 9.03 eV, IP2 = 11. 46 eV; hence b ~ 2.4 eV

Note that b parameter cannot be transferred from spectroscopy to thermodynamic properties

Note the build-up of a large LUMO amplitude at the Cb

position to the O=C group in acrolein (Michael addition)

de3

a - 1.49 b

de4

0.851

0.65

-0.58

p2,CC

*

DE = 0.38 b

de4= 0.44 b

*

*

pOC - pCC

H = 0.60 b

c* = 0.73

p1,OC

*

de4

a - 0.37 b

DE = 2.62 b

de3= 0.05 b

*

pOC - pCC

de2

H = 0.37 b

c* = 0.14

*

*

y3 = pOC + 0.73 pCC - 0.33 pCC - 0.03 pOC

DE = 1.62 b

de2= 0.19 b

*

pOC - pCC

H = 0.60 b

c* = 0.33

*

*

y2 = pCC - 0.47 pOC + 0.33 pOC - 0.00 pCC

DE = 0.62 b

de1= 0.18 b

de1

pOC - pCC

p2,CC

de2

H = 0.37 b

c* = 0.47

a + 0.99 b

0.526

The EPMO-estimated p-energy levels

may be compared to the exact HMO-

energies given on slide 2 of this Chapter

de1

p1,OC

de3

a + 1.85 b

net p-energy stabilization: ~ 2 · (de2 + de3) = 0.48 b

thus, essentially the same as for 1,3-butadiene;indeed, the trans → cis torsional barrier for acrolein is essentially the same as for 1,3-butadiene.


Oc iv

a -2b

a -2b

a - b

a - b

a

a

1

1

1

1

√2

√2

√2

√2

a + b

a + b

a +2b

a +2b

*

sCC

1,3-butadiene: from symmetry-adapted group orbitals

0.372

0.602

-

-

y4 = jin - 0.62 jout

A

a - 1.62 b

-0.372

+

+

y3 = jout - 0.62 jin

A

0.602

j- = (f2 - f3)

j- = (f1 – f4)

a - 0.62 b

in

S

out

A

DE = 1.0 b

de2= 0.62 b

H = 1.0 b

c* = 0.62

S

A

j+ = (f2 + f3)

j+ = (f1 + f4)

a + 0.62 b

in

out

0.602

S

-

-

y2 = jout + 0.62 jin

-0.372

a + 1.62 b

S

+

+

y1 = jin + 0.62 jout

0.372

0.602

chemical association: thermal ring opening of cyclobutene occurs in conrotatory mode

*

sCC

A

S

y4

S

*

pCC

A

175 ºC

y3

pCC

j-

S

out

j+

A

out

175 ºC

S

y2

C2

pCC

A

y1

A

*

pCC

C2

sCC

sCC

S

conrotatory

ring opening


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