OCIV. 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 mathis@org.chem.ethz.ch
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Orbital Concepts
and
Their Applications in Organic Chemistry
Klaus Müller
Script
ETH Zürich, Spring Semester 2009
Chapter 5
psystems
HMO and extended PMO method
Lecture assistants:
Deborah Sophie MathisHCI G214 – tel. 24489mathis@org.chem.ethz.ch
Alexey FedorovHCI G204 – tel. 34709 fedorov@org.chem.ethz.ch
Sik
Hik
interaction energy between
adjacent pp orbitals:bCC = b uniform interaction parameter for Catoms in psystem
Hij
pp interaction energy
involving heteroatoms:in simplest approach bCX = b
Him
interaction energy between
nonadjacent 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 selectron density modulated by pelectron density
in simplest HMO approach:Hii = Hkk = aC=aa : uniform energy parameter for ppAO of Catoms in psystems
in refined HMO models:Hii = ai + hi bai: dependent on local topology
hi : numerical parameter (small)
b : all energy corrections in b units
S1,2 ~ 0.25
poverlap 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 bparameters 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 xpolynomial:
1
x
1
0
x4 – x3 – 3x2 + 2x + 1 = 0
the solutions x1, x2, …, xn ofthe polynomial in x provides the
eigenvalues (pCMO 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
backsubstitution 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
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 2center 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 multiorbital system:
occ
p(pCC) = 1
p(pCO) = 0.895
2
Nel = ∑ niyi dv
∫
i
occ
= ∑ ∑ni qK
= ∑ qK
i
i
qK : partial pAO population in yi at center K
i
K
K
qK : total pelectron population at center K
occ
i
pKM =∑ ni pKM
i
pKM : partial pbond order in yi between centers K and M
pKM : total pbond order between centers K and M
i
2center C=X psystem 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 psystem 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 highlying occupied orbital, e.g., the nNdominated 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 DEHOMOp* and increasing p*orbital amplitude at the C center of the C=X system.
Protonation (or complexation by a Lewis acid) of the Oatom in the splane of the C=O system results in a marked lowering the fO level and concomitant
increase of the pelectrophilicity of the C=O system.
The pMO systems of the C=X units are useful orbital building blocks for the derivation of the porbital structures of more complex psystems
using the extended perturbation MO (EPMO) method.
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
fCinduced 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: buildup of amplitude of
equal absolute size at allylic center
fCinduced 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 )
symmetryadapted
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
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
2center3el
sytem in ZOA
repulsion in
2center4el
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=Cassisted solvolysis
(45°C, H2O/EtOH):
C=Cassistedhomolytic bond cleavage:
C=CpromotedCH acidity:
94.5
kcal/mol
82.3
kcal/mol
CH acidity (DHº, gas):CH3CH2H 420.1
CH2=CHH 407.5
CH2=CHCH2H 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,
CX 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 CX
sCC
sCC
+
+
no inter
action by
symmetry
sCX
disrotatory
ring opening
conrotatory
ring opening
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 psystems
*
de2
a  1.19 b
fNinduced 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
fNinduced 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
pcc
the enol ether psystem
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
HOMOcontrolled electrophilic attack (by soft electrophile) occurs at Cbof enol ether.
Note that the large amplitude at Cb in the HOMO of the enol ether psystem arises
from polarization of the C=C double bond by the Op lonepair, not from pel.transfer!
(see next 2 slides)
pcc
..
the enol ether psystem
how much pcharge transfer from X into CC psystem?
generalized orbital interactions and mixing effects
assuming fX to lie below pCClevel
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 pcharge transfer arises only from the interaction of the
doubly occupied fX with the unoccupied p*CC orbital;
hence, net pcharge transfer can be estimated to be ≤ 2d*2 .
For a more quantitative estimate, the atomic pcharges from
the normalized porbitals y1 and y2 have to be considered:
*
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 pcharge 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 pcharge in CCpunit:
≈
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 pcharge transfer is estimated to be
.
.
p
dq (X→CC)≤ 2 0.2182 / (1 + 2 0.5182) = 0.062; hence, not more than ca. 3%
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 psystem
y3 = pCC  0.52 fC + 0.18 pCC
*
0.71
c*pp* = 0.52
2.0
fCinduced 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
fCinduced 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 HMOsolution 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
fCinduced 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
fCinduced 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.
a 2b
a  b
a  b
a
a
a + b
a + b
a +2b
a +2b
comparison: amide and ester psystems
net p energy stabilization: ~ 2 · 0.3 b = 0.6 b
net p chargeshift from fN to CO:~ 0.13
the CN torsion barrier disrupting N…C=O p conjugationis typically 1820 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
fNinduced 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
fNinduced 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
..
fNinduced 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
fNinduced 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
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
1,3butadiene: from 2 conjugated ethylene psystems
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 penergy stabilization: ~ 2 · 2 de2 = 0.47 b
induced
mixing
Note that the closedshell (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 1,3butadiene: 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 buildup 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 EPMOestimated penergy 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 penergy stabilization: ~ 2 · (de2 + de3) = 0.48 b
thus, essentially the same as for 1,3butadiene;indeed, the trans → cis torsional barrier for acrolein is essentially the same as for 1,3butadiene.
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
1,3butadiene: from symmetryadapted 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