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1
Variational principle
The complete set of exact eigenfunctions of H define an orthogonal complete basis set for the total space of wave
functions.
H is Hermitic. Let be a and b two normalized wavefunctions (associated with two different values Ea and Eb ). We have therefore (due to hermiticity)
< a |H| b > = Ea< a | b > < a |H| b > = Eb < a | b >
Wherefrom (Ea - Eb) < a | b > = 0and since Ea Eb
< a | b > = 0
If so, it is possible to express any function as a linear combination of the exact eigenfunctions, .
Statement 1:
demonstration:
Consequence:
2
Variational principle
The energy associated with a function is always above that of the eigenfunction of lowest energy: E0.
is not a eigenfunction; it is associated with an energy <E> that is an average energy (mean value)
A mean value is always intermediate relative to extreme:Greater than the smallest!
<E> > E0
Statement 2:
3
Mean value
• If 1 and 2 are associated with the same eigenvalue o: O(a1 +b2)=o(a1 +b2)
• If not O(a1 +b2)=o1(a1 )+o2(b2)
we define <O> = (a2o1+b2o2)/(a2+b2)
Dirac notations
4
Variational principle
The energy associated with a function is always above that of the eigenfunction of lowest energy: E0.
and are two eigenfunctions < |H| = < | E and |H| > = E| >
wherefrom < |H| > = E From statement 1, is a linear combination of s
| >= c| >
let multiply the left by < |; this leads to only one term c= < | > and similarly c*= < | >
then | >= < | > | >
Statement 2:
5
Variational principle
and are two eigenfunctions associated to E and E
From statement 1, is a linear combination of s
| >= c| >
then
< | > = c < |> c
< | > = c 2
< |> = c 2
= 1
Statement 2, normalization:
6
< |H | > = c < |H | > c
< |H | > = E < | > c
< |H | > = E c c
< |H | > = E c 2 > Ec
2 = E
E > E
Variational principleStatement 2, Demonstration:
An non-exact solution has always a higher energy than the lowest exact solution
7
and are two eigenfunctions < |H| = < | E and |H| > = E| > wherefrom < |H| > = E
From statement 1, is a linear combination of s | >= c| >
Let multiplie the left by < |; this leads to only one term c= < | >
and similarly c*= < | > then < | > = c < |> c
< | > = < | > < |> < | > = < | > < | >
< | > = |< | >| 2 =1
Demonstration < |H | > = < | > < |H | > < | >
< |H | > = E < | > < | > < | >
< |H | > = E < | > < | >
< |H | > = E |< | >| 2 > E|< | >| 2 = E
< |H | > > E
Using Dirac notation
8
Variation principleGiven an approximate expression that depends on
parameters, we must determine the parameters to minimize the energy.
Within LCAO, an MO is a linear combination of AOs: We have then to chose the coefficients so to minimize the energy.
Note that the variational principle is not restricted to cases of linear combinations.
10
Slater exponent for He
where Z* is a parameter
<E> = - Z*2/2 u.a. <T> = + Z*2/2 u.a.
<V> = <(1,Z*)|-Z*/r|(1,Z*)>
using <(1,Z*)|1/r|(1,Z*)> = Z* u.a.
<V> = -Z*<(1,Z*)|1/r|(1,Z*)> = -Z*2u.a.
Next slide
14
LCAO
is a linear combination of N s
| >= c| >
an approximate function
Using parameters: c
Atomic orbitals
Worse description than
There are N coeffcients to determineLet minimize E with respect to each of themTherefore, there is a set of N equations, E/c = 0
This seems fine; however there is a problem; Which is the problem?
15
LagrangianThere is a constraint due to normalization: the
N coefficients are not independent.
Joseph Louis de Lagrange French1736 - 1831
We search the minimum of a function L of N+1 variablesWhich “makes no difference with H”
L = < | > - E [< | > - 1 ]
There is a set of linear expressions to derive
L/c = 0 Should be 0
16
Secular determinant
Solving | Hij-ESij| =0, we find the Eis.
This is a N degree equation of E to be solved.
From N AOs, we find N energies and N MOs.
We find simultaneously E and (eigenfunction and eigenvalues).
17
LCAO
Assuming no change in the AOs *, the MOs correspond to a unitary change of the basis set of AOs.
MOs are orthogonal and normalized.
Neglecting overlaps, rc2ir=1 for a MO, i
and ic2ir=1 for a AO, i ; the AOs are distributed
among the MOs
* This is not the case for self consistent methods.
Sum over r atoms
for a given iorbital
Sum over iorbitals for a given r atom
18
Schrödinger equation for LCAO
Erwin Rudolf Josef Alexander SchrödingerAustrian 1887 –1961
H = E
H icii = Eicii Let multiply on the left by j, integrate and develop
ici< |H | i > = Eici < j |i >
ici (Hij-E Sij ) = 0This is the set of linear equation solved for | Hij-E S ij |= 0 secular determinant
The complete set of MOs are solution (not only the lowest).Our first approach from the variational principle emphasized the solution of lowest energyEvery extreme (derivative=0) is a physical solution!
19
Hückel Theory
Erich Armand Arthur Joseph Hückel German 1896-1980Never awarded the Nobel prize!Also known for Debye-Hückel theory of electrolytic solutions
In the 1930's a theory was devised by Hückel to treat the electrons of conjugated systems such as aromatic hydrocarbon systems, benzene and naphthalene. Only electron MO's are included because these determine the general properties of these molecules and the electrons are ignored. This is referred to as sigma-pi separability. The extended Hückel Theory introduced by Lipscomb and Hoffmann (1962) applies to all the electrons.
20
Hückel Theory
Erich Armand Arthur Joseph Hückel German 1896-1980Never awarded the Nobel prize!Also known for Debye-Hückel theory of electrolytic solutions
We consider only 2pZ orbitalsTwo parameters:The atomic 2p energy level:
E(2pZ) = (~ -11.4 eV)The resonance integral for adjacent 2p orbitals
Hij(C=C) = (~ -3 eV)
E = + x x= ( – E)/
We chose defining as negative
Sij = ij (the overlap is neglected).
21
SLATER KOSTER
Slater Koster for metal atoms : 9 orbitals s, p et d
Several overlaps
ss, sp, sd,
pp, pp, pd, pd,
dd, dd, dd.
Phys. Rev B 94 (1954) 1498
22
EHT Theory
• Only valence orbitals are described• AOs are Slater orbitals
– Parameters are the Hii (atomic energy levels)– The Slater exponents
• The Overlap, Sij, are rigorously calculated from the geometry and the AOs
• The resonance integrals, Hij, are derived from the Sij. Wolfsberg-Helmoltz formula
Hij = 1.75 (Hii+Hjj)/2 Sij
23
For linear and cyclic systems (with n atoms), general solutions exist.
Linear:
Cyclic: For linear polyenes the energy gap is given as:
Polyenes
Charles Alfred Coulson (1910-1974), English
24
Extension to heteroatomsparameter Coulombic
IntegralResonance
Integral
Sulfurwith 2 electrons
S = + 0,5
CS = 0,4
Sulfurwith 1 electron
S = + 0,2
CS = 0,6
Fluorine F = + 3
CF = 0,7
Chlorine Cl = + 2
CCl = 0,4
Bromine Br = + 1,5
CBr = 0,3
CH3 grouphyperconj
ugation
Me = + 2
CMe = 0,7
parameter CoulombicIntegral
Resonance
Integral
Carbon C = CC =
Oxygen 1 electron
O = +
CO =
Oxygen 2 electrons
O = + 2
CO = 0,8
Nitrogen 1 electron
N = + 0,5
CN =
Nitrogen 2 electrons
N = + 1,5
CN = 0,8
Sulfur 2 electrons
S = + 0,5
CS = 0,4
Sulfur 1 electron
S = + 0,2
CS = 0,6
25
| Hij-E Sij |= 0
1 2 3 4 5 6 …
1 -x 1 or 0 1 or 0 1 or 0 1 or 0 1 or 0
2 1 or 0 -x 1 or 0 1 or 0 1 or 0 1 or 0
3 1 or 0 1 or 0 -x 1 or 0 1 or 0 1 or 0 = 0
4 1 or 0 1 or 0 1 or 0 -x 1 or 0 1 or 0
5 1 or 0 1 or 0 1 or 0 1 or 0 -x 1 or 0
6 ::
1 or 0 1 or 0 1 or 0 1 or 0 1 or 0 -x
1 means that the atoms 2 and 4 are connected, The determinant is symmetric relative to the diagonal
26
| Hij-E Sij |= 0
1 2 3 4 5 6 …
1 -x 1 or 0 1 or 0 1 or 0 1 or 0 1 or 0
2 1 or 0 -x 1 or 0 1 or 0 1 or 0 1 or 0
3 1 or 0 1 or 0 -x 1 or 0 1 or 0 1 or 0 = 0
4 1 or 0 1 or 0 1 or 0 -x 1 or 0 1 or 0
5 1 or 0 1 or 0 1 or 0 1 or 0 -x 1 or 0
6 ::
1 or 0 1 or 0 1 or 0 1 or 0 1 or 0 -x
This is an N degree equation of x
27
| Ei-x |= 0solving is a diagonalization problem
1 2 3 4 5 6 …
1 E1-x 0 0 0 0 0
2 0 E2-x 0 0 0 0
3 0 0 E3-x 0 0 0 = 0
4 0 0 0 E4-x 0 0
5 0 0 0 0 E5-x 0
6 ::
0 0 0 0 0 E6-x
A set of first degree equations: E=Ei associated with i
28
The diagonalization is a unitary transformation
OMs are orthogonal thus rcricrj= ijIf i=j, the sum of the coefficient’s squares for a given atom is 1 (normalization)If i≠j, the sum of the products of coefficients is 0 (orthogonality)
The matrix of coefficients is unitary
wherefrom = icricsj= ijIf i=j, the sum of the coefficient’s squares for a given orbital is 1AOs are distributed among all the MOsIf i≠j, the sum of the products of coefficients is 0
If all the orbitals were filled, there would be no interaction: all the r-s bond indices should be zero.
29
C2H4 2 e in 2 orbitals as in H2
We have solved the problem using symmetry
and without solving the secular equation.
1 2
1 -x 1 = 0
2 1 -x
Inputs are what is in the secular determinant : and connectivity
Outputs are orbitals, energies, total energies, charges and bond indices
30
C2H4 without overlap
1 2
1 -x 1 = 0
2 1 -x
u Bonding g antibonding
Energy x=1 x=-1
coefficients
-x c1 + c2 = 0 C=1/√2
-c1 + c2 = 0
c1 = c2
c1 + c2 = 0
c1 = -c2
Charge on atoms:
1 – i i cri2
0 0
Bonding
lrs= i i cricsi
1 for u2 0 for ug
-1 for u2
N-1 linear equations; the last one is redundant.
31
C2H4 including overlap 4e repulsion
1 2
1 -E -ES = 0
2 -ES -E
u Bonding g antibonding
Energy / (1+S) / (1-S)
coefficients c1 = c2 = c
c = 1 /√(2+2S)
c1 = -c2 = c
c = 1 /√(2+2S)Charge on atoms:
1 – i i (cri2 –s
cri csi )
0 0
Bonding
OPrs= i i cricsiSrs
S/(1+S)
for u2
S/(2+2S)- S/(2-2S)
= -4S2/(4-4S-4S22) ~ - S2
for ug
E2 = (-ES)2 E = ± (-ES)
Mulliken
Overlap populations
32
2 OA interaction modifying
1 2
1 /2 -E = 0
2 /2 -E
(/2 -E) (-/2 -E) = 2 E2 - (2 /4) = 2
E2 = 2 + (2 /4)
E = ±√[2 + (2 /4)] •If , E+ =
•If <<, E+ = (/2) (1 + 4 2/ 2) 0.5
E+ = (/2) (1 + 2 2/ 2) = /2 (1 + 2 2/
E+ - E = 2/ 2nd order Perturbation
term
2/
2/
The geometric mean
of and
33
ButadieneC1
C2
C3
C4
The topology is C1 bond C2, C2 bond to C3 and C3 bond to C4
A linear model contains the information with more symmetry (the topology does not distinguish between cis and trans)
1 2 3 4
1 -x 1 0 0
2 1 -x 1 0 = 0
3 0 1 -x 1
4 0 0 1 -x
C1 C2 C3 C4
34
Conservation of Orbital Symmetry
H C Longuet-Higgins E W Abrahamson
Hugh Christopher Longuet-Higgins1923-2004
The Molecular orbitals are solution of the symmetry operators of the molecule.MOs from different symmetry groups do not mix.
36
ButadieneSymmetric C1 C2 C3 C4
S
x2 - x - 1 = 0 x = (1 ± √5)/2 Golden numbers 1.618 and -0.618Coefficients - (1 ± √5)/2 c1 + c2 = 0 and normalization c1
2 + c22 = ½
c = 0.3717 and 0.6015
Mind that symmetry 1 is for all the atomsNot the reduced part
-0.6181.618
= 0
40
x2 = 1+x
The medial right triangle of this "golden" pyramid (see diagram), with sides is interesting in its own right, demonstrating via the Pythagorean theorem the relationship or
41
Fibonacci recursion,irrational numbers (incommensurable)
In the "zeroth" month, there is one pair of rabbitIn the first month, the first pair begets another pairIn the second month, both pairs of rabbits have another pair, and the first pair dies.In the third month, the second pair and the new two pairs have a total of three new pairs, and the older second pair.
Rabbit population, assuming that:
42
Spirale constructions,
in nature,in music..
Related ?The Doctrine of the Mean (中庸 ,
py Zhōngyōng) is one of the Four Books, part of the Confucian
canonical scriptures.
44
ButadieneSymmetric C1 C2 C3 C4
S
Recipe to build the same determinant
Express along the AOs and gather interactions with equivalent atomsMind that symmetry reduction makes the normalization incomplete.
45
ButadieneAntisymmetric C1 C2 C3 C4
A
x2 + x - 1 = 0 x = - (1 ± √5)/2 Golden numbers 0.618 and -1.618Coefficients (1 ± √5)/2 c1 + c2 = 0 and normalization c1
2 + c22 = ½
c = 0.3717 and 0.6015
Mind that symmetry 1 is for all the atomsNot the reduced part
0.618 -1.618
46
ButadieneAntisymmetric C1 C2 C3 C4
A
Recipe to build the same determinant
Express along the AOs and gather interactions with equivalent atomsMind that symmetry reduction makes the normalization incomplete.
47
ButadieneSmall 0.3717
Large 0.6015Small 0.3717
Large 0.6015
Large 0.6015
Small 0.3717
The number of nodes increases, the amplitude is large at the middle for the extreme; it is large on the edges for the Frontier orbitals.
480.3717
0.3717
0.6015
0.6015
0.6015
-.60150.6015
0.6015
-.6015
0.6015
-.3717
-.3717
-.3717
0.3717
-.3717
1.618
0.618
-.618
-1.618
Butadiene
49
Bonding orders Ground state
Butadiene
0.894 0.8940.447
No initial statement that distinguishes 1-2 from 2-3
50
Bonding orders Ground state
Butadiene
1.35Ả
No initial statement that distinguishes 1-2 from 2-3Matches the Lewis formula and indicates the stabilization due to delocalization
1.35Ả1.46Ả
C=C – C=C
52
It is no use to iterate
The Hückel method is powerful since predictive.It is possible to make the parameterization
dependent on the differences in bond lengths choosing 12 = 34 larger than 23 or modifying according to the charge.
This does not make the calculation predictive.It is possible to make iterations ( technique) to
make the results “self consistent”; this is not recommended. This does not allow controlling the parameters.
53
Annulenes
1 2 3 N-1 N-1
1 -x 1 0 0 0 12 1 -x 1 0 0 0
3 0 1 -x 1 0 0 = 0
0 0 1 -x 1 0
N-1 0 0 0 1 -x 1
N 1 0 0 0 1 -x
cr-1 -E cr + cr+1 = 0Secular equation:
55
Annulenessolutions for CN
R (r) = r+s angle = 2s/N
An annulene is a periodic system;
Solutions for annulenes are those of any periodic system: crystals (Bloch sums)
The new coefficient for r is the old one multiplied by a constant
56
The system allows solutions Rj with angle (j=2j/N).
There is N rotations possible including identity (j=0 ou N) : the « simple » rotation and the « multiple» rotations j
Functions satisfying to rotations j=2j/N:
We build linear combinations of AOs,(crr) satisfying rotation j
(crr) = e-ijj (crr) with cr (j) = N-1/2 eijrj = e –i2jr . There are N solutions, with a quantum number: j..
i is the square root of -1; i2=-1
annulenes
57
r=00
r=5
r=4
r=1
r=2
r=3
r=40
r=3
r=2
r=5
r=0
r=1
N
N exp(i2 2/6)
N exp(i2 4/6)
N
N exp(i2 8/6)
N exp(i2 10/6)
C5 x exp(-ijj)=
N exp(i2 10/6) x exp(-i2 2/6) =
exp(i2 2/6)
Rotation by 120° = 2/6(j=2, orbital 2)
The 5th AO takes the place of the first
C5 x exp(i2 2/6)
N
N exp(i2 8/6)
N exp(i2 10/6)
N
N exp(i2 4/6)
58
They are the LCAO functions satisfying the rotations j
There are j solutions; the eigenvalues are e-ijs
annulenes
j= crr
59
Benzene
The complexe solutions are associated with rotations only; they are not eigenfunctions of the other symmetry operators and must be the degenerate
annulenes
+
+
Two real solutions and 4 complexesDegenerate by pairs
60
j = 3, AS
j = ± 2, AA j = ± 2, SS
j ± 1, SA j = ± 1, AS
j = 0, SS
annulenesReal solutionsSatisfying two mirror symmetry
61
j = 3, AS
j = ± 2, AA j = ± 2, SS
j ± 1, SA j = ± 1, AS
j = 0, SS
annulenesOrbitals from symmetry only
All the coefficients are equal c=1/√6
E=< 1/√61+.. .IHI1/√61+…>= 12*(< 1/√61IHI1/√62>+…) = 2
All the coefficients are equal c=1/2
E = 4*(< 1/22IHI1/23>+…) =
62
j = 3, AS
j = ± 2, AA j = ± 2, SS
j ± 1, SA j = ± 1, AS
j = 0, SS
annulenesOrbitals from symmetry only
The total density in E orbital must respect symmetry
Degenerate orbitals: energy must be
02+c2=2/6 → c= 1/√3
1/22+c2=2/6 → c= 1/√12
Normalization is satisfied 4*1/12+2*1/3=1
63
For any annulene of large size …the number of OMs remains in the gap
from 2 to -2
2
2
-2j=3
j=±2
j=±1
j=0
64
j = 3, AS
j = ± 2, AA j = ± 2, SS
j ± 1, SA j = ± 1, AS
j = 0, SS
annulenesChosing another set of planes; rotation by 60°
Combining orbitalsS’A’ = ½ [SA + √3 AS ]A’S’ = ½ [SA - √3 AS ]New symmetry appearsWhile old ones desappears
65
Hückel determinant for CyclobutadieneNotations: x = ↔ E = + x units , origin 12 3 4
1 -x 1 0 12 1 -x 1 0 = 03 0 1 -x 14 1 0 1 -x
Two possibilities of using mirror symmetries
68
The Jahn-Teller Theorem 1937 "any non-linear molecular system in a degenerate electronic state will be unstable and will undergo distortion to form a system of lower symmetry and lower energy thereby removing the degeneracy"
There is a symmetry reduction (a geometry distortion) when the a set of HOMOs is not completely filled.
Edward Teller 1908-2003Hungarian-American
Hermann Arthur Jahn 1907-1979 English
69
4e-8e 0-6e 2e-8e3e-5e-9e 3e-9e
In an octahedral crystal field, the t2g orbitals occur at lower energy than the eg orbitals. The t2g are directed between bond axes while the eg point along bond axes.
The full octahedral symmetry takes place only when the t2g set is fully occupied.
70
C4H4 is unstable, it exists stabilized by asymmetric ligand (one eg orbital is stabilized).It dimerizes easily.
Cyclobutadiene: C4H4
Chain of Vn-benzenen+1: The HOMO gap vanishes for n=4; then there is a rotation of the successive benzene loosing the full D5h symmetry.
V V V
If the rings are eclipsed, there is no gap; the rotation of ring opens a gap.
72
Jahn-Teller effects are grouped into two categories. The first arises from incomplete shells of degenerate orbitals. It includes the first-order Jahn-Teller effect and the pseudo Jahn-Teller effect. The second arises from filled and empty molecular orbitals that are close in energy and is the second-order Jahn-Teller effect. The two categories have quite different physical bases. As a result, geometric distortions produced by the first are quite small and normally lead to dynamic effects only.
In favorable cases, the second-order Jahn-Teller effect produces very large distortions, including complete dissociation of a molecule. This can occur even when the relevant molecular orbitals are separated in energy by as much as 4 eV.
73
CuCl2 Cu2+ 9eNiCl2 Ni2+ 8e
The shielding effect this has on the electrons is used to explain why the Jahn-Teller effect is generally only important for odd number occupancy of the eg level. The effect of Jahn-Teller distortions is best documented for Cu(II) complexes (with 3 electrons in the eg level) where the result is that most complexes are found to have elongation along the z-axis.
The Jahn-Teller Theorem 1937
74
Apparent exceptions to the theorem are probably examples of what has been called the "dynamic Jahn-Teller effect". In these cases either the time frame of the measurement does not allow the distortion to be seen because of the molecule randomly undergoing movement or else the distortion is so small as to be negligible.
For one of the copper complexes, the bond lengths are apparently identical. If the X-ray structure of the sample is redone at varying temperatures it is sometimes possible to "freeze" a molecule into a static position showing the distortions.
75
Alternant conjugated hydrocarbons
Definition: Atoms of conjugated molecules could be divided into two sets of atoms (starred and unstarred atoms) so that a member of one set is formally bonded only to members of the other set. Many compounds are alternant; they are not when their structure contains an odd-member ring.
76
Alternant conjugated hydrocarbons
-E cr + s cs = 0
if j = r* cr r + s° cs s is an eigenfunction
‘j = r* cr r - s° cs s is another one with Ej'=-Ej
•Orbital energy levels are symmetric relative to a level.
•If there is an odd number of orbital, one of them is nonbonding
77
Alternant conjugated hydrocarbonsNon bonding orbital, SOMO
The nonbonding orbital has non-zero coefficient only at the starred atoms, and the sum of the coefficient of the starred atoms attached to a given unstarred atom must equal zero.
The secular equation -E cr + becomes s cs = 0 with E=0.
Applied on a starred atoms s cs = 0 it gives again that cs = 0
Applied on a unstarred atoms r cr = 0 it tells that the unstarred atoms belong to nodal planes.It is easy to find the coefficients from there.
78
Allyle radical
The SOMO is localized on the terminal atoms
Allyle anionThe charge is -1/2 on each terminal atom
Allyle cationThe charge is +1/2 on each terminal atom
a = 1/√
H2C CH2
a = -1/√
species):
79
Dewar method for estimating aromaticity
a
-a
Two bonds close the cycleOne bond makes the polyene
S
A
If the SOMO is symmetric, it is better to make 2 bonds:
If the SOMO is antisymmetric, it is better to make only one bond:
4n+2 electrons aromatic 4n electrons antiaromatic
SS
S
Singlecarbon
S or A
S 2n+1A 2n-1
Polyeneradical
80
Dewar method for estimating aromaticity
-3a
-b
-2a
3a
-a
a
-a
a
0
b- b
b -b
b
- b
5a
pyrène
There is an energy gainby forming a new cycle
Two bonds close the cycleOne bond makes the polyene
81
The net charge of alternant hydrocarbons is zero.The electron density on r is 1
| coeff2 of occupied MOs| = | coeff2 of unoccupied MOs | wherefrom ii cri
2=1
demonstration :
occ cr,occ2 + nb cr,nb
2 + vac cr,vacc2 =1
occ cr,occ2 + nb cr,nb
2 = 1 ii cr,i
2 = 1
82
The bond indices between atoms of the same set is zero.(first order perturbation between atoms of the same set is zero).
OMs are orthogonal thus rcricrj= ijIf i=j, the sum of the coefficient’s squares for a given atom is 1 (normalization)
If i≠j, the sum of the products of coefficients is 0 (orthogonality)The matrix of coefficients is unitary
wherefrom = icricsj= ijIf i=j, the sum of the coefficient’s squares for a given orbital is 1 If i≠j, the sum of the products of coefficients is 0
occ cr,occcs,occ + nb cr,nbcs,nb
+ vac cr,vacccs,vacc
=rs occ cr,occcs,occ
+ nb cr,nbcs,nb = 0
occ cr,occcs,occ + nb cr,nbcs,nb
= 0 ricricsj=
830.3717
0.3717
0.6015
0.6015
0.6015
-.60150.6015
0.6015
-.6015
0.6015
-.3717
-.3717
-.3717
0.3717
-.3717
1.618
0.618
-.618
-1.618
Butadiene is an alternant
84
j = 3, AS
j = ± 2, AA j = ± 2, SS
j ± 1, SA j = ± 1, AS
j = 0, SS
annulenesBenzene is
alternant
E = 2
E = -2
E = 1
E = -1
85
Importance of the non bonding orbital
• for radical species, it represents the unpaired electron
•For anions or cations it monitors the charged= ii cr,i2
For anions d= occ cr,occ2 + cr,nb
2
d= (occ cr,occ2 + cr,nb
2 ) + cr,nb2
d= 1 + cr,nb2
For cations d= occ cr,occ2
d= (occ cr,occ2 + cr,nb
2 ) - cr,nb2
d= 1 - cr,nb2
86
Heteroatoms:CO
parameter CoulombicIntegral
Resonance
Integral
Carbon C = CC =
Oxygen 1 electron
O = +
CO =
Oxygen 2 electrons
O = + 2
CO = 0,8
1 electron for CO
C O
C -x 1 = 0
O 1 -x+1
x2-x-1=0x= (1±√5)/2
0
1
1.618
-0.618
0.8506 -0.5257
0.5257 0.8506
C O
Bonding, large amplitude on O
Antibonding, large amplitude on C
O more electronegative
87
allyle1 2 3
1 -x 1 0
2 1 -x 1
3 0 1 -x
x3-2x=0 Solutions:x=- √2x=0x= √2
Sym 1 2
1 -x 1
2 2 -x
Sym 1/√2(1+3) 2
1/√2(1+3) -x √ 2
2 √ 2 -x
AntiSym 1
1 -x
88
Allyle coefficients
x=0
-xc1+c2=0 c2=0
c1-xc2+c3=0 c3=-c1 this orbital is found antisymm
c2-xc3=0 c2=0 no new informationx=√2
-√2 c1+c2=0 c2= √2 c1
c1-√2 c2+c3=0 c3 = c1 this orbital is found symm
c2-xc3=0 1/√2 c3= 1/√2 c1 = c2
These expressions have be normalized
89
-√2
0
√2
-1/√2 1/√2
1/2 1/√2 1/2
1/2 -1/√2 1/2
radical cation anion
q11-2(1/2)2-1(1/√2)2 = 0 1-2(1/2)2
= 1/2
1-2(1/2)2-2(1/√2)2
= -1/2
q21-2(1/√2)2 = 0 1-2(1/√2)2
= 01-2(1/√2)2 = 0
Charges are controlled by the filling of the nonbonding MO.They appear on the terminal atoms.The sum of charges is the global charge
90
-√2
0
√2
-1/√2 1/√2
1/2 1/√2 1/2
1/2 -1/√2 1/2
radical cation anion
l12=l232(1/2)(1/√2) = 0.707 = 0.707 = 0.707
bond indices are independent from the filling of the nonbonding MO.
91
Exercises
Find, using symmetry, the orbitals for
• the trimethylenemethane H2C=C(CH2)2
• the pentadiene*
• the cyclopentadiene*
*Explain the origin of the solutions x=±1 for the pentadiene*Explain the origin of the golden number solutions for the cyclopentadiene
92
Trimethylene-methane
Symmetry orbitals : MOs
0 1/√2 [0 + 1/√3 (1+ 2+3)]
1/√3 (1+ 2+3) 1/√2 [0 - 1/√3 (1+ 2+3)]
1/√2 (2-3) 1/√2 (2-3)
√(2/3) 1+ 1√6 ( 2+3) √(2/3) 1+ 1√6 ( 2+3)
Energies 0 energies ± √3
}
C1 C0
C2
C3
93
Trimethylene-methaneC1 C0
C2
C3
1/√2 [0 - 1/√3 (1+ 2+3)]
1/√2 [0 + 1/√3 (1+ 2+3)]
√(2/3) 1+ 1√6 ( 2+3) 1/√2 (2-3)
94
Cyclopentadienyl radical and ions
Sym 1 2 3
1 -x +2 0
2 1 -x 1
3 0 1 -x+1
-x2(1-x)+2(1-x)+x=0 x3-x2-x+2=0 x=2 solution (x-2) (x2-x-1) x= (1±√5)/2
The antisymmetric solutions are those from butadiene
2
3
1
5
4
98
Exercise: naphtalene
2) Solve the SS secular determinant 3) Deduce from the SS energy levels those for AS 4) What must be the levels for SA and AA solutions ?5) Draw the non-bonding level of the pentadienyl radical (C5 chain)?
1) Write the Huckel determinant for each symmetry group of the naphatlene.
6) Explain from there what are the levels SS and AS at x=±1
99
naphtaleneSS 1 2 3
1 -x+1 +1 0
2 1 -x 1
3 0 2 -x+1
SS -x (1-x) 2-3(1-x)=0 (1-x)(-x2+x+3)=0 x=1 solution (-x2+x+3)=0 x= (1±√13)/2x = 2.30277 and -1.30277AS solutions have opposite energies (alternant) -1, x=-2.30277 and +1.30277The SA and AA solutions are the 4 MOs from butadiene ± 0.618 and ± 1.618The levels at ±1 are duplication of the non bonding MO from the pentadienyle in-phase and out of-phase.