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Theoretical background
1. Early orbital concepts for the description of
saturated and unsaturated compounds
- Quantum Chemistry
- Molecular orbital (MO) description of
the allyl cation, radical and anion
- Resonance descripton of the allyl cation, radical and anion
- Theories of conjugative stabilization: butadiene
- Electrophilic addition to conjugated dienes
2. The Schrödinger equation and the Hamiltonian operators
- The central field Hamiltonians
- The molecular Hamiltonian
3. The Born-Oppenheimer approximation and the concept of the
potential energy surfaces
Orbital concepts
Orbital concepts for the description of saturated and unsaturated compounds
- Quantum Chemistry
computational chemistry:
the numerical implementation of the principles of quantum mechanics to chemical problems
For the “one electron” atom the solution is the corresponding one-electron Schrödinger
equation: Y  Y, of the H-atom or of H - like ions (He+, Li2+, Be3+, B4+ ,...etc.),
is a one-electron wave equation which can be solved analytically.
The solutions:
the atomic orbitals of the H - atom (or H - like ions)
the associated orbital energies (E = ).
Warning: atomic orbitals are mathematical functions and not physical objects
Example: the 1s orbital of H has the following form
1
1s = 1s

e-r
r := the distance of the electron
measured from the nucleus
An example: the hydrogen atom (H) 1 electron 1 occupied AO
E(RHF/3-21G)= -0.4962 Hartree
2pz
AO 3 (+1.069)
2py
2px
4 times
Degenerated
AOs
2s
AO 3 (+1.069)
AO 3 (+1.069)
AO 2 (+1.069)
z
x
1s
AO 1 (-0.4962)
y
Orbital concepts
Orbital concepts
In a many electron atom (e.g. Be) the solutions of the corresponding
many-electron Schrödinger equation is more complicated
constructed from one-electron wave functions, i.e., from orbitals.
Example 1: Be with 4 electrons:
4 - electron state-wave function constructed as two doubly occupied atomic orbitals (1s and 2s)
the double occupancy is stipulated to involve one electron with  spin and one with  spin.
(1,2,3,4) ~ 
1s(1)(1)  1s(2)(2)  2s(3)(3)  2s(4)(4)
1s2
2s2
This double occupancy is frequently denoted
in terms of the energy levels 1s and 2s.
Example 2: Ne with 10 electrons:
state-wave function: Y
constructed from five doubly occupied
atomic orbitals (1s, 2s, 2px, 2py, 2pz)
corresponding to the 1s2 2s2 2p6 occupancy scheme
2p
s
s
1s
2
2s 2
2s
s
Orbital concepts
An example: the neon atom (Ne) 10 electron 5 occupied AO
E(RHF/3-21G)= -127.8038245 Hartree
z
3 times degenerated
AOs
-0.79034 Hartree
2px AO 3
2py AO 4
2s AO 2 -1.86515 Hartree
1s AO 1 -32.56471 Hartree
2pz AO 5
x
y
many-electron molecular systems
analogous to that of many-electron atoms:
i.e. the many-electron state wave functions are constructed
from many one-electron wave functions.
Orbital concepts
For atoms: one-centered atomic orbitals are appropriate
For molecules: multicentered one-electron functions (molecular orbitals) are needed
Molecular orbitals are constructed by the linear combination of atomic orbitals (LCAO).
Orbital concepts
An example: the hydrogen molecule (H2)
2 electron 1 double occupied MO, = 0 Debye
EPS
-0.007  töltés. 0.007
E(RHF/3-21G)= -1.12295984 Hartree
(LUMO)
anti-bonding MO
* MO 2 (0.26570)
(HOMO)
 MO 1 (-0.59424)
bonding MO
1s
AO 1 (-0.4962)
The AO of H atom
the functional form
2pz = Nz exp(r)
the shape
A 2pz atomic orbital
Two types of MO can be constructed
from a pair of 2p type AOs
depending on their orientations:
a  - type and a  - type
a parallel orientation yields
a  - bonding and
a  * anti-bonding orbital pair.
Orbital concepts
memo: the + and - represent the sign
of the mathematical function denoted by 2pz.
An axial orientation yields
a  - bonding and
a * anti-bonding orbital pair
Orbital concepts
Linear combination is a mathematical process of Linear Algebra
where vectors are subjected to linear combination i.e.
multiplication with weighing coefficients and subsequent addition and/or subtraction.
In this sense, atomic orbitals may be regarded as vectors.
This is obvious in the case of px, py and pz orbitals
shape
vector model
the 2p atomic orbitals.
memo:
1s or 2s orbitals have no directionality
but can also be regarded as vectors.
Therefore, the electron distribution of C, N, O, F and Ne can be represented by 5 vectors:
1s, 2s, 2px, 2py, 2pz.
These five vectors will be part of the so-called 'basis vectors' or 'basis sets'
used for a molecular orbital calculations.
Example 1: the formaldehyde (H2C=O)
The LCAO transformation of
the 12 AO basis set to 12 MO
8 doubly occupied an
4 empty (denoted by *)
10*
Orbital concepts
9 *
8 *
anti-bonding MO
a number of empty
(antibonding or virtual) MO’s
*
1  - electron pair
8
7
O
C
1s, 2s, 2px, 2py, 2pz
1s, 2s, 2px, 2py, 2pz
H
1s
H
1s
6
two n – type
lone
electron pairs (lp)
LCAO
5
4
3
2
1
3  - bonds:
a C - O and
two C - H bonds
2 atomic cores:
1s2 for C and
1s2 for O
12 AO represent the 'basis set' for the MO calculation for formaldehyde
Orbital concepts
10*
9*
8 double occupied MOs
8*
12 AO
 2*
1s

2p
2p

lp (n)
6
5
4
2s
-bonds
2s
3
1s
2
core
1
core
1s
H
H
H
C
C
H
O:
O
The set of MO’s obtained
from the LCAO-MO
calculation reflects
the symmetry of the molecule.
These MO’s are called
canonical MO’s (CMO’s);
they are fully delocalized
They are not directed
along the traditional
representation of chemical bonds.
Orbital concepts
Example 2: water (H2O)
The LCAO transformation of
the 7 AO basis set to 5 MO
No canonical MO’s,
obtained from the LCAO calculation,
correspond to the two equivalent
bonds directed along the O - H lines.
Instead, an anti-symmetric (σantisym)
and
a symmetric (σsym)
delocalized -type CMO
the linear combination (i.e. addition and subtraction)
of these CMO’s leads to localized molecular orbitals
(LMO’s) which show the directionality of the two O - H bonds
Orbital concepts
they are directly computable
from the AO basis set
and
they faithfully represent
the symmetry of the molecule
The CMO representations of molecules
have many advantages:
The CMO representations of molecules
have disadvantages:
they do not have a directionality,
which coincides with the
direction of the chemical bonds.
The three most important structural motifs in organic chemistry
180 o
120
o
120
o
109.5 o
120 o
linear
trigonal planar
tetrahedral
Cartesian coordinates do not match all three structural motifs
The directionality of the AO is not a prerequisite for the formation of a MO, for a qualitative or
pictorial description of bonding, chemists frequently hybridize the Cartesian AO. As a result,
chemical problems are often discussed in terms of hybrid atomic orbitals (HAO).
Orbital concepts
hybridization is an orthogonal transformation of a given basis set
(1 = 2s; 2 = 2px; 3 = 2py; 4 = 2pz)
to an equivalent basis set {i}.
Since the hybridization may involve the mixing (i.e. linear combination)
of two (2s, 2px), three (2s, 2px, 2py) or four (2s, 2px, 2py, 2pz) AO’s,
it is possible to derive 3 types of hybridization, referred to as sp, sp2 and sp3 respectively.
sp3 hybrids
1
( 1   2
2
1
2 = ( 1   2
2
1
3 = (  1   2
2
1
4 = ( 1   2
2
1 =
sp2 hybrids
 3  4 )
sp hybrids
1 = 1/ 3 ( 1   2   3 )
1 = 1/ 2 (  1   2 )
 3  4 )
2 = 1/ 3 ( 1  0.366 2  1.366 3 )
 3  4 )
3 = 1/ 3 ( 1  1.366 2  0.366 3 )
3 =  3
 3  4 )
4 =  4
4 =  4
2 = 1/ 2 ( 1   2 )
The computed CMO’s are delocalized over the whole molecule
whether they belong to the  or the  representations.
CMO’s are symmetry adapted.
The geometrical equivalents of localized molecular orbitals (LMO)
are governed by the stereochemistry of the molecular bonding.
.
Orbital concepts
sp orbital
Example: Methane may be used to illustrate this point
sp2 orbital
A schematic representation of the shapes and
orientation of the four valence CMO of CH4.
sp3 orbital
Approximate shapes of
hybrid atomic orbitals
(HAO) of carbon
Orbital concepts
EPS
-0.007  charge. 0.007
schematic representation of the change of energy levels upon localization of CH4
Computed CMOs
Of CH4
Degenerated MOs
MO 3
MO 4
MO 5
(-0.54483)
(-0. 54483)
(-0. 54483)
RHF/3-21G
MO 2
(- 0.94587)
MO 1
(-11.14448)
closed shell
Orbital concepts
Localization involves the addition or subtraction of the four  (CMO) to obtain the four  (LMO)
(1,  2,  3,  4)
Transformation
i.e. Localization
(1,  2,  3,  4)
The shapes and orientation of the four valence LMO of CH4
obtained by localization of the four valence CMO’s
Orbital concepts
For CH4, the four C - H bonds { YI }, may also be formed from the four sp3 hybrid AO {i}
and the 1s orbitals { Hi } of the four hydrogen atoms.
1 = a1 + bH1
2 = a2 + bH2
3 = a3 + bH3
4 = a4 + bH4
The linear combination of carbon HAO and hydrogen AO to form methane LMO.
However, { i }, the HAOs also have their own expression in terms of the unhybridized
AO of carbon. Take for example the first of the series 
emember that 1 = 2s, 2 = 2px, 3 = 2py, 4 = 2pz and H1 = 1sH1).
Y1  a( 1   2   3   4 )  b(  H 1 )
n order to see the set of (LMO) expressed in terms of the AO
we have to specify the expressions for the set of (CMO).
1 = C11 + C2 (H1 + H2 + H3 + H4)
2 = C12 + C2 (H1 - H2 + H3 - H4)
3 = C13 + C2 (H1 - H2 - H3 + H4)
4 = C14 + C2 (H1 + H2 - H3 - H4)
Orbital concepts
The first LMO (1) may be written
as the following linear combination of the four CMO:
 and
 1 = ½ 1 + ½ 2 + ½ 3 + ½ 4
= ½ {1 + 2 + 3 + 4}
= ½ {C11 + C2 (H1 + H2 + H3 + H4)
+ C12 + C2 (H1 - H2 + H3 - H4) + C13 + C2 (H1 - H2 - H3 + H4)
+ C14 + C2 (H1 + H2 - H3 - H4)}
When the addition is performed,
the terms involving H2, H3 and H4 cancel
and we are left with the expression
1
Y1  C1{ ( 1   2   3   4 )}  2C 2  H 1
2
The hybrid structure is clearly recognizable in the first term
of the above expression and
we may rewrite it conveniently in the form
Y1  C11  2C2 H1
Compare this to the equation given at the beginning of this section for the first LMO,
i.e. one of the C – H,
Y1  a11  b1  H1
Conclusion: the CMO and LMO representations are equivalent
Consider H2O again.
The LMO picture of the 4 valance electron pairs related to an sp3 hybridized oxygen
The two lone electron pairs (lp) point to the back
and
the two bond pairs (O - H) point towards the front.
The two bond pairs have been discussed earlier in this chapter
LMO for the water:
in the ideal case a = 109.5o = b;
in reality a < 109.5o < b.
Orbital concepts
Orbital concepts
Fact is that both LMO and CMO representations
of the 2 lone pairs of oxygen in H2O lead
to the same 4-electron density.
LMO
4 -electron density
CMO
Two orthogonal planes containing the 4 bond electron density
and the 4 lone-pair electron density.
Orbital concepts
Molecular Orbital (MO) Description of the Allyl Cation, Radical and Anion
Orbital concepts
Orbital concepts
Orbital concepts
Theoretical background
1. Early orbital concepts for the description of
saturated and unsaturated compounds
- Quantum Chemistry
- Molecular orbital (MO) description of
the allyl cation, radical and anion
- Resonance descripton of the allyl cation, radical and anion
- Theories of conjugative stabilization: butadiene
- Electrophilic addition to conjugated dienes
2. The Schrödinger equation and the Hamiltonian operators
- The central field Hamiltonians
- The molecular Hamiltonian
3. The Born-Oppenheimer approximation and the concept of the
potential energy surfaces
Schrödinger equ. & Hamiltonian
Theoretical background
1. Early orbital concepts for the description of
saturated and unsaturated compounds
- Quantum Chemistry
- Molecular orbital (MO) description of
the allyl cation, radical and anion
- Resonance descripton of the allyl cation, radical and anion
- Theories of conjugative stabilization: butadiene
- Electrophilic addition to conjugated dienes
2. The Schrödinger equation and the Hamiltonian operators
- The central field Hamiltonians
- The molecular Hamiltonian
3. The Born-Oppenheimer approximation and the concept of the
potential energy surfaces
Born-Oppenheimer approx.