OC-IV

1
OC-IV
Orbital Concepts and Their Applications in
Organic Chemistry
Klaus Müller
Script ETH Zürich, Spring Semester 2009
Lecture assistants Deborah Sophie MathisHCI
G214 tel. 24489mathis_at_org.chem.ethz.ch Alexey
FedorovHCI G204 tel. 34709 fedorov_at_org.chem.et
hz.ch
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Chapter 1
some basic elements of orbitals
see also documentBasic Features of MO
Theory on http//www.chen.ethz.ch/course/
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Quantum Mechanics
neglect of electron correlation
Orbital Models
free, resting electron
E 0
discrete stationary states for bound electrons
(Elt0) in an atom or molecule
yi ei
i-th state described by orbital yi
electron energy in yi (relative to E0) given by
orbital energy ei
spatial distribution of the electron (electron
density ri) is given by yi2.
approximation of the MOs by linear combinations
of AOs ? LCAO MO approximation
for atomsAOs
fk, ek
for moleculesMOs
yi, ei
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Typical result of an LCAO MO calculation for the
electronic ground state of a molecule with an
even number of n electrons using m AOs for the
LCAO MO approximation.


alternative set of empty localized MOs (LMOs)
fi to represent the same virtual electronic
space.
(m - n/2) antibonding or partially antibonding
unoccupied MOs yi (ei gt 0).
transformation
alternative set of doubly occupied localized
MOs (LMOs) fi to represent the same
electronic structure.
n/2 bonding or partially bonding, doubly
occupiedMOs yi (ei lt 0).


properties of LMOs
properties of canonical MOs
- delocalized over whole molecule, adapted to
the symmetry of the molecule

• fully localized, corresponding to
• elements of the molecular structure

- correct representation of the total electron
density distribution rtot
total electronic density distribution
etot ? bi ei
sum of orbital energies as an estimate of
relative energies
- correct representation of the total orbital
electron energy etot
i
taking into account the interactions between LMOs
construction of approximate LMOs
? orbital conjugative effects
approximate LMO representation of the molecular
electronic structure
? qualitative estimates of energy
differences between isomeric molecular
systems
Hybrid-AO method
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UV-Photoelectron Spectroscopy gives qualitative
insight into the molecular orbital structure of
closed-shell molecules
E
Ekin hn - IPn
photoelectron count as a function of Ekin
HeI resonance line
hn 21.22 eV
e-
e 0 (free electron at rest)
e1
IP1 -e1
photo- emission
Koopmans theorem
e2
IP2 -e2
Frozen Orbital Approximation
e3
IP3 -e3

gas phase (ultra-high vacuum)
sCH/sCC-dom
sCC-dom
schematic representations of a few canonical MOs
of cyclohexane (symmetry D3d)
el.counts/sec
sCH/sCC-combined ribbon orbitals (eg)
sCH-dom
sCC-dom
p-type
pCH2-dominant orbital (a1g)
el.counts/sec
sCH-dom
sCC-dom
sCH/sCC-combined orbitals (eu)
p-type
el.counts/sec
sCC-dominant orbital (a1u)
UV-Photoelectron Spectra of cyclohexane,
cyclohexene, and 1,4-cyclohexadiene
P Bischof, J A Hashmall, E Heilbronner, V
Hornung, Helv Chim Acta 52, 1745 (1969)
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canonical LCAO MOs
obtained as solutions from the eigenvalue problem
canonical LCAO molecular orbitals
- extend over the whole molecule

• are symmetric or antisymmetric with respect to
  all molecular symmetry elements
• (note this is strictly true only for
  non-degenerate orbitals however, every set of
• degenerate orbitals represents the true
  molecular symmetry)

- are the starting points for the discussion of
spectroscopic properties
- can be used for the approximate calculation of
any other molecular property
- are the starting points for calculations of
electron correlation effects (going beyond the
Hartree-Fock limit)
- however, canonical MOs are quite inconvenient
for transparent rationalization of effects
in organic chemistry
LCAO LMOs
by rigorous localization schemes followed by
truncation of orbital tails
eg
a1g
6 equivalent sCC-LMOs
eu
6 equivalent axial sCH-LMOs
a1u
D3d
6 equivalent equatorial sCH-LMOs
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the LMO representation of the molecular
electronic structure and its relations to
the elements of the classical structure formula
in organic chemistry
(a)
From the individual sets of occupied and
unoccupied canonical MOs, obtained by rigorous
quantum chemical calculations for the electronic
ground state of a closed-shell molecule and
transformed by unbiased localization
schemes, followed by truncation of orbital tails
and renormalization, one obtains for each single
bond in a classical molecular structure a
(s,s)-LMO pair
Eel

• unoccupied,
• energetically high-lying orbital

s - LMO
AB

• essentially localized in A-B bond domain,
• axially symmetric w.r.t. A-B bond axis

• antibonding between atoms A and B the s-LMO
  has a nodal plane
• orthogonal to the bond A-B axis

A
B

• doubly occupied
• energetically low-lying orbital

s - LMO

• essentially localized in A-B bond domain,
• axially symmetric w.r.t. A-B bond axis

AB

• bonding between atoms A and B

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the LMO representation of the molecular
electronic structure and its relations to
the elements of the classical structure formula
in organic chemistry
(b)
From the individual sets of occupied and
unoccupied canonical MOs, obtained by rigorous
quantum chemical calculations for the electronic
ground state of a closed-shell molecule and
transformed by unbiased localization
schemes, followed by truncation of orbital tails
and renormalization, one obtains for each double
bond in a classical molecular structure a
(s,s)-LMO pair and a (p,p)-LMO pair
Eel

• characteristic features of th s-LMO as in (a)

s - LMO
AB
p - LMO

• unoccupied, energetically high-lying orbital,
• but lying below s

AB

• essentially localized in A-B bond domain,
• antisymmetric w.r.t. AB double bond plane

• antibonding between A and B,
• but less antibonding than s-LMO the p-LMO
  has two orthogonal nodal planes,
• one orthogonal to the bond A-B axis, and
• one in the AB double bond plane

A
B
AB plane

• doubly occupied, energetically low-lying
• orbital, but higher than s-LMO

p - LMO
AB

• essentially localized in AB bond domain,
• antisymmetric w.r.t. AB bond plain

• bonding between A and B,
• but less bonding than s-LMO

• characteristic features of the s-LMO as in (a)

s - LMO
AB
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the LMO representation of the molecular
electronic structure and its relations to
the elements of the classical structure formula
in organic chemistry
(c)
From the individual sets of occupied and
unoccupied canonical MOs, obtained by rigorous
quantum chemical calculations for the electronic
ground state of a closed-shell molecule and
transformed by unbiased localization
schemes, followed by truncation of orbital tails
and renormalization, one obtains for each triple
bond in a classical molecular structure a
(s,s)-LMO pair and two mutually orthogonal
(p,p)-LMO pairs
Eel

• characteristic features of th s-LMO as in (a)

s - LMO
AB

• two energetically degenerate
• mutually orthogonal p-LMOs
• characteristic features as in (b)

p - LMOs
AB
A
B

• two energetically degenerate
• mutually orthogonal p-LMOs
• characteristic features as in (b)

p LMOs
AB
s - LMO

• characteristic features of th s-LMO as in (a)

AB
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the LMO representation of the molecular
electronic structure and its relations to
the elements of the classical structure formula
in organic chemistry
(d)
From the individual sets of occupied and
unoccupied canonical MOs, obtained by rigorous
quantum chemical calculations for the electronic
ground state of a closed-shell molecule and
transformed by unbiased localization
schemes, followed by truncation of orbital tails
and renormalization, one obtains for each lone
electron pair in a classical molecular
structure a doubly occupied n-LMO pair
Eel

n - LMO

• doubly occupied non-bonding n-LMO

A

• essentially localized at atom A

• energy depending on nature of A as well as
  amount of valence s-character
• typically higher in energy than s-LMOs

Eel

B

• two energetically degenerate doubly occupied
  non-bonding n-LMOs

• essentially localized at atom B

• energy depending on nature of B as well as
  amount of valence s-character
• typically higher in energy than s-LMOs

n LMOs
B
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AO basis sets in LCAO MO approaches
general aspects STOs versus GTOs
Based on the exact quantum-mechanical solutions
(eigenfunctions) for the H-atom,similar slightly
simplified atomic orbitals 1s, 2s, 3s, , 2p, 3p,
, 3d, etc. can be defined for the heavier atoms.
These are Slater AOs or Slater-type orbitals
(STOs).These AOs are characterized by an
exponential decay (e-ar) of the radial amplitude,
which is an important aspect of proper AOs.
However, this results in considerable
computational efforts in solving the various
integrals involving differential overlaps between
AOs located at different centers. Gaussian
functions (e-ar ) are much more convenient.
Thus, Gaussian-type orbitals (GTOs) are
typically used nowadays, which results in a
massive computational saving. However, for a
proper description of the orbital tails more
distant from a nucleus, at least two GTOs of the
same type, but different exponentials, have to be
combined. This has led to the notion of STO-nG
which indicates that a linear combination of n
GTOs of a given type are used to represent one
STO of this type.
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minimal basis and extended basis sets
The minimal basis set of AOs includes the 1s-AO
for the H-atom and the 1s, 2s, and 2p-AOs for
the first-row heavy atoms. While minimal-basis
set calculations can provide a good, albeit rough
orientation about the electronic structure of
organic molecules, larger basis sets are
typically used nowadays. The advantage of
minimal-basis set calculations lies in their more
direct correlation with structural concepts of
organic chemistry.
Extended orbital basis sets typically include
several GTOs, a most prominent approach
consisting in using a so-called split-valence
basis set, in which each valence orbital (e.g.,
2s, 2p for the 1st-row heavy atoms) is
represented by 2 (valence double-zeta), 3
(valence triple-zeta), or 4 (valence
quadruple-zeta) orbitals of the corresponding
type. The notation 6-31g indicates that the
core AOs is approximated by 6 GTOs, while for
the valence AOs, a double-zeta split is used
with 3 GTOs in fixed combination for the first
component of the valence orbital and a single GTO
as a variable second component.
An increase in the basis set is often balanced
with the addition of polarization functions
(e.g., p-AO for H-atom, d-AOs for 1st-row heavy
atoms) such basis sets are denoted with an
asterisk, e.g., 6-31G.
With a properly balanced increase of the basis
set the computational results ultimately converge
to the Hartree-Fock limit, which can then be
used for calculations of electron correlation
effects (post-Hartree-Fock calculations). Note
that correlation calculations are of little
meaning when medium-size or highly unbalanced
extended basis sets are used. Correlation-consist
ent extended basis sets have been derived and
are typically denoted as cc-pVDZ, cc-pVTZ, where
cc-p stands for correlation-consistent
polarized, and VDZ, VTZ, etc., stand for,
respectively, valence-double-zeta and
valence-triple-zeta.
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approximate LMOs by means of the Hybrid-AO
approach
The Hybrid-AO approach used in organic chemistry
is primarily restricted to a minimal valence-AO
basis set, i.e.
number of AOs
symbolic representation
atom
1s
H
1
1st-row(B,C,N,O,F)
4
2s
2px
2py
2pz
Such symbols are used throughout. They denote in
gross qualitative terms the contour surfaces of
constant absolute amplitude for a given AO, with
the shading indicating positive amplitude
domains. s-AOs are spherically symmetric, 1s
and 2s differing mainly in their radial
extensions however, for reasons of
orthogonality to the core-1s AO, the 2s valence
AO has a spherical nodal surface, separating a
small inner (negative domain) from the dominant
outer domain. The nodal aspect of the 2s-AO is
ignored in qualitative hybrid-AO
treatments. The p-AOs are axially symmetric
with one nodal plane orthogonal to the orbital
axis. A p-AO is antisymmetric with respect to its
nodal plane. The three p-AOs are
mutually orthogonal. They form a Cartesian vector
system, i.e., a p-AO in any specific
spatial orientation can be vectorially decomposed
into its px-, py-, pz-components. Also note
that the three p-AOs complement each other to
the full spherical symmetry, i.e., the combined
density distributions of px, py, pz, constitute a
spherical density distribution.
2
2
2
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core- and (averaged) valence orbital energies
from X-ray PES for the free atoms (using Al Ka1,2
X-rays at 1486.6 eV) by D. A. Shirley et al.,
Phys. Rev. B 15, 544 (1977)
Electrons in 2p-AOs are more shielded by the
core electrons from the positively charged
nucleus than electrons in 2s-AOs. Accordingly,
2s-AO are lower in energy than 2p-AOs. The
energy gap DE2p-2s increases with increasing
nuclear charge. Similar trends are seen for the
3p and 3s-AOs of the 2nd-row heavy atoms.The
approximate DE2p-2s for C, N, O atoms are,
respectively, 8 eV, 11 eV, and 16 eV.