Transcript Document

Electronic Structure of
Organic Materials
-
Periodic Table of Elements
Rayleigh-Ritz Principle
Atomic Orbitals (AO)
Molecular Orbitals (MO - LCAO)
Hybridization
Example: Benzene
Periodic Table of Elements
Mendeleyev: Order by weight and chemical properties
Quantum mechanics: Order by electron number and nature of orbitals!
Atomic Orbitals
Atomic Orbitals represent a solution of the time-independent
Schrödinger equation:
Where
is the Hamiltonian.
Thus the Schrödinger equation becomes:
Y are the eigenfunctions to the
operator H, and the energy levels E are the corresponding
Here the wavefunctions
eigenvalues of the solution.
Atomic Orbitals
Rayleigh-Ritz principle / method
The best guess for the solution of the Schrödinger equation
is the trial wave function
expectation value:
Ya which minimizes the energy
This is THE workhorse of all computational quantum chemistry
and computational material sciences.
Analogy: Eigenvector problem of finite (symmetric) matrices
Ax = lx  (xTAx) / (xTx) is extremum (minimal)
Atomic Orbitals
Atomic quantum
numbers:
n : main
l : angular
m : magnetic
For the most simple atom, the H-atom, the orbitals with n > 1
are energetically (almost) degenerated.
Atomic Orbitals
3D visualization of atomic orbitals
Atoms that constitute typical organic compounds such as
H, C, N, O, F, P, S, Cl have outermost (valence) electrons in s and p orbitals.
The orbitals are ordered according to their angular momentum: s=0, p=1 & d=2.
(The coloration differentiates between positive and negative parts of the wave functions.)
Molecular Orbitals
Commonly molecular orbitals are derived by the method of
Linear Combination of Atomic Orbitals (LCAO).
The ansatz for the Schrödinger equation is a linear
combination of atomic single-electron wavefunctions. Using
the Rayleigh-Ritz method, Ya is the initial guess, from which
the energy can be calculated as:
Now
Ya has to be
varied to minimize
Ea.
Molecular Orbitals
Formation of bonding and anti-bonding(*) levels:
Molecule AB
Atom A
LUMO
Atom B
*
“band gap“
HOMO
If more atoms are used to construct the molecule,
HOMO and LUMO consist of the corresponding number
of levels. (overlap)
Molecular Orbitals
-bands in conjugated polymers
Isolated
atomic
orbital
Isolated
molecular
orbital
2 “interacting”
molecular
orbitals
many
“interacting”
molecular
orbitals
Very many
“interacting”
molecular
orbitals
Molecular Orbitals
Example: The most simple molecule: H2
+
Good approximation, as the
nuclei have a much higher mass
than the electron.
Molecular Orbitals
Single atom wavefunctions ja
and jb, and linear combination
for the molecule:
Y is the ansatz for the Schrödinger equation / Ritz Principle:
Approximate solution can be calculated according from
Molecular Orbitals
C – Coulomb integral (<0)
D – Resonance integral (<0)
S – Overlap integral (0<S1)
<Ya| Vb| Ya>
<Ya| Va| Yb>
<Ya|Yb>
Molecular Orbitals
Solutions for Y:
A bond is formed as the electron has an increased probability
between the two nuclei (top):
The kinetic energy is lowered as the electron is spread over a
larger spatial region.
In the anti-bonding state, the kinetic energy is increased as the
probability is going to zero between the two nuclei.
Molecular Orbitals
Taking the Coulomb repulsion between the two nuclei at distance
Rab into account, yields for H2+ a binding energy of 1.7 eV:
E [eV]
anti-bonding
bonding
Rab [10-10m]
Molecular Orbitals
If more than one electron are to be considered, the Pauli
principle has to be obeyed, i.e. one orbital can be populated by
maximal two electrons with opposite spin.
The energetic degeneracy is lifted by the exchange interaction
(Spin orbit coupling):
Esinglet
Etriplet
Hybridization: sp3
Hybridization of atomic orbitals allow optimized geometries for bonds:
p-orbital in carbon
Hybridization: sp3
3
4
Hybridization: sp3
The sp3 hybridization leads to s-type bonds (“direkt bonds“).
Hybridization: sp3
Nitrogen in ammonia shows sp3-hybridization as well (note: free electron pair!)
Hybridization: sp2
Carbon-carbon double bonds are described by sp2-hybridization
Hybridization: sp2
For the sp2-hybridization two p orbitals are mixed with one s
orbital:
h1= s
+21/2
py
h2= s + (3/2)1/2 px - (1/2)1/2 py
h3= s - (3/2)1/2 px - (1/2)1/2 py
gives rise to three sp2-orbitals
in the plane and
one singly occupied pz-orbital
perpendicular to that plane
Hybridization: sp2
ethylene
Formation of -bonds from two
pz orbitals
Bond Length
The following generalizations can be made about bond length:
1. Bond lengths between atoms of a given type decrease with the amount
of multiple bonding. Thus, bond lengths for carbon-carbon bonds are in the
order C-C > C =C > C=C
2. Bond lengths tend to increase with the size of the bonded atoms. This
effect is most dramatic as we proceed down the periodic table. Thus, a CH
bond is shorter then a C-F bond, which is shorter then a C-Cl bond. Since
bond length is the distance between the center of bonded atoms, it is
reasonable that larger atoms should form longer bonds.
3. When we make comparisons within a given row of the periodic table,
bonds of a certain type (single, double, or triple) between a given atom and
a series of other atoms become shorter with increasing electro negativity.
Thus, the C-F bond in H3C-F is shorter then the C-C bond in H3C-CH3.
This effect occurs because a more electronegative atoms has a greater
attraction for the electrons of the bonding partner, and therefore ‘pulls it
closer,’ than a less electronegative atom.
Quoted from Organic Chemistry by G.M. Loudon
Benzene
To find a solution, the Hückel method is applied:
1.) Electrons in the s-bonds are not considered as influence for the
-electrons
2.) Ansatz for the wavefunction:
where ji are the pz wavefunctions of individual C-atoms at pos. i
Benzene
Benzene has degenerate molecular orbitals!
Benzene
Benzene
From benzene to polyacenes: red-shift in absorption
Thus the larger the -conjugated system, the smaller the
optical band gap! Compare with particle in a box again...
Benzene
Particle in a box:
Organic Semiconductors: Basics
Most simple conjugated polymer: Polyacetylene (sp2-hybridized)
C
H
H
H
H
C
C
C
C
C
C
H
H
C
H
H
Compare with Polyethylene (sp3-hybridized):
H
C
C
H
H
H
H
C
C
H
H
H
H
C
C
H
H
H
H
C
C
H
H
H
Organic Semiconductors: Basics
Polyacetylene: dimerization
unit cell a and 2a
( )
metall?
a
2a
(
Band structure (a) and density of states (b) for
trans-(CH)x. The energy gap of 20 opens up at
k=2ð/a due to the Peierls distortion
)
semiconductor!
The total energy (electronic plus lattice
distortion) as a function of u. Note the
double minimum associated with the
spontaneous symmetry breaking and the
twofold degenerate ground state.
Organic Semiconductors: Basics
Polyacetylene
(a) undimerized structure
(b) dimerized structure due to
the Peierls instability.
(c) cis-polyacetylene
(d) degenerate A and B phases
in trans-polyacetylene
(e) soliton in transpolyacetylene
(f) ...again a bit more realistic.