Download s - Bond - Friedrich-Schiller

4. Molecular many electron systems: electronic & nuclear movement
 Hamiltonian for a polyatomic molecule treated as Coulomb system with N nuclei
(coordinates {R}) and n electrons (coordinates {ri}) :
In atomic units i.e. ~ = qe = me = 1
Kinetic energy operator for nuclei
Kinetic energy operator for electrons
Nuclei-electron interaction operator
Electron-electron interaction operator
Nuclei-nuclei interaction operator
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IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
 (3N + 3n)-dimensional problem:
Born-Oppenheimer Approximation: separate treatment of electronic and nuclear
motion allows the total wavefunction of a molecule to be broken into its electronic
and nuclear
components:
Does not depend on {ri} =
constant for given nuclear
geometry
Decomposition of Hamiltonian:
= adiabatic potential energy surfaces
Schrödinger equation for complete problem:
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IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Multiplication with
and integration over electron coordinates
 Schrödinger equation for nuclear motion:
C describe coupling between nuclear and electron motion thus the resulting
coupling of electronic states (non-adiabatic coupling)
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IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
 Born-Oppenheimer approx. neglects coupling between nuclear and electron motion

C = 0

Electrons adjust immediately or adiabatically to any nuclear motion:



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displays the potential for nuclear motion
Born-Oppenheimer:
nuclear motion is described on adiabatic potential energy surfaces
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
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IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Atmomic Orbitals:
main quantum number (n)
1s
2s
orbital quantum
number (l=s,p,d,f)
2p
http://en.wikipedia.org/wiki/Atomic_orbital
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IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Atomic Orbitals:
m= -l ... l
Energy
sequence:
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http://en.wikipedia.org/wiki/Atomic_orbital
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Bond types (orbital symmetry):
Orbitals:
2 x 2p = pp-p
Bonds:
plane of symmetry
Free Pairs: n
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http://en.wikipedia.org/wiki/Pi_bond
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Binding orbitals to explain the structure of molecules:
Hybridization (Valence Bond Theory)
useful for molecules with C, N, O, (S, P)
limits noticeable for d-orbitals involved in binding
Carbon Ground state: 1s2 2s2 2px1 2py1
Carbon usually binds 4 hydrogen (Methane). Why?:
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http://en.wikipedia.org/wiki/Hybrid_orbital
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Influence of the Hydrogen:
linear mix of s and p: sp3 hybrid orbitals
sp3 orbital binding with 1s:
s - Bond
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http://en.wikipedia.org/wiki/Hybrid_orbital
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
C – C double Bonds (Ethene):
p - Bond
s - Bond
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http://en.wikipedia.org/wiki/Hybrid_orbital
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
C – C double Bonds (Ethene), sp2:
s - Bond
p - Bond
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http://en.wikipedia.org/wiki/Double_bond
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
 Molecular orbital or electronic configuration (z.B. Formaldehyd)
 Energetic order of transitions:
p* ← n < p* ← p < s* ← n < p* ← s < s* ← s
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IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
 spin multiplicity
LUMO
HOMO
• Total spin quantum number S = ∑ si
with si = +½ or - ½
• Multiplicity M = 2S + 1
• M = 1: Singulet
• M = 2: Dublet
• M = 3: Triplet
Triplet T1: usually lower energy than S1
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IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Molecular
orbital
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Electronic
configuration
Electronic
states
UV/Vis-absorption spectrum
IPC Friedrich-Schiller-Universität Jena
4. Molecular many electron systems: electronic & nuclear movement
Jablonski-Scheme
Excitation [10-15 s]
v=1
Rotational levels
J=4
v=0
Microwavespectroscopy
J=3
J=2
J=1
J=0
UV-VIS-spectroscopy
S4
S3
Internal conversion
[10-14 s]
Tn
S2
IR- & NIRspectroscopy
Intersystem crossing
v=4
v=3
Vibrational levels
S1
T1
Fluorescence Phosphorescence
[10-9 s]
[10-3 s]
v=1
v=0
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S0
IPC Friedrich-Schiller-Universität Jena
5. UV-Vis-Absorption
5.1 Franck-Condon principle
 Interpret electronic absorption spectra based on
||2 of the vibrational levels
 electronic transitions (~10-16s)
are much faster than the vibrational
period (~10-13s)
of a given molecule
thus nuclear coordinates do not
change during transition
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IPC Friedrich-Schiller-Universität Jena
5. UV-Vis-Absorption
5.1 Franck-Condon principle
 Transition dipole moment for a transition between the states |i and |f:
 For excitation
follows:
B.O.-approximation
 Electronic transition dipole moment
is developed in a rapidly converging Taylor
expansion about nuclear displacements from the equilibrium position
 Condon approximation neglects higher order terms i.e. electronic transition dipole
moment
is assumed to be constant i.e. nuclear coordinates correspond to
equilibrium geometry
 Condon approximation:
Transition dipole moment:
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IPC Friedrich-Schiller-Universität Jena
5. UV-Vis-Absorption
5.2 Franck-Condon principle


= degree of redistribution of electron density during transition
= degree of similarity of nuclear configuration between vibrational
wavefunctions of initial and final states.

 Intensity of a vibronic transition is direct proportional to the square modulus of the
overlap integral between vibrational wavefunctions of the two electronic states =
Franck-Condon-Factor:
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IPC Friedrich-Schiller-Universität Jena
5. UV-Vis-Absorption
5.1 Franck-Condon principle
|f
|f
|i
|i
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IPC Friedrich-Schiller-Universität Jena