In atomic units i.e. ~ = q e = m e = 1

1. 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

2. 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:

3. 4. Molecular many electron systems: electronic & nuclear movement Multiplication with and integration over electron coordinates cn(R)  Schrödinger equation for nuclear motion: C describe coupling between nuclear and electron motion thus the resulting coupling of electronic states (non-adiabatic coupling)

4. 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: • displays the potential for nuclear motion • Born-Oppenheimer: nuclear motion is described on adiabatic potential energy surfaces

5. 4. Molecular many electron systems: electronic & nuclear movement

6. 4. Molecular many electron systems: electronic & nuclear movement Atmomic Orbitals: main quantum number (n) orbital quantum number (l=s,p,d,f) 2s 1s 2p http://en.wikipedia.org/wiki/Atomic_orbital

7. 4. Molecular many electron systems: electronic & nuclear movement Atomic Orbitals: m= -l ... l Energysequence: http://en.wikipedia.org/wiki/Atomic_orbital

8. 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 http://en.wikipedia.org/wiki/Pi_bond

9. Carbon Ground state: 1s2 2s2 2px1 2py1 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 usually binds 4 hydrogen (Methane). Why?: http://en.wikipedia.org/wiki/Hybrid_orbital

10. linear mix of s and p: sp3 hybrid orbitals sp3 orbital binding with 1s: s - Bond 4. Molecular many electron systems: electronic & nuclear movement Influence of the Hydrogen: http://en.wikipedia.org/wiki/Hybrid_orbital

11. p - Bond s - Bond 4. Molecular many electron systems: electronic & nuclear movement C – C double Bonds (Ethene): http://en.wikipedia.org/wiki/Hybrid_orbital

12. 4. Molecular many electron systems: electronic & nuclear movement C – C double Bonds (Ethene), sp2: s - Bond p - Bond http://en.wikipedia.org/wiki/Double_bond

13. 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

14. Triplet T1: usually lower energy than S1 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

15. 4. Molecular many electron systems: electronic & nuclear movement Molecular orbital Electronic configuration Electronic states UV/Vis-absorption spectrum

16. 4. Molecular many electron systems: electronic & nuclear movement Jablonski-Scheme v = 1 UV-VIS-spectroscopy J = 4 Microwave-spectroscopy S4 Internal conversion[10-14 s] S3 J = 3 Rotational levels Excitation [10-15 s] J = 2 J = 1 Tn J = 0 v = 0 S2 IR- & NIR- spectroscopy Intersystem crossing S1 T1 Vibrational levels v = 4 Fluorescence[10-9 s] Phosphorescence[10-3 s] v = 3 v = 1 v = 0 S0 IPC Friedrich-Schiller-Universität Jena 16

17. 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

18. 5. UV-Vis-Absorption 5.1 Franck-Condon principle • Transition dipole moment for a transition between the states |i and |f: • For excitation follows: • 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: B.O.-approximation

19. 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 directly proportional to the square modulus of the overlap integral between vibrational wavefunctions of the two electronic states = Franck-Condon-Factor:

20. 5. UV-Vis-Absorption 5.1 Franck-Condon principle |f |i |f |i