Introduction to Theories of Chemical Reactions Graduate Course Seminar Beate Flemmig FHI
I. Overview What kind of reactions? gas phase / surface unimolecular / bimolecular thermal / photochemical
What kind of information? structure of reactants, products ΔE react mechanism TS, E a k τ, lifetime of intermediates
What kind of theoretical approach? MO and VB theory almost without calculation, based on symmetry arguments Quantum Chemistry Born-Oppenheimer approximation solution of electronic Schrödinger equation optimization of stationary points of PES follow IRC TD data TS theory
Molecular Dynamics semiempirical potential or calculation of forces on the fly (CPMD) T 0 beyond BO (wave-packet dynamics) more than one PES (photochemistry!) solution of nuclear SE real-time evolution of a chemical reaction related to pump-probe spectroscopy or femtosecond chemistry statistical mechanics many particles
An attempt to summarize the approaches a: number of particles x: electronic coordinates X: nuclear coordinates t: time R. Daudel, Quantum Chemistry, John Wiley, p. 299
II. Example Reaction: Chelate Ring Inversion Organometallics, 22 (2003) 1196.
experimentally ΔE a 62 kj/mol from 1 H NMR H. Köpf, Angew. Chem. 83, 146-147, (1971) Δν between signals of H at the two C 5 H 5 -rings k(t C ) = π 2 Δν approximate formula coalescence temp. T C (~ 20 ºC) k(t C ) = RT C N A h e ΔE a RT C M. Hesse, H. Meier, B. Zeeh Spektroskopische Methoden in der organischen Chemie Thieme Verlag 1991
Lewis structure Stabilizing and destabilizing effects of folding
III. Application of MO Theory Ring opening and closing of ozone - A forbidden reaction
molecular plane σ preserved in the reaction levels of MOs with different symmetry w. r. t. σ are crossing
computed* kinetic persistence of the cyclic isomer * K. Ruedenberg et al. J. Chem. Phys. (1991) 94 8054.
life time of the intermediate k = Ae E a RT Arrhenius equation τ ~ 1 k
unimolecular reaction first-order rate law d x d t = k c 0 x [ ] ln c 0 c 0 x = kt x := 1 2 c 0 t :=τ half-life time τ = ln2k 1
Fill in numbers assume preexponential factor of 10 15 s -1 (unimolecular reaction) calculated barrier 23 kcal/mol = 95.7 kj/mol (the lower one) room temperature 25ºC k = Ae E a RT k =10 15 s 1 e -1 95.7kJmol 8.31JK 1 mol 1 298K k = 0.64 10 2 s -1 τ = ln2k 1 τ = 44 s
Use cyclic ozone as a ligand 6 π electron donor
Some Predicted Complexes applied an 18-electron strategy Transition Metal Complexes of Cyclic and Open Ozone and Thiozone Flemmig, B.; Wolczanski, P. T.; Hoffmann, R. J. Am. Chem. Soc. 2005 ASAP Article
Woodward - Hoffmann Rules to explain stereoselectivity of cycloadditions
different symmetry elements remain, depending on the mechanism
correlation diagram: level-crossing for disrotatory path see also: R. Hoffmann, Angew. Chem. Int. Ed. (2004) 43, 6586-6590
coefficients of Hückel solutions (π systems) correspond to particle-in-abox solutions
VB-Theory S. Shaik, A. Shurki, Angew. Chem. Int. Ed. (1999) 38, 586-625.
IV. Application of Quantum Chemistry use Born-Oppenheimer approximation and solve the electronic Schrödinger equation (for fixed R)
ideally: determine electronic properties as functions of nuclear coordinates - i. e. determine the PES Potential Energy Surface, PES: governs nuclear motion, forces on the nuclei F x = - E/ x stationary points correspond to (meta)stable species (local or global minima) and to transition states (saddle points) shape of PES around stationary points determines vibrational spectra electronic transitions correspond to transitions from one PES to another minimum energy path corresponds to reaction coordinate
in reality: Calculated PES only available for very small systems, for example H+H2 in colinear arrangement without Coulomb interactions with Coulomb interactions H. Eyring, M. Polanyi, 1931 M. Karplus et al. 1968
in practice: stationary points of PES, their characterization (frequency calculations), and maybe the reaction path (IRC calculation)
Transition State Theory Eyring equation H. Eyring, J. Chem. Phys. (1935) 3, 107 k = k b T h Q TS Q R e E a k b T partition function Q = g j levels ε vib = n + 1 2 e ε j k b T hν
Example: Study of a Radical Clock Rearrangement on a Surface FTIR: ring modes at1393 cm -1 and 1434 cm -1 vanish and C=C stretch mode emerges at 1645 cm -1 C. M. Friend, I. Kretzschmar, JACS (2000) 122 12395.
Cyclopropylmethoxide and 3-butenyloxide on Mo(110) ΔEtotal = 31 kj/mol Flemmig, B.; Kretzschmar, I.; Friend, C. M.; Hoffmann, R. J. Phys. Chem. A. 2004; 108(15) 2972-2981
Alternative mechanisms already for the isolated molecule
Which is the rate-determining step? the one with the highest activation energy (not the one with the highest barrier) see also: H. Eyring et al. The Theory of Rate Processes, McGraw Hill, New York 1941.
V. Molecular Dynamics nuclei have kinetic energy T K > 0 H = T K + U T K = 1 2 P I 2 M I U : T e (r) + V ek (r,r) + V ee (r) + V KK (R) P I : classical momentum classical equations of motion H R I = P I H P I = R I solutions R I (t) and P I (t) are the trajectories of the nuclei
VI. Wave-Packet Dynamics treat also the nuclei quantum-mechanically HΦ(R,t) = ih t Φ(R,t) H = T K + U + [H extern ] T K = h2 2 2 M I simulations only for intervals of a few picoseconds very fast reactions: e.g. laser-induced isomerizations
ϕ R R
References/Acknowledgement C. Engler (Uni Leipzig) Überblick über die Näherungshierarchie und Lösungsansätze in der Quantentheorie, Graduate Course H.-J. Werner (Uni Stuttgart) Computational Chemistry in Catalysis, Graduate Course R. Hoffmann (Cornell) Bonding in Molecules, CHEM798 W. J. Moore, D. O. Hummel, Physikalische Chemie, de Gruyter Berlin 1986.