Introduction to Theories ofChemical Reactions
Graduate Course SeminarBeate Flemmig
FHI
I. Overview
What kind of reactions?
• gas phase / surface• unimolecular / bimolecular• thermal / photochemical
What kind of information?
• structure of reactants, products• ΔEreact• mechanism• TS, Ea• k• τ, lifetime of intermediates
What kind of theoretical approach?
• MO and VB theoryalmost without calculation, based on symmetryarguments
• Quantum ChemistryBorn-Oppenheimer approximationsolution of electronic Schrödinger equationoptimization of stationary points of PESfollow IRCTD data → TS theory
• Molecular Dynamicssemiempirical potentialor calculation of forces ‘on the fly’ (CPMD)T ≥ 0
• beyond BO (wave-packet dynamics)more than one PES (photochemistry!)solution of nuclear SEreal-time evolution of a chemical reactionrelated to “pump-probe“ spectroscopy or ”femtosecondchemistry”
• statistical mechanicsmany particles
An attempt to summarize the approaches …
R. Daudel, Quantum Chemistry, John Wiley, p. 299
a: number of particles
x: electronic coordinates
X: nuclear coordinates
t: time
II.ExampleReaction:Chelate RingInversion
Organometallics, 22 (2003) 1196.
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k(TC ) =RTCNAh
e−ΔEa
RTC
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k(TC ) =π
2Δν
experimentally ΔEa 62 kJ/mol from 1H NMR
H. Köpf, Angew. Chem. 83, 146-147, (1971)
M. Hesse, H. Meier, B. ZeehSpektroskopische Methoden in derorganischen Chemie Thieme Verlag 1991
approximate formulaΔν between signalsof H at the twoC5H5-rings
coalescence temp.TC (~ 20 ºC)
Stabilizing and destabilizing effects of folding
Lewis structure
III. Application of MO TheoryRing opening and closing of ozone - A forbidden reaction
molecular plane σ preserved in the reactionlevels 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
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k = Ae−EaRT
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τ ~ 1k
Arrheniusequation
half-life time
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d xd t
= k c0−x[ ]
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ln c0c0−x
= kt
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x := 12c0
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t :=τ
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τ = ln2k−1
unimolecular reaction
first-order rate law
Fill in numbersassume preexponential factor of 1015s-1 (unimolecular reaction)
calculated barrier 23 kcal/mol = 95.7 kJ/mol (the lower one)
room temperature 25ºC
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k = Ae−EaRT
k =1015s−1e−95.7kJmol-1
8.31JK−1mol−1298K
k =0.64 ⋅10−2s-1
τ = ln2k−1
τ = 44 s
Use cyclic ozone as a ligand
6 π electron donor
applied an ‘18-electron strategy’
Some Predicted Complexes
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 ofHückel
solutions(π systems)
correspond toparticle-in-a-box solutions
VB-Theory
S. Shaik, A. Shurki, Angew. Chem.Int. Ed. (1999) 38, 586 - 625.
IV. Application of QuantumChemistry
use Born-Oppenheimer approximation
and solve the electronic Schrödinger equation (for fixed R)
ideally: determine electronic properties as functions ofnuclear coordinates - i. e. determine the PES
Potential Energy Surface, PES:
• governs nuclear motion, forces on the nuclei Fx = -∂E/∂x
• stationary points correspond to (meta)stable species (local orglobal minima) and to transition states (saddle points)
• shape of PES around stationary points determinesvibrational spectra
• electronic transitions correspond to transitions from onePES 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 (IRCcalculation)
Transition State Theory
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k =kbTh
QTS
QR e−Ea
k bTEyring equationH. Eyring, J. Chem. Phys. (1935) 3, 107
partition function
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Q = g jlevels∑ e
−ε j
k bT
εvib = n +12
hν
Example: Study of a Radical ClockRearrangement on a Surface
FTIR: ring modes at1393 cm-1and 1434 cm-1 vanish andC=C stretch mode emerges at 1645 cm-1
C. M. Friend, I. Kretzschmar, JACS (2000) 122 12395.
Cyclopropylmethoxide and 3-butenyloxide on Mo(110)
Flemmig, B.; Kretzschmar, I.; Friend, C. M.; Hoffmann, R. J. Phys. Chem. A. 2004; 108(15) 2972-2981
ΔEtotal = 31 kJ/mol
Alternative mechanisms already for the isolated molecule
Which is the rate-determining step ?
… the one with the highest activation energy (not the onewith the highest barrier)
see also: H. Eyring et al. The Theory of Rate Processes, McGraw Hill, New York 1941.
V. Molecular Dynamics
classical equations of motion€
H = TK +U
TK =12
PI2
MI∑
nuclei have kinetic energy TK > 0
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U :Te (r) + VeK (r,R) + Vee (r) + VKK(R)PI : classical momentum
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RI•
=∂H∂PI
PI•
= −∂H∂RI
solutions RI(t) and PI(t) are the trajectories of the nuclei
VI. Wave-Packet Dynamicstreat also the nuclei quantum-mechanically
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HΦ(R,t) = ih ∂∂tΦ(R,t)
H = TK +U+ [Hextern ]
TK =h2
2∇2
MI∑
simulations only for intervals of a few picoseconds
very fast reactions: e.g. laser-induced isomerizations
R
R
ϕ
References/Acknowledgement
• C. Engler (Uni Leipzig) Überblick über dieNäherungshierarchie und Lösungsansätze in derQuantentheorie, Graduate Course
• H.-J. Werner (Uni Stuttgart) Computational Chemistry inCatalysis, Graduate Course
• R. Hoffmann (Cornell) Bonding in Molecules, CHEM798
• W. J. Moore, D. O. Hummel, Physikalische Chemie, deGruyter Berlin 1986.