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.