Reaction

1
Reaction Transition State
Computational Chemistry 5510 Spring 2006 Hai Lin
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Bridge the Gap

• Experimental rate constatnts (macroscopic)

• Quantum states (microscopic)

Boltzmann Distribution
Macroscopic rate constant is an average over all
microscopic rate constants weighted by the
probability of finding a molecule with a given
set of quantum numbers (e.g., electronic,
vibrational, rotational, translational, nuclear
spin ...)
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Transition State Theory
Saddle Point

• A reaction proceeds along a reaction path from
  one minimum to another via an intermediate
  maximum (first-order saddle point).

E
Product

• The transition state (TS) passes through the
  maximum and devide the hyperspace into reactant
  and product.

x2
x1
Reactant
Reaction Path
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Classical Reaction Energy Profile
Transition state DE
Energy
DE classical energy
DE 0
Perpendicular coordinates
s 0
Reaction coordinate

• Reaction coordinate leads the system from
  reactant to product along a minimum energy path
  (MEP).
• Transition state theory places the transition
  state at the maximum of MEP.

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Free Energy Profile
Generalized transition state DG
Energy
DG free energy
DE classical energy
DG 0
Perpendicular coordinates
s 0
s
Reaction coordinate
Variational effect

• Variational transition state theory (VTST) places
  the transition state at the maximum of free
  energy curve.
• Transition state theory (TST) ignores the
  variational effect.

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A Semi-classical Theory

• Motion along the reaction coordinate is treated
  classically.

AB C
Reactive Trajectory
Minimum Energy Path

• Motions perpendicular to the reaction coordinate
  are treated quantum mechanically.

Non-reactive Trajectory
R(B-C)
A BC

• All trajectories originated from reactant and
  passing through TS go to product.

Transition State
R(A-B)
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TST Rate Constant

• There is an equilibrium energy distribution among
  all possible quantum states along the reaction
  path including at the transition structure point.

• The canonical TST rate constant k at a given
  temperature T is

k (kBT/h) Exp(-DG /RT)
or
k (kBT/h) Exp(DS /R) Exp(-DH /RT)
Entropy contribution
Enthalpy contribution
kB is the Boltzmann constant, h is the Planck
constant, and R is the gas constant.
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TST Rate Constatnt (2)

• Canonical TST rate constant can also be expressed
  as

k (kBT/h) (Q/QR) Exp(-DE /RT)
QR is the partition function for reactant. Q is
the generalized partition function for TS
(because the motion along the reaction coordniate
is an imaginary-frequency vibration). Please
note that we calculate the electronic partition
function for TS differently from the textbook
(Equ. 12.21) in that the Exp(-DE /RT) is
explicitly taken out.
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Partition Functions

• Partition function for a molecule

q Si gi Exp(-Dei /kBT)
qtot qelec ? qtrans ? qrot ? qvib etot eelec
etrans erot evib

• Partition function for M molecules

Qtot q1 ? q2 ? ... ? qM
for distinguishable particles
Qtot qM/M!
for non-distinguishable particles
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Partition Functions (2)

• Electronic partition function

qelec gGS Si gESi Exp(-DeESi /kBT)
where GS and ESi denote ground state and the i-th
excited state, respectively. Normally DeESi gtgt
kBT, and qelec gGS.

• Translational partition function

qtrans (2pMkBT/h2)3/2V
where M is the molecular mass, V is usually taken
to be the volume of one mole of ideal gas.
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Partition Functions (3)

• Rotational partition function for linear molecules

qrot (8p2I0 kBT) / (sh2)
where I0 is the moment of inertia.

• Classica rotational partition function for
  nonlinear molecules

qrot p½ (8p2 kBT / sh2)3/2 (I1I2I3)3/2
where I1, I2, and I3 are moments of inertia along
the principal axes of inertia. (Using the
principal axes of inertia, the matrix of moment
of inertia is diagonal.) Symmetric number s is
determined by the molecular symmetry.
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Partition Functions (4)

• Vibrational partition function

qtot qvib1 ? qvib2 ? ... ? qvibF
At minima, F 3N 6(5) for nonlinear (linear)
molecule At transition state, F 3N 7(6) for
nonlinear (linear) molecule

• Harmonic oscillator approximation

qvibi 1 Exp(-hni /kBT)-1
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Reaction Coordinate Motion

• Classical picture

Energy
Energy
0
0.5
1
Reaction coordinate
Transmission probability
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Transmission Coefficients

• A semi-classical treatment to improve the rate
  constants

kcorr k kTST
Account for all quantum contributions
Rate constant by TST calculation
Corrected rate constant with quantum contributions

• Value of k is usually between 0.5 and 2, but can
  also be very large in some cases, e.g., at very
  low T.
• At high T, recrossing is important.
• At low T, tunneling is important.

Note Transmission coefficient here is not the
transmission probability in the previous slide!
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Variational Transition State Theory

• Optimize the location of transition state (not
  necessarily at the saddle point).
• Include multi-dimensional tunneling contributions
  (can cut the corner at the concave side)

AB C
Sample Trajectory
Minimum Energy Path
Multi-dimensional Tunneling Path
Optimized Generalized Transition State
R(B-C)
A BC
Saddle Point
R(A-B)
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How Well Can We Do?

• Errors in DE 10 kcal/mol
• Meaningless rate constant calculations
• Refine the energies, please.

• Errors in DE 1 kcal/mol
• Good accuracy, difficult to get
• Harmonic oscillator and rigid rotor
  approximations used commonly
• Error in DG can be several kcal/mol.
• Errors in rate constants with a factor of 10 at T
  300 K

• Errors in DE 0.1 kcal/mol
• Excellent accuracy, only possible for limited
  cases
• Anharmonicity to be considered
• Error in DG can be 0.2 kcal/mol.
• Recrossing and tunneling become important, and
  advanced dynamics treatment are desirable
• Errors in rate constants with a factor of 2 at T
  300 K

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Optimize the Transition Structure

• The key a good initial guess for the geometry

• Newton-Raphson methods
• Check convergency

• Vibrational analysis to identify the TS
• one and only one imaginary-frequency mode that
  corresponds to the motion of the reaction
  coordinate

• Verify that the minimum energy path passing
  through the TS indeed connects the reactant and
  product
• Intrinsic reaction path (IRC) calculation

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Obtain a Good Guess

• Many ways to generate a reasonable initial guess
  of the transition structure
• None of these ways guarantees a good initial
  guess
• Some commonly used tricks
• Interpolate between the reactant and product
  using internal coordinates

• Impose symmetry when possible

• Surface scan by constrainted optimization

H
C
N
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Constrainted Surface Scan

• Imagine that the OH is approaching the CH4
  slowly, how does the CH4 continuously adjust
  itself accordingly?
• Successively change r1 by a small amount in the
  optimizarion where r1 is constrainted, and
  monitor the energy and r2.

E
r2
r1
Reaction coordinate
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Constrainted Surface Scan (2)

• Use lower-level of theory and larger step size to
  get preliminary energy profile, and refine the
  region close to the maximum with higher-level of
  theory and smaller step size.
• Scan for small model, and use the result to build
  TS for large system.

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Summary

• Reaction Energy Profiles
• Classical reaction energy profile
• Free energy profile
• Variation effect
• Transition State Theory
• Basis assumptions
• Partition functions
• Quantum contributions
• Optimize Transition Structure
• Initial guess
• Vibrational analysis at TS
• IRC verification

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Your Homework

• Read the slides.
• Read textbook (Take notes when you read.)
• 12
• 14.8
• Questions
• What assumptions does Transition State Theory
  make?
• What is the symmetric number for H2O and HOD?
• Which coordinate can be used as the reaction
  coordinate for reaction Cl HF (1) rCl-H, (2)
  rH-F, (3) dr rCl-H - rH-F, (4) R rCl-H
  rH-F, or none of the above?
• What is the recrossing and what is tunnelling?
  Under what circumstance does one should consider
  their contribution?