Understanding Chemical Reaction Mechanisms with Quantum Chemistry

JN Harvey (Leuven),Winter School on
Computational Chemistry, 2015
1
Understanding Chemical Reaction
Mechanisms with Quantum Chemistry
Jeremy Harvey
Department of Chemistry
KU Leuven
Belgium
jeremy.harvey@chem.kuleuven.be

2
Understanding Chemical Reaction
Mechanisms with Quantum Chemistry
• Computation is now a routine tool to probe reaction
mechanisms, which can be used to:
• Characterize (structure and properties) elusive
intermediates to assist identification
• Calculate relative (free) energies of the species
• Locate and characterize (bonding, substituent effect, etc.)
transition states connecting different species
• Calculate rate constants

3
Overview of material covered
I.  Structural characterization
II.  Local and global minima: the conformer problem
III.  Transition state searching
IV.  Understanding solvent effects
V.  Statistical mechanics, transition state theory (including
observations on barrierless reactions)
VI.  From the (free) energy surface to mechanism
Studies of reaction mechanisms using computational chemistry require studying the struc-
ture of reactants and intermediates (by default in vacuum), and also the structure and energy
of TSs. One also needs to take solvation into account (at least approximately using e.g. a
continuum).To study mechanisms one also needs to consider equilbria and rates for
elementary steps – and this requires the use of statistical mechanics and rate theories such
as transition state theory. Finally, one needs to see how the different steps fit together to
form the mechanism.

4
I. Structures: Geometry Optimization
One way to
find the
geometry
is to calculate
the energy
at all
possible
geometries.
Only possible
with 2 or 3
atoms!!

5
Geometry Optimization – 2
Efficient methods use
the gradient and the
Hessian matrix):
R
V
g
∂
∂
=
'2
1 2
RR
V
H
∂∂
∂
=
• The gradient g is easy to calculate once the wavefunction is known
(cost ≈ same as energy), and is implemented for many methods in
many codes.
• Analytical Hessian matrices can also be calculated, but this is more
expensive (≈ 10 times energy, and needs more disk and memory)
• Typically, the gradient and an approximate Hessian are used
• Optimization requires ~ N steps (N is number of atoms).
( ) 1
min
−
×−= HRgRR

6
What is a Geometry?
Computationally = the structure at the minimum of the
potential energy surface (re)
Microwave spectrometry: zero-point average (r0 not re)
X-Ray crystallography: thermal average + packing effects.
Can be quite different from re!
For ‘stiff’ degrees of freedom (e.g. C–C bond lengths),
computation and experiment usually in good agreement.
For soft degrees of freedom, comparison may not always
be meaningful.

7
II. Global and Local Minima
• Local minimization is most
common with QM methods
• Global optimization is much more difficult.
• Energies of conformers can be very different, which can lead to
incorrect conclusions.

8
Strategies with Conformers
• Ignore them (multiple identical
minima, that affect partition
function only slightly)
• Brute force + chemical intuition.
3 rotatable bonds; 3 local minima each: 27 conformers. But many
are unreasonable, and a skilled researcher can just calculate all the
reasonable ones (e.g. JACS 2005, 13468)
• Use experiment e.g. X-ray data
• Conformational searching at a low level
(e.g. MM) then re-optimization
• Recognize that in some cases, the lowest
energy species cannot be found

9
III. Locating Transition States
As for minima, efficient
methods use gradient + Hessian
Algorithm needs to “know” what to
do in each direction:
• “Go down” (3N – 7 directions)
• “Go up” (1 direction)
Approximate Hessian used to locate
minima does not work usually.

10
Transition States. 1. Symmetry
In some cases, the TS has a different symmetry than the reactants and products, and
normal optimization (‘opt’ in Gaussian) can be used, provided one starts with a
structure that has the higher symmetry. E.g. inversion of ammonia (TS is planar, D3H vs.
C3V reactants), or H + H2 à H2 + H (TS has a plane of symmetry, D∞H) or
intramolecular proton transfer in enol of acetylacetone (again, extra symmetry, C2V in
TS).This is rarely useful for ‘real’ reactions but it is conceptually interesting.

11
Transition States. 2. Hessian
In a limited region around the TS, the Hessian has a negative eigenvalue
With chemical intuition (or other calcs), a geometry within this
region may be guessed and used as a starting
point.The keyword ‘opt(ts,calcfc)’ is used.
In some cases, it
may be faster to
calculate
the Hessian at
lower level (HF,
semiempirical)
which requires a
two-step job

12
Transition States. 3. QST.
The reaction coordinate at the TS resembles the difference between
reactant and product geometries.This is the basis of the QST
(Quasi-Synchronous Transit) method.
Reactant
Minimum
Product
Minimum
TS
TS guess
Structure A
Structure B
Search Direction
2 geometries (do not have
to be reac and prod) are given.
The program uses them to guess
TS geom and run optimization.
No exact Hessian is needed.
Keyword: ‘opt(qst2)’ or ‘opt(qst3)’

13
QST Method example
Reactant
Minimum
Product
Minimum
TS
TS guess
Structure A
Structure B
Search Direction
Chair conformer
is closer to TS.
For bimolecular reactions,
there is not a single
‘geometry’ for the reactants
build constrained
structure with
forming bonds
~ 2Req in length

14
Transition States. 4. Mapping.
• If the reaction involves a change in a bond length (angle),
carry out several calculations with different values for the distance
(or angle), while freezing this coordinate.
• Use the resulting profile to identify the TS region.
• Use methods 2 or 3 above.

15
Locating Transition States –
Concluding Points
• The ‘right’ strategy depends on the problem at hand
• Provided a search is successful, the result will not depend on which
method was used – pragmatism is key
• There are typically many TSs in a system. It is important to check
that one has the one that one thinks one has – inspection of the
structure / inspection of the imaginary mode / optimization from
distorted structure / IRC optimization
• Systems with many atoms typically have many conformers – this can
be true for TSs also and requires caution
• Chemical experience is important

16
IV. Reactions in Solution
Gas-phase system
Solvated ion+–
+–
+
–
+
–
+ –
+ –
+
–+
–
+–
Polarisable continuum
Structures, and especially (free) energies
can be very different in vacuum and in
solution – and this usually needs to be taken
into account

17
Solvent effects: Qualitative Aspects
CH3
ClCl
–
CH3
ClCl
–
CH3 ClCl
–
It is very helpful to have a qualitative
intuition of the importance of
solvation for different species. By and
large, solvation free energy is largest
for species with:
•  Greater partial charges
•  More solvent exposure
•  More polar solvent
•  For anions, hydrogen-bonds
donated by solvent
•  For cations, hydrogen bonds
donated to solvent
The identity reaction of chloride +
methyl chloride is a good model to
develop this qualitative under-
standing.To do this, it is helpful to
work backwards from the known
solvated free energy surface.This
model reaction is simple; products
and reactants have the same free
energy, there are no
intermediates, and there
is just one barrier

18
The solvation interactions for separated
reactants, reaction partners in close contact,
and TS, are all strong since all are anionic.
However, there are differences due to
different solvent exposure of charge.
Reactants: chloride (red)
is highly
solvent (green) exposed.
CH3Cl (blue) also (but
neutral, so interacts less
with solvent)
Reactants in close
contact: the charge on
chloride is now partially
desolvated
In the TS, the
charge is smeared
over both Cl
atoms, and
remains less
solvent-exposed
than in separate
chloride

19
CH3
ClCl
–
CH3 ClCl
–
Good
solvation
Less good
solvation
CH3
ClCl
–
Such reasoning allows one to construct a rough mental model of the vacuum-phase (free)
energy surface, reasoning backward from the (intuitively known) solution-phase (free)
energy surface. Note that in vacuum there is an intermediate!
Solution-phase PES
Vacuum PES

20
Continuum solvent models
•  For a more quantitative approach!
•  Continuum models are straightforward to use, and yield good accuracy(within
a few kcal/mol of experiment) for free energy of solvation at a given fixed
geometry.
•  The good accuracy does not mean that the models are perfect. Indeed, even a
very simplistic model performs quite well: “[If] one makes the assumption that all ions
have the same solvation free energy (-65.0 kcal/mol), that all nonaqueous neutrals have the
same solvation free energy (-5.38 kcal/mol), and that all aqueous neutrals have the same
solvation free energy (-2.99 kcal/mol), […] then the [mean unsigned errors] MUEs for this
“three parameter model” are 8.6, 1.5, and 2.7 kcal/mol, respectively. In only 6 of 15 cases do
the [continuum] models show smaller MUEs than the three-parameter model.” Cramer and
Truhlar, Acc. Chem. Res. 2008, 41, 760.
•  Note that some cases require explicit solvent modelling, when solute-solvent
interaction is strong.This is straightforward to do from the point of view of
electronic structure theory but raises issues of conformational searching and
statistical mechanics.

21
V. Rate and equilibrium constants
• Once one has characterized the structure of reactants,
intermediates and TSs, one can seek to model the rates and
positions of equilibrium for each step in a mechanism.
• This can be done informally & qualitatively
• Or more quantitatively – this requires statistical theories (or in
principle reaction dynamics/molecular simulation theories, though
their use in mechanistic studies is in practice rare)
• The key approximation used is the rigid rotor/harmonic oscillator/
ideal gas form of statistical mechanics, with construction of overall
molecular partition functions for each species

22
Ideal Gas Statistical Mechanics
• 1 mole of ideal gas:
• Where the partition function is:
• q can be calculated (approximately) from molecular properties
derived from computations: mass, structure, vibrational freqs
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=−
AN
q
RTKGG ln0
elvibrottrans qqqqq ×××=
V
h
Tmk
q B
trans
2
3
2
2
⎟
⎠
⎞
⎜
⎝
⎛ π
=
2
1
2
3
1
⎟
⎠
⎞
⎜
⎝
⎛ π
⎟
⎠
⎞
⎜
⎝
⎛
σ
=
ABChc
Tk
q B
rot
1
exp1
−
∏ ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ν−
−=
i B
i
vib
Tk
h
q
The rigid rotor approximation is not exact, but not very wrong even for ‘floppy’ modes < 50-100
cm-1. It is important however to check that the frequency analysis yields the correct number of real
and imaginary frequencies, and to check that the small real frequencies are not too badly affected by
numerical errors (e.g. by testing the effect of using tight DFT integration grids).Where needed, soft
frequencies that correspond e.g. to hindered rotation can be treated differently.

23
-16
-15
-14
-13
-12
-11
-10
-16 -15 -14 -13 -12 -11 -10
log (kexp) / cm3
molecule-1
s-1
log
kcomp
/
cm
3

molecule
-‐1

s
-‐1
Transition State Theory
TST is not exact
(barrier re-crossing,
tunneling, …)
Can often provide k to within
one order of magnitude,
provided E0 very accurate
RT
G
B
e
h
Tk
k
‡
Δ−
=
A + B
C + D
AB
E0
Tk
E
B B
e
qq
q
h
Tk
k
0
‡
BA
AB −
×=
As one (among many!) examples of the accuracy of
simple TST, see Petit & Harvey, PCCP, 2012, 14, 184-191.
H atom abstraction from many species by OH radical: k in
gas phase is obtained within a factor of three of experiment.

24
Solution Statistical Mechanics
If one can calculate ΔG° (for equi-
libria) or ΔG‡ (for rate constants)
in the gas phase, and can calculate
ΔG for solvation using continuum
model, this thermodynamic cycle
provides ΔG° or ΔG‡ in solvent:
There are some approximations that need paying attention to here.
•  Most important, ΔGsolv is usually obtained from continuum models as “ΔG*”, the free energy for
solvation without change in standard state. ΔG‡ in the gas phase is usually computed with a p = 1
atm standard state, whereas the correct standard state in solution would be 1 M (solute) or X
M (pure solvent). Converting from one standard state to another is easy, just add a term
RT ln (C/C’) where C and C’ are the corresponding concentrations.
•  Also, it is important to obtain an accurate gas phase ΔG‡.
•  For species that interact strongly with solvent, the continuum model may not capture or ΔGsolv
accurately enough – and a microsolvated gas phase model may be preferable (careful with
standard state!)
ΔGsolv
A + B C + D
ΔG‡
gas-phase
ΔG‡
solution
Asol + Bsol Csol + Dsol
ΔGsolv

25
• 3 T
• 3 R
• 3N – 6V
Barrierless Reactions
•  Not all elementary steps involve a saddle-point on the potential energy surface
(“Transition State”). E.g. ligand dissociation/addition or recombination of two radicals. In
each case, as fragments B and C come together to form A, potential energy V goes down.
•  All elementary steps do however involve a maximum in free energy with respect to some
generalized reaction coordinate, as bringing two fragments together leads to the loss of
rotational and translational degrees of freedom, hence a decrease in entropy.This is shown
schematically here:
B + C
A
• 6 T degrees of freedom
• 6 R degrees of freedom
• 3N – 12VTS:
• 3 T ; 3 R
• 3N – 12V
• 6 “transitional modes” which are essentially
vibrations, hence have low entropy contribution
V
G

26
Barrierless Reactions – B
•  In this case, computation can be used to map out energy surface and prove that there is
no barrier on the potential energy surface. But calculating free energy is harder, as there is
no stationary point (saddle point) corresponding to the TS.
•  Variational forms of TST can be used to obtain the critical free energy.
•  One pragmatic alternative approach to calculating rate constants is simpler. Most
barrierless reactions in solution, in the forward direction, have a “diffusion controlled”
rate constant k ~ 109 - 1010 M-1 s-1.
•  More precisely, k depends on T and the solvent viscosity η:
•  For typical T and η, this is equivalent to ΔG‡ of 4 – 5 kcal/mol
B + C
A
V
G 4-5
~10
η3
8 Tk
k B
≅

27
VI. Free energy surfaces and Mechanism
Most mechanisms involve multiple steps. How do the potential energy and free energy
surfaces map to the predicted kinetics (reaction order, rate- or turnover-limiting step, rate
constant)? And how, more generally, should the dialogue with experiment happen?
G
R
TSA
TSB
PB
PA
One simple multi-step mechanism: R can react
competitively to give PA or PB. Reaction over
the lower TS is favoured.
In case of many subsequent steps,
one needs to take into account
the correct free
energy gap to
characterize
reactivity

28
Free energy surface and kinetics
•  Trying to predict observed behaviour for a given computed free energy profile
of a multi-step mechanisms is essentially a problem in applied chemical kinetics.
Depending on one’s experience, skill in chemical kinetics, and preferences, the
following techniques can be used to assess mechanism based on a computed
free energy surface:
•  Instinct & hand-waving (OK if you know what you are doing, and in simple
cases, but can easily be wrong!)

•  The energy span approach of Kozuch and Shaik (Acc. Chem. Res., 2011, 44, 101–
110) – much more reliable, but assumes standard conditions, and is tricky to
apply in the case of multiple branches in the reaction scheme

•  Explicit kinetic modelling, using simulation software such as Copasi (
http://www.copasi.org/)

•  The author’s chapter titled “Free-Energy Surfaces and Chemical Reaction
Mechanisms and Kinetics” in the book “Modeling of Molecular Properties” edited by
Peter Comba, Wiley-VCH, Weinheim, 2011 may also be helpful.

29
Models and Mechanisms
•  Right at the start of modelling a reaction mechanism, one chooses a microscopic
model of the reacting system.
•  Sometimes this choice is obvious and unproblematic, but sometimes it is not.
Typical problems occur in the case of solvent participation and proton transfer
steps (also when counterions, dimers, phase separation, are involved).
•  Solvent participation: a solvent molecule may plays a major role in a mechanism,
but if is not suspected when building the microscopic model, the latter may
completely omit the solvent.
•  Proton transfer: especially in polar solvents such as water, many mechanisms
involve acid-base equilibria – these need to be treated correctly (e.g. protons
should not be added/removed without regard to the free energy implications).
•  The same applies to electron transfer in redox chemistry.

30
Studying Reaction Mechanisms
•  One must be careful to distinguish (a) characterization of, b) support of and (c) proof of a
reaction mechanism.
•  (a) Can be done by calculating structures and properties of intermediates for a
proposed mechanism that is assumed to be correct.
•  (b) Can be done by showing that these properties (including the all-important TS
energies) are consistent with the assumed mechanism
•  Just as it is for experimentalists, (c) is much harder!
•  Need to reproduce experimental rate laws, rate constants, and their dependence on
substituents, temperature, solvent, and show that all possible alternative mechanisms
are not possible.
•  This places extreme demands on accuracy.
•  Also requires imagination, to generate then characterize computationally all the possible
mechanisms.
•  Given the errors present in computation, even though one may be able to find a
mechanism that one convinces oneself is “consistent with” experiment, it should be
remembered that this is not the same as proving that this mechanism is the correct one.

31
Conclusions
•  In this e-presentation, I covered a number of topics relevant to
using quantum chemistry to study chemical reaction
mechanisms
•  I also did not cover many other topics (e.g. the all-important
aspect of method accuracy, or indeed anything about electronic
structure and related methods such as QM/MM)
•  Some aspects were more introductory: how to find a TS? How
does energy minimization work? What are continuum solvent
models? How does transition state theory work?
•  Some are less well known, and are important: importance of
conformational searching; barrierless reactions; assessment of
mechanism in complex kinetic schemes; building a model for a
reaction.

Understanding Chemical Reaction Mechanisms with Quantum Chemistry

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