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Unformatted text preview: Chapter P4: Transition–State Geometries and Activation
Up until this point, we have concerned ourselves with quantum chemical
calculations on stable molecules, that is, molecules that at least in principle
can be directly observed and characterized experimentally. There is,
however, another class of “molecules” of great importance to chemists that
cannot be observed, let alone directly characterized. These are transition
states and correspond to locations on a potential energy surface that lie at the
top of pathways connecting stable molecules.
There is no reason why quantum chemical calculations should perform any
poorer (or any better) for transition states than they do for stable molecules.
The only difference is that we can not lean (directly) on experiment to gain
confidence. However, we can use our prior experience with stable molecules
to prejudge how well the calculations are likely to describe the geometries,
energies and other properties of transition states. In one sense, calculations
take on added value and become the primary (and sometimes only) means
A Transition State in One Dimension
In a one-dimension (a diatomic molecule), a transition state is a point on the
potential energy curve where the energy is at a maximum, that is, a point for
which the first derivative of the energy with respect to the bond distance, R,
is zero and the second derivative (curvature) is negative. For comparison, a
stable diatomic molecule is a minimum on the energy curve, that is, a point
for which the first derivative is also zero but the second derivative is
positive. transition state: dE/dR=0, d2E/dR2<0 minimum: dE/dR=0, d2E/dR2>0 The difference between the two will be reflected in the molecule’s infrared
spectrum. Recall from the discussion in Chapter P2, that the vibrational
frequency of a diatomic molecule is proportional to the square root of the
ratio of curvature and reduced mass. This means that the frequency for a
1 transition state is an imaginary number (square root of a negative number).
Of course, it is not possible to actually record the infrared spectrum of a
transition state, but it is no more difficult and no less reliable to calculate
such a spectrum than it is to calculate the spectrum of a normal molecule.
A Transition State in Many Dimensions
A non-linear molecule made up of N atoms is fully characterized by its
location on a (3N-6) dimensional potential energy surface. (The origin of the
coordinate system and orientation of the molecule in the coordinate system
are arbitrary, removing six of the 3N Cartesian coordinates.) While it is not
possible to actually visualize such a multi-dimensional surface (at least we
don’t know how to visualize it), it is possible to represent it mathematically
and to identify “interesting” points on the surface. These are points where
the first derivative is zero for each of the coordinates, and are referred to as
stationary points. The procedure required to decide whether a particular
stationary point is a minimum or a maximum with regard to each of the
coordinates has already been detailed in Chapter P2, and will only be
briefly outlined here. First, the full set of 3N x 3N second energy derivatives
(∂2E/∂xi∂xj) with respect to the Cartesian coordinates, x, are calculated. This
is the costly (in terms of computer time) step in the overall procedure.
Second, the second derivatives are mass weighted; diagonal terms (∂2E/∂xi2)
are divided by the mass of the atom i, Mi, and off-diagonal terms
(∂2E/∂xi∂xj) are divided by the product of the square root of the masses of
the atoms i and j. Third, the Cartesian coordinates are replaced by a new set
of normal coordinates, ζ, defined such that the matrix of mass-weighted
second energy derivatives is diagonal.
(∂2E/∂ ζ i∂ ζ j)/(√Mi√Mj) = δij (∂2E/∂ ζ i2)/Mi δij is 1 if i=j and 0 otherwise. Finally, the six normal coordinates corresponding to the three translations
and three rotations of the molecule relative to its center of mass are
removed, leaving 3N-6 coordinates corresponding to vibrational motions.
This procedure can be fully automated and is no more demanding in terms of
human time than energy calculation. However, it is an order of magnitude
more demanding in terms of computer time.
A transition state in one dimension corresponds to a maximum of the energy.
Is it also an energy maximum in the general 3N-6 dimensional case? Yes
and no. We hypothesize that a transition state is a stationary point on a 2 multidimensional potential energy surface for which only one of the second
derivatives in normal coordinates is negative, and the second derivatives for
all the remaining normal coordinates are positive. That is to say, a transition
state is an energy maximum in only one dimension, the so-called reaction
coordinate, and an energy minimum in the 3N-7 remaining dimensions.
Liken a chemical reaction to driving across a mountain range (a twodimensional system). The “goal” is the same, to go from reactants to
products in the case of the reaction and to go from one valley to another in
the case of the drive. The difference is that the road traverses a twodimensional surface (of the mountain) whereas the chemical reaction may
involve hundreds of dimensions. The important point is that there is no need
to climb to the top of a mountain (a maximum in both dimensions) to move
between two valleys, but it is sufficient to go through a valley between
mountains (a maximum in one dimension but a minimum in the other).
Similarly, it is only necessary to pass through an energy maximum in only
one coordinate (the reaction coordinate) to move between reactants and
products of a chemical reaction.
Not all stationary points on a multidimensional potential energy surface that
satisfy this requirement are likely to be transition states for the process that
is actually of interest. In fact, it is likely that very few are. An additional
“chemical” requirement needs to be imposed, mainly that the point lies on a
path that actually connects the reactants and products. Finally, note that there
may be more than one point on the energy surface that satisfies both the
mathematical and chemical criteria. Without identifying and examining each
and every one of them it will generally not be possible to say which is the
lowest in energy and most likely to be pathway that is actually followed.
While transition states cannot be directly observed, there are, however,
indirect ways to suggest how their structures differ for those of the reactants.
Most informative are kinetic isotope effects, that is, rate changes in
response to changes in the masses of one or more atoms in the reactants. A
large change in rate implies that the affected atoms are intimately involved
in the reaction, whereas a negligible change implies that they are not
significantly involved. Most common are isotope effects associated with
substitution of deuterium for hydrogen, where the rate ratio is referred to as
kH/kD. The effects are small but measureable, typically in the range of 1.03
to 1.07. kH/kD for a process that involves complete CH (CD) bond 3 dissociation is ~1.2. Effects resulting from mass changes for other elements,
for example, 13C for 12C, are much smaller and are only rarely considered.
What is the origin of kinetic isotope effects? While a full treatment is
beyond the scope of this text, the simplified description provided here
conveys the essential point. It is that the Born-Oppenheimer approximation
removes nuclear mass from the Schrödinger equation (see Chapter xx), and
the energy and wavefunction are not affected by changes in mass. However,
vibrational frequencies change with mass; the heavier the atoms involved in
a particular vibration the smaller the frequency (see Chapter P2). Properties
that depend on frequencies also change with mass. This includes the zeropoint vibrational energy (ZPE), which needs to be added to the energy
obtained from a quantum chemical calculation to account for the residual
(0K) vibrational energy of a molecule (see Chapter P4). ZPE is positive and
is proportional to the sum of the vibrational frequencies, meaning that it
decreases with increasing mass, for example, as hydrogen atoms are
replaced by heavier deuterium atoms. ZPEH ZPED To good approximation, a kinetic isotope effect may be seen to arise from
differences in zero-point energies of different isotopic variants between the
transition state and the reactants, ∆ZPE, for example, kH/kD. The expression
below assumes energies in atomic units (hartrees) and that the imaginary
frequency is not included in the calculation of ZPE for the transition states.
kH/kD = exp[-1060(∆ZPEH – ∆ZPED)]
∆ZPE = ZPEtransition state – Σ ZPEreactants
Do we want to give the “correct” expression from statistical mechanics?
Ene Reaction: The ene reaction involves addition of a electron-poor double bond to an
alkene with a allylic hydrogen. The hydrogen is transferred and a new carbon-carbon
bond formed, for example, in the addition of maleic anhydride and propene. 4 A large kH/kD for the allylic hydrogen would suggest that its environment in the transition
state is quite different from that in the reactant. On the other hand, a nearly unit kH/kD
would suggest either that the hydrogen has yet to move from the reactant or has fully
moved to the product.
Structures and vibrational frequencies for the two reactants and for the transition state for
the ene reaction of maleic anhydride and propene obtained from HF/6-31G* calculations
are provided in ene reaction under the Chapter xx directory. Does the calculated
transition-state structure show that bond making and bond breaking occurs
simultaneously? Specifically, is the migrating hydrogen partially bonded to two carbons?
Examine the vibrational motion corresponding to the imaginary frequency in the infrared
spectrum of the transition state. This corresponds to the reaction coordinate. Would you
describe this reaction as a concerted process? Elaborate.
Calculate ∆ZPE for the reaction. (Do not count the imaginary frequency for the transition
state in your calculation.) Next, change the “migrating hydrogen” to a “migrating
deuterium” and repeat the calculation. This is very rapid as it requires only recalculation
of the frequencies from the existing second energy derivatives and the new set of atomic
masses. Calculate kH/kD. Is it consistent with a concerted process?
13 C/12C Isotope Effects on CH Bond Dissociation in Methane: How much does the CH
bond dissociation energy in methane change as a result of changing the mass of carbon
(from 12C to 13C)? Use the B3LYP/6-31G* model to calculate equilibrium geometries and
vibrational frequencies for methane and methyl radical and evaluate the change in zeropoint energy for bond dissociation. Change the mass of carbon to (13C) for both
molecules, rerun the calculation and recalculate the change in zero-point energy for bond
dissociation. 5 Finding a transition state is likely to be more difficult than finding an equilibrium
structure. One reason for this is that, in contrast to the enormous number of experimental
equilibrium geometries (see Chapter P2), there are (and can be) no experimental
transition-state geometries on which to base a guess. Another reason is that methods for
locating transition states are less well developed than methods for finding minima. A
third reason is that the energy surface in the vicinity of a transition state is likely to be
more shallow than that in the vicinity of an energy minimum. After all, a transition state
reflects a delicate balance between bond breaking and bond making, whereas an
equilibrium structure reflects a situation in which bonds are formed to maximum extent.
The reason that this may be a problem is that shallow energy surfaces are not likely to be
properly described by a quadratic function, the form that is assumed in all common
optimization algorithms. As a consequence, a search for a transition state will probably
require more steps than a search for an equilibrium structure.
The key to finding a transition state is to provide a good guess at its structure. Transitionstate geometries, like equilibrium geometries, are expected to exhibit a high degree of
uniformity among closely-related systems, and the best way to do this is to base the guess
on the transition state for a closely-related system. Spartan Student provides a database
of transition-state geometries for a wide variety of simple reactions.
A transition state is a stationary point and the procedure used to locate it is identical to
that used to find an equilibrium structure. The only difference is that it is a stationary
point which is an energy maximum in one dimension (and an energy minimum in each of
the other dimensions). This needs to be enforced. The underlying algorithm, like that for
finding an equilibrium geometry, involves an iterative process, which is terminated only
when all energy first derivatives closely approach zero and all geometrical variables
reach constant values. In practice, finding a transition state may be two or three times
more costly in terms of computer time as finding an equilibrium geometry, although it is
no more costly in terms of human time.
After locating a transition state, it may be desirable to obtain its “infrared spectrum” in
order to verify that it contains a single imaginary frequency, and to establish that the
motion corresponding to this frequency is appropriate for the reaction of interest. 6 Limiting Behavior of Hartree-Fock, B3LYP and MP2 Models for
Transition-State Geometries and Activation Energies
We first set out to establish the limiting behavior of Hartree-Fock, B3LYP
and MP2 models with regard to the geometries of transition states and
energy differences between transition states and reactants (activation
energies). This serves two purposes. First, it provides a set of “reference”
transition-state geometries and associated activation energies with which to
compare geometries and activation energies obtained from simpler and more
practical models. Second, it separates any effects arising from the LCAO
approximation (use of a finite basis set) from effects arising from
replacement of the exact many-electron wavefunction by an approximate
Hartree-Fock, B3LYP or MP2 wavefunction. In practice, it is not possible to
actually reach the basis set limit of any of these models. However, it is
possible to use a sufficiently large basis set such that the addition of further
functions to the basis will have only small (and quantifiable) effects on
calculated transition-state geometries and activation energies. We employ
the cc-pVTZ basis set for this purpose. This is about as large a basis set that
can presently be routinely applied for transition-state geometry and
activation energy calculations on any but the simplest chemical reactions,
that is, involving molecules with more than a few non-hydrogen elements.
The cc-pVTZ is smaller and less flexible than the cc-pVQZ basis set
previously used to establish limits for equilibrium geometries (Chapter P2)
and reaction energies (Chapter P3). However, we have seen that HartreeFock, B3LYP and MP2 models using the cc-pVTZ basis set generally yield
nearly identical equilibrium geometries and reaction energies with those
obtained from the corresponding models with the cc-pVQZ basis set. There
is no reason to expect that this behavior will not carry over to transition-state
geometries and activation energies.
“Key” bond distances in transition states for a few simple chemical reactions
obtained from the Hartree-Fock, B3LYP and MP2 models with the cc-pVTZ
basis set are provided in Table P4-1. These include an intramolecular
rearrangement, a pyrolysis reaction, a Diels-Alder reaction and examples of
the Cope and Claisen rearrangements. Taken together, these represent a
number of broad classes of important chemical reactions. A number of
observations may be made. 7 i) Transition states, like stable molecules, show consistent structures,
although variations in geometrical parameters appear to be larger. This is
consistent with the expectation that the energy surface in the vicinity of a
transition state is likely to be “flatter” than that in the vicinity of an
ii) Where reaction leads to a change in bonds from single to double (or vice
versa), the bonds seen in the transition state are intermediate in length.
For example, transition states transforming single to double carboncarbon bonds typically show lengths around 1.4Å, intermediate between
“normal” CC double bonds (1.30-1.32Å) and “normal” CC single bonds
(1.50-1.54Å). Note that these lengths are quite similar to those found in
typical aromatic compounds. 8 Table P4-1: “Limiting” (cc-pVTZ basis set) Hartree-Fock, B3LYP and MP2 Bond
Lengths in Transition States (Å)
state reaction a
c MP2 a
d MeCN MeNC 1.37
2.20 H ! tOCHO H2C CH2 + HCO2H b + a
d c a
O O a d e O
c f 9 iii) The lengths of forming (or dissociating) single bonds show wide
variations. For example, forming (dissociating) CC bonds range from
1.80Å (in the methyl isocyanide rearrangement) to 2.36Å (in the
Claisen rearrangement of allyl vinyl ether) according to the
B3LYP/cc-pVTZ model. Most commonly, the lengths of forming
(dissociating) single bonds obtained from the MP2/cc-pVTZ model
are shorter than those obtained from the other two models.
Forming Bonds in Transition States and van der Waals Radii: Van derWaals radii for
hydrogen, carbon, nitrogen and oxygen are 1.2Ǻ, 1.92Ǻ, 1.55Ǻ, and 1.52Ǻ, respectively.
How do the forming (dissociating) single bonds in the transition states given in Table P41 compare with the sum of the van der Waals radii for the two atoms involved in the
bond? Does your conclusion alter with theoretical model? Activation Energies
The rate of a chemical reaction is given by the product of a rate constant, k,
and the reagent concentrations [A], [B] ... (a, b, … are integers or fractions).
rate = k [A]a [B]b … The rate constant is given by the Arrhenius equation.
k = A exp[∆E‡/RT] The pre-exponential factor, A, accounts for the efficiency of collisions
among molecules and is generally assumed to be constant for reactions
involving a single set of reactants going to different products or for reactions
involving closely-related reactants. The activation energy, ∆E‡, is the
energy of the transition state connecting the reactants and products of the
reaction, referenced to the energy of the reactants.
∆E‡ = Etransition state – Ereactants Activation energies will always be positive numbers, meaning that the
transition state will always be higher in energy than the reactants. R is the
gas constant and T is the temperature (in K).
A good rule of thumb is that reactions with activation energies >200 kJ/mol
will not occur at normal temperatures, while those with activation energies
<100 kJ/mol will be unstoppable under the same conditions. Since the rate
also depends on temperature (through the rate constant), this means that a
10 reaction with a low activation energy can be slowed down or “stopped” by
lowering the temperature, while a reaction with a high activation energy can
be accelerated by raising the temperature.
Some reactions proceed without energy barriers and discernible transition states. Radicals
combine without a barrier, for example, two methyl radicals combine to form ethane, and
radicals add to multiple bonds with little or no barrier, for example, methyl radical adds
to ethylene to form 1-propyl radical.
H3C• + •CH3 →H3C–CH3 H3C• + H2C=CH2 → CH3CH2CH2• In the gas phase, SN2 reactions involving anionic nucleophiles, for example, reaction of
ethoxide anion and ethyl iodide leading to diethyl ether and iodide anion, proceed without
an energy barrier.
EtO- + EtI → EtOEt + IThe barrier observed in solution is a consequence of the solvent. Charge is already more
spread out in the transition state than in the reactants, meaning that it the solvent has less
of a role to play. The Arrhenius equation is assumes that all molecules pass through a single
transition state on their way from reactants to products. While this is not
unreasonable, it is also not necessarily true. While a reactant without
sufficient energy will be unable to reach the transition state (and proceed to
products) reactants with excess energy will be able to proceed to products
over alternative (higher-energy) pathways. The previous analogy to a road
through a mountain valley is also applicable here. Other roads that cross
above the mountain valley could also be followed. Despite its limitations,
the Arrhenius equation has proven of great value in interpreting measured
reaction rates and in connecting variations in rates with changes in molecular
Activation energies from Hartree-Fock, B3LYP and MP2 models with the
cc-pVTZ basis set for the same set of reactions used previously for bond
distance comparisons are provided in Table P4-2. Activation energies
estimated from experimental rate data by way of the Arrhenius equation are
available for all but one of these reactions and have also been provided. For
all the reasons, described above, comparisons between calculated and
“experimental” activation energies must be viewed with healthy skepticism. 11 Table P4-2: “Limiting” (cc-pVTZ basis set) Hartree-Fock, B3LYP and MP2 and
G3(MP2) Activation Energies (kJ/mol)
cc-pVTZ Reaction MP2 G3(MP2) expt. H2C CH2 + HCO2H 170
208 99 34 70 84 242 !tOCHO 186
184 MeCN MeNC 159
167-184 147 107 136 151 212 122 106 121 130 + O O 12 Also provided are activation energies obtained from the G3(MP2) recipe
(based on equilibrium and transition-state geometries from the MP2/6-31G*
A HF/6-31G* frequency calculation is part of the G3(MP2) recipe and, for the case of a
transition state, leads to an imaginary frequency. We ignore this frequency in calculation
of zero-point energy and temperature corrections terms. While the number of examples is too few to permit generalizations to be
made from them, it is worthwhile to point out the obvious
i)Activation energies obtained from the “limiting” (cc-pVTZ basis set)
Hartree-Fock model are significantly larger than those from either the
corresponding B3LYP or MP2 models or from the G3(MP2) recipe, as
well as from activation energies obtained from experimental rate data using
the Arrhenius equation. This suggests that electron correlation in (these)
transition states is larger than that in corresponding reactants. The reason
for this is not obvious, but suggests that correlation in delocalized systems
is larger that in localized systems.
ii)B3LYP and MP2 models and the G3(MP2) recipe yield smaller activation
energies. The former are in good accord with activation energies obtained
from experimental rates. On the other hand, activation energies from
MP2/cc-pVTZ calculations for the Diels-Alder reaction, and the Cope and
Claisen rearrangements appear to be much too small. That for the DielsAlder reaction is less than half of the estimated experimental value (and a
third of the activation energy from B3LYP/cc-pVTZ calculations).
iii)With one exception (isomerization of methyl isocyanide) results from
limiting B3LYP calculations are closer to G3(MP2) results than results
from limiting MP2 calculations.
iv)In very general terms, the results for activation energies show much wider
variation than those for reaction. Part of this is due simply to the fact that
the activation energies dealt with here are typically larger numbers than
the reaction energies dealt with in Chapter P3. However, part of the
problem reflects the greater sensitivity to change in theoretical model for a
transition state than for an equilibrium structure. 13 Practical Hartree-Fock, B3LYP and MP2 Models for Transition-State
Geometries and Activation Energies
Except for very small molecules, Hartree-Fock, B3LYP and MP2 models
with large basis sets such as cc-pVTZ are not likely to be practical for
routine calculations of transition-state geometries and activation energies.
The G3(MP2) model is also not practical for any but very small molecules.
While these and other models which involve very large basis sets and/or
extended treatments of correlation are certainly of value to judge “limits”,
simpler models smaller are needed for routine calculations. Hartree-Fock
and B3LYP calculations with the 6-311+G** basis set may routinely be
applied to molecules with molecular weights up to 400 amu, although MP2
calculations are more restricted. The 6-31G* basis set is significantly smaller
and lacks diffuse functions. It may be routinely used for calculations on even
larger molecules. The question to be addressed is whether they are able to
match bond lengths from the corresponding cc-pVTZ basis set models to
within ±0.02Å and activation energies to within ±4 kJ/mol.
“Key” bond distances in transition states for a few simple chemical reactions
obtained from Hartree-Fock, B3LYP and MP2 models with 6-311+G** and
6-31G* basis sets are provided in Table P4-3. Mean absolute deviations for
all six models relative to the corresponding models with the cc-pVTZ basis
set are on the order of only 0.01A. Individual deviations for B3LYP and
MP2 models in particular are significantly larger in some cases (see Table
P4-1). The higher sensitivity of transition-state geometries obtained from
correlated models relative to Hartree-Fock models to basis set is consistent
to previous results for equilibrium geometries (see Chapter P2), and is not
unexpected. Bond distances obtained with the 6-311+G** basis set are
closer to those obtained from the cc-pVTZ basis set values than those
obtained using the 6-31G* basis set, for both B3LYP and MP2 models. The
differences are insignificant for Hartree-Fock models.
The close agreement between “limiting” Hartree-Fock, B3LYP and MP2
models and the corresponding “practical” models for transition-state
geometries is an important result as it means that quantum chemical methods
can be applied to real problems and not just idealized systems. 14 Table P4-3: Bond Lengths in Transition States from Practical Hartree-Fock,
B3LYP and MP2 Models (Å)
6-31G* 6-311+G** 6-31G* 6-311+G** 6-31G* 6-311+G**
d MeCN MeNC 1.39
2.22 H !tOCHO H2C CH2 b + a
d c a
O O a d e O
c f 15 Transition States for Related Pyrolysis Reactions: Do transition states
forcloselyrelated reactions have very similar geometries? Were this the case, it would
mean that the transition state calculated for a simple reaction could be used as a guess at
the transition state for a more complex but related reaction. A good example is provided
by comparison of transition states for pyrolysis of tert-butyl formate (leading to formic
acid and isobutene) and of formate (leading to formic acid and ethylene). Use the HF/631G* model to calculate the transition state for the former and compare it to the
analogous transition structure for ethyl formate pyrolysis given in Table P4-3. Are the
“key” bond lengths very similar? Rationalize any significant differences that you find.
Transition States for Related Intramolecular Rearrangements: Another example is
provided by rearrangements of methylisocyanide to acetonitrile and tert-butylisocyanide
to tert-butyl cyanide. Use the HF/6-31G* model to calculate the transition state for the
latter and compare lengths of both the CN bond that is being broken and the CC bond that
is being formed with those in the analogous transition state for methylisocyanide (the
geometry of which is given in Table P4-3. Rationalize any significant differences that
Transition States for “Chemically-Related” Reactions: The ene reaction of 1-pentene
is closely related to the pyrolysis reaction of ethyl formate. Here the products are propene
and ethylene rather than formic acid and propene.
+ Calculate the transition state for the ene reaction of 1-pentene and compare with that for
ethyl formate pyrolysis given in Table P4-3. Use the HF/6-31G* model. In particular,
focus on the CH bond length of the migrating hydrogen. Has it shortened, elongated or
remained roughly the same from the ene reaction to the pyrolysis. If it has changed, is the
direction of the change consistent with the Hammond Postulate? Elaborate. Do whatever
additional calculations that may be required to answer the question.
Localized vs. Delocalized Bonding: Diels-Alder cycloadditions as well as Cope and
Claisen rearrangements all involve transition states with delocalized bonding The HF/631G* model typically overestimates activation energies for by 30-100 kJ/mol. The
transition states for all of these reactions show “delocalized” bonding (single and double
bonds are replaced by bonds of intermediate length), whereas the reactants exhibit normal
single and double bonds. Does this suggest that the 6-31G* model always favors
molecules with localized bonds?
Use the HF/6-31G* model to obtain equilibrium geometries for toluene (a molecule with
a delocalized π system) and its isomer, cycloheptatriene (a molecule with discernable
single and double bonds). Is the energy difference smaller or larger than the known
difference in heats of formation (toluene is 133 kJ/mol lower in energy)?
16 Activation Energies
Activation energies for the same set of reactions obtained from HartreeFock, B3LYP and MP2 models with both 6-31G* and 6-311+G** basis sets
are given in Table P4-4. Signed deviations from the corresponding models
with the cc-pVTZ basis set are provided in parentheses alongside each entry.
“Experimental” activation energies are also given.
Significant differences appear between results with 6-31G* and cc-pVTZ
basis sets for all three classes of models. The worst cases are the Diels-Alder
reaction of cyclopentadiene and ethylene and (except for Hartree-Fock
models) the ethyl formate pyrolysis reaction. These differences are
significantly reduced upon moving from the 6-31G* to 6-311+G** basis
sets. With one exception (the ethyl formate pyrolysis reaction from the MP2
model) deviations between activation energies calculated from the
corresponding 6-311+G** and cc-pVTZ models are 4 kJ/mol or less. As
with results for transition-state geometries, this is an important result and
suggests that practical quantum chemical models are able to provide
descriptions that are quite close to those from large-basis-set (“limiting”)
models. 17 Table P4-4: Activation Energies from Practical Hartree-Fock, B3LYP and MP2
6-31G* 6-311+G** reaction B3LYPa
6-31G* 6-311+G** MP2a
6-31G* 6-311+G** expt. 191 (5) 187 (1)
297 (14) 282 (-1) 171 (1)
221 (16) 167 (-3)
207 (2) 178 (6)
252 (24) 166 (-18) 180 (-4) 83 (-16) 97 (-2) 49 (-15) 37 (-3) 84 237 (-5) 241 (-1) 144 (-3) 148 (1) 119 (12) 110 (3) 151 205 (-7) 209 (-3)
121 (-1) 120 (-2)
a) deviations from corresponding “limiting” (cc-pVTZ basis set) values are given in parenthesis 130 MeCN MeNC !tOCHO H2C CH2 + HCO2H 170 (-2)
236 (8) 167-184 + O O 18 Vinyl Alcohol: Even though vinyl alcohol, H2C=C(H)OH, is significantly less stable than
its isomer acetaldehyde (see Chapter P3), it can be stored for long periods. This suggests
that the activation energy for isomerization to acetaldehyde is likely to be substantial.
Use the B3LYP/6-31G* model to calculate equilibrium geometries for acetaldehyde and
vinyl alcohol. What is the room-temperature Boltzmann ratio of acetaldehyde and vinyl
alcohol according to this model? Is vinyl alcohol likely to be detectable in an equilibrium
mixture? Obtain the transition-state geometry for the isomerization and calculate the
activation energy. Is it large enough that once formed, vinyl alcohol is likely to persist for
long times? Elaborate.
Dichlorocarbene Addition to Ethylene: Singlet dichlorocarbene adds to ethylene to
CCl 2 + H2C Cl CH2 Obtain an equilibrium geometry for singlet dichlorocarbene using the HF/6-31G* model
and display an electrostatic potential map. This shows the distribution of charge on the
accessible surface. Where is the electrophilic site, in the ClCCl plane or out of the plane?
How would you expect dichlorocarbene to approach ethylene? Does this lead to a product
with the proper three-dimensional geometry? Elaborate.
Use the HF/6-31G* model to obtain the transition state for addition of dichlorocarbene to
ethylene. Is the calculated structure consistent with the conclusions reached above
regarding the electrophilic character of dichlorocarbene? Elaborate.
Hydroxycarbene: Singlet hydroxycarbene (HCÖH) is a proton transfer isomer (or
tautomer) of formaldehyde. The best experimental estimate is that it lies ~220 kJ/mol
higher in energy than formaldehyde, which suggests that it would not be detectable were
it in rapid equilibrium with formaldehyde. The only way that hydroxycarbene would be
detectable were if the energy barrier leading to formaldehyde was sizable (>100 kJ/mol).
Additionally, it is necessary that the barrier to dissociation to separated hydrogen and
carbon monoxide molecules also be sizable.
HCÖH H2 + CO
Use the B3LYP/6-311+G** model to calculate both the equilibrium geometries of all
reactants and products of the two reactions above. Do the calculations reproduce the
relative energies of formaldehyde and hydroxycarbene? Is the barrier to isomerization to
formaldehyde sufficiently high to permit hydroxycarbene to be detected? Is dissociation
to hydrogen and carbon monoxide competitive with isomerization?
Dimerization of Borane: Borane (BH3) dimerizes to diborane.
BH3 + BH3 B2H6 19 Is there an activation energy associated with dimerization? To see if there is, first obtain
the equilibrium geometry of diborane and the elongate two “opposite” BH bonds (marked
in the drawing above) from 1.3Ǻ (close to the equilibrium value) to 2.5Ǻ in steps of
0.1Ǻ. Use the B3LYP/6-31G* model. If you find an energy barrier, obtain the transition
state. 20 Using Approximate Geometries for Activation Energy Calculations
Finding a transition state geometry is likely to be more costly in terms of
computation time than finding an equilibrium geometry. However, since the
potential energy surface in the vicinity of a transition state would be
expected to very flat, the small differences in transition-state geometries
seen among different theoretical models are not likely to lead to significant
changes in activation energies. Therefore, it may not always be necessary to
utilize “exact” transition-state geometries in carrying out activation energy
calculations. Hartree-Fock models may replace B3LYP and MP2 models
and small basis sets may replace larger basis sets.
Problems … 21 Relative Activation Energies
It will not always be necessary to formulate reaction rate comparisons in
terms of absolute activation energies. Differences in activation energies
among closely-related reactions will often be sufficient to answer the
question at hand. Important examples include rate changes due to
substituents or to changes in regiochemistry or stereochemistry of the
Table P4-5 compares activation energies for Diels-Alder reactions of 1,3butadiene with a series of cyanoalkenes, relative to the activation energy for
reaction of 1,3-butadiene and acrylonitrile. Experimental relative rates for a
series of closely-related Diels-Alder reactions involving cyclopentadiene
instead of 1,3-butadiene are also provided and show both an increase in rate
with increasing number of cyano groups and a sensitivity to the location of
the groups. While both Hartree-Fock and MP2 models with both 6-31G*
and 6-311+G** basis sets reproduce the ordering of reaction rates, neither of
the B3LYP models is successful in this regard. In stead both show an
increase in activation energy (suggesting a decrease in rate) from 1,1dicyanoethylene, to tricyanoethylene to tetracyanoethylene. The reason for
the apparent failure is unclear.
Discussion of Table P4-6 22 Table P4-5:Activation Energies for Diels-Alder Reactions of 1,3-Butadiene and
Cyanoalkenes Relative to the Reaction of 1,3-Butadiene and Acrylonitrile from
Practical Hartree-Fock, B3LYP and MP2 Models (kJ/mol)a
6-31G* 6-311+G** reaction
acrylonitrile 0 0 B3LYP
6-31G* 6-311+G** MP2
6-31G* 6-311+G** expt. b 0 0 0 0 0 cis-1,2-dicyanoethylene -7 -9 0 -1 -14 -14 1.9 trans-1,2-dicyanoethylene -11 -13 -4 -6 -17 -18 1.9 1,1-dicyanoethylene -27 -29 -25 -26 -22 -21 4.6 tricyanoethylene
a) energies of reactions: -31
7.6 b) log10 rate relative to reaction of cyclopentadiene acrylonitrile as a standard. Table P4-6: Room Temperature Boltzmann Ratios of Regio and Stereo Products
from Practical Hartree-Fock, B3LYP and MP2 Models
6-31G* 6-311+G** 6-31G* 6-311+G** 6-31G* 6-311+G**
regiochemistry dimethylborane + propene C2:C1 100 100 1,3-butadiene + acrylonitrile
cyclopentadiene + acrylonitrile
cyclopentadiene + maleic anhydride stereochemistry
97 23 99 99 98 98 18
99 Isomerization of Ethyl Isocyanide: The activation energy for isomerization of methyl
isocyanide to acetonitile (methyl cyanide) is 191 kJ/mol according to the HF/6-31G*
model, somewhat larger than the value of 159 kJ/mol obtained from experimental rate
data. Use the same model to calculate the activation energy for the analogous reaction of
ethyl isocyanide. Assuming that the error is the same as for the methyl isocyanide
isomerization, what is your best guess for activation energy for reaction of ethyl
isocyanide? Which reaction appears to be faster? Offer an explanation of your result.
Hydroboration of Alkenes vs. Alkynes: Alkylboranes add not only to carbon-carbon
double bonds but also to carbon-carbon triple bonds. Use the B3LYP/6-31G* model to
obtain structures and energies for reactants and transition states for addition of
dimethylborane (Me2BH) to both ethylene and acetylene. Which reaction is faster? Offer
Hydroboration of propene leads predominately to a product derived from attack of
hydride onto the more highly substituted alkene carbon. Use the B3LYP/6-31G* model
to obtain structures and energies for the two transition states for addition of
dimethylborane to propyne. Which is lower in energy? Is the difference in transition-state
energies small enough that you would expect to see both products? Elaborate. 24 Thermodynamic vs. Kinetic Control of Chemical Reactions
We now have the computational tools in investigate both thermochemical
and kinetic preferences of chemical reactions. For reactions where the two
are different, this leads to the possibility changes in reaction conditions will
lead to changes in product distributions. Long reaction times and high
temperatures should favor thermodynamic products whereas short reaction
times and low temperatures favor kinetic products.
Consider the products resulting from heating 6-bromohexene in the presence
of tri-n-butyl tin hydride (a radical initiator). The first step involves
abstraction of bromine leading to hex-5-enyl radical. This can either pick up
hydrogen giving 1-hexene, or rearrange either to cyclohexyl radical or to
cyclopentylmethyl radical, which in turn may pick up hydrogen giving
cyclohexane and methylcyclopentane, respectively.
Br ! •
hex-5-enyl radical 17% •
rearrangement cyclopentylmethyl radical
• cyclohexyl radical 81% 2% Certainly, cyclohexyl radical is more stable than cyclopentylmethyl radical,
because six-membered rings are more stable than five-membered rings, and
because 2° radicals are more stable than 1° radicals. However, the dominant
(rearranged) product is methylcyclopentane. This suggests that the reaction
is kinetically controlled, and that rearrangement to cyclopentylmethyl radical
requires less energy than rearrangement to cyclohexyl radical. 25 Ring Opening of Hex-5-enyl Radical: Verify that cyclohexyl radical is lower in energy
than cyclopentyl radical. Use the HF/6-31G* model. Were the ring opening reaction
thermodynamically controlled, what would be the ratio of methylcyclopentane and
cyclohexane at room temperature?
Verify that the transition state leading from hex-5-enyl radical to cyclopentylmethyl
radical is lower in energy than that leading to cyclohexyl radical. Were the ring opening
reaction kinetically controlled, what would be the ratio of methylcyclopentane and
cyclohexane at room temperature?
Polymerization of Cyclopentadiene: Cyclopentadiene undergoes a Diels-Alder
reaction with itself yielding either an endo or exo dimer. + + endo dimerization exo dimerization The process would be expected to continue, giving rise either to an endo or exo polymer
(exo polymer depicted below), but it actually stops around the 20-mer. Space-filling models for short strands of exo (left) and endo (right) polymers of
cyclopentadiene suggest why. The exo polymer is helical and there is nothing stopping it
from continuing to grow, whereas the endo polymer closes back on itself (in about 20
units) and cannot continue. Short strands of both exo and endo polymers of cyclopentadiene are provided in
cyclopentadiene polymers in the Chapter P4 directory.
Use the HF/6-31G* model to establish whether the exo or endo cyclopentadiene dimer is
lower in energy. Rationalize your result. Were the dimerization thermodynamically
26 controlled, what would be the ratio of exo and endo dimers at room temperature? Given
the fact that polymerization stops at the 20-mer, would you conclude that the reaction is
thermodynamically controlled? Elaborate.
Next, establish which transition state is lower in energy. Rationalize your result. Were the
dimerization kinetically controlled, what would be the ratio of exo and endo dimers at
room temperature? Given the fact that polymerization stops at the 20-mer, would you
conclude that the reaction is kinetically controlled? Elaborate.
Suggest reaction conditions that would lead to polymerization beyond the 20-mer. 27 Charge Distributions in Transition States
Electrostatic potential maps may be used to describe charge distributions in
transition states just as they may be employed portray charge distributions in
ordinary molecules (see discussion in Chapter P1). Pyrolysis of ethyl
formate (leading to formic acid and ethylene) provides a good example of
the kinds of information that may result.
O O O O H H O + OH Here, the electrostatic potential map (center image) clearly shows that the
hydrogen being transferred from carbon to oxygen is positively charged (it is
an electrophile). An even clearer picture results by using a surface of much
larger electron density in order to map the electrostatic potential (right hand
image). 28 Charge Transfer in Diels-Alder Reactions: The most common Diels-Alder reactions
are between electron-rich dienes and electron-deficient dienophiles, for example, between
cyclopentadiene and tetracyanoethylene.
NC CN CN +
CN CN Do transition states for Diels-Alder reactions show evidence for charge transfer from the
diene to the dienophile? If so, does the extent of transfer correlate with reaction rate?
Use the HF/6-31G* model to obtain geometries for reactants and transition states for
Diels-Alder reactions of cyclopentadiene as a diene, and ethylene and tetracyanoethylene
as dienophiles. For each reaction, compare electrostatic potential maps of reactants and
transition state. Is there a noticeable transfer of charge? Is it in the expected direction,
that is, does the π system of the diene lose electrons and the double bond of the
dienophile gain electrons in moving from reactants to the transition state? For which
reaction is charge transfer greater? Is your result consistent with the observed change in
rate with change in dienophile: tetracyanoethylene >> ethylene?
SN2 Reaction of Chloride and Methyl Bromide: Electrostatic potential maps can be
used to help explain why bromide is a better SN2 leaving group than chloride. Use the
HF/6-31G* model to obtain an energy profile for reaction of chloride with methyl
bromide, from separated reactants to separated products. Start with chloride 3.5Å from
the carbon in methyl bromide and move it in steps of 0.1Å to 1.7Å. Plot both the energy
and the energy corrected for the effect of aqueous environment vs. the CCl distance (the
“reaction coordinate”). Is the reaction
Cl– + CH3Br → CH3Cl + Br– endothermic or exothermic in the gas phase? How do the solvent corrections affect
overall endo or exothermicity? Are your results consistent with the observation that
iodide is a better leaving group than chloride? Elaborate.
Examine electrostatic potential maps for the structures along the reaction profile, with
particular focus on the reactants and products and on the transition state. For which
(reactant, product or transition state) is the charge most localized? For which is it most
delocalized? What does this say about the relative abilities of Cl and Br to act as leaving
groups? Does the energy along the reaction pathway appear to correlate with the extent to
which charge is delocalized? Elaborate. Does the change in energy due to the solvent
correction correlate with the extent to which charge is delocalized? Elaborate.
SN2 Reactions of Alkyl Bromides: The rates of SN2 reactions involving alkyl bromides
decrease with each successive replacement of hydrogen by an alkyl group.
C H H Br > C
Br > C
H H 29 CH3 CH3
Br > C
CH3 CH3 Br slow The usual explanation centers around the increase in coordination of carbon from four in
the reactants to five in an SN2 transition state, for example, for reaction of bromide and
Br– + CH3Br Br C
H Br CH3Br + Br– H The resulting increase in (unfavorable) steric interactions should be least for reaction of
the methyl halide and greatest for the tert-butyl halide, consistent with a decrease in rate
with increased substitution.
Obtain transition states for addition of bromide to both methyl bromide and to tert-butyl
bromide using the HF/6-31G* model, and examine as space-filling models. Is the
transition state for addition to tert-butyl bromide noticeably more crowded than that for
addition to methyl bromide? If not, why not? Hint: compare the carbon-bromine bond
distances in the two transition states.
Calculate and compare electrostatic potential maps for the two SN2 transition states.
These convey the distribution of charge on the accessible surface. Is the extent of charge
separation comparable or is it significantly different between the two transition states? If
the latter, is their a relationship between the carbon-bromine distance and degree of
charge separation? Speculate on the origin of the change in rate of SN2 reactions with
change in substitution at carbon. 30 Reaction Pathways
As detailed above, a transition state connecting stable molecules (the
reactants and products in a chemical reaction) is a well-defined location on
the potential energy surface. Of course, there may be several transition states
connecting the same reactants and products, just as there may be several
passes leading across a mountain range, but each is a well-defined location.
On the other hand, the reaction pathway leading upward from the reactants
to the transition state and then downward to the products is not unique. Just
as there are many possible roads leading up to and down from a particular
mountain pass, there are many ways the reactants can reach a transition state
and many ways to move away from the transition state to products. It is not
obvious which pathway a reaction will follow or even that it will follow only
Several ways have been proposed to provide reaction pathways, but all are
arbitrary in the sense that there are many (an infinite number) reaction
pathways. A common procedure referred to as an intrinsic reaction
coordinate smoothly connects reactants, transition state and products. While
examination of the motion along an intrinsic reaction coordinate may be
satisfying in that it “leaves an impression” of how a reaction proceeds, there
is no guarantee that it at all reflects what is really happening.
A simpler procedure is to follow the motion corresponding to the vibration
with the imaginary frequency back from the transition state toward reactants
and forward toward products. While this approach does not actually lead to
either reactants or products, it is usually sufficient to tell whether the
transition state is that for the reaction of interest. 31 Testing the Hammond Postulate
The Hammond Postulate states that transition state in a one-step reaction
more closely “resembles” the side of the reaction that is higher in energy.
Thus, the transition state of an exothermic reaction more closely resembles
the reactants while the transition state of an endothermic reaction more
closely resembles the reactants. Thus, the closer the energy of a transition
state to that of the reactant (or product), the greater its structure will
resemble the structure of the reactant (or product). In the limit of an
exothermic reaction with no energy barrier, the “transition state” is the
reactant (and vice versa for an endothermic reaction).
transition state transition state product
Energy reactant endothermic reaction Energy exothermic reaction product reactant reaction coordinate reaction coordinate The Hammond Postulate should not be invoked for a reaction that is only
slightly exo or endothermic. On the contrary, it should be valid for a reaction
involving a high-energy reactive intermediate. Here, the reaction
connecting the intermediate to product would be expected to be highly
exothermic and have a low energy barrier. The transition state should closely
resemble the intermediate. Because modeling allows chemists to explore the
structures of both reactive intermediates and transition states, it is now
possible to test the limits of the Hammond Postulate. 32 Kinetic Isotope Effects and Hydrogen Abstraction: Reaction of chlorine with d1dichloromethane can either involve hydrogen abstraction (leading to HCl) or deuterium
abstraction (leading to DCl).
Cl• D + HCl hydrogen
abstraction C• H + DCl deuterium
abstraction C• Cl C
D Cl Cl
Cl The two reactions follow the same pathway, pass through the same transition state and
lead to the same products. The only difference is whether a hydrogen or deuterium atom
is abstracted. This affects the zero-point energies of the transition states and products, and
leads to kinetic and equilibrium isotope effects, respectively.
Obtain equilibrium geometries and infrared spectra for dichloromethyl radical and
hydrogen chloride using the HF/6-31G* model. Use calculated zero-point energies to
evaluate the equilibrium isotope effect, that is, the energy of the reaction.
• Cl2CH + DC l • Cl2CD + HC l Use the Boltzmann equation to calculate the equilibrium distribution of products at 25°C.
Obtain the structure of the transition state for hydrogen abstraction from dichloromethane
by chlorine atom. Again use the HF/6-31G* model. Calculate zero-point energies for the
two possible monodeuterated transition-states. Use the Boltzmann equation (with
transition state zero-point energies instead of reaction product zero-point energies) to
calculate the kinetic distribution of products at 25°C. 33 ...
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This note was uploaded on 02/22/2010 for the course CHEM N/A taught by Professor Head-gordon during the Spring '09 term at University of California, Berkeley.
- Spring '09