Computational chemical investigation
of factors affecting the reactivity of
the hetero Diels-Alder reaction
Master of Science Degree Project
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF CHEMICAL SCIENCE AND ENGINEERING
APPLIED PHYSICAL CHEMISTRY Jonas Ståhle
Supervisor: Professor Tore Brinck 2012-03-06
2 Abstract
Recent research has shown that small hydrogen bonding catalysts can catalyze the hetero Diels-Alder reaction. In this thesis such hydrogen bonding catalysts in conjunction with varying functional groups and their effect on the hetero Diels-Alder reaction have been investigated. The influence of the different solvents has been investigated as well. The activation barriers for the different region- and stereo isomeric pathways have been compared in order to determine the stereo specificity of the reactions. These calculations have been done using the B3LYP functional for the geometry
optimizations and then M06-2X for single point calculations. For the solvated cases the cPCM model and the M06-2X functional were used. It was shown that for the catalyzed systems bulkier groups in the endo position tend to have a lower activation barrier, allowing for control over the
stereoselectivity. Electron withdrawing groups have an activating effect and are also synergistic with the hydrogen bonding catalysts. The solvent with the lowest dielectric constant gave the lowest activation barrier.
3
Table of Contents
Theory Overview ... 4
The Schrödinger Equation ... 4
Atomic Units ... 4
The Born-Oppenheimer Approximation ... 4
The Variational Principle ... 5
Slater Determinants ... 6
Hartree Fock method ... 7
Roothan equations ... 8
Self Consistent Field ... 10
Density Functional Theory ... 10
Transition state theory ... 10
Introduction ... 12
Computational Details ... 13
Results and Discussion ... 17
HOMO-LUMO gap ... 21
Solvated reactions ... 23
Conclusions ... 24
Appendix A ... 25
Appendix B ... 26
References ... 29
4
Theory Overview
The Schrödinger Equation
The main problem in most quantum chemical approaches revolves around solving the approximate, time independent, non-relativistic Schrödinger equation.
(1)
Where is the Hamiltonian operator for a system consisting of M nuclei and N electrons. The nuclei and electrons are described by the spatial vectors and . Distance between electron i and j is denoted as , between electron i and nucleus A as and between nuclei A and B as .
The Hamiltonian is expressed as follows, in atomic units:
(2)
The first term corresponds to the electron kinetic energy, the second to the nuclei kinetic energy. The Last three terms are the coulomb attraction between nuclei and electrons and the repulsion between electrons and nuclei respectively.
Atomic Units
The equation above is expressed in atomic units. The values for all units, mass, charge, etc, are set to unity to simplify all expressions. They are converted to SI units according to the following table:
Physical quantity name conversion factor
length a0 5.2918*10-11 m
mass me 9.1095*10-31 kg
charge e 1.6022*10-19 C
energy εa 4.3598*10-18 J
angular momentum h 1.0546*10-34 Js
Table 1
The Born-Oppenheimer Approximation
To simplify the Schrödinger equation further, the Born-Oppenheimer approximation can be applied.
Since the comparative mass of the nucleus much greater than that of the electrons, over 1800 times for a simple hydrogen atom, the nuclei move considerably slower. As such, the electrons can be considered to move in a field of fix nuclei to a good approximation, this is called the Born-
Oppenheimer approximation. Since the nuclei are fixed in space, we can ignore the kinetic energy contribution from the nuclei, and the repulsion between the nuclei is constant. This reduces the Hamiltonian to the so called electronic Hamiltonian:
(3)
5
(4)
The total energy is the sum of the electronic energy and the constant nuclear repulsion energy.
(5)
The Variational Principle
As mentioned before, we’re interested in determining an approximate solution to the Schrödinger equation. It will be shown that for a normalized trial function, the parameters can be varied until a minimum is reached, this minimum corresponds to the approximate ground state energy. This is called the variational principle. For the Hamiltonian operator , the Schrödinger equation has an infinite number of exact solutions:
(6)
And
is a Hermitian operator and so the eigenvalues are real and it’s corresponding eigenfunctions are orthonormal
(7)
It is assumed that the eigenfunctions of form a complete set and we can consequently form any function from a linear combination of the eigenfunctions .
(8)
(9)
Due to the normalization condition, we have
(10)
However, it can also be expanded as follows
(11)
Since
(12)
And
6
(13)
Then the expectation value of any trial function is always an upper bound to the ground state energy
(14)
Slater Determinants
So far we’ve only been concerned with a one electron problem. In reality, we must deal with many electron systems. The easiest way to describe this system would be to first assume that the electrons don’t interact (which is, however, a rather drastic approximation), that is . If so each one electron Hamiltonian would be independent and we could form the full Hamiltonian as a sum of one electron Hamiltonians.
(15)
Where is the electronic Hamiltonian. The corresponding eigenfunction of this sum is the so called Hartree-Product and it’s a product of the individual eigenfunctions of the one electron Hamiltonians.
(16)
Here the new concept of a spin orbital has also been introduced. Only using three spatial coordinates r can not sufficiently describe our system, the electrons spin must also be taken into consideration.
The spin orbital is attained simply by forming a product of a spatial orbital and a spin function and corresponding to a spin up and down respectively. The spin function is dependent on an undefined parameter and is assumed to be an orthonormal function:
(17)
Returning to the Hartree-product, one fundamental flaw of the expression still needs to be
addressed, the fact that the Hartree-product does not satisfy the antisymmetry principle. To satisfy the antisymmetry principle, the expression needs to not distinguish between electrons and be antisymmetric with respect to the interchange of two coordinates. In the two-electron case:
(18)
Putting electron one in orbital two and electron two in electron one would create the following expression:
(19)
7
Clearly they distinguish between electrons. However, if a linear combination of these Hartree products were to be formed, an expression that does not distinguish between electrons and is antisymmetric with respect to the interchange of two coordinates would be obtained:
(20)
Where the is a normalization factor. An interchange of two coordinates would change the symbol of the expression.
(21)
This expression is called the Slater-determinant. As the name suggest, it can also be written as a determinant. In the previous two electron example as:
(22)
And in a more general way as:
(23)
If two rows or two columns were to be interchanged, the sign would also switch. Also if two rows were to be the same, the resulting determinant would be zero. I.e. two electrons can not occupy the same spin orbital, the Pauli Exclusion Principle.
Hartree Fock method
The best Slater determinant, i.e. the one yielding the lowest energy, is then obtained by using the variational principle as described earlier. By varying the spin orbitals and minimizing the energy, one ends up with the Hartree-Fock equation. The rather extensive derivation will not be included here and only the final expression is given.
(24)
is a one electron operator called the Fock-operator and it can be described more clearly as follows:
(25)
8
in turn is the Hartree-Fock potential consisting of one coulomb part and one correlation part.
(26)
The clearest way to describe these two operators is by showing how they operate on a spin function:
(27)
(28)
The Hartree-Fock potential can be described as an average field experienced by the i:th electron.
Note that thus far it is still a matter of spin orbitals. In order to proceed with the actual Hartree-Fock calculations, the spatial orbitals must first be obtained, i.e. the spin orbitals must be integrated over their spin thus yielding spatial orbitals. After such a procedure the Hartree-Fock equation takes the following form:
(29)
Were the new Fock operator looks slightly different:
(30)
Since when going from a spin orbital to a spatial orbital the number of coulomb interacting spatial orbitals are two but only one exchange effect exists, since they have different spin.
Roothan equations
The Hartree-Fock equation can only be solved through tedious numerical solutions. In order to overcome this problem, Roothan suggested that instead of using unknown spatial orbitals, a linear combination of a set of known spatial basis functions will turn the aforementioned differential equation in to a set of algebraic equations with only a set of constants unknown. This may in turn be solved through matrix calculations1.
(31)
Substituting this linear expansion into the Hartree-Fock equation will give the following:
9
(32)
Note that it is the spatial orbital form. From there multiplying with on the left and integrating gives:
(33)
To simplify this somewhat two new matrices are defined, the first one is the so called overlap matrix,
(34)
which describes how much the orbitals overlap. The diagonal elements are unity and the off-diagonal range from 0 to 1. The second matrix is the Fock matrix.
(35)
which is just the matrix representation of the Fock operator. With these two new matrices the previous equation can be simplified as
(36)
and in even more compact form as:
(37)
This is the Roothan equation. Matrix C contains all the expansion coefficients
(38)
and contains the orbital energies
(39)
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Self Consistent Field
The field itself depends on the spin orbitals, the system must be solved iteratively. A starting guess generates a field that in turn can yield an improved approximation for the spin orbitals to be used as a new starting guess. Once the field does not change notably between iterations the spin orbitals can be assumed to be optimal. This is called the self consistent field method.
A disadvantage of all Hartree-Fock based methods are that they are in essence an approximation and can never be solved exactly.
Density Functional Theory
The DFT methods, which have recently been employed with increasing frequency, however, differ in this aspect. They are in theory exact provided that certain functionals can be defined. The DFT methods are based on the papers by Kohenberg and Kohn2 and one year later another one by Kohn- Sham3. In essence the DFT methods revolve around the fact that the external potential, and thus also the ground state energy, is a unique functional of the electron density
(40)
which can be described as the probability of finding any one electron with an arbitrary spin in the volume element with all other electrons in arbitrary positions and spin. Also since the electrons are indistinguishable the chance to find any such one electron is just N times the one electron expression.
Most density functional methods still determine the electron density from molecular orbitals though.
An approach similar to the Hartree-Fock method called Kohn-Sham is used.
Despite their frequent use the DFT methods still have some drawbacks and their usefulness can vary depending on the system at hand. This report is concerned only with main group elements and the B34LYP5 used for optimization and the recently developed M06-2X6 used for energy computations should be well sufficient.
Transition state theory
A transition state is a stationary point in between a set of reactants and products. What sets it apart from a regular geometry optimization where the energy is minimized with regard to the geometry parameters is that the transition state corresponds to a maximum rather than a minimum. We will see that this will give it some interesting properties. First of all, much like a normal geometry optimization, it is a stationary point, i.e. the first derivative of the energy with regard to geometry parameters is zero. Worthy of note is that these geometry parameters are not necessarily limited to bond lengths but bond angles and dilateral angles are of interest as well. The first derivative of a stationary point is zero for all geometry parameters.
In order to decide whether a stationary point is a minimum or a maximum the character of the second derivative must be decided. A minimum will have a positive second derivative for all parameters and a maximum will have a positive second derivative for all but one parameter, which must be negative. This geometry parameter is along the reaction coordinate. What does this one negative second derivative tell us then? First of all let’s consider the physical interpretation of the
11
second derivative. The bonds in a molecule vibrate constantly and can be approximated as a harmonic oscillator. The potential energy inherent to a harmonic oscillator may be described by the following equation:
(41)
Where k is a force constant describing the stiffness of the bond and q is the displacement. This approximation only holds true close to the equilibrium distance where a quadratic approximation is close to the true energy. In other words a frequency calculation does not give any relevant results outside of a stationary point. The second derivative of the potential energy with regard to the geometry displacement is the force constant only.
(42)
This is also referred to as the Hessian. Since the bonds are treated as harmonic oscillators, they also have a frequency they vibrate at which can be described as follows:
(43)
The frequency is obviously dependent on the square root of the force constant, what this means in other words is that a negative force constant, that is the reaction coordinate for a transition state since it corresponds to a maximum, will have an imaginary frequency. Oftentimes though in chemical programs it’s denoted as a negative frequency rather than the true imaginary frequency. Since the whole approximation relies on the fact that the bond vibration can be treated as a harmonic oscillator and approximated to a quadratic function, when running a frequency calculation it’s imperative that it is done with the same parameters as the geometry optimization. If not the Hessian may differ at the stationary point and the results are meaningless.
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Introduction
The Hetero Diels Alder reaction is a variation of the well known Diels Alder reaction using a hetero atom on the dienophile. This is one of the most prominent ways of synthesizing a six-membered ring with a hetero atom. Various hetero atoms are viable in hetero Diels-Alder reaction, the most
common being oxygen, nitrogen and sulphur, for this paper the emphasis will be on oxygen however.
The hetero Diels-Alder is also, very much like the regular Diels-Alder, a rather stereo specific reaction making it a very attractive reaction to investigate further. It has also been shown that this stereo specificity as well as the reactivity may be further enhanced by simple Hydrogen-bonding catalysts, be it in the form of smaller molecules like chloroform7 or chiral alcohols like TADDOL89, hereon referred to as H-bonding. These are very interesting due to their low costs and availability. The catalyst is, for all systems involving catalysts in this thesis, methanol (1).
The hetero Diels-Alder has not been as thoroughly investigated as its regular counterpart, although especially recently more and more research has been put into understanding the hetero Diels-Alder further10.
However especially mechanistic understanding is still lacking in some areas. One such area is how the reactivity and stereo specificity is affected by introducing different functional groups to the
dienophile with and without an H-bonding catalyst. Recent similar studies for the regular Diels-Alder reaction has been carried out by Brinck and Linder11 but to the best of our knowledge similar
research for the hetero Diels-Alder has not been published.
In order to see how the reactivity changes with different functional groups, five different variations of the dienophile were investigated. In the Diels-Alder reaction, the HOMO of the diene and the LUMO of the dienophile are the reacting orbitals. The opposite, the LUMO of the diene reacting with the HOMO of the dienophile, is called a reverse electron demand reaction. In none of the systems investigated in this thesis could any reverse electron demand be detected. By introducing electron withdrawing groups to the dienophile, the LUMO will be stabilized and lower in energy. This will decrease the gap to the HOMO, consequently facilitating the reaction. Apart from the hydrogen case (2a), i.e. no functional group, three electron withdrawing groups (EWG) and one electron donating group (EDG) were investigated. The electron donating group is assumed to have a destabilizing effect by raising the LUMO of the dienophile. The EWGs are CN in alpha position (2b), CN on the carbonyl carbon, also known as pyruvonitrile (2c) and trifluoromethyl (2d). The EDG is an amide (2e).
The diene is dimethoxybutadiene (3) throughout all computations.
13
In the same way that lowering the LUMO for the dienophile, raising the HOMO on the diene would also increase reactivity but no such effects were investigated in this study. It has been shown that the so called Danishefsky’s diene12 as well as Brassard’s diene13 are excellent dienes in the hetero Diels- Alder reaction. In order to decrease computational costs the similar but slightly simpler
dimethoxybutadiene was used instead. This has been done successfully in other investigations as well5.
Computational Details
In order to determine the reaction barrier and reaction energy, the sum of the reactant energies and the product energy respectively were subtracted from the transition state energy.
(44)
(45)
(46)
In the case of the catalyzed reaction the energy of the methanol H-bonded to the dienophile was used.
(47)
14
The geometry optimizations were carried out in the Gaussian 03 and Gaussian 09 programs. All reactants, transitions structures products and the catalyst were optimized and frequency calculations were calculated with the B3LYP functional and the 6-31G* basis set.
Only one imaginary frequency was found for each transition structure, corresponding to the forming C-C and C-O bonds. Single point calculations were then done on all optimized structures with the M06-2X functional and the 6-311++G(2d,2p) basis set. This has recently been shown to increase the accuracy of the computed energies. All calculations for the solvated reactions were done in the cPCM model with the M06-2X functional and the 6-31G* basis set.
Each uncatalyzed reaction has four different geometries: meta-endo, meta-exo, ortho-endo and ortho-exo. Meta and ortho refers to the two possible regioisomers. The meta channel corresponds to the O1-C3 and C2-C6 forming bond reaction while the ortho corresponds to the O1-C6 and C2-O3.
The naming convention for the atoms is clarified in figure 1 and a comparison between the meta- and ortho channel can be seen in figure 2.
Figure 1 Clarification of atom naming convention.
Figure 2 The meta- and ortho regioisomers.
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Endo and exo refers to the two possible stereoisomers, that is the orientation of the acetaldehydes methyl group relative to the diene. This is further clarified in figure 3.
Figure 3 The endo- and exo stereoisomers.
In the catalyzed reaction the catalyst will also have two possible orientations to the dienophile, trans and cis with regard to the methyl group, see figure 4. This description will be used throughout to facilitate comparison between reactions, trans always meaning that the structure skeleton and the catalyst is oriented in one fashion.
Figure 4 Cis- and trans orientations of the catalyst to the dienophile. In substituted cases the cis- and trans orientations will still refer to the relative position to the methyl group.
At first the uncatalyzed reaction for the unsubstituted entities were investigated. In order to get a good starting guess for the transition state geometry optimization, the forming bonds were
elongated in a stepwise fashion, ranging from the product to fully separated reactants. For each step a single point calculation were then made in order to obtain the best guess for the transition state;
the point lowest in energy. These calculations were done with a fast but rather imprecise semi- empirical method, AM1, since only the relative minimum was of interest. Starting from these points the transition states were optimized.
It became clear after the first optimization on the uncatalyzed, unsubstituted reaction that the meta channel is much higher in energy and therefore not a feasible path. This is in agreement with
computational and experimental results for similar reactions. This result can be easily understood from frontier orbital theory. As can be seen in figure 5, during a presumed stepwise reaction only the ortho channel will have a stabilizing resonance structure.
16
Figure 5 Resonance structures for the meta- and ortho regioisomers. The meta channel (to the left) can’t fulfill the ring closing and won’t benefit from the resonance structure.
The resonance structure shown is not able to complete the reaction and is therefore not a possible stabilizing step. For this reason, the meta channel is discarded in the consequent calculations. A complete scheme listing all the computations done can be found in appendix A.
When investigating whether the reaction is stereospecific or not, the following expression can be used to determine the enatiomeric excess:
(48)
and are the reaction constants for the two reactions. Their exact values are not of great interest here, but knowing that a difference in reaction rate of a factor 10 gives a rough ee of 81% is enough for a qualitative understanding. A difference in factor 10 in turn corresponds to roughly 1.36 kcal, according to an expression derived from the Arrhenius equation:
(49)
For the solvated reactions, the four solvents presented in table 2 were considered. The solvents were chosen to cover a range of dielectric constant values. Methanol and acetonitrile are close in dielectric constant but represent a protic and an aprotic solvent, respectively. Only the unsubstituted reaction was considered. Noteworthy is that only single point calculations were carried out for the solvated reactions. Some studies of similar systems point towards a shift from a concerted reaction to a stepwise reaction714. This has, however, not been investigated further here.
Solvent Dielectric Constant Toluene 2.38
Methanol 33 Acetonitrile 37.5
Water 80
Table 2
17
Results and Discussion
The reaction is assumed to proceed in a concerted manner, giving one transition state and one product. Studies on the differences between concerted and stepwise Diels-Alder reactions show that the concerted is generally energetically favored15 and the stepwise is usually not to be found. Also, as previously mentioned, it has been reported that the Hetero Diels-Alder reaction has shown an ability to proceed through a stepwise reactions when carried out in solvent, possibly due to the solvent being able to stabilize the diradical intermediate. More activated dienes, such as Danishefsky’s diene and Brassard’s diene have also sometimes given indication of a stepwise reaction path13. For the systems at hand, however, there were no evidence for a stepwise reaction. Judging from the charge transfer of the different systems, ranging from 0.26 to 0.68, at most it can be said to have a
zwitterionic character. It appears to be a weak connection between the charge transfer and the activation barrier in the uncatalyzed systems but any such connection disappears for the catalyzed systems.
Table 3 Relevant geometries and energies for all systems. All computations were done with the M06-2X functional and the 6-311++G(2d,2p) basis set.
catalyzed uncatalyzed catalyzed uncatalyzed
2a cis trans 2b cis trans
endo exo endo exo endo exo endo exo endo exo endo exo
∆d1 (Å) 1,81 1,77 1,80 1,79 1,90 1,89 ∆d1 (Å) 1,84 1,70 1,76 1,72 1,87 1,81
∆d2 (Å) 2,51 2,53 2,52 2,59 2,28 2,32 ∆d2 (Å) 2,72 2,63 2,70 2,82 2,44 2,53
∆d12 (Å) 0,70 0,76 0,72 0,80 0,38 0,43 ∆d12 (Å) 0,89 0,94 0,93 1,11 0,57 0,73
∆dcatchange (Å) 0,33 0,33 0,34 0,37 - - ∆dcatchange (Å) 0,32 0,21 0,36 0,38 - -
∆dH-bond (Å) 1,74 1,70 1,70 1,72 - - ∆dH-bond (Å) 1,77 1,69 1,70 1,72 - -
∆E‡ (kcal/mol) 12,14 14,97 14,29 13,04 18,64 19,94 ∆E‡ (kcal/mol) 3,24 5,37 5,51 3,00 9,96 11,84
∆E‡cchange (kcal/mol) -6,50 -4,97 -4,35 -6,90 - - ∆E‡cchange (kcal/mol) -6,72 -6,47 -4,45 -8,84 - -
catalyzed uncatalyzed catalyzed uncatalyzed
2c cis trans 2d cis trans
endo exo endo exo endo exo endo exo endo exo endo exo
∆d1 (Å) 1,85 1,74 1,79 1,82 1,93 1,86 ∆d1 (Å) 1,88 1,77 1,79 1,86 1,90 1,88
∆d2 (Å) 2,60 2,64 2,57 2,78 2,31 2,45 ∆d2 (Å) 2,74 2,72 2,68 2,78 2,43 2,47
∆d12 (Å) 0,75 0,90 0,78 0,96 0,38 0,60 ∆d12 (Å) 0,85 0,95 0,89 0,92 0,54 0,59
∆dcatchange (Å) 0,37 0,30 0,40 0,36 - - ∆dcatchange (Å) 0,32 0,36 0,36 0,33 - -
∆dH-bond (Å) 1,77 1,72 1,75 1,78 - - ∆dH-bond (Å) 1,78 1,71 1,74 1,79 - -
∆E‡ (kcal/mol) 3,54 3,59 5,98 2,01 12,20 11,61 ∆E‡ (kcal/mol) 1,27 2,72 3,32 1,49 9,70 10,28
∆E‡cchange (kcal/mol) -8,66 -8,02 -6,22 -9,60 - - ∆E‡cchange (kcal/mol) -8,43 -7,56 -6,38 -8,79 - -
catalyzed uncatalyzed
2e cis trans
endo exo endo exo endo exo
∆d1 (Å) - - - - 1,89 2,02
∆d2 (Å) - - - - 2,12 1,99
∆d12 (Å) - - - - 0,23 -0,03
∆dcatchange (Å) - - - - - -
∆dH-bond (Å) - - - - - -
∆E‡ (kcal/mol) - - - - 35,34 34,27
∆E‡cchange (kcal/mol) - - - - - -
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The transition state is asynchronous with a lengthening of the oxygen – carbon bond and conversely a shortening of the carbon – carbon bond. This is in good agreement with similar studies14. The reaction energy, reaction barrier and bond lengths of interest for the different optimized transition states are displayed in table 3 and the corresponding geometries can be viewed in appendix B.
As noted earlier the meta pathway is much higher in energy, hence its exclusion in calculations after the uncatalyzed systems. One thing worthy of note is that these meta configurations are also much less asynchronous. This is in close agreement with various studies pointing towards a relation between activation barrier and asynchronicity11. The other systems also point towards the same conclusion, more asynchronous transition state have lower activation energy. Graphs comparing the B3LYP calculated energies against asynchronicity and the M06-2X calculated energies against
asynchronicity can be seen in figure 6 and figure 7 respectively. The correlation is less obvious for the results from the M06-2X calculations. However, these still use the geometries from the B3LYP
optimizations. Since these geometries are optimized in B3LYP they account less for long distance interactions that suddenly the single point calculations in M06-2X more accurately describe, thus some deviating results are to be expected. As a matter of fact, Linder et .al has shown that said asynchronicity is less pronounced for systems optimized in M06-2X.
Figure 6 The B3LYP calculated activation energy plotted against the asynchronicity. The asynchronicity is expressed by the distance d12.
0,400 2,400 4,400 6,400 8,400 10,400 12,400 14,400 16,400 18,400
0,600 0,700 0,800 0,900 1,000 1,100 1,200
Activation energy / kcal/mol
∆d12 / Å
Trans/Exo Trans/Endo Cis/Exo Cis/Endo
19
Figure 7 The M06-2X calculated activation energy plotted against the asynchronicity. The asynchronicity is expressed by the distance d12.
Another geometry parameter that could have been noteworthy is the H-bond distance between the catalyst and the dienophile. However, no apparent correlation was found.
For all the catalyzed systems there are four configurations of interest. Two of them vary the position of the functional groups and the other two vary the position of the catalyst. The most pronounced difference is the positioning of the catalyst, the orientations having the catalyst under the structure, as seen in table 3, consistently yielding lower energies. This is possibly due to the fact that the orbitals of the catalyst overlap with the conjugated π-orbital system, having a stabilizing effect.
Another possibility could have been that the catalyst not oriented under the diene is more exposed and consequently has a bigger surface in vacuum. This sometimes gives higher energy results.
Comparing the reactions in solvation to the unsolvated ones the difference between configurations with the catalyst in different positions lessens.
Regarding actual synthesis, however, the difference between the exo and endo stereoisomers are of much more interest since they directly influence which product is formed (at least which one is the kinetic product). The difference is not as pronounced as the one between different catalyst
orientations but it’s still recognizable. A more detailed listing of the exo/endo difference can be seen in figure 8.
0,400 2,400 4,400 6,400 8,400 10,400 12,400 14,400 16,400
0,600 0,700 0,800 0,900 1,000 1,100 1,200
Activation energy / kcal/mol
∆d12 / Å
Trans/Exo Trans/Endo Cis/Exo Cis/Endo
20
Figure 8 The difference in activation energy between the exo and the endo stereoisomers. If the endo stereoisomer is higher in energy the value will be positive. Note that due to naming conventions 2a will have a positive value when the bulkier group is in endo position since the substituent is hydrogen.
The absolute values range from 0.22kcal/mol to 1.89kcal/mol, detailed data can be seen in table 4 below. It is not a large difference but is still enough to make the reaction stereospecific
dienophile uncatalyzed(kcal/mol) catalyzed (kcal/mol)
2a 1.299 0.898
2b 1.889 -0.242
2c -0.597 -1.534
2d 0.576 0.222
2e -1.069 -
Table 4 Difference in activation energy between the exo and endo stereoisomers. See also figure 8.
The corresponding ee for these values, using equation 47 and 48 would be 18.4% and 92.1%. These numbers are however to be taken with some skepticism since the margin of error of the
computational method is about the same as the difference between enantiomers. Still, the systems should be stereospecific.
What’s interesting to note is that the selectivity is somewhat reduced in the catalyzed systems.
Regarding which conformation that’s actually preferred the results once again differ slightly between the uncatalyzed and the catalyzed systems. The catalyzed systems tend towards an endo preference for the bulkier group. This is the case for all but the system with mehtylfluoride as the functional group, which could be explained by the comparatively large steric hindrance of the methylfluoride group. In the other cases, however, a similar effect as normal Diels-Alder could explain the endo preference. In normal Diels-Alder reactions a clear endo preference can usually be seen. This is due to the conjugated pi orbitals of the dienophile orienting to overlap with the pi orbitals of the diene, thus stabilizing the system. For the hetero Diels Alder systems at hand, some have similar conjugated pi-orbitals and some have other orbitals that can overlap with the pi-orbitals of the diene when in endo position.
-2,000 -1,500 -1,000 -0,500 0,000 0,500 1,000 1,500 2,000 2,500
Uncat Cat
endo exo difference / kcal/mol
2a 2b 2c 2d 2e
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As for the actual catalytic effect, the values for all systems can be seen in the previously mentioned table 3. It ranges from -4.3kcal/mol kcal to -9.6kcal/mol. It also appears like there is a synergetic effect between the activating groups and the catalyst, the catalytic effect tends to be greater for already more stable configurations. The results are rather promising, seeing that such a simple molecule can catalyze the reaction to such an extent.
HOMO-LUMO gap
As mentioned, the roles of the electron withdrawing functional groups have been to lower the LUMO of the dienophile, closing the HOMO-LUMO gap and thus increasing the reactivity. The HOMO-LUMO gap is not the only attribute determining the reactivity but very often it has a correlation. When closely inspected the systems at hand also follow this trend with some minor deviations. The activation energy plotted against the HOMO-LUMO gap is displayed in figure 9 for the uncatalyzed systems and figure 10 for the catalyzed systems.
Figure 9 The activation energy plotted against the HOMO-LUMO gap for the uncatalyzed systems. The system with the smallest gap, 2c, is contrary to expectation not the lowest in energy, possibly due to the resonance effect displacing electrons giving less overlap.
2a
2d
2c 2b
2e
0,000 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000
0,150 0,170 0,190 0,210 0,230 0,250 0,270 0,290 0,310
Activation energy / kcal/mol
HOMO-LUMO gap / eV
exo endo
22
Figure 10 The activation energy plotted against the HOMO-LUMO gap for the catalyzed systems.
The reason for the deviations can be explained by more closely inspecting how the functional groups are electron withdrawing. The cyanide group is electron withdrawing mainly through resonance effects while the methylfluoride group is electron withdrawing only by inductive effects. The cyanide group is the more electron withdrawing of the two and pyruvonitrile also has a smaller HOMO-LUMO gap than CF3, i.e. it lowers the LUMO more by withdrawing electrons. The CN is also electron
withdrawing but due to the placement it can’t reach full effect. The reason why CF3 is lower in energy than pyruvonitrile is due to the nature of the electron withdrawing effect. The resonance effect displaces the electrons somewhat leading to less overlap, which is not the case for the inductive effects. Other studies have also shown indications that functional groups with inductive effects tend to be lower in energy than groups with resonance effects of somewhat equal electron withdrawing ability11.
2a
2c 2b
2d
0,000 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000
0,150 0,170 0,190 0,210 0,230 0,250 0,270
Activation energy / kcal/mol
HOMO-LUMO gap / eV
cis/endo cis/exo trans/endo trans/exo
23
Solvated reactions
For the solvated reactions a rather clear trend can be observed. The solvents with lower dielectric constants facilitate the reactions better. Differences between the different systems are very small though and only the one with the lowest dielectric constant, toluene, is clearly lower. A list of all values can be seen in table 5.
Cat Uncat
cis trans
endo exo endo exo endo exo Unsolv
∆E‡ (kcal) 9,149 11,970 11,367 10,103 16,354 17,685
∆E‡catchange (kcal) -7,205 -5,715 -4,986 -7,582 - -
Water
∆E‡ (kcal) 11,243 12,386 12,133 11,548 17,760 18,552
∆E‡catchange (kcal) -6,516 -6,166 -5,626 -7,004 - -
Acetonitrile
∆E‡ (kcal) 11,220 12,411 12,147 11,545 17,741 18,547
∆E‡catchange (kcal) -6,521 -6,136 -5,594 -7,002 - -
Toluene
∆E‡ (kcal) 10,342 12,439 11,991 11,017 17,147 18,224
∆E‡catchange (kcal) -6,805 -5,786 -5,157 -7,207 - -
Methanol
∆E‡ (kcal) 11,215 12,415 12,149 11,544 17,738 18,546
∆E‡catchange (kcal) -6,522 -6,131 -5,589 -7,002 - -
Table 5 cPCM computed solvated activation energies for 2a.
One thing worthy of note is that regular Diels-Alder reactions have been proven to react faster in water, not due to the solvent acting in a stabilizing fashion but rather by acting as an antisolvent pushing the reactants together. This kind of interaction, if applicable to the current system, may not be accounted for in these calculations.
Studies comparing how well solvation models, for water in particular, correspond to the actual increase in reactivity have also been conducted. Since the water molecules may act as a catalyst as well by H-bonding, calculating the water molecules explicitly will yield a rather different result from calculations from a solvation model. In one study the PCM model has been compared to systems ranging from one to three explicit water molecules as well as combined systems of PCM with two explicit water molecules16. In the systems in this study, some of the solvents may very well act as an H-bonding catalyst, making the cPCM results unreliable. However an understanding how the reaction is facilitated in general by solvents of different dielectric constants is still obtainable.
24
Conclusions
The reactions proceed through asynchronous concerted pathways. There is also a a clear correlation between activation energy and asynchronicity were more asynchronous reactions have a lower activation energy. The meta pathway also proved to be much higher in energy and can be discarded as a possible reaction path.
For the stereoisomeric orientation of the dienophile and the catalyst in the transition state, the most favored orientation of the catalyst was to have it pointing in under the diene and the most favored orientation of the dienophile was to have the bulkier group in endo position unless it was sterically hindered.
The catalytic effect ranges from -4.3kcal/mol to -9.6kcal/mol and there is a synergetic effect between the catalyst and activation functional groups. The functional groups yielding the lowest activation energies are also lowered the most when catalyzed.
There is a clear correlation between the HOMO-LUMO gap and the activation energy, smaller gaps giving the lower activation energies. However there is a deviation, the functional group giving the smallest gap does not give the lowest activation energy. This is due to the nature of the electron withdrawing group. It is electron withdrawing by resonance and therefore it displaces the electrons slightly, giving a worse overlap.
The effect of the solvent is most pronounced for toluene, the solvent with the lowest dielectric constant, which gives the lowest activation energies. How well the other solvents facilitate the reaction can also be ordered by dielectric constant although the differences are very small. Only the case with toluene is clearly lower. The effects from the solvents are also just an effect of the
dielectric constant, no explicit bonding effects were accounted for. Therefore solvents that could form H-bonds and catalyze the reaction will not yield lower reaction energies.
25
H
uncatalyzed
ortho
endo
exo
meta
endo
exo
catalyzed
cis ortho
endo
exo
trans ortho
endo
exo
Pyruvonitrile
uncatalyzed ortho
endo exo
catalalyzed
cis ortho
endo exo
trans ortho
endo
exo
alpha position CN
uncatalyzed ortho endo
exo
catalalyzed
cis ortho endo
exo
trans ortho
endo
exo
trifluoromethyl
uncatalyzed ortho
endo exo
catalalyzed
cis ortho endo
exo
trans ortho
endo exo
amide uncatalyzed ortho
endo
exo
Appendix A
26
Appendix B
endo/cis endo/trans
2a
2b
2c
2d
27
exo/cis exo/trans
2a
2b
2c
2d
28
endo exo
2a ortho channel
2a meta channel
2b
2c
2d
2e
29
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