Transition States in Chemical Reactions: Tutorial and Assignments

Transition States

Chemical Reactions Transition States

Chemical reactions occur by the rearrangement of nuclear configurations from the reactant state to the product state. For polyatomic molecules, there is an enormously large number of possible rearrangement paths that take reactants to products. Reactant molecules that have lots of energy could follow a path that involves high energy configurations, reactants with less energy will follow a path that involves configurations with lower energy. A complete description of a chemical reaction dynamics would include all these paths. However, such a complete description is challenging because of the need to map out a multidimensional potential energy surface. Instead, a simplified approach, termed the transition state theory, is commonly employed.

The simplest versions of the transition state theory assume that reactants behave like very tired mountain climbers who are trying to get from one valley to another and have to cross a mountain range. Such mountain climbers will seek out the easiest path, one that avoids steep climbs and raises minimally in altitude. They will seek out a gorge that takes them over the ridge. Like a group of tired mountain climbers, the reactant molecules in the transition state theory will follow a unique path that connects the reactant basin and the product basin. The highest point on this path is the col, or saddle point that separates the reactant basin from the product basin. The saddle point is the point of highest energy along the reaction path and is also the point of lowest energy in the direction perpendicular to the reaction path (lowest point of the ridge that separates reactants and products.

Optimization of Transition States

Transition states correspond to saddle points on the potential energy surface. Strictly speaking, a transition state of a chemical reaction is a first order saddle point. Like minima, the first order saddle points are stationary points with all forces zero. Unlike minima, one of the second derivatives in the first order saddle is negative. The eigenvector with the negative eigenvalue corresponds to the reaction coordinate. Transition state search thus attempts to locate stationary points with one negative second derivative. The basic recipe is: identify the reaction mode and maximize energy along this mode while minimizing energy in all other directions. One reason why transition state optimization is more difficult than the search for a minimum is that a successful search should start off in a region where the reaction coordinate already has a negative curvature. In other words, search for a transition state should start near the transition state.

Here are some recipes to locate transition state structures for chemical reactions:

Manually build a guess structure and optimize it using first and second derivatives. This approach is also called eigenvector following because the search will follow the eigenvector with a negative eigenvalue. If the starting structure has one and only one negative second derivative then the search will converge to the closest saddle point. If the starting structure has several negative eigenvalues, the search either fails or follows the direction with the largest negative eigenvalue. The main shortcoming of methods that require second derivatives is high computational cost and in practice algorithms that calculate the Hessian once and then update it during the search (Requested by Opt=CalcFC in Gaussian)are faster than methods methods that calculate Hessian at each step. Starting with a guess transition state structure is often successful for simple reactions for which chemical intuition provides reasonable transition state guesses. For example, transition states for S_N2 reactions involving alkyl halides can be located using this approach because we know that the transition state has penta-coordinate nature with the incoming nucleophile and the leaving group roughly in-line.
Build the structures of the reactant and the product and use a synchronous transit-guided quasi-Newton approach to generate an initial guess and optimize it to the transition state. If the interpolation is carried out along the straight line that connects reactants and products, the method is called the linear synchronous transit (LST). If the interpolation is carried out along the parabola that connects reactants and products, the method is called the quadratic synchronous transit (QST2). By default, the initial guess will be a structure exactly halfway between the reactants and the products complex. This approach works well for bimolecular reactions where structures of the reactant complex and the product complex can be specified. However, QST2 fails for many unimolecular isomerization reactions because the interpolation between the reactant and the product gives a poor guess structures.
Build the structures of the reactant complex, the product complex, and guess for the transition state, and use a synchronous transit-guided quasi-Newton approach (QST3) to optimize the transition state. This approach is more robust than QST2 as it overcomes problems die to poor interpolated guess structures. The advantage of transit-guided quasi-Newton approaches over the eigenvector following method is that a full evaluation of second derivative matrix is not needed. Thus, QST2 and QST3 can be employed with methods such as CCSD(T) for which analytic second derivatives are not available.
Scan the reaction path or slice of a potential energy surface to identify saddle points. This is very robust approach that eventually will lead to transition state, but because it requires a large number of calculations, it is also time-consuming. The scanning approach is effective when there is only one reaction coordinate, as in the case of transitions between conformational isomers. For example, to locate the transition states for the internal rotation in hydrogen peroxide, the HO-OH dihedral angle can be scanned in 20 degree increments to identify maxima along this internal rotation path. The scanning approach becomes less effective as the number of geometric variables that change as the reaction occurs increases. However, mapping out potential energy surfaces is sometimes valuable as it allows to identify alternative reaction paths.

It is clear from the above discussion that the choice of the method depends largely on the nature of the problem. Furthermore, because transit-guided quasi-Newton approaches use approximate guess Hessians that might differ significantly from the true Hessian, it is important to perform full second order derivative calculation (Freq keyword in Gaussian) to verify that the structure from QST2 or QST3 optimization is indeed a transition state with one and only one negative eigenvalue.

In practice, it is often faster to pre-optimize the transition state with an affordable method. For example, if one is interested in the CCSD/cc-pVTZ transition state structure, it is a good idea to first find the transition state at a MP2/cc-pVTZ level. This has two key advantages. First, the relatively fast MP2/6-311+G(d) optimization most likely yields a structure that is a good starting point for the more coupled cluster expensive optimization. Second, the Hessian at the MP2/6-311+G(d) level can be used as the initial guess Hessian for the CCSD/cc-pVTZ optimization. (Opt=ReadFC keyword in Gaussian.

Tutorial

The Menshutkin Reaction: Gas Phase Reaction Path

The reaction between tertiary amines and alkyl halides is known as the Menshutkin reaction. This reaction provides an useful route to quaternary ammonium salts. The Menshutkin reaction involves nucleophilic substitution of halide by the nucleophilic amine and can be thus classified as an S_N2 process. The Menshutkin reaction converts two neutral reactants into a pair of charged products while the more familiar S_N2 displacements involve an anionic nucleophile and an anionic leaving group. A computationally efficient model for the Menshutkin reaction is a reaction between ammonia and methyl chloride.

The transition states of Menshutkin reactions can be located readily using synchronous transit-guided quasi-Newton approach (QST2) approach. One of the features of S_N2 reactions in the gas phase is the formation of a reactant complex before the transition state is reached. The reactant complex in the Menshutkin reaction is stabilized by dipole-dipole interactions and dispersion attraction. In case of an energetically favorable back-side attack, the incoming nucleophilic nitrogen is almost in-line with the leaving halogen group. The products of the Menshutkin reaction are a pair of oppositely charged ions that are attracted to each other via Coulombic forces which are balanced by the exchange repulsion. Thus, in the gas phase, a product complex where the halide is near the methyl group is expected to form. The structures of such reactant and product complexes are suitable starting points for QST2 search for the transition state.

Using MOLDEN, build structures of the reactant complex and the product complex for a reaction between ammonia and methyl chloride. Save these structures as reactant.dat and product.dat. Close MOLDEN and use Unix command cat (cat reactant.dat product.dat > nh3_ch3cl_qst2_ts_search.dat) to combine these two files into one input file. Edit the input file to add appropriate computational directives (HF/3-21G Opt=QST2 is suitable for a quick pre-optimization) and separate the two molecule definitions by an empty line. The structure of the input file thus is:

%Mem=16MW
%Chk=nh3_ch3cl_ts_qst2_search.chk
# HF/3-21G Opt=QST2

Menshutkin reaction between NH3 and CH3Cl: Reactant complex

0 1

[Reactant Complex Z Matrix]

Menshutkin reaction between NH3 and CH3Cl: Reactant complex

0 1

[Product Complex Z-matrix]

Submit this calculation. After completion (the calculation should finish in few minutes) examine the convergence of optimization with MOLDEN. The transition state located by QST2 method at HF/3-21G level has the C-Cl distance of 2.416 and the C-N distance of 1.948.

Build a guess transition state for the Menshutkin reaction between ammonia and methyl chloride using MOLDEN. Write this guess structure out as nh3_ch3cl_hess_ts_search.dat and edit the input to add directives for optimization at the HF/3-21G level using analytical second derivatives at the initial point (Opt=(TS,CalcFC) in Gaussian). Perform calculation and compare the structure obtained via this approach to the structure obtained with the synchronous transit-guided quasi-Newton approach.

Homework

Level 1

Manually build the transition state for the reaction between ammonia and methyl chloride and optimize it at the HF/3-21G, HF/6-31G, HF/6-31+G(d,p), MP2/6-31+G(d,p), and B3LYP/6-31+G(d,p) levels. Build structures of the two separated reactants and optimize these at the same levels. Build structures of the two separated product and optimize these at the same levels. Perform frequency analysis for each of the structures at each level to verify the minima and saddle points. Calculate reaction energy, reaction enthalpy, and reaction free energy in the gas phase at each level. Calculate activation energy, activation enthalpy, and activation free energy in the gas phase at each level. Discuss which levels of theory seems to be reliable in describing this reaction. Hint: using at least Cs symmetry is appropriate and will speed up calculations.

Level 2

Perform a reaction path scan along the reaction coordinate for a Menshutkin reaction between N-methylamine and methyl chloride using a method and basis set that seems reliable and efficient based on Level 1 calculations. In the scan, decrease the distance between the nucleophilic nitrogen and the electrophilic carbon in 0.2 angstrom steps from 3.5 angstroms to 1.5 angstrom. The relaxed potential energy scan can be performed in Gaussian using the Opt=Z-matrix directive. Locate and verify (i.e. perform frequency calculation) the transition state and the reactant complex along this path. Calculate the energy difference, enthalpy difference, and the free energy difference between the reactant complex and the transition state. Hint: using at least Cs symmetry is appropriate and will speed up calculations.

Level 3

Compare the activation barriers in the Menshutkin reaction of methyl chloride with four different nuleophiles: ammonia, N-methylamine, N,N-dimethylamine, and quinuclidine (1-Azabicyclo[2.2.2]octane) using MP2/6-311+G(d,p) energies at HF/6-31+G(d,p) optimized geometries. Also, calculate the gas phase proton affinity for these four bases at the same level. Is there a correlation between the gas phase proton affinity and the gas phase nucleophilicity?

You may use a computational chemistry program of your choice. If you prefer using Windows, then PC Gamess is a good choice (however, you still want to build your molecules in MOLDEN).