Prediction of Rate Constants for Conformational Interconversion using Quantum Mechanical Calculations
Introduction
Conformational interconversion between conformers occurs via internal rotation around single or partial double bonds. The height of the free energy barrier between conformers determines the rate of interconversion while the relative free energies of two conformers determine their populations. Rate constants for such processes can be can be obtained by a variety experiments, including NMR line shape analysis and spin saturation transfer measurements. This tutorial illustrates how to obtain the rate constant for internal rotation around the C-N bond in N,N-dimethylformamide.
The typical steps required to find rate constants using computational chemistry are:
- Create the initial structure for the ground state using a molecular editor
- Optimize the geometry of the ground state such that it corresponds to the energy minimum
- Perform frequency analysis to obtain the partition function and free energy for the ground state
- Create the initial structure for the transition state using a molecular editor
- Optimize the geometry of the transition state such that it corresponds to the saddle point
- Perform frequency analysis to obtain the partition function and free energy for the ground state
- Calcululate activation free energy
- Calculate the rate constant
Background
Thermodynamic formulation of transition state theory (TST) allows calculating rate constants from the activation free energy using the Eyring equation:
The temperature-dependent prefactor evaluates to 6.21x1012 s-1 at room temperature.
In order to use the Eyring equation, activation free energy must be determined. In general, the activation free energy can be obtained as the difference of standard state free energies between the transition state (TS) and the ground state (GS):
The standard state free energies of molecules can be estimated readily for the gas phase reactions by evaluating the electronic, translational, rotational, and vibrational partition functions. Computational chemistry programs typically also print out the enhalpies and entropies which are evaluated as sums of different contributions:
Relatively simple formulas for translational, rotational, and vibrational contributions to enthalpy and entropy can be derived assuming that molecules move at non-relativistic speeds, rotate as rigid bodies, and vibrate as harmonic oscillators. The relevant formulas are given in many stastistical mechanics text and online via Wikipedia or Gaussian Thermochemistry White Paper. Contributions form hindered internal rotation, vibrational anharmonicity, and rovibrational coupling are more difficult to evaluate, and are often ignored. Rigorous calculation of activation free energies in solution is significantly more complex because one needs to consider the mutual interaction of solvent molecules and the reacting system.
Building Molecules
You can use a molecular builder such as Molden to build initial guess structures for the ground state and the transition state. Follow these steps to generate a initial structure suitable for optimisation
- Sketch on the paper the structures of the ground state and the transition state for the insomerization between two equivalent conformers of N,N-dimethylformamide using common chemical notation. Decide what are the main differences between the ground state and transition state structure. For example, in conformational isomerization, the transition state typically has one dihedral angle value that is significantly different from the value in the ground state.
- You will need to build the molecule using the connectivity description known as the Z-matrix. In the Z-matrix notation, the atoms in the molecule are defined by bond a length to one of the the preceding atoms, by a bond angle made to two preceding atoms and by a dihedral angle made to three preceding atoms. Decide about the order of atoms based on the guidelines below and write the atom order on the structures you drew earlier.
- The few rules that you should keep in mind when working with Z-matrizes are:
- it is often best to start from a atom relatively close to the geometrical center of the molecule and work outwards. For example, when building N,N-dimethylformamide, starting with carbonyl carbon is not a bad choice.
- when adding a new atom, click on the atom it is connected to, then on the atom it makes an angle with, then on the atom it forms dihedral angle with.
- it is better to define dihedral angles of a atom via the atom it is directly connected and by the two atoms that are directly connected to the latter atom. Thus, it is best to define the dihedral angle of the two methyl hydrogens in methanol as +120 and -120 degrees via atoms C, O, and H(C) rather than +60 and -60 degrees via atoms C, O, H(O)
- as you go, you may adjust bond lenghts and bond angles to reflect the local geometry of the molecule. For example, the sp2 hybridized carbonyl carbon in N,N-dimethylformamide is planar and the dihedral angle defined by C, O, N,and H should be 180 degrees.
- all bond angles should be less that 180 degrees. However, dihedral angles can have any value. For symmetric molecules with linear groups, such as acetonitrile, use of dummy atoms, or XYZ representation is needed to give proper symmetry.
- use symmetry whenever appropriate to shorten the computational time. Students who have studied group theory should find it fairly easy to symmetric molecules. However, a common pitfall is to specify higher symmetry because of fallacies of our 'chemical intuition'. For example, you may be tempted to draw N,N-dimethylformamide in the CS symmetry thinking that this is a planar molecule. However, the nitrogen center in N,N-dimethylformamide may not be perfectly planar in reality.
- Start MOLDEN by typing the command molden into Unix shell. Two windows, graphical area and the Molden Control appear. Click on the ZMAT editor to open another window that allows building molecules from scratch.
Molden Graphical Window |
MOLDEN Z-Matrix editor |
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- The Zmatrix editor in Molden is pretty simple to learn. The command Add Line adds a new atom in the molecule and creates a new line in the Z-matrix. The Substitute Atom by Fragment allows replacement of atoms with common functional groups. For example, you can easily add the two methyl groups when building N,N-dimethylformamide by first building formamide, then substituting the two hydrogens on N with methyl groups. You may occasionally need to use Reorder Z-matrix to achieve desired local or global symmetry.
- Build the structure of the ground state using the Z-matrix editor. Click on the Add Line, and click on the element. The atom appears in the Z-matrix editor and in the MOLDEN graphical window. Continue building until you have the complete molecule with in nearly planar arrangement of the heavy atom framework and nearly symmetrical methyl groups.
- Save the structure in a format acceptable for the computational chemistry program you will be using. The Cartesian format (XYZ) is almost universally understood but minor modification in line labels may be needed when the Cartesian format is used with your favourite program. We will be using the commercial program Gaussian for this task because it allows calculation of analytical second derivatives both at the HF and MP2 levels. To save the structure in Gaussian Z-matrix format, click on Gaussian radio-button and give a name for the molecule (dmf_min.dat). Last, hit Write Z-matrix to save the Z-matrix file in your directory.
- Open another Unix shell and verify that the file was successfully written in your directory. Minimize, but do not close the Molden window.