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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:

Background

Thermodynamic formulation of transition state theory (TST) allows calculating rate constants from the activation free energy using the Eyring equation:
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):
DeltaG = G(TS) - G(Min)
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:
H = Eqm + Eel + Etr + Evib + Ehindrot + RT
S = Sel + Str + Srot + Svib + Shindrot
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

  1. 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.
  2. 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.
  3. The few rules that you should keep in mind when working with Z-matrizes are:
  4. 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
  5. 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.
  6. 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.
  7. 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.
  8. Open another Unix shell and verify that the file was successfully written in your directory. Minimize, but do not close the Molden window.

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Course materials by Dr. Kalju Kahn, Department of Chemistry and Biochemistry , UC Santa Barbara. 2006