Chemistry 453 Lecture 17: Intramolecular Interactions and Rigid Rotors - Prof. Gabriele Va, Study notes of Biochemistry

Download Chemistry 453 Lecture 17: Intramolecular Interactions and Rigid Rotors - Prof. Gabriele Va and more Study notes Biochemistry in PDF only on Docsity! University of Washington - Department of Chemistry Chemistry 453 - Spring Quarter 2008 Lecture 17 05/09/08 Text Reading Chap 9 p. 493-497; 442-444 Intramolecular structure C. Intramolecular Interactions: Rigid Rotors and Harmonic Oscillators Thus far we have considered intermolecular interactions in systems of molecules that have no internal motions. However, except for monatomic systems, all molecules display internal motions. For example, diatomic molecules undergo whole-molecular rotation (i.e. rigid body rotation) and changes in the length of the bond (i.e. bond vibration). When biological molecules interact with each other or as we change temperature or denature them, their chemical structure (configuration) does not change, however, their three-dimensional structure can and often does change. Their structure is determined by the rotation about chemical bonds whose length and angles are fixed by covalent properties of the chemical bonds that hold these molecules together. The conformation of a molecule is dictated by the value of each of the angles that are more or less free to rotate (C-C double bonds, for example, are not free to rotate). For a large molecules like protein or DNA, there are very many possible conformations, but many are not populated because energetically unfavorable. For example, for proteins, only certain regions of the conformational space in the peptidic backbone are populated due to steric clashes between amino acid side chains (Ramachandran’s plot) and similar considerations apply to nucleic acids as well. Bond lengths and bond angles are determined by interactions of electron and nuclei that define the chemical bonds, and are eminently of quantum mechanical nature. However, the forces that change them can be treated by simple classical physical models. The energy of moving atoms so as to stretch or compress a bond, or to change a bond angle, depends (close to the value which is most favored, or equilibrium value) on the square of the change in bond length or the square of the change in bond angles: 2)( eqvibvib rrkU −= 2)( eqrotrot kU ϑϑ −= Here the differences represent deviations from the equilibrium value that is the bond length and angle that are quantum mechanically most favored. The energy increases when the bonds are perturbed from their equilibrium positions and, as it happens, it takes far more energy to change a bond length than a bond angle. Changing the bond length by as little as 0.1 A requires about 10 kJ/mol, a double bond requires twice that amount, while changing a bond angle by 10 degrees only requires about 5 kJ/mol for a tetrahedral bond. These expressions apply only when the changes are small, large deviations would lead to bond breakage and cannot be described by these simple models, which are analogous to classical harmonic oscillators or springs. Rotations about single bonds cause large changes in molecular conformations and dictate the three-dimensional structure of biological molecules. These rotations do not require much energy (unless there are double bonds involved, in which case the energy are substantial); for example, the benzene ring in phenylalanine in proteins rotate rapidly around the bond connecting it to the polypeptide chain at room temperature, the barrier is only about 2.5 kJ/mole. Torsion energies will have various minima corresponding to most favored states, and can be represented by a potential that has the following form: ))3cos(1( 2 φ+= tor V U Where φ is the torsion angle. In order to rotate a double bond, it takes much more energy because the p bond has to be broken, while for a bond with partial double character, like the C-N bond in formamide or in peptides, the energy is intermediate. Rigid Rotators Let us consider rotations and vibrations in a more formal context. A homonuclear diatomic molecule (e.g. H2, N2, etc.) may be classically represented as a dumbbell. Consider the dumbbell as composed of two spheres with masses m, which are intended to represent atoms, connected by a mass-less bar, which is intended to represent a chemical bond. The bond length is r. If the dumbbell does not interact with other dumbbells, the total energy is purely kinetic, as usual, i.e. E E Etrans rotate= + where Etrans is the kinetic energy of translational motions and Erotate is the kinetic energy associate with rotational motions. The translational energy is, in Cartesian coordinates E p p p mtrans x y z= + +2 2 2 2