page_title: Energy profile (chemistry)
section_title: Degrees of freedom
hierarchical_section_title: Energy profile (chemistry) / Potential energy surfaces / Degrees of freedom
caption_attribution_description: English: The figure shows the energy minimum corresponding to optimized molecular structure for water- O-H bond length of 0.0958nm and H-O-H bond angle of 104.5
context_page_description: For a chemical reaction or process an energy profile is a theoretical representation of a single energetic pathway, along the reaction coordinate, as the reactants are transformed into products. Reaction coordinate diagrams are derived from the corresponding potential energy surface, which are used in computational chemistry to model chemical reactions by relating the energy of a molecule to its structure. The reaction coordinate is a parametric curve that follows the pathway of a reaction and indicates the progress of a reaction.
  Qualitatively the reaction coordinate diagrams have numerous applications. Chemists use reaction coordinate diagrams as both an analytical and pedagogical aid for rationalizing and illustrating kinetic and thermodynamic events. The purpose of energy profiles and surfaces is to provide a qualitative representation of how potential energy varies with molecular motion for a given reaction or process.
context_section_description: An N-atom system is defined by 3N coordinates- x, y, z for each atom. These 3N degrees of freedom can be broken down to include 3 overall translational and 3 (or 2) overall rotational degrees of freedom for a non-linear system (for a linear system). However, overall translational or rotational degrees do not affect the potential energy of the system, which only depends on its internal coordinates. Thus an N-atom system will be defined by 3N-6 (non-linear) or 3N-5 (linear) coordinates. These internal coordinates may be represented by simple stretch, bend, torsion coordinates, or symmetry-adapted linear combinations, or redundant coordinates, or normal modes coordinates, etc. For a system described by N-internal coordinates a separate potential energy function can be written with respect to each of these coordinates by holding the other (N-1) parameters at a constant value allowing the potential energy contribution from a particular molecular motion (or interaction) to be monitored while the other (N-1) parameters are defined.
  Consider a diatomic molecule AB which can macroscopically visualized as two balls (which depict the two atoms A and B) connected through a spring which depicts the bond. As this spring (or bond) is stretched or compressed, the potential energy of the ball-spring system (AB molecule) changes and this can be mapped on a 2-dimensional plot as a function of distance between A and B, i.e. bond length.
  The concept can be expanded to a tri-atomic molecule such as water where we have two O-H bonds and H-O-H bond angle as variables on which the potential energy of a water molecule will depend. We can safely assume the two O-H bonds to be equal. Thus, a PES can be drawn mapping the potential energy E of a water molecule as a function of two geometric parameters, q₁= O-H bond length and q₂=H-O-H bond angle. The lowest point on such a PES will define the equilibrium structure of a water molecule.
  The same concept is applied to organic compounds like ethane, butane etc. to define their lowest energy and most stable conformations.
caption: Figure 3: PES for water molecule: Shows the energy minimum corresponding to optimized molecular structure for water- O-H bond length of 0.0958nm and H-O-H bond angle of 104.5°
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