Energy profile

For a chemical reaction or process an energy profile (or reaction coordinate diagram) 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 (PES), which are used in computational chemistry to model chemical reactions by relating the energy of a molecule(s) to its structure (within the Born–Oppenheimer approximation). The reaction coordinate is a parametric curve that follows the pathway of a reaction and indicates the progress of a reaction. For a chemical reaction or process an energy profile (or reaction coordinate diagram) 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 (PES), which are used in computational chemistry to model chemical reactions by relating the energy of a molecule(s) to its structure (within the Born–Oppenheimer approximation). 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 (one-dimensional energy surfaces) 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. In simplest terms, a potential energy surface or PES is a mathematical or graphical representation of the relation between energy of a molecule and its geometry. The methods for describing the potential energy are broken down into a classical mechanics interpretation (molecular mechanics) and a quantum mechanical interpretation. In the quantum mechanical interpretation an exact expression for energy can be obtained for any molecule derived from quantum principles (although an infinite basis set may be required) but ab initio calculations/methods will often use approximations to reduce computational cost. Molecular mechanics is empirically based and potential energy is described as a function of component terms that correspond to individual potential functions such as torsion, stretches,bends, Van der Waals energies,electrostatics and cross terms. Each component potential function is fit to experimental data or properties predicted by ab initio calculations. Molecular mechanics is useful in predicting equilibrium geometries and transition states as well as relative conformational stability. As a reaction occurs the atoms of the molecules involved will generally undergo some change in spatial orientation through internal motion as well as its electronic environment. Distortions in the geometric parameters result in a deviation from the equilibrium geometry (local energy minima). These changes in geometry of a molecule or interactions between molecules are dynamic processes which call for understanding all the forces operating within the system. Since these forces can be mathematically derived as first derivative of potential energy with respect to a displacement, it makes sense to map the potential energy E of the system as a function of geometric parameters q1, q2, q3 and so on. The potential energy at given values of the geometric parameters (q1, q2,…, qn) is represented as a hyper-surface (when n >2 or a surface when n ≤ 2). Mathematically, it can be written as- E= f(q1, q2,…, qn) For the quantum mechanical interpretation a PES is typically defined within the Born–Oppenheimer approximation (in order to distinguish between nuclear and electronic motion and energy) which states that the nuclei are stationary relative to the electrons. In other words, the approximation allows the kinetic energy of the nuclei (or movement of the nuclei) to be neglected and therefore the nuclei repulsion is a constant value (as static point charges) and is only considered when calculating the total energy of the system. The electronic energy is then taken to depend parametrically on the nuclear coordinates meaning a new electronic energy (Ee)need to be calculated for each corresponding atomic configuration. PES is an important concept in computational chemistry and greatly aids in geometry and transition state optimization. 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.