Energy profile (chemistry)
In theoretical chemistry, an energy profile is a theoretical representation of a chemical reaction or process as a single energetic pathway as the reactants are transformed into products. This pathway runs along the reaction coordinate, which is a parametric curve that follows the pathway of the reaction and indicates its progress; thus, energy profiles are also called reaction coordinate diagrams. They are derived from the corresponding potential energy surface, which is used in computational chemistry to model chemical reactions by relating the energy of a molecule to its structure.
File:Reaction Coordinate Diagram.png|thumb|300px|right|Figure 1: Reaction Coordinate Diagram: Starting material or reactant A convert to product C via the transition state B, with the help of activation energy, after which chemical energy is released
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.
Potential energy surfaces
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 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 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. 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 of the system as a function of geometric parameters,, and so on. The potential energy at given values of the geometric parameters is represented as a hyper-surface or a surface. Mathematically, it can be written asFor the quantum mechanical interpretation, a PES is typically defined within the Born–Oppenheimer approximation which states that the nuclei are stationary relative to the electrons. In other words, the approximation allows the kinetic energy of the nuclei to be neglected and therefore the nuclei repulsion is a constant value 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 must be calculated for each corresponding atomic configuration.
PES is an important concept in computational chemistry and greatly aids in geometry and transition state optimization.
Degrees of freedom
An -atom system is defined by coordinates: for each atom. These degrees of freedom can be broken down to include 3 overall translational and 3 overall rotational degrees of freedom for a non-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 -atom system will be defined by or 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 -internal coordinates a separate potential energy function can be written with respect to each of these coordinates by holding the other parameters at a constant value allowing the potential energy contribution from a particular molecular motion to be monitored while the other parameters are defined.Consider a diatomic molecule AB which can macroscopically visualized as two balls connected through a spring which depicts the bond. As this spring is stretched or compressed, the potential energy of the ball-spring system 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 bonds and bond angle as variables on which the potential energy of a water molecule will depend. We can safely assume the two 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, bond length and 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.
Characterizing a PES
The most important points on a PES are the stationary points where the surface is flat, i.e. parallel to a horizontal line corresponding to one geometric parameter, a plane corresponding to two such parameters or even a hyper-plane corresponding to more than two geometric parameters. The energy values corresponding to the transition states and the ground state of the reactants and products can be found using the potential energy function by calculating the function's critical points or the stationary points. Stationary points occur when the 1st partial derivative of the energy with respect to each geometric parameter is equal to zero.Using analytical derivatives of the derived expression for energy, one can find and characterize a stationary point as minimum, maximum or a saddle point. The ground states are represented by local energy minima and the transition states by saddle points.
Minima represent stable or quasi-stable species, i.e. reactants and products with finite lifetime. Mathematically, a minimum point is given as
A point may be local minimum when it is lower in energy compared to its surrounding only or a global minimum which is the lowest energy point on the entire potential energy surface.
Saddle point represents a maximum along only one direction and is a minimum along all other directions. In other words, a saddle point represents a transition state along the reaction coordinate. Mathematically, a saddle point occurs when
for all except along the reaction coordinate and
along the reaction coordinate.
Reaction coordinate diagrams
The intrinsic reaction coordinate, derived from the potential energy surface, is a parametric curve that connects two energy minima in the direction that traverses the minimum energy barrier passing through one or more saddle point. However, in reality if reacting species attains enough energy it may deviate from the IRC to some extent. The energy values along the reaction coordinate result in a 1-D energy surface and when plotted against the reaction coordinate gives what is called a reaction coordinate diagram. Another way of visualizing an energy profile is as a cross section of the hyper surface, or surface, long the reaction coordinate. Figure 5 shows an example of a cross section, represented by the plane, taken along the reaction coordinate and the potential energy is represented as a function or composite of two geometric variables to form a 2-D energy surface. In principle, the potential energy function can depend on N variables but since an accurate visual representation of a function of 3 or more variables cannot be produced a 2-D surface has been shown. The points on the surface that intersect the plane are then projected onto the reaction coordinate diagram to produce a 1-D slice of the surface along the IRC. The reaction coordinate is described by its parameters, which are frequently given as a composite of several geometric parameters, and can change direction as the reaction progresses so long as the smallest energy barrier is traversed. The saddle point represents the highest energy point lying on the reaction coordinate connecting the reactant and product; this is known as the transition state. A reaction coordinate diagram may also have one or more transient intermediates which are shown by high energy wells connected via a transition state peak. Any chemical structure that lasts longer than the time for typical bond vibrations can be considered as intermediate.A reaction involving more than one elementary step has one or more intermediates being formed which, in turn, means there is more than one energy barrier to overcome. In other words, there is more than one transition state lying on the reaction pathway. As it is intuitive that pushing over an energy barrier or passing through a transition state peak would entail the highest energy, it becomes clear that it would be the slowest step in a reaction pathway. However, when more than one such barrier is to be crossed, it becomes important to recognize the highest barrier which will determine the rate of the reaction. This step of the reaction whose rate determines the overall rate of reaction is known as rate determining step or rate limiting step. The height of energy barrier is always measured relative to the energy of the reactant or starting material. Different possibilities have been shown in figure 6.
Reaction coordinate diagrams also give information about the equilibrium between a reactant or a product and an intermediate. If the barrier energy for going from intermediate to product is much higher than the one for reactant to intermediate transition, it can be safely concluded that a complete equilibrium is established between the reactant and intermediate. However, if the two energy barriers for reactant-to-intermediate and intermediate-to-product transformation are nearly equal, then no complete equilibrium is established and steady state approximation is invoked to derive the kinetic rate expressions for such a reaction.