# Reaction Coordinates in Potential Energy Diagrams

- Page ID
- 9830

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Reaction potential energy diagrams are graphs that show the energy of a process as a function of the extent to which that process has occurred. As these are graphs showing mathematical functions, there must be a numerical coordinate axis that shows the independent variable. This coordinate is called the *reaction coordinate*, and it reflects the geometry of the system. Very often, the reaction coordinate reflects extent to which a reaction has progressed from reactants to products, starting with reactants near the y-axis (the energy coordinate) and progressing toward products.

## Reaction Coordinates for Diatomic Systems

The simplest reaction coordinate is found for a diatomic system. In this case, the geometry in the molecular frame is completely described by the intermolecular distance, r. Therefore, the potential energy diagram for the dissociation of a simple diatomic molecule consists of a plot of the energy of the system as a function of separation.

## Reaction Coordinates for Polyatomic Systems

For systems of 3 or more atoms, the geometry coordinates get more complicated. A non-linear molecule consisting of N atoms will have 3N-6 different geometry coordinates, and there are 3N-5 for linear molecules (this also applies to N = 2). That means that a depiction of the potential energy for a non-linear molecule isa 3N - 6 dimensional surface (3N - 5 dimensions for a linear molecule).

Formally, the coordinate axes consist of the mathematical normal coordinates that describe motion of the atoms in the molecule, although more physical meaningful coordinates can also be utilized.

### Triatomic Systems

A non-linear triatomic molecule can be described with three coordinates. The best example is water, HOH. The three coordinates in water are nominally the two O-H bond lengths and the H-O-H bond angle. Mathematically, these are described as symmetric and asymmetric OH stretching motion, and the HOH bend.

Considering the challenge in representing a 4-D system (3 geometry coordinates and an energy coordinate), there are simplifications and/or approximations that can be made. For example, it is to create a plot where one or two of the variables are fixed. For example, the energy can be plotted against the HOH bond angle, holding the OH bonds at their equilibrium bond lengths. Because the optimal OH bond lengths do not vary significantly as a function of bond angle, a plot of energy vs bond angle in water will be very close to the lowest energy part of the potential energy surface. Alternatively, it is possible to fix the bond angle and one of the OH bond lengths, and look at the energy as a function of the length of the other OH bond. Although this is likely a reasonable approximation of the lowest energy path for breaking the O-H bond, it does not account for changes in the preferred bond angle that might occur, or, to a lesser extent, the change in the OH bond length between water and hydroxyl radical.

Alternatively, it is possible to plot a *relaxed energy surface* for O-H breaking where the other variables (HOH bond angle, other O-H bond length) are always at the optimal values as a function of the breaking O-H bond length. In this plot, the energy does account for the changes in the preferred bond angle and the change to the hydroxyl bond length. If there is a readily available numerical coordinate to use, then a relaxed surface scan is very convenient and is an accurate reflection of the surface, since it refers to the lowest energy pathway in that coordinate.

### Bond Rotation

The potential energy surface for bond rotation is an example of a relaxed surface scan. The reaction coordinate is the dihedral angle between groups on the two atoms, which can be easily observed in a Newman projection. Potential energy surfaces for bond rotation are commonly used for conformational analysis of molecules like ethane and butane.

## More complicated systems

Most organic chemical reactions cannot be described by a single simple coordinate. Even when they can, it may be more convenient to use a more complicated coordinate. For example, chloride undergoes S_{N}2 reaction with methyl bromide through a 5-centered transition state.

In principle, the energy of the process can be plotted vs the Cl-C bond length. Therefore, the energy increases gradually as chloride approaches from long distance until the bond length reaches that in the transition state, which is approximately 200 pm. From there, the energy will decrease precipitously as the Cl-C bond length approaches the value in CH_{3}Cl, 175 pm. A more convenient coordinate might utilize the Cl-C bond length until the formation of the transition state, and switch to the C-Br bond length after the transition state. Alternatively, if using the Cl-C bond length, it might be better to not to use a linear scale, or even consistent scaling.

Consequently, most potential energy surfaces in organic chemistry are not drawn with any single, numerical coordinate, but with a generalized reaction coordinate that reflects the geometry changes in the reaction. Instead of numerical labels on the axis, the positions are labeled with their structures, and the regions in-between are the paths that connect the indicated geometries. Because of this, it is possible to reduce the highly-dimensional surfaces of complicated reactions to 1-dimensional curves.

## Important Things to Remember

**1. Potential energy diagrams are graphs.**

Energy is on the y-axis, and the x-axis indicates geometry. Energy is a function of geometry.

**2**. **The geometry changes refer to changes in bonding (changes in atom positions).**

Each point on the diagram has the same molecular formula (same atoms and electrons). The bonding can change, however. For example, in the S_{N}2 process shown above, the reactant is Cl^{-} + CH_{3}Br, and the product is CH_{3}Cl + Br^{-}. Even if this is not explicitly shown, it is implicit in the diagram. It is not possible to compare energies of different atoms.

**3. The reaction coordinate in a complex reaction is the projection of a multidimensional surface into a single coordinate**

A lot of important details about the reaction can be lost in the projection. It can be helpful to think of different steps in the reaction as happening in orthogonal directions - for example, if one occurs in the plane of the drawing, the next step might come out toward you, or away from you.