# 3: Characteristics of Energy Surfaces

- Page ID
- 11553

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Learning Objectives

In this Chapter, you will learned about the following things:

- Characteristics of Born-Oppenheimer energy surfaces, and how to find local minima, transition states, intrinsic reaction paths, and intersection seams on them.
- The harmonic normal modes of vibration extracted from the mass weighted Hessian matrix, and how symmetry can be used to simplify the problem.

Born-Oppenheimer energy surfaces (or the empirical functions often used to represent them) possess important critical points that detail the properties of stable molecular structures, transition states, intersection seams, and reaction paths, all of which play central roles in the theoretical description of chemical reactions and molecular properties. In this Chapter, you will learn about these special points on the surfaces, how to find them, and what to do with them once you know them.

- 3.1: Strategies for Geometry Optimization and Finding Transition States
- The extension of the harmonic and Morse vibrational models to polyatomic molecules requires that the multidimensional energy surface be analyzed in a manner that allows one to approximate the molecule’s motions in terms of many nearly independent vibrations. In this Section, we will explore the tools that one uses to carry out such an analysis of the surface, but first it is important to describe how one locates the minimum-energy and transition-state geometries on such surfaces.

- 3.2: Normal Modes of Vibration
- Having seen how one can use information about the gradients and Hessians on a Born-Oppenheimer surface to locate geometries corresponding to stable species and transition states, let us now move on to see how this same data is used to treat vibrations on this surface.

- 3.3: Intrinsic Reaction Paths
- There is a special path connecting reactants, transition states, and products that is especially useful to characterize in terms of energy surface gradients and Hessians. This is the Intrinsic Reaction Path (IRP). The general procedure to construct an IRP is outlined in this module.

## Contributors and Attributions

Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry

Integrated by Tomoyuki Hayashi (UC Davis)