Potential Energy Surface (PES)

Points on the PES where the first derivative (gradient) of the energy with respect to all atomic coordinates is zero, meaning the net force on every atom vanishes.

• Global Minimum: The absolute lowest energy point on the entire surface, representing the most stable conformation of the molecule.
• Local Minimum: A basin or well on the surface corresponding to a metastable conformer or isomer. A molecule can be trapped here.
• Saddle Point: A point that is a minimum in all directions except one, where it is a maximum. The highest energy point along the minimum energy path connecting two minima.
• Transition State: A first-order saddle point that represents the critical bottleneck structure through which reactants must pass to become products.

A continuous, one-dimensional parameter that tracks the progress of a chemical reaction from reactants to products along the minimum energy path (MEP). It is not a simple bond length but a collective coordinate involving the concerted motion of many atoms.

• The energy profile plotted along this coordinate is the familiar reaction profile diagram.
• The maximum along this coordinate corresponds to the transition state.
• The difference in energy between the transition state and the reactants defines the classical barrier height.

The Hessian is the matrix of second derivatives of the energy with respect to atomic coordinates. Its eigenvalues and eigenvectors reveal the local curvature of the PES.

• At a Minimum: All eigenvalues of the Hessian are positive, indicating the energy increases in every direction.
• At a Transition State: Exactly one eigenvalue is negative, corresponding to the reaction coordinate. The associated eigenvector points toward reactants and products.
• Harmonic Vibrational Frequencies: Calculated from the mass-weighted Hessian eigenvalues. An imaginary frequency (reported as a negative number) is the hallmark of a transition state.

The steepest-descent path connecting a transition state to its two adjacent minima. It represents the kinetically most favorable route for a reaction if the system has negligible momentum.

• Often calculated using the Intrinsic Reaction Coordinate (IRC) method, which follows the gradient downhill in mass-weighted coordinates.
• The MEP confirms that a transition state correctly connects the intended reactants and products.
• The curvature of the MEP can reveal dynamical effects, where the reaction path tunnels through a barrier or bypasses a shallow intermediate.

A multidimensional seam where two electronic potential energy surfaces become exactly degenerate (equal in energy). These are not isolated points but funnels that allow extremely fast, non-radiative transitions between electronic states.

• Critical for understanding photochemistry, such as vision, photosynthesis, and DNA photoprotection.
• At a conical intersection, the Born-Oppenheimer approximation breaks down completely.
• Characterized by the branching space defined by two vectors: the gradient difference and the non-adiabatic coupling vector.

A region where two potential energy surfaces of the same symmetry approach each other very closely but do not touch, repelling each other due to the non-crossing rule for diatomic molecules or states of the same symmetry in polyatomics.

• The energy gap at the closest approach is a measure of the electronic coupling between the two states.
• The character of the wavefunction swaps rapidly across the avoided crossing, leading to abrupt changes in molecular properties like the dipole moment.
• A key signature in electron transfer and charge separation processes.