Related Experiment Video
Updated: May 29, 2026

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
Published on: April 8, 2020
Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation
1Department of Physics, Syracuse University, Syracuse, New York 13244, USA. dbmehta@syr.edu
Abstract:
The stationary points (SPs) of a potential-energy landscape play a crucial role in understanding many of the physical or chemical properties of a given system. However, unless they are found analytically, no efficient method is available to obtain all the SPs of a given potential. We present a method, called the numerical polynomial-homotopy-continuation method, which numerically finds all the SPs, and is embarrassingly parallelizable. The method requires the nonlinearity of the potential to be polynomial-like, which is the case for almost all of the potentials arising in physical and chemical systems. We also certify the numerically obtained SPs so that they are independent of the numerical tolerance used during the computation. It is then straightforward to separate out the local and global minima. As a first application, we take the XY model with power-law interaction, which is shown to have a polynomial-like nonlinearity, and we apply the method.
Related Concept Videos
Stability of Equilibrium Configuration: Problem Solving
Problem-solving in the context of the stability of equilibrium configuration...
Potential-Energy Criterion for Equilibrium
Energy Diagrams - II
The point in the energy diagram at which the system’s potential energy is the lowest is known as the local minima. The system tends to stay in this position indefinitely unless acted upon by a net force. The slope of the potential energy diagram at the local minima is zero, indicating that zero net force is acting on the system. The slope...
One-Degree-of-Freedom System
A one-degree-of-freedom system is defined by an independent variable that determines its state and behavior. One example of a one-degree-of-freedom system is a simple harmonic oscillator, such as a...
Linearization and Approximation
Newton’s Method