Exploring Model Energy and Geometry Surfaces Using Sum of Squares Decompositions

Area of Science:

• Computational Chemistry
• Optimization Theory
• Materials Science

Background:

• Exploring nonconvex potential energy surfaces is challenging due to numerous local minima and high energy barriers.
• Existing methods often struggle with disconnected configurational spaces and a priori sampling of minima.

Purpose of the Study:

• To develop a method for obtaining the global minimum on model potential energy surfaces without prior sampling.
• To transform nonconvex problems into convex ones for efficient and guaranteed solutions.
• To provide physical and geometric insights into potential energy surfaces.

Main Methods:

• Deriving a convex problem guaranteed to yield the same solution or a lower bound to the true global minimum.
• Utilizing sum of squares decomposition and semidefinite programming (SDP) for efficient problem solving.
• Applying semidefinite duality for verification and geometric interpretation.

Main Results:

• Successfully obtained the global minimum or a lower bound for model potential energy surfaces.
• Demonstrated efficient solution of a large class of nonconvex problems via SDP.
• Provided a systematic approach for improving solutions and gaining physical insights.

Conclusions:

• The developed method efficiently solves nonconvex optimization problems, including finding global minima of potential energy surfaces.
• Semidefinite programming and duality offer a powerful framework with theoretical guarantees and physical interpretability.
• The approach is applicable to various problems, including geometric calculations and molecular dynamics simulations.