[Paper Review] An index-resolved fixed-point homotopy and potential energy landscapes

Tianran Chen, Dhagash Mehta|arXiv (Cornell University)|Apr 24, 2015

Molecular spectroscopy and chirality11 references3 citations

TL;DR

This paper introduces an index-resolved fixed-point homotopy method that directly targets stationary points (SPs) of a specific Morse index in potential energy landscapes, without requiring prior SPs as starting points. The method successfully locates SPs of index 0 and 1 in Lennard-Jones clusters, even when trajectories pass through degenerate SPs, demonstrating robustness and global sampling capability for chemical physics applications.

ABSTRACT

Stationary points (SPs) of the potential energy landscapes can be classified by their Morse index, i.e., the number of negative eigenvalues of the Hessian evaluated at the SPs. In understanding chemical clusters through their potential energy landscapes, only SPs of a particular Morse index are needed. We propose a modification of the "fixed-point homotopy" method which can be used to directly target stationary points of a specified Morse index. We demonstrate the effectiveness of our approach by applying it to the Lennard-Jones clusters.

Motivation & Objective

• To develop a homotopy continuation method that directly targets stationary points of a specified Morse index in potential energy landscapes.
• To overcome the limitation of existing homotopy methods that do not distinguish between SPs of different indices.
• To enable global sampling of index 0 (minima) and index 1 (transition states) SPs without requiring known starting SPs.
• To handle degenerate SPs effectively, which are challenging for Newton-based methods.
• To provide a robust, globally convergent approach for finding SPs of interest in complex energy landscapes, particularly in chemical clusters.

Proposed method

• The method modifies the fixed-point homotopy by introducing a target index $ m $, ensuring that the homotopy path preserves the Morse index along trajectories.
• It uses a homotopy system $ H^{(m)}({\mathbf{x}}, t) = \mathcal{H}(\hat{V}_t^{(m)}) \cdot \mathbf{F}^{(m)}({\mathbf{x}}, t) $, where $ \mathbf{F}^{(m)} $ is a fixed-point formulation tailored to index $ m $.
• The homotopy is defined such that $ \hat{V}_t^{(m)} $ interpolates between a simple potential and the target potential $ V_N $, with the Hessian structure preserved to maintain index during continuation.
• The method starts from random initial points in $ \mathbb{R}^n $, avoiding reliance on precomputed SPs, and follows the homotopy path using numerical continuation.
• It leverages the continuity of eigenvalues of the Hessian to preserve index along non-degenerate paths, and shows resilience even when encountering degenerate SPs.
• The approach is validated on Lennard-Jones clusters for $ N = 7 $ to $ 15 $, with successful recovery of SPs of index 0 and 1.

Experimental results

Research questions

• RQ1 Can a homotopy continuation method be designed to directly target stationary points of a specific Morse index in potential energy landscapes?
• RQ2 Does the proposed method preserve the Morse index along homotopy paths even when passing through degenerate SPs?
• RQ3 Can the method achieve global sampling of index 0 and index 1 SPs without requiring known SPs as starting points?
• RQ4 How does the method perform on complex systems like Lennard-Jones clusters with respect to accuracy and robustness?
• RQ5 What is the role of degeneracy in index preservation during homotopy continuation, and can the method remain effective under relaxed conditions?

Key findings

• The index-resolved fixed-point homotopy successfully locates stationary points of index 0 and 1 in Lennard-Jones clusters for $ N = 7 $ to $ 15 $, with all recovered SPs matching the target indices.
• At least 98% of the SPs found for $ N = 10 $ to $ 15 $ passed through degenerate SPs during their homotopy paths, yet index preservation was maintained.
• The method achieved high accuracy, with eigenvalues of the Hessian at SPs showing numerical precision up to $ 10^{-15} $, indicating reliable convergence.
• Degenerate SPs in Lennard-Jones clusters were numerically observed, with up to 6 zero eigenvalues (excluding rotational/translational modes), confirming their presence and relevance.
• The method demonstrated robustness by preserving the target index even across turning points and degeneracies, suggesting that index preservation may hold under significantly relaxed conditions.
• The approach enables global sampling of index 0 and 1 SPs without requiring prior knowledge of any SPs, overcoming a key limitation of locally convergent methods.