Cone-Shaped Constrained Quasi-Newton Method: Efficient and Robust Single-Ended Transition State Optimization Algorithm

Efficient transition state location is a central challenge in heterogeneous catalysis. While single-ended methods are more efficient than double-ended methods, their convergence is often highly sensitive to the quality of the initial guess. Here, we propose a Cone-shaped Constrained Quasi-Newton (CCQN) method, which introduces a cone-shaped constraint to restrict the search direction, thereby effectively guiding the system from potential well regions toward saddle regions. After crossing the inflection curve, the optimization switches to the partitioned rational function optimization algorithm for further refinement. This curvature partitioned optimization strategy reduces the sensitivity to the quality of the initial guess while maintaining the efficiency of single-ended methods. Across 150 transition state optimization tasks with varying initial guess qualities, CCQN achieves an overall success rate of 93.3%, requiring only approximately 50 energy-force evaluations on average. The method exhibits strong robustness and convergence efficiency, offering a new tool for high-throughput transition state searches and mechanistic studies of complex catalytic reactions.