AbstractThe chain‐of‐states (CoS) constant advance replicas (CAR) method and its climbing image variant (CI‐CAR) for locating minimum energy paths (MEPs) and transition states are reported. The CAR algorithm applies the Lagrange multiplier method for imposing holonomic constraints on a chain‐of‐replicas, aiming to maintain equal mass‐weighted/scaled root‐mean‐square (RMS) distances between the adjacent replicas by removing the sliding‐down displacements contributed by the potential gradients during path optimization. Two contextual regularization schemes with clear geometrical interpretations are implemented to jointly promote high convergence and numerical robustness of the CAR algorithm. We show that the constrained reaction path can be solved normally within 5 steps of Lagrange multiplier updates with remarkably high numerical precision via the CAR approach. The efficacy of the CAR methods is demonstrated by testing on multiple analytical, classical, and quantum mechanical transition paths: the Müller potential, the alanine dipeptide isomerization, the helix unwinding of the VIVITLVMLKKK 12‐mer peptide, and the Baker set of reactions. We also explore the potential of applying adaptive momentum (AdaM) optimizers for locating optimal transition paths under complex conformational changes. Most importantly, we discuss extensively the differences and connections between our newly proposed CAR methods and several related methods, with focuses on the reaction path with holonomic constraints (RPCons) approach of Brokaw et al. [J. Chem. Theory Comput. 2009, 5 (8), 2050–2061] and the state‐of‐the‐art string method (SM) of E et al. [J. Chem. Phys. 2007, 126 (16), 164103]. The CAR approach represents a latest update to the general theoretical framework of reaction path finding algorithms in the two‐ended CoS regime.