• Ṙ(t) measures the reactive flux across the dividing surface (with δ(R) as the Dirac delta function), and ⟨...⟩ represents the canonical ensemble average (subject to a constraint on the dividing surface which is enforced by δ[R(t) -R ‡ ] inside F(t)). Further, Ṙ ‡ (0) represents the initial velocity of the nuclei on the dividing surface. The above flux-side formalism of the reaction rate can be derived from Onsager's regression hypothesis, with derivations presented in standard textbooks (e.g., Chapter 8.3 in Ref. 386). Numerical examples of κ can be found in Ref. 69 and Ref. 375 for the VSC problems. Alternatively, κ can also be computed using the Grote-Hynes (GH) rate theory, or equivalently the multi-dimensional transition state theory (MTST), 388 that treats all degrees of freedoms classically. Within GH theory, the transmission coefficient is given by