, so 1/𝒵 cancels the 𝑒 -𝛽ℏ𝜔c factor that arises from the integral. The above theoretical results also justify the commonly used single mode approximation in investigating VSC reactivities because only the mode with frequency 𝜔 c survives. Using the expression of 𝒜(𝜔 0 ) (Eq. 23) in the rate constant expression of Eq. 19 and taking the limit of 𝑁 = 1, one obtains the previous re-sult of 𝑘 VSC (see Eq. 32) for a single molecule coupled to a single mode in Ref. .