where cos 2 ϕ =1/3 for fully isotropic case. As a result, all of the approximate FGR rate expressions, including Eq. 43 and Eq. 44, will be modified by multiplying a factor of cos 2 ϕ =1/3. Note that this is a unique feature of the quantum FGR theory, in which the η 2 c feature is the key to surviving the isotropic averaging. Previous classical GH theory will not survive isotropic averaging and will give zero modification of the rate constant.