where J(ω) is the spectral density defined in Eq. 3. In the above expression (Eq. 21), we denote all of the C j→k (t, s) as C D→D (s) because they are all identical. We further define the single-site friction kernel 74 γ(s), γ(s) as well as the finite-temperature friction kernel γ β (s) are expressed as