where R(ω) and R(-ω) are the excitation and de-excitation transition rates, β = 1/k B T , with k B the Boltzmann constant and T the temperature of the system, and ⟨ μ(0) μ(t)⟩ qm is the quantum dipole autocorrelation function. The energy flux can be calculated by classical electromagnetic theory, and in the end, the IR spectra can be obtained with the Fourier transform of the quantum dipole autocorrelation function