# MBD expressions
I present three different yet equivalent expressions for the MBD energy (MBD1–MBD3 below). By MBD energy, I mean the interaction energy of a system of Drude oscillators under dipole approximation. (In our Chemical Reviews paper and my thesis, I started from the fluctuation–dissipation theorem for the electrons, but I start directly with oscillators here to simplify the discussion.)
First two general formulas that I'll use. For any Hamiltonian of the form $\hat H=\hat T+\hat U+\hat V$ (kinetic energy, one-particle potential, particle interaction), the interaction energy can be obtained via adiabatic connection by generalizing the Hamiltonian as $\hat H(\lambda)=\hat T+\hat U+\lambda\hat V$ and using the Hellmann–Feynman theorem,
$$
E_\text{int}=E(1)-E(0)=\int_0^1\mathrm d\lambda\frac{\partial E}{\partial\lambda}=\int_0^1\mathrm d\lambda\langle\Psi(\lambda)|\hat V|\Psi(\lambda)\rangle
\label{eq:adiabatic-connection}
$$
The zero-temperature fluctuation–dissipartion theorem for multiple quantities, $\hat A_i$, connects the correlations in their spontaneous fluctuations to the corresponding generalized susceptibilities, $\partial\langle\hat A_i\rangle/\partial f_j(u)$, which desribe response to the generalized external forces $f_j$ at freqeuncy $u$,
$$
\langle\Psi|(\hat A_i-\langle\hat A_i\rangle)(\hat A_j-\langle\hat A_j\rangle)|\Psi\rangle=-\frac1\pi\int_0^\infty\mathrm du\operatorname{Im}\frac{\partial\langle\hat A_i\rangle}{\partial f_j(u)}\equiv-\frac1\pi\int_0^\infty\mathrm du\frac{\partial\langle\hat A_i\rangle}{\partial f_j(\mathrm iu)}
\label{eq:fluctuation-dissipation}
$$
Now going to MBD, the Hamiltonian is
$$
\hat H_\text{MBD}=\hat T+\hat U+\hat V=\sum_i\frac{\mathbf{\hat p}_i^2}2+\sum_i\frac12m_i\omega_i^2|\mathbf{\hat r}_i-\mathbf R_i|^2+\sum_{i