Here, from the perspective of electronic structure, only the electronically adiabatic state energies E α and dipole matrix elements µ αβ are required as input. Note that in the last line of Eq. 18, we have explicitly inserted the dependence on the nuclear positions R for clarity, but in general we choose to neglect writing such dependence for the sake of clarity. The photonic basis is in principle infinite, since the harmonic oscillator has infinite eigenstates, so, similar to the electronic subspace which is truncated at N el (Eq. 7), we also introduce a truncation of the photonic Hilbert space including only the lowest N F Fock states. The polaritonic Hamiltonian can be easily constructed via Kronecker products (e.g., μ ⊗ â) of the sub-space operators; however, it is worth noting the extreme sparsity afforded by the Fock basis. To make this clear from a visual perspective, the block-like nature of the matrix in this basis can be written as,