Linked-Cluster Formulation of Electron–Hole Interaction Kernel in Real-Space Representation without Using Unoccupied States

Electron–hole or quasiparticle representation plays a central role in describing electronic excitations in many-electron systems. For charge-neutral excitation, the electron–hole interaction kernel is the quantity of interest for calculating important excitation properties such as optical gap, optical spectra, electron–hole recombination, and electron–hole binding energies. The electron–hole interaction kernel can be formally derived from the density–density correlation function using both Green’s function and time-dependent density functional theory (TDDFT) formalism. The accurate determination of the electron–hole interaction kernel remains a significant challenge for precise calculations of optical properties in the GW+BSE formalism. From the TDDFT perspective, the electron–hole interaction kernel has been viewed as a path to systematic development of frequency-dependent exchange–correlation functionals. Traditional approaches, such as many-body perturbation theory formalism, use unoccupied states (which are defined with respect to Fermi vacuum) to construct the electron–hole interaction kernel. However, the inclusion of unoccupied states has long been recognized as the leading computational bottleneck that limits the application of this approach for larger finite systems. In this work, an alternative derivation that avoids using unoccupied states to construct the electron–hole interaction kernel is presented. The central idea of this approach is to use explicitly correlated geminal functions for treating electron–electron correlation for both ground and excited state wave functions. Using this ansatz, it is derived using both diagrammatic and algebraic techniques that the electron–hole interaction kernel can be expressed only in terms of linked closed-loop diagrams. It is proved that the cancellation of unlinked diagrams is a consequence of linked-cluster theorem in real-space representation. The electron–hole interaction kernel derived in this work was used to calculate excitation energies in many-electron systems, and results were found to be in good agreement with the EOM-CCSD and GW+BSE methods. The numerical results highlight the effectiveness of the developed method for overcoming the computational barrier of accurately determining the electron–hole interaction kernel to applications of large finite systems such as quantum dots and nanorods