A tale of two vectors: A Lanczos algorithm for calculating RPA mean excitation energies

The experimental and theoretical determination of the mean excitation energy, I(0), and the stopping power, S(v), of a material is of great interest in particle and material physics and radiation therapy. For calculations of I(0), the complete set of electronic transitions in a given basis set is required, effectively limiting such calculations to systems with a small number of electrons, even at the random-phase approximation (RPA)/time-dependent Hartree-Fock (TDHF) or time-dependent density-functional theory level. To overcome such limitations, we present here the implementation of a Lanczos algorithm adapted for the paired RPA/TDHF eigenvalue problem in the Dalton program and show that it provides good approximation of the entire RPA eigenspectra in a reduced space. We observe rapid convergence of I(0) with the number of Lanczos vectors as the algorithm favors the transitions with large contributions. In most cases, the algorithm recovers RPA I(0) values of up to 0.5% accuracy at less than a quarter of the full space size. The algorithm not only exploits the RPA paired structure to save computational resources but also preserves certain sum-over-states properties, as first demonstrated by Johnson et al. [Comput. Phys. Commun. 120, 155 (1999)]. The block Lanczos RPA solver, as presented here, thus shows promise for computing mean excitation energies for systems larger than what was computationally feasible before.