Accelerating Resonant Scattering Simulations with the Multi-Shifted Biconjugate Gradient Method

Resonant spectroscopies such as RIXS and resonant Raman scattering provide essential insight into collective excitations in quantum materials that are inaccessible to non-resonant probes like ARPES or neutron scattering. However, theoretical modeling of these resonant processes is often limited by the computational cost of evaluating Kramers–Heisenberg–type response functions, which require solving large linear systems for many incident photon energies and intermediate states with finite core-hole lifetimes. To overcome this challenge, we develop a multishifted biconjugate gradient (MSBiCG) algorithm that exploits the shared Krylov subspace structure across different energy shifts. This approach eliminates redundant matrix–vector multiplications by recycling residuals from a single seed system through a collinearity relation, reducing the computational complexity to that of linear spectroscopies. Mathematical analysis and numerical benchmarks demonstrate that the MSBiCG method achieves constant scaling with respect to the number of incident energies while maintaining high accuracy and stability. This advancement provides a scalable and efficient framework for simulating resonant spectroscopies in correlated quantum materials.