Relativistic hyperpolarizabilities for atomic H, Li, and Be$^+$ systems

For atoms in external electric fields, the hyperpolarizabilities are the coefficients describing the nonlinear interactions contributing to the induced energies at the fourth power of the applied electric fields. Accurate evaluations of these coefficients for various systems are crucial for improving precision in advanced atom-based optical lattice clocks and for estimating field-induced effects in atoms for quantum information applications. However, there is a notable scarcity of research on atomic hyperpolarizabilities, especially for the relativistic realm. Our work addresses this gap by establishing a novel set of alternative formulas for the hyperpolarizability based on fourth-order perturbation theory. These formulas offer a more reasonable regrouping of scalar and tensor components compared to previous formulas, thereby enhancing their correctness and applicability. To validate our formulas, we perform the calculations for the ground and low-lying excited pure states of few-electron atoms H, Li, and Be$^+$. The highly accurate results obtained for the H atom could serve as benchmarks for further development of other theoretical methods.