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We analyze second-order $M1$-$M1$ and $M1$-$E2$ effects to the hyperfinestructure (HFS) of the lowest energy $P$ states of alkali-metal atoms arisingfrom the coupling of the two ($J=1/2,3/2$) fine-structure levels through thehyperfine interaction. We find these effects to be especially sizable in Li,leading to a $9\sigma$ correction to the most accurate reported experimentalvalue of the $A(P_{1/2})$ HFS constant of $^7$Li [D. Das and V. Natarajan, J.Phys. B \textbf{41}, 035001 (2008)]. For the remaining alkali-metal systems,the results tabulated within may be referenced as higher precision is sought inexperimental determination of the HFS constants.

Second-order effects on the hyperfine structure of $P$ states of alkali-metal atoms