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We propose a method for calculating the hyperfine structure (hfs) of multielectron atoms based on a combination of configuration superposition and many-body perturbation theory. The latter is used to construct an effective Hamiltonian and an effective hfs operator in configurational space. The method can be applied in calculations of the matrix elements of any one-electron operators. By way of an example we calculate the magnetic hfs constant A for several lowest levels of neutral thallium. We show that the method achieves a calculation accuracy of about 1%, which earlier was possible only for atoms with a single valence electron.