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We present a new second-order representation of the relativistic Hartree-Fock equation, which can be solved by the standard Hartree-Fock technique. An alternative reduction for the magnetic part of the Breit interaction is presented in an explicit expression. A corresponding program has been developed, which improves significantly the scaled linear mesh introduced by Herman and Skillman. The structures for a number of atoms and ions are calculated and the agreement of our results with those published is excellent. We evaluate the fine-structure intervals of nd(n = 3-40) Rydberg series for sodium. The inverted fine-structure splitting values are obtained directly as the differences of eigenvalues obtained from a self-consistent field procedure. Taking into account the Gaunt effect enables the accuracy of the calculation to be substantially improved. The complete treatments reproduce very well the inverted fine structures along the Rydberg series and the relative difference between the present results and the experiments does not exceed 4.4%.

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In this paper, we consider the two-dimensional compressible magnetohydrodynamic system with Coulomb force. We apply the method of relative entropy to establish the weak-strong uniqueness property of this system.