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A systematic comparative study has been performed to better understandDFT$+U$ (density functional theory + $U$) method. We examine the effect ofchoosing different double counting and exchange-correlation functionals. Thecalculated energy distribution and the Hund-$J$ dependence of potential profilefor representative configurations clearly show the different behaviors of eachDFT$+U$ formalism. In particular, adopting spin-dependent exchange-correlationfunctionals likely leads to undesirable magnetic solution. Our analyses arefurther highlighted by real material examples ranging from insulating oxides(MnO and NiO) to metallic magnetic systems (SrRuO$_3$ and BaFe$_2$As$_2$). Thecurrent work sheds new light on understanding DFT$+U$ and provides a guidelineto use the related methods.

The effect of double counting, spin density, and Hund interaction in the different DFT+$U$ functionals