Calculations of Hubbard<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>U</mml:mi></mml:math>from first-principles

The Hubbard $U$ of the $3d$ transition metal series as well as $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_{3}$, $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_{3}$, Ce, and Gd has been estimated using a recently proposed scheme based on the random-phase approximation. The values obtained are generally in good accord with the values often used in model calculations but for some cases the estimated values are somewhat smaller than those used in the literature. We have also calculated the frequency-dependent $U$ for some of the materials. The strong frequency dependence of $U$ in some of the cases considered in this paper suggests that the static value of $U$ may not be the most appropriate one to use in model calculations. We have also made comparison with the constrained local density approximation (LDA) method and found some discrepancies in a number of cases. We emphasize that our scheme and the constrained local density approximation LDA method theoretically ought to give similar results and the discrepancies may be attributed to technical difficulties in performing calculations based on currently implemented constrained LDA schemes.