Calculating and Characterizing the Charge Distributions in Solids

Accurate estimation of the partial atomic charges on metal centers is useful for understanding electronic and catalytic properties of materials. However, different methods of calculating these charges may give quite different results; this issue has been more widely studied for molecules than for solids. Here we study the charges on the metal centers of a test set of 18 solids containing transition metals by using density functional theory with several density functionals (PBE, PBE+U, TPSS, revTPSS, HLE17, revM06-L, B3LYP, B3LYP*, and other exchange-modified B3LYP functionals) and four charge models (Bader, Hirshfeld, CM5, and DDEC6). The test set contains 12 systems with nonmagnetic metal centers (eight metal oxides (MO₂), two metal sulfides (MS₂), and two metal selenides (MSe₂)) and six ferromagnetic transition metal complexes. Our study shows that, among the four types of charges, Bader charges are the highest and Hirshfeld charges are the lowest for all the systems, regardless of the functional being used. The CM5 charges are bigger than DDEC6 charges for MX₂ with M = Ti or Mo and X = S or Se, but for the other 14 cases they are lower. We found that the most of the systems are sensitive to the Hubbard <i>U</i> parameters in PBE+U and to the percentage <i>X</i> of Hartree-Fock exchange in exchange-modified B3LYP; as we increase <i>U</i> or <i>X</i>, the charges on the metal atoms in MX₂ increase steadily. Testing different density functionals shows charges calculated with higher Hubbard <i>U</i> parameters in PBE+U are comparable to B3LYP (with 20% Hartree-Fock exchange). Among four meta-GGA functionals studied, the charges with HLE17 have the closest agreement with B3LYP. The variation of charges with choice of charge model is greater than the variation with choice of density functional.