1e:T44e,The formal relationship between ultrasoft (US) Vanderbilt-type pseudopotentials and Bl\"ochl's projector augmented wave (PAW) method is derived. It is shown that the total energy functional for US pseudopotentials can be obtained by linearization of two terms in a slightly modified PAW total energy functional. The Hamilton operator, the forces, and the stress tensor are derived for this modified PAW functional. A simple way to implement the PAW method in existing plane-wave codes supporting US pseudopotentials is pointed out. In addition, critical tests are presented to compare the accuracy and efficiency of the PAW and the US pseudopotential method with relaxed core all electron methods. These tests include small molecules $({\mathrm{H}}_{2}{,\mathrm{}\mathrm{H}}_{2}{\mathrm{O},\mathrm{}\mathrm{Li}}_{2}{,\mathrm{}\mathrm{N}}_{2}{,\mathrm{}\mathrm{F}}_{2}{,\mathrm{}\mathrm{BF}}_{3}{,\mathrm{}\mathrm{SiF}}_{4})$ and several bulk systems (diamond, Si, V, Li, Ca, ${\mathrm{CaF}}_{2},$ Fe, Co, Ni). Particular attention is paid to the bulk properties and magnetic energies of Fe, Co, and Ni.1f:T4ee,In this study, a ratiometric electrochemical sensor based on metal-organic frameworks (MOFs) was developed for sensing of multiplex metal ions. The bifunctional MOFs were prepared in a way to integrate two signal tags and a detection probe. In the presence of target metal ions, the target metal ions can replace the framework metal-ion center in the original MOFs through an ion-exchange reaction, leading to ratiometric electrochemical signals under different applied potentials. One consisted of the Cu²⁺ signal generated from electroactive MOFs selected as internal reference signals. The other consisted of the signal induced by other target metal ions. Using the I_{metal ions}/I_Cu²⁺ signal as the output, the prepared ratiometric probe was able to eliminate disturbance caused by the sensing environment. Moreover, the large surface area and abundant active sites in MOFs produced a multiplex ratiometric electrochemical sensor with improved characteristics in terms of reproducibility, stability, and sensitivity. The sensor was also simple without sophisticated instrumentation, amplification processes, or an acid dissolution/preconcentration procedure, hence promising for practical applications.

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