Density-functional theory (DFT) is an exact alternative formulation of quantum mechanics, in which it is possible to calculate the total energy, the spin, and the charge density of many-electron systems in the ground state. In practice, it is necessary to use uncontrolled approximations that can mainly be verified against experimental data. Atoms and ions are simple systems, where the approximations of DFT can be easily tested. We have calculated within DFT the total energies, spin, and higher ionization energies of all the ions of elements with 1⩽Z⩽29. We find the calculations in close agreement with experiment, with an error of typically less than ca. 1% for 1⩽Z⩽29. Surprisingly, the error depends on the electronic configuration of the ion in both local spin density approximation and Perdew-Burke-Ernzerhof general gradient approximation and independent of both self-interaction correction and relativistic corrections. Larger errors are found for systems in which the spin-spin correlation is significant, which indicates the possible benefit from an orbital-dependent formulation of the correlation energy functional.
In the exact Kohn-Sham density-functional theory, the total energy versus the number of electrons is a series of linear segments between integer points. However, commonly used approximate density functionals produce total energies that do not exhibit this piecewise-linear behavior. As a result, the ionization potential theorem, equating the highest occupied eigenvalue with the ionization potential, is grossly disobeyed. Here, we show that, contrary to conventional wisdom, most of the required piecewise linearity of an arbitrary approximate density functional can be restored by careful consideration of the ensemble generalization of density-functional theory. Furthermore, the resulting formulation introduces the desired derivative discontinuity to any approximate exchange-correlation functional, even one that is explicitly density dependent. This opens the door to calculations of the ionization potential and electron affinity, even without explicit electron removal or addition. All these advances are achieved while neither introducing empiricism nor changing the underlying functional form. The power of the approach is demonstrated on benchmark systems using the local density approximation as an illustrative example.
Transition voltage spectroscopy (TVS) has become an accepted quantification tool for molecular transport characteristics, due to its simplicity and reproducibility. Alternatively, the Taylor expansion view, TyEx, of transport by tunneling suggests that conductance–voltage curves have approximately a generic parabolic shape, regardless of whether the tunneling model is derived from an average medium view (e.g., WKB) or from a scattering view (e.g., Landauer). Comparing TVS and TyEx approaches reveals that TVS is closely related to a bias-scaling factor, V0, which is directly derived from the third coefficient of TyEx, namely, the second derivative of the conductance with respect to bias at 0 V. This interpretation of TVS leads to simple expressions that can be compared easily across primarily different tunneling models. Because the basic curve shape is mostly generic, the quality of model fitting is not informative on the actual tunneling model. However internal correlation between the conductance near 0 V and V0 (TVS) provides genuine indication on fundamental tunneling features. Furthermore, we show that the prevailing concept that V0 is proportional to the barrier height holds only in the case of resonant tunneling, while for off-resonant or deep tunneling, V0 is proportional to the ratio of barrier height to barrier width. Finally, considering TVS as a measure of conductance nonlinearity, rather than as an indicator for energy level spectroscopy, explains the very low TVS values observed with a semiconducting (instead of metal) electrode, where transport is highly nonlinear due to the relatively small, bias-dependent density of states of the semiconducting electrode.