Binding energy calculation
At the DFT level of theory, the core-level electron BEs can be obtained by applying initial state or final state methods.19The final state method includes the core-hole in the electronic structure calculation to get the total energy of the corresponding excited state (E exc). A separate calculation gives the energy of the ground state (E gs). The BE is then obtained as the difference between these two quantities:
BE = E excE gs (1)
On the one hand, since only the energy differences are used, the final state method takes advantage of DFT’s high level of accuracy concerning energy differences and neglects inaccuracies arisen by choice of a basis set or a functional used. When applied to small molecules, this approach shows mean absolute errors in the order of 0.2–0.3 eV with respect to experiments.20,21 On the other hand, many calculations must be run to obtain BE values for every excited state. Still, the final state method is computationally less expensive than alternatives like the third-order algebraic diagrammatic construction (ADC(3)) method and GW approximation.18,22,23 For comparison, the GW method, applied to small molecules, gives mean absolute errors below 0.1 eV.24
The initial state method is computationally even less expensive, as it accounts only for the energy level of the core electron in the ground state. For example, according to Janak’s theorem,25the 1s electron BE can be approximated by a negative Kohn–Sham orbital eigenvalue:
BE ≈ −ε(1s) (2)
Although appealing for its simplicity, the initial state method is sensitive to the choice of the DFT functional and considerably overestimates the BEs.21
Besides explicit DFT calculations, the BE can be estimated from a purely electrostatic model. Considering a core electron to be localised entirely at its mother-atom, one may assume that its orbital energy is determined by the electrostatic potential near the atomic nucleus. Consequently, the corresponding BE value (via Eq. 2) depends on the charge distribution, at first approximation, given by the local charges of the mother-atom and its neighbouring atoms:13
BE ≈ V (qi ,qj ) =kqi +l Σji Vj +m (3)
where the qi is the atomic charge on the giveni -th atom, k is proportionality constant, l = 14.4 eV·Å/e , the sum is an estimate for the electrostatic potential of the other atoms, where Vj =qj /Rij , and m is a constant determined by choice of the reference value. For this work, we obtained the k value of 13.45 eV/e by linear fitting the 1s Kohn–Sham eigenvalues of C+, C, and C vs their charge. The value for m was calculated for each ion pair so that the aliphatic C(1s) electron BE value equals to the reference value of 285.0 eV.
Below we refer to the above-described methods of BE calculation as ΔKS (Eq. 1), ε(1s) (Eq. 2), and V (q ) (Eq. 3) methods. In the literature, the delta Kohn–Sham method is also known as delta self-consistent field (ΔSCF);21 Janak’s theorem is commonly viewed as an analogue of Koopman’s theorem.26Only a few codes can run calculations with the core-hole required for the ΔKS method. The ε(1s) method is more accessible than the ΔKS method, yet the basis set used must describe the 1s orbital, which is somehow problematic for the plane wave basis sets. The V (q ) method is probably the most universal, yet it depends on the type of atomic charges used in Eq. 3.
The results of the described ΔKS, ε(1s), and V (q ) methods can be improved in several ways. In principle, they should converge upon increasing the model size. The smallest IL model is an ion; adding a counter-ion to it creates a solvate shell and introduces ion-ion interactions – from the weak dispersion and hydrogen-bonding to the much stronger ion-ion Coulomb interaction. Taking more than one ion into account affects the results but also increases the cost of the calculations. Similarly, at the computational cost, the absolute BE values can be more accurately calculated by applying an asymptotically correct exchange-correlation potential, for example, via hybrid functionals.27 Knowing all that, we made a pragmatic choice in favour of a simpler model and a common functional to save resources for calculating a more extensive set of ILs.