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 exc − E 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 Σj ≠i 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.