Chapter 2.13. Core-hole potentials and related effects

The full core-hole (FCH) and related excited-electron and core-hole (XCH) approaches (with the latter being discussed later) attempt to model core-excited final states by focusing on the resulting core-excited electron density. In an effective single-particle approximation, this includes the effects of a core-level vacancy. Ignoring the excited electron itself, the resulting core-hole density, computed self-consistently by the use of an orbital occupancy constraint, can be associated with an effective core-hole potential which can be considered to be screened by the surrounding valence electron density and impacts valence electron states, both occupied and unoccupied.

Some of the earliest core-hole calculations were used to model the energy differences between core-excited states of the same atom in different material or chemical contexts, thereby probing chemical shifts. Pehlke & Scheffler (1993) employed the FCH approach to study chemical shifts in silicon and germanium at surfaces and in the bulk. Making use of the plane-wave pseudopotential formalism to describe such extended systems, thereby without the explicit inclusion of core orbitals within the valence-only electron density, they employed core-excited pseudopotentials to define the atomic core contribution to the screened core-hole potential for the valence electrons. The extension and development of this pseudopotential formalism for X-ray absorption spectra was made by Mauri, Cabaret and coworkers (Taillefumier et al., 2002) and was applied to several insulators and semiconductors. These authors referred to this approach as including core-hole screening to all orders, in contrast to the linear-response approaches that already existed (and are discussed later in Section 2.2). Unlike linear-response methods, in a calculation with periodic boundary conditions constrained-occupancy approaches require supercells that are constructed from multiple primitive cells, yet contain a single core-hole `defect' in order to eliminate interaction between core holes. Hence, this is strongly analogous to the defect calculations prevalent in the study of semiconductors.

The FCH potential (the self-consistent field resulting from the core-hole orbital constraint) defines a full spectrum of effective single-particle orbitals, which can be computed explicitly up to high energies in order to determine the transition probabilities, or the associated core-excited spectral function (related to the imaginary part of the Green's function) can be used to determine the excitation spectral intensity more efficiently.

The FCH approach neglects the contribution of the excited electron within the core-excited electron density. In this way it treats all core-excited states equally, without emphasis on one final state over any other. However, it was noted that for systems with particularly localized orbitals, such as molecular solids or liquids, this had the effect of insufficiently screening the core hole, leading to overestimated absorption intensity because of the deeper attractive potential for the unoccupied orbitals. Explicit inclusion of the excited electron within the first available orbital was used to ameliorate this effect in water and ice (Prendergast & Galli, 2006). The effect of inclusion of the excited electron also leads to almost complete screening at relatively short distances in molecular condensed phases via Gauss's law.

The so-called excited-electron and core-hole (XCH) approximation had, as an obvious choice, been employed by multiple researchers before the FCH work just discussed. Mo & Ching (2000) used it to describe periodic solids within a localized atomic orbital framework both for X-ray absorption and inner-shell electron energy-loss spectra. Prior to this, it had been used in a study of the 1s core exciton in diamond by Jackson & Pederson (1991). In fact, this (only slightly) more expensive ΔSCF approach (which considers changes in the total energy of two SCF calculations with and without core excitation) is exactly what Slater was aiming to avoid through the transition-state approach.

The XCH approach, which makes use of a neutral excited-state approximation, facilitates easier comparison of the total energies determined from core-excited electron densities, particularly for cases within different periodic lattices or inhomogeneous systems, such as surfaces, defects etc. In contrast, the longer-ranged (partially screened) Coulomb interactions of the full core-hole approach make such comparisons difficult, particularly for systems with different periodicities (and associated Madelung constants). The example of water indicated above highlighted the need for accurate relative alignment of core-excited states for atoms of the same element at different sites or in different contexts. The large configurational space of hydrogen-bonding environments found in liquid water at room temperature, sampled using molecular dynamics within a finite supercell, presented significant variations in the local on-site potential for the excited O atoms and in the polarization response of the valence electron density of the surrounding water molecules, depending on whether they are donating, accepting or lacking hydrogen bonds.

In the Slater transition-state method (Slater & Johnson, 1972; Slater, 1972) the excitation energy is estimated as the orbital energy difference between the half-occupied initial and final levels of a variationally determined (transition) state; this corresponds to taking the excitation halfway and using this state as a reference. Because only orbitals i and f change occupation, we can write the total energy of the system as E(n_i, n_f), where the reference state is given by n_i = n_f = 1/2. Compared with this reference, the energy of the initial (ground) state can be written as E(n_i + 1/2, n_f − 1/2) and correspondingly for the excited state E(n_i − 1/2, n_f + 1/2). Expanding to second order in the occupation numbers, we use Janak's theorem (Janak, 1978) to replace the derivative of the Kohn–Sham energy with respect to the occupation number n_i by the orbital energy ɛ_i. We thus obtain for the ground state (with all derivatives taken at the variationally determined reference state):For the desired excited state we instead obtainFor the transition energy E_fi = E_f − E_i = E(n_i − 1/2, n_f + 1/2) − E(n_i + 1/2, n_f − 1/2) we then find (to second order) E_fi = ɛ_f(1/2, −1/2) − ɛ_i(1/2, −1/2) + , with the last term indicating corrections third order in orbital occupancy changes.

The derivation assumes that the second-order response of the orbitals to the change in the occupation can be neglected, i.e. the half-occupied orbitals represent both the initial and final state characters; the first-order contribution is eliminated through self-consistency and the second-order contribution through a strict cancellation under this assumption. The same orbitals can then be used to represent both initial and final states. This simplifies the evaluation of transition moments to single-electron transitions, because it avoids contributions from the other occupied orbitals.

Each transition in the transition-state approach requires a separate calculation. Even for an isolated molecule, this makes it impractical to generate a complete X-ray absorption spectrum including valence, Rydberg and continuum states, where hundreds of states can be required for a reasonable representation. The transition-potential (TP) approach of Triguero et al. (1998) was introduced as an approximation to the Slater transition state and eliminates the state-by-state calculations by simply neglecting the half-excited electron. The variationally relaxed density of the resulting molecular ion core with the half-occupied core hole provides a potential from which all excited states are obtained in one global diagonalization; this includes all interactions between the excited electron and the molecular ion, with the only approximation being that the electronic density of the molecular ion is kept frozen and is not allowed to relax when interacting with the excited electron.

The TP approximation to computing XAS spectra builds on the fact that the partially occupied core level is very isolated both in space and energy, and is furthermore common to all excited states; in cases where several atoms of the same element are present, the core level can still be made unique by replacing the other cores by effective core potentials or pseudopotentials or by localizing and freezing these orbitals so they cannot mix with the desired state (Leetmaa et al., 2010).

Removing an electron from the 1s level is the largest perturbation on the remaining electron density. The half-electron in the excited levels typically interacts only weakly with the remaining molecular ion core, so that neglecting it is normally a very good approximation. When the excitation is into unoccupied, localized valence states, however, the interaction cannot be neglected. The choice of 0.5 electrons as the fractional core-occupation number of the 1s level minimizes the error by approximating the excitation energy as an orbital-energy difference, but it is strictly valid only when the upper level is also half-occupied and the same orbitals can describe both the initial and the final state. The half-core-hole, TP approximation is therefore not rigid and other core occupations/core potentials may be considered (Cavalleri et al., 2005; Nyberg et al., 1999).

Suppose that one can regard the core-excited atom or molecule as an impurity in the band structure of the remaining system and describe the resulting spectra following the Newns–Anderson impurity model (Anderson, 1961; Newns, 1969) as illustrated in Fig. 1(a). The core-hole potential then pulls down the local atomic orbitals of the core-excited molecule, which changes the electronic interactions with the surrounding condensed phase. In this picture one can consider the core-excited atom as a local impurity with a different set of atomic orbitals interacting with the surrounding bands. If the interaction is with a broad featureless continuum of electronic states, this results in a resonance that simply corresponds to a broadening of the atomic level, with the width depending on the strength of the interaction (Fig. 1).

Figure 1 (a) Illustration of the Newns–Anderson impurity model applied to (a) a single atomic level interacting with a featureless continuum of electronic states and (b) the XAS of ice, where the gas-phase excited states interact strongly with the continuum DOS. Reprinted with permission from Nilsson & Pettersson (2011).

If there are more specific features in the density of states (DOS) the resulting resonances can be more complex, as illustrated in Fig. 1(b) for the specific case of a core-excited water molecule in hexagonal ice (Nilsson & Pettersson, 2011). The water molecular orbitals will be down-shifted because of the core-hole potential where one 1s electron has been removed. The unoccupied DOS of ice has a well defined strong resonance corresponding to the conduction band (Chen et al., 2010), but is otherwise rather featureless, decreasing in intensity towards lower energies. The strength of the core-hole potential determines the energy position of the antibonding 4a₁ and 2b₂ levels of the core-excited molecule and consequently how they will be broadened into resonances; the 2b₂ level will always interact more strongly, since there is a higher DOS at similar energies in comparison to the energy region close to the lower-lying 4a₁. The hybridization with the strong peak of the conduction band (post-edge in Fig. 1) leads to a new resonance that will mostly be of band-structure character involving the surrounding molecules. How the strength of the core-hole potential affects the hybridization is seen in Fig. 2, which shows computed XAS spectra using the transition-potential method (Triguero et al., 1998), but varying the core-hole potential from half core hole to full core hole in steps of 0.1 (Leetmaa et al., 2010). The spectra have been energy calibrated using the `ΔK-S' approach of Kolczewski et al. (2001) and therefore have exactly the same onset in terms of DOS, but not in terms of the intensity distribution (Leetmaa et al., 2010). With increasing core-hole potential both the pre-edge and main-edge intensities shift to lower energy, consistent with the simple picture of an increased energetic separation of the `impurity' states from those of the condensed phase in the Newns–Anderson model. Because of weaker hybridization with the conduction band, there will be less local 2p character and the intensity of the post-edge also diminishes. There is also a shift of the post-edge towards lower energy as the isolated core-excited orbitals are pulled down with increasing core-hole potential. Within the transition-potential approximation (Triguero et al., 1998), the best agreement with the experimental ice spectrum is with the half core-hole potential (Leetmaa et al., 2010; Iannuzzi, 2008), which also has been shown to work well for many other systems (Aziz, Freiwald et al., 2006; Cavalleri et al., 2002, 2006; Damian Risberg et al., 2007, 2009; Kolczewski & Hermann, 2003; Kolczewski et al., 2001; MacNaughton et al., 2006; Näslund et al., 2003; Nyberg et al., 2003; Ogasawara et al., 2002; Öström et al., 2004, 2006, 2007; Öström, Triguero, Nyberg et al., 2003; Öström, Triguero, Weiss et al., 2003; Pettersson et al., 1999; Schiros et al., 2006, 2007; Wilks et al., 2006; Aziz, Zimina et al., 2006; Mijovilovich et al., 2009).

Figure 2 Computed spectra of hexagonal ice using the transition-potential approach (Triguero et al., 1998) with different core-hole potentials from half core hole (0.5 CH) to full core hole (1.0 CH) in steps of 0.1; for details, see Leetmaa et al. (2010). The full black line gives the experimental spectrum from Nordlund et al. (2004) and the three main spectral regions, pre-edge, main-edge and post-edge, are indicated.

By the use of a ΔK-S approach, an absolute energy scale can be determined by variationally computing the difference in total energy between the ground state and the first fully relaxed core-excited state for each configuration or each non-equivalent center (Kolczewski et al., 2001). The lowest energy state in the TP-XAS spectrum is set equal to the ΔK-S energy, and all other states are shifted accordingly; this gives a quite reliable absolute energy scale for the onset of each spectrum (Kolczewski et al., 2001; Odelius et al., 2006). Considering relative energy differences within each spectrum, the accuracy relies on the fact that the relaxation effect of the state used to compute the shift (usually the lowest core-excited state) is similar in character for all shifted spectra. Because the complete spectrum contains excitations to valence, Rydberg and continuum states, there may be different effects depending on the character of the excited state. Additional core-excited states may thus need to be determined variationally by applying the ΔK-S shifting procedure explicitly to the required number of low-lying excited states and shifting the remaining states according to the shift of the highest of the ΔK-S corrected states (Kolczewski et al., 2001; Damian Risberg et al., 2007, 2009; Mijovilovich et al., 2009). In a Kohn–Sham framework, a sequence of orthogonal core-excited states can be determined variationally by requiring that the core level should be singly occupied and the system should be neutral, which defines the lowest unoccupied molecular orbital (LUMO). Removing this orbital from the basis set and repeating the calculation gives the next state and defines LUMO+1. This procedure is well defined for a few low-lying, well separated discrete transitions, but becomes impractical when the DOS becomes too high (Leetmaa et al., 2010).

The ΔK-S energy correction also corrects for the neglect of relaxation because of the interaction of the excited electron with the full core-hole state. However, the exchange-correlation functionals used in density-functional theory (DFT) are only approximate, and the absolute energy scale thus also depends on which functional is used (Takahashi & Pettersson, 2004). The main contribution to the functional dependency is because of the core level, which is where the electron density is the highest, while the relative energies of core-excited states are much less affected (Takahashi & Pettersson, 2004). This opens the possibility of calibrating the calculations further if a suitable reference can be found where the error in the functional can be evaluated against some other higher level calculation or experiment. In such cases the ΔK-S shifted spectrum may be corrected to within a few tenths of an electronvolt of experiment through a computationally derived empirical shift for functional-dependent and relativistic effects (Leetmaa et al., 2006, 2008, 2010).

This empirical correction depends on the chemical environment in a molecule, such that it is in general not possible to define a unique and generally applicable correction for each element and functional (Takahashi & Pettersson, 2004); a sufficiently similar system must be found for the calibration in each case. Still, even in the absence of a reliable empirical correction to the absolute energy scale, chemical shifts between, for example, different O atoms at an oxide surface (Kolczewski & Hermann, 2003, 2004; Kolczewski et al., 2007; Cavalleri et al., 2007, 2009), different waters or OH groups on a metal (Schiros et al., 2006, 2007) or inequivalent atoms in a molecule can be very reliably obtained through just the ΔK-S correction, because the functional dependency largely cancels out when taking differences (Takahashi & Pettersson, 2004).