Chapter 6.17. The OCEAN suite: core excitations

The atomic electronic structure program performs all-electron and norm-conserving pseudopotential calculations. Other options include relativistic versus nonrelativistic calculations and Hartree–Fock versus LDA calculations. In pseudopotential calculations, electron partial waves with low angular momentum l, up to l = 2 or l = 3, experience a different potential in the core region and are solutions of the radial Schrödinger equation with accurate scattering properties in chemically relevant energy ranges. Because OCEAN works within a pseudopotential context, users must provide pseudopotentials in the form V_l(r) to the atomic program, which can calculate corresponding pseudized and all-electron partial waves at any energy.

All-electron (ae) wavefunctions are reconstructed from pseudized (ps) wavefunctions using `optimized projector functions' (OPFs), which are described elsewhere (Shirley, 2004) and inspired by the projector-augmented-wave method (Blöchl, 1994). The atomic program finds a few (about four) normalized OPFs that best span partial waves within a user-specified energy range and cutoff radius R_c that should enclose pseudized regions and pertinent core orbitals. Spherical harmonics times ae and ps projector functions, {f_ae(ν, l, r)} and {f_ps(ν, l, r)}, form a basis for band states inside r < R_c. For a site at the origin, ae and ps band states and coefficients for basis functions are related byReconstruction is performed when calculating transition-matrix elements, the electron–core-hole interaction and (as of version 2.0) calculating core-hole screening, the latter typically increasing screening near the nucleus, presumably because the reintroduction of radial nodes also reintroduces radial antinodes.

OCEAN obtains ground-state electronic structure and related quantities by using a plane-wave/pseudopotential program [for example ABINIT (Gonze et al., 2009) or Quantum ESPRESSO (Giannozzi et al., 2009)]. Bloch states are calculated on a regular grid throughout the full Brillouin zone. The grid can be displaced to preclude degeneracies among k points and accelerate convergence versus grid density.

The largest part of the electron–core-hole interaction is the spherically symmetrical central potential. It is important to screen the central potential accurately. This differs from the valence-hole case, where a model dielectric function (Levine & Louie, 1982) with local field effects (Hybertsen & Louie, 1988) suffices. Obviously, the atomic core environment is very different from a nearly free-electron gas. OCEAN includes screening by the excited site's core electrons in the atomic program and by valence electrons in situ. If the Green's function iswhere E_nk is a band energy and ±iη reflects state occupancy, then the bare and full ALDA polarization functions areHere x and x′ are a real-space coordinate, K_xc is a local vertex correction and E_F is the Fermi energy. The RPA omits K_xc, retaining only the Coulomb interaction v. The function χ screens the core hole (already screened by core electrons) plus a sphere of charge −e with radius ∼0.2 nm. A model screens a compensating sphere of charge +e. Quadrature facilitates radial and angular integrations (Shirley, 2006).

Multipole interactions lead to multiplet effects and a repulsive exchange part. Slater F_k and G_k integrals are calculated in the atomic program for all-electron OPFs. A scaling-factor parameter can be affixed, for example 0.83 improves the results in first-row TM systems. It may be possible to compute solid-state (although not atomic) contributions to this factor using screening calculations within OCEAN.

DFT band gaps and widths differ from those revealed using photoemission and inverse photoemission, which is a quite systematic effect (Shirley, 1998a). OCEAN allows one to amend the band structure suitably, typically by relating quasiparticle (qp) and DFT electron-level energies according to the formfor occupied (v) and unoccupied (c) states, respectively, allowing rigid shifts Δ_v, Δ_c and stretches s_v, s_c of bands. The actual band shifts and stretches can be measured or calculated (Hedin, 1999). State-dependent and energy-dependent self-energy corrections can also be incorporated into an OCEAN calculation (Vinson & Rehr, 2012).

OCEAN considers electron–hole pair states of the formThe operators act on the ground state to create the electron–hole pair. N, L, m_h and σ_h are standard quantum numbers for an atomic core level, while n, k and σ_e denote electron band index, crystal momentum and spin, respectively. By including core orbital angular momentum and all spin momentum degrees of freedom, multiplet and spin–orbit effects are included naturally. Thus, multiplet spectra can be computed for nominally d⁰ systems such as Ti⁴⁺ in rutile, because there is no other electron in the 3d subshell. Also, DFT includes the ligand or crystal field automatically, unlike model multiplet calculations.

A spectrum can be computed using Fermi's golden rule. A site contributes to the imaginary part of the dielectric function at X-ray wavevector q and energy ω according towhere Ω is the unit-cell volume. In XAS the operator that acts on a core electron isThe electron coordinate r is relative to the nucleus, while e and q are the X-ray electric field and momentum. For NRIXS and EELS, one has O ∝ exp(iQ · r) for momentum transfer Q, and the spectrum is related to the dynamic structure factor. Spectra can be evaluated efficiently within the Haydock recursion method (Vinson et al., 2011). Treating RIXS requires evaluatingsuch as by using the generalized minimal residual (GMRES) approach (Saad & Schultz, 1986). The subsequent electron–valence-hole pair can also be treated with OCEAN (Shirley et al., 2001).

The effective Hamiltonian is H_BSE = H_e + H_h + H_s.o. + H_C + H_M. H_e includes the electron-band energy, H_h the average core-level binding energy, H_s.o. core-level spin–orbit coupling, H_C the central potential of the core hole and H_M multipolar terms. The action of multipolar terms and spatially varying parts of H_C inside the OPF cutoff radius are performed within the OPF basis. The remainder of H_C is smooth, and its action is evaluated on a real-space grid using fast Fourier transform techniques. All of the above allow fast evaluation of H_BSE acting on an arbitrary state.

OCEAN employs periodic boundary conditions. Systems lacking translational invariance (for example molecules, liquids and surfaces) require the use of a large periodic box, possibly with multiple atomic configurations (Vinson et al., 2012; Niskanen et al., 2017; Petitgirard et al., 2019; Spieker­mann et al., 2019). Additionally, nominally periodic systems lose periodicity owing to thermal or zero-point motion (Vinson et al., 2014). However, judicious sampling of atomic displacements can render an accurate averaging of spectra with only a few calculations.

Diamond is a simple system offering high-quality spectra. It is easy to calculate spectra up to about 100 eV above the C 1s edge including only 60 conduction bands with good Brillouin-zone sampling. The most reliable spectroscopies at the C 1s near edge are NRIXS and EELS, followed by XAS, which might rely on electron-yield or fluorescence-yield signatures of absorption and suffer from instrument carbon build-up. Fig. 1 shows near-edge spectra obtained using OCEAN, NRIXS (Galambosi et al., 2007) and, above 315 eV, XAS (B. N. Ravel, private communication). Despite the prediction of a low-lying s-symmetry core-hole exciton (Jackson & Pederson, 1991), EELS (Batson, 1993), NRIXS (Galambosi et al., 2007) and calculations using the predecessor of OCEAN (Shirley, 1998b) suggested otherwise.

Figure 1 C 1s near-edge spectrum of diamond as calculated by OCEAN (top curve, red) and as measured using NRIXS (bottom, blue points; Galambosi et al., 2007) and XAS (bottom, solid green line; B. N. Ravel, private communication). As discussed in the text, NRIXS should have fewer artefact-related effects than XAS, especially in the near-edge region.

Vinson et al. (2011) and Vinson & Rehr (2012) consider a wide range of 3d TM systems at the 2p edge of the TM species. SrTiO₃ and CaF₂ serve as standard systems and allow comparison to all-electron core BSE results (Laskowski & Blaha, 2010; Gulans et al., 2014). Vinson and Rehr also consider other systems, including metallic calcium. Results are shown in Fig. 2.

Figure 2 The near-edge structure at the Ca 2p edge in metallic calcium and calcium fluoride. Calculations are shown by the red solid curve, and measurements are shown by blue dashed curves for calcium fluoride (de Groot et al., 1990) and metallic calcium (Fink et al., 1985).

Spectra of TM compounds can be of interest at a TM 1s pre-edge. OCEAN can treat such systems, which can feature dipole-allowed 1s→np and quadrupole-allowed 1s→md transitions, such as in rutile (Shirley, 2004). Perovskites such as SrTiO₃ and PbTiO₃ are of interest (Woicik et al., 2007) because 3d–4p mixing causes E_g-symmetry Ti 3d states to acquire partial 4p character, giving rise to strong absorption cross sections that help to reveal local atomic geometries.

Vinson et al. (2012) modelled the O 1s spectra of liquid water and two ice phases as presented in Fig. 3. Calculations for the liquid sample multiple molecular configurations. Care is required to estimate core-level shifts for inequivalent sites, such as oxygen sites in water, as discussed elsewhere (Pasquarello et al., 1996).

Figure 3 O 1s near-edge spectra in the Ih phase of ice and liquid water. Calculated results are shown by red solid lines. Measured results (blue dashed lines) were obtained using NRIXS (Pylkkänen et al., 2010) and XAS (Tse et al., 2008) in ice and scanning transmission X-ray microscopy in liquid water (Nilsson et al., 2010).

Direct RIXS calculations are more involved because they entail core-excited and valence-excited state calculations. Diamond was an early subject of RIXS measurements (Ma et al., 1992; Carlisle et al., 1999) and calculations (Shirley, 2000). OCEAN is able to capture several of the changes in X-ray emission for incident energy from 5 to 25 eV above the C 1s edge.

LiF demonstrates the ability of NRIXS to probe excitations of different symmetries. A small peak below the dipole-allowed F 1s edge was attributed to a vibrationally allowed, s-symmetry core-hole exciton. OCEAN and NRIXS (Hämäläinen et al., 2002; Vinson et al., 2011) at various momentum transfers confirmed this, as shown in Fig. 4. Finite-temperature molecular-dynamics simulations provide snapshots of atomic positions that make the pre-edge feature optically allowed (Pascal et al., 2014). Others (Tse et al., 2014) have also compared measured and calculated NRIXS of pressurized silicon to study the pressure-induced insulator-to-metal transition in this material.

Figure 4 F 1s near-edge spectra obtained by NRIXS for several momentum transfers in LiF, as calculated (top) and measured (bottom; Hämäläinen et al., 2002).

Debye–Waller (DW) effects are ubiquitous in core spectra, affecting electron-scattering processes because of displacements of atoms from equilibrium positions, as has been reviewed elsewhere (Rehr & Albers, 2000). Others (Story et al., 2014) show how spectra can include some vibrational effects to all orders using a cumulant approach and system-specific knowledge of vibrational properties that is obtained elsewhere. Near-edge features are also affected for reasons outside the above DW treatment. One can also vary atomic coordinates by sampling phonon modes or using molecular-dynamics amenable to disordered systems (Vinson et al., 2014; Brouder et al., 2010; Nemausat et al., 2015; Pascal et al., 2014; Prendergast & Galli, 2006; Niskanen et al., 2017). Others (de Groot et al., 1990) cite vibrational effects in d⁰ transition-metal compounds for broadening in cases of strong ligand–TM hybridization, for example E–e Jahn–Teller effects couple E_g electron states and e_g modes. OCEAN can help to determine the coupling strength to use in effective Jahn–Teller Hamiltonians (Tinte & Shirley, 2008; Gilmore & Shirley, 2010) to amend computed spectra. Still others (Zacharias & Giustino, 2016) have analysed valence edges, and core edges should also be treatable.

Multi-electron excitations broaden spectral features whenever an electron and/or hole state is far from the Fermi level. The `electron–hole continuum' part of the loss function smoothens the onset of broadening (Soininen et al., 2003; Kas et al., 2007; Fister et al., 2011). Multi-electron excitation also transfers spectral weight to satellites, as largely captured by a cumulant approach related to Hedin's GW self-energy (Hedin, 1999). This improves calculated photoemission (Guzzo et al., 2011; Gumhalter et al., 2016; Lischner et al., 2015) and near-edge (Kas et al., 2015) spectra. Others (Kas et al., 2016) have presented a method that allows the inclusion of all losses, including interference between intrinsic and extrinsic losses because of the coupling of all particles to the valence electron density. Including such effects should become a standard aspect of calculations performed using tools such as OCEAN.

The systems that OCEAN can treat are limited by our use of the BSE. Many-electron effects are included only with a more complete description of departures from the ground-state wavefunction. However, even Coster–Kronig decay (Coster & Kronig, 1935) and charge-transfer effects can be studied in limited cases if response-theory analysis of the environment facilitates an enhanced description of on-site excitations, with adequate separability of excitations near a site versus at longer range.