Explicit-Core ΔSCF Scheme

• Explicit-Core ΔSCF Scheme is a rigorous DFT-based method that computes core-electron binding and transition energies by explicitly simulating a core-hole state.
• It employs sequential self-consistent field calculations with constrained orbital occupations to enable full electron relaxation and incorporate spin-orbit and multiplet effects.
• The method achieves quantitative agreement with experimental XPS/XAS spectra in both molecules and solids while effectively mitigating finite-size and charge compensation issues.

The explicit-core Delta Self-Consistent Field ($\Delta$SCF) scheme is a rigorous total-energy approach for predicting absolute and relative core-electron binding energies and transition energies in molecules and solids. In this method, core-level spectroscopy signatures such as X-ray photoelectron spectra (XPS) and X-ray absorption spectra (XAS) are simulated by performing constrained electronic structure calculations with an explicit core-hole occupation constraint. The method is based on density functional theory (DFT) and encompasses a sequence of self-consistent-field (SCF) calculations in which the occupation of a specific core orbital is manipulated, allowing full relaxation of all other electrons. The explicit-core $\Delta$SCF approach incorporates final-state relaxation, multiplet effects (to the extent of single-determinant DFT), and spin-orbit splitting, enabling highly quantitative agreement with experiment for core-level energetics and lineshapes in both molecules and condensed phases.

1. Fundamental Principles and Formulation

The explicit-core $\Delta$SCF scheme defines the core-level binding energy ($E_B$) as the total-energy difference between a neutral ground state and a core-hole state. In the most general form,

$$E_{\mathrm{binding}} = E_{\mathrm{total}}[N-1;\ \mathrm{core\ hole}] - E_{\mathrm{total}}[N]$$

where $E_{\mathrm{total}}[N]$ is the total DFT energy of the neutral, closed-shell $N$-electron system and $E_{\mathrm{total}}[N-1;\ \mathrm{core\ hole}]$ is the total energy after removing one electron from a particular core orbital and allowing the system to fully relax subject to this occupation constraint (Kahk et al., 2021, Klein et al., 2020, Kahk et al., 2021, Lee et al., 2016, Susi et al., 2014, Kahk et al., 2018).

The method realizes the Gunnarsson–Lundqvist constrained search for core-excited states: the lowest core-excited eigenstate is found via minimization of the many-body Hamiltonian in the subspace where the core-occupation operator $\hat{n} = |\varphi_c\rangle \langle \varphi_c|$ is zero. This is cast as a constrained-search DFT minimization with a penalty functional or direct occupation constraint (An et al., 22 Nov 2025).

Corrections for spin-orbit coupling can be included either directly via fully relativistic pseudopotentials (yielding split $j$-manifolds in p- and d-levels) or post hoc by subtracting a portion of the experimental splitting (e.g., $\Delta_{SO}/3$ for $2p_{3/2}$ levels) (Kahk et al., 2021, Lee et al., 2016).

2. Algorithmic Steps, SCF Protocols, and Occupation Constraints

The explicit-core $\Delta$SCF protocol involves two primary SCF calculations:

• Ground-state calculation: Standard SCF optimization with all electrons occupying the lowest-energy orbitals as per Aufbau principle.
• Core-hole state calculation: SCF optimization in which a chosen core orbital (e.g., $1s$ or $2p$) is forcibly depopulated by imposing occupation $n=0$ (or $n=1$ for a single spin channel in open-shell cases) (Susi et al., 2014, Kahk et al., 2021).

Core-hole localization is enforced using projection operators in the occupation constraint, maximum-overlap methods (MOM) that track the hole throughout SCF cycles, and symmetry breaking where necessary, such as small distortions or temporary increase of nuclear charge (Kahk et al., 2021, Klein et al., 2020). In practice, the occupation constraint is implemented either via explicit projector operators (e.g., FHI-aims "force_occupation_projector") or via penalty functionals added to the Hamiltonian (An et al., 22 Nov 2025).

In periodic systems, a compensating uniform background is used to maintain charge neutrality in the presence of a core hole. The explicit-core approach is applicable in both finite clusters and periodic supercells, with supercell convergence and spurious charge interactions treated via Makov–Payne corrections or extrapolation schemes (Kahk et al., 2021).

3. Extensions: Spin-Orbit Splittings, Spectra, and Combinations with Other Methods

For L/M edges (e.g., $2p$, $3d$) and systems with significant spin-orbit coupling, the occupation projector must resolve the full set of spinor (Clebsch–Gordan) components to target specific $j$ manifolds. The penalty operator generalizes to act in the non-collinear space, allowing distinct treatment of $2p_{1/2}$ vs. $2p_{3/2}$ (or $3d_{3/2}$ vs. $3d_{5/2}$), yielding the correct experimental splitting (An et al., 22 Nov 2025, Lee et al., 2016, Kahk et al., 2021).

For core-level absorption (XAS, NEXAFS), the delta SCF scheme is frequently combined with linear-response time-dependent DFT (TDDFT): the lowest core-to-virtual transition energy is calculated by $\Delta$SCF and used to align the corresponding TDDFT spectrum by a rigid shift, ensuring absolute edge alignment and experimental comparability (Annegarn et al., 2022, Klein et al., 2020). Dipole matrix elements are reformulated into a single-determinant overlap using selection rules and Gram–Schmidt completion, reducing computational cost from $\mathcal{O}(N^4)$ to $\mathcal{O}(N^3)$ (An et al., 22 Nov 2025).

4. Basis Sets, Relativistic Treatment, and Numerical Settings

Accurate explicit-core $\Delta$SCF calculations require basis sets capable of describing localized, highly contracted core orbitals. All-electron codes (e.g., FHI-aims, GPAW, NWChem) employ tight atomic basis functions on the core-hole atom, often with augmented exponents (Kahk et al., 2021, Susi et al., 2014, Kahk et al., 2018, Klein et al., 2020). Plane-wave implementations use ultrasoft pseudopotentials with custom channels for core holes.

Relativistic effects are incorporated via scalar relativistic approximations (e.g., ZORA) for light elements and fully relativistic pseudopotentials for heavier elements and transition metals. Spin-orbit effects are either included in SCF (where the code permits) or adjusted post hoc based on experimental splitting (Kahk et al., 2021, Lee et al., 2016).

SCF convergence thresholds are typically $10^{-6}$ eV for energy, with additional occupational and density criteria. Geometries can be fixed or relaxed with hybrid BFGS routines, with forces converged below $5\times10^{-3}$ eV Å$^{-1}$ (Kahk et al., 2021, Susi et al., 2014).

5. Error Analysis, Corrections, and Comparisons to Experiment

The explicit-core scheme yields mean absolute errors (MAE) for core-level binding energies within $\sim$0.2–0.4 eV of experiment across a wide range of systems. For first-row transition metals, raw $\Delta$SCF energies exhibit element-dependent systematic shifts, which are corrected by empirical offsets ($a_Z$) for each metal, reducing MAE from $0.73$ eV to $0.20$ eV (Kahk et al., 2021).

For solids, supercell finite-size effects and spurious Coulomb interactions are mitigated via Makov–Payne corrections and $1/L$ extrapolation (Kahk et al., 2021). Cluster-based relative energy shifts to gas-phase references provide cancellation of functional bias (Kahk et al., 2018), yielding binding-energy shifts in agreement with experimental XPS.

In core-level spectra (XPS/XAS), explicit-core $\Delta$SCF approaches reproduce both energy positions and lineshapes, resolving discrepancies between DOS-based assignments and experiment by capturing final-state screening effects, multiplet splitting, and polarization anisotropies in the spectra (Lee et al., 2016, An et al., 22 Nov 2025, Klein et al., 2020, Annegarn et al., 2022).

6. Practical Implementation, Limitations, and Best Practices

Effective implementation necessitates large supercells for periodic systems or sufficiently large clusters in aperiodic cases to minimize spurious interactions. Core-hole localization must be carefully verified (via orbital visualization) and enforced, especially in symmetric molecules or delocalized bands. Maximum overlap methods (MOM) and occupation projection are essential to avoid variational collapse (Kahk et al., 2021, Klein et al., 2020).

The scheme is robust for both metals and insulators, but large-gap ionic systems may show inaccuracies due to functional limitations in valence band placement; hybrid or GW-based corrections are proposed for materials with strong ionic character (Kahk et al., 2021, Klein et al., 2020). Dipole selection rules and projector formalism enable scalable computation of intensity distributions for XAS (An et al., 22 Nov 2025).

Explicit-core $\Delta$SCF yields computational cost comparable to ground-state DFT, though full all-electron schemes can be 10–30× slower than efficient frozen-core alternatives (Susi et al., 2014). Plane-wave schemes require specialized core-hole pseudopotentials; the all-electron approach recovers relaxation missing in frozen-core variants.

7. Applications and Benchmark Results

Explicit-core $\Delta$SCF approaches have been systematically validated for transition-metal $2p$ edges (Kahk et al., 2021), first-row K-edge shifts in molecules and solids (Kahk et al., 2021, Susi et al., 2014, Kahk et al., 2018, Klein et al., 2020, An et al., 22 Nov 2025), and core spectra in complex systems including catalytically relevant adsorbates (Kahk et al., 2018). Typical MAEs relative to experiment are $0.2-0.4$ eV (absolute energies) and $0.1$ eV (relative shifts).

Combined TDDFT + ΔSCF simulations accurately align excitation onsets and reproduce polarization-dependent spectral features, with RMS deviations below $0.3$ eV across test sets (Annegarn et al., 2022, Klein et al., 2020, An et al., 22 Nov 2025).

The explicit-core Delta SCF methodology is now established as a reference approach for core-level binding energy prediction, absolute edge alignment in XAS, and quantitative spectral assignment in molecular and condensed-phase systems.