Six-Orbital f-d-p Model

• Six-orbital f-d-p model is a compact, orbital-resolved Hamiltonian that retains key f, d, and ligand p states to describe mixed localized–itinerant systems.
• It emphasizes optimized basis selection based on orbital resolution, local symmetry, and covalency, which directly affects occupancy and interaction strengths.
• The formulation guides practical applications in materials such as actinide oxides and pyrochlore iridates by balancing crystal-field splitting, Hund coupling, and hybridization effects.

A six-orbital $f$-$d$-$p$ model is most plausibly understood as a reduced multi-orbital Hamiltonian in which a small set of active $f$, $d$, and ligand $p$ states is retained explicitly while more extended, strongly bonding, or higher-energy combinations are downfolded. The literature does not define a single canonical six-orbital basis. Instead, it establishes a common principle: the physically relevant manifold must be selected by orbital resolution, local symmetry, and covalency, because basis choice changes correlated-shell occupancy, Hund competition, effective crystal fields, and Fermi-surface topology even when the noninteracting bands near the Fermi level are reproduced identically [(Parragh et al., 2013); (Warda et al., 22 Aug 2025); (2206.13030)].

1. Definition and range of meanings

The expression “six-orbital $f$-$d$-$p$ model” does not denote a unique standard Hamiltonian. In practice, it denotes a compact orbital-resolved description of mixed localized–itinerant systems containing correlated $f$ states, transition-metal $d$0 states, and explicit ligand $d$1 states. The compactness is crucial: the most relevant papers repeatedly warn that full atomic $d$2, $d$3, and $d$4 shells are often not the right low-energy manifold, because crystal-field splitting and covalent mixing separate localized submanifolds from strongly bonding ones (Warda et al., 22 Aug 2025).

This point is explicit in mixed actinide–transition-metal oxides. In ternary monouranates $d$5, the best-performing orbital-resolved setup retains only the $d$6-site $d$7 orbitals, the four least-occupied U-$d$8 occupation-matrix eigenstates $d$9–$p$0, and one O-$p$1 orbital per oxygen site, namely O1-$p$2 and O2-$p$3. That setup is not literally a six-orbital model per formula unit, but it is conceptually very close to a reduced $p$4-$p$5-$p$6 manifold built from selected localized $p$7, $p$8, and $p$9 channels rather than entire shells (Warda et al., 22 Aug 2025).

A related but more explicitly “six-orbital-style” inference appears for doped pyrochlore iridates. The experimental analysis directly motivates localized $f$0-$f$1, $f$2-$f$3 states dominated by the $f$4 manifold, and oxygen $f$5 ligand states, especially basal O-$f$6. The paper does not define a unique minimal basis, but it states that a plausible six-orbital interpretation would contain one effective $f$7-$f$8 degree of freedom, the three $f$9-$d$0 orbitals, and two symmetry-relevant oxygen ligand orbitals, with the basal $d$1-$d$2 sector more strongly justified than the apical one (Kumar et al., 2020).

For EuNi$d$3P$d$4, the low-energy spectrum is dominated by Ni $d$5 conduction bands and Eu $d$6 flat multiplet-derived features, while P $d$7 is chemically present but not directly resolved as a separate low-energy quasiparticle sector. A plausible six-orbital reduction was therefore described as two effective Eu $d$8 orbitals or doublets, two Ni $d$9-derived orbitals or bands, and two P $p$0-derived ligand orbitals. This is an inference rather than a spectroscopically fixed basis, but it reflects the observed hierarchy of localized $p$1, itinerant $p$2, and implicit ligand $p$3 sectors (2206.13030).

At the same time, multi-band $p$4-$p$5 work on vanadium perovskites shows why aggressive truncation can fail. There, a nominally $p$6 problem still requires all five V $p$7 orbitals plus explicit O $p$8, because the total $p$9 occupancy per V is close to $f$0 electrons in LaVO$f$1, and the nominally empty $f$2 sector can be more populated than the third $f$3 orbital. In that sense, six-orbital reduction is not a default simplification but a carefully justified projection (Rościszewski et al., 2018).

2. Orbital resolution, projectors, and basis selection

The central methodological issue is not merely which atoms are present, but which local orbitals remain sufficiently localized to serve as correlated degrees of freedom. In orbital-resolved DFT$f$4 for mixed $f$5-$f$6 compounds, the occupation matrix is diagonalized and the actual corrected orbitals are the occupation-matrix eigenstates

$f$7

not bare atomic harmonics. The paper emphasizes that fractional occupations may arise either from genuine self-interaction error or from projector mismatch. In the latter case, the correction punishes covalency rather than correlation, and the resulting Hubbard forces can drive the lattice toward better overlap with the chosen projector set rather than toward the physical minimum (Warda et al., 22 Aug 2025).

This is why orbital resolution matters. In $f$8, the $f$9-site $d$0-like orbitals are the strongly $d$1-$d$2 hybridized sector, whereas $d$3 is the more localized sector. The U-$d$4 shell is likewise nonuniform: the least-overlapping orbital is associated with $d$5, while orbitals with lobes aligned with bond axes are more strongly ligand mixed. Oxygen $d$6 is also split into strongly hybridized $d$7-like channels and a more weakly hybridized lone-pair-like channel. The result is that a viable $d$8-$d$9-$p$0 model is typically built from localized submanifolds rather than from whole shells (Warda et al., 22 Aug 2025).

A closely related lesson comes from the comparison of $p$1-only and $p$2 orbital models. There, a four-band Hamiltonian with two $p$3 orbitals and two ligand $p$4 orbitals is downfolded by Löwdin projection to an effective two-band $p$5-only model whose noninteracting Fermi-level bandstructure is identical by construction. Yet the orbital character is not identical: the $p$6 model retains explicit ligand weight, while the $p$7-only model compresses that physics into effective $p$8-like Wannier orbitals. The paper’s point is not that either basis is formally illegitimate, but that basis choice changes occupancy and therefore changes the many-body regime (Parragh et al., 2013).

The same argument underlies the preference for Wannier-like projectors or for restricted manifolds composed only of the most localized orbitals. A six-orbital $p$9-$f$0-$f$1 model built from projector functions whose spatial extent does not match the real coordination geometry is liable to misassign covalent states as localized Hubbard orbitals, thereby distorting both interaction strengths and structural response (Warda et al., 22 Aug 2025).

3. Hamiltonian structure

The reusable architectural core is a block-partitioned one-particle Hamiltonian plus local interactions on the chosen correlated blocks. In the explicit four-band $f$2 model, the local Hamiltonian after Brillouin-zone integration is written as

$f$3

which separates correlated local levels, ligand levels, and local hybridization. The same block logic is directly described as useful for any $f$4-$f$5-$f$6 extension (Parragh et al., 2013).

A natural extension of this architecture to an $f$7-$f$8-$f$9 problem was formulated as

$d$00

This form includes onsite $d$01, $d$02, and $d$03 levels; crystal-field terms in the $d$04 and $d$05 sectors; hybridizations among $d$06-$d$07, $d$08-$d$09, and $d$10-$d$11; local interactions on the correlated sectors; and a double-counting term when the model is derived from density-functional methods. The same discussion explicitly notes that a realistic $d$12-shell generally requires strong spin-orbit coupling and a full rotationally invariant Slater interaction rather than a simple two-orbital Kanamori form (Parragh et al., 2013).

For the $d$13 sector, the standard local interaction written in the comparison paper is rotationally invariant Kanamori, including intraorbital $d$14, interorbital $d$15, Hund exchange $d$16, spin-flip, and pair-hopping terms. The vanadate $d$17-$d$18 study likewise retains a full multi-orbital interaction structure and, importantly, includes on-site oxygen interactions $d$19 and $d$20. Its explicit warning is that neglecting $d$21 does not yield a neutral simplification; it yields an effective model with renormalized parameters, especially an artificial reduction of the metal $d$22 (Rościszewski et al., 2018).

In DMFT implementations, the lattice problem is solved on the full orbital basis and then projected to the correlated subspace. For the $d$23 model this is written as projection of the full local Green’s function to the $d$24 block,

$d$25

followed by construction of the Weiss field. The text states directly that this projection-plus-impurity procedure generalizes to $d$26-$d$27-$d$28 models by projecting onto the chosen correlated $d$29 and/or $d$30 block and solving the corresponding impurity problem (Parragh et al., 2013).

4. Filling, Hund competition, and renormalized level alignment

The most transferable lesson for six-orbital $d$31-$d$32-$d$33 modeling concerns occupancy. In the two-orbital $d$34-only model, the total $d$35 occupancy is fixed to $d$36, which is quarter filling. In the corresponding four-band $d$37 model, the total occupancy is fixed to $d$38, and $d$39-$d$40 hybridization increases the actual $d$41 occupancy to $d$42–$d$43. The enlarged basis therefore drives the correlated shell toward half filling, even though the noninteracting bands at the Fermi level are matched to the $d$44-only model (Parragh et al., 2013).

That filling shift qualitatively changes the balance between crystal-field physics and Hund’s-rule physics. Orbital polarization is defined as

$d$45

and the Fermi-surface-relevant effective splitting is

$d$46

At quarter filling, correlations enhance the initial crystal-field splitting. Near half filling, Hund exchange favors a larger local moment and more even orbital occupancy, so correlations reduce orbital polarization. The paper further states that the trends are more sensitive to $d$47 than to moderate changes in $d$48, and that in a $d$49 model double counting is not merely an irrelevant constant but renormalizes the $d$50-$d$51 level offset and hence the charge-transfer energy (Parragh et al., 2013).

The same logic is directly stated to be transferable to $d$52-electron systems. This suggests that in a six-orbital $d$53-$d$54-$d$55 model one must monitor $d$56, $d$57, and possibly total correlated occupancy $d$58, because hybridization can move the system between crystal-field-dominated, Hund-dominated, mixed-valent, Kondo-like, or Mott-like regimes. In $d$59-electron compounds the effect may be more dramatic, since valence fluctuations, charge-transfer energies, and multiplet structure are often highly sensitive to occupancy (Parragh et al., 2013).

EuNi$d$60P$d$61 adds an important refinement. There, low-temperature ARPES reveals Eu $d$62–Ni $d$63 hybridization consistent with a periodic Anderson description, but the temperature evolution above and below the Kondo coherence temperature near $d$64 K is opposite in sign and therefore not captured by a simple hybridization-only picture. The paper argues that both $d$65-$d$66 hybridization and an $d$67-conduction-electron Coulomb interaction of Falicov–Kimball type are imperative. For a six-orbital $d$68-$d$69-$d$70 Hamiltonian, this makes explicit $d$71-type terms plausible whenever the observed spectra show non-monotonic spectral-weight transfer rather than a purely coherent–incoherent crossover (2206.13030).

5. Representative realizations

In mixed actinide–transition-metal oxides, the compact $d$72-$d$73-$d$74 logic is clearest. The monouranates $d$75, $d$76, and $d$77-$d$78 contain transition-metal $d$79, uranium $d$80, and oxygen $d$81 states in a rutile-related orthorhombic structure with strongly anisotropic uranyl-like $d$82 bonding. Group-theoretical analysis assigns the $d$83-site to $d$84 and U to $d$85, producing nondegenerate orbital sectors in which $d$86 is more localized than $d$87, and only a subset of U-$d$88 eigenstates remains weakly hybridized. The best-performing orbital-resolved setup excludes the strongly bonding $d$89-$d$90-O-$d$91 sector and the most ligand-admixed $d$92 channels, while retaining localized $d$93, localized $d$94, and nonbonding $d$95. The same study also reports that structural symmetry breaking is controlled mainly by $d$96-site $d$97-orbital chemistry, while explicit oxygen treatment becomes decisive in strongly hybridized cases such as $d$98 (Warda et al., 22 Aug 2025).

Pyrochlore iridates supply a different realization, centered on exchange and orbital-selective ligand hybridization. In $d$99, the indispensable sectors are localized $p$00, $p$01 states dominated by the $p$02 manifold, and O $p$03 orbitals. The paper assigns the composition-dependent O $p$04-edge shift specifically to increased hybridization between $p$05-$p$06 and basal O-$p$07, while the apical-related peak changes little with $p$08. It further interprets the suppression of magnetic order and the emergence of Kondo-like behavior in Pr-rich samples in terms of exchange between localized $p$09-$p$10 and itinerant $p$11-$p$12 electrons. A six-orbital $p$13-$p$14-$p$15 model for this class is therefore constrained to include both $p$16 and an orbital-selective $p$17, with the basal oxygen channel especially prominent (Kumar et al., 2020).

EuNi$p$18P$p$19 demonstrates the same compact logic in a valence-fluctuating intermetallic rather than an oxide. The low-energy Fermi surface is stated to be mainly constructed by Ni $p$20 orbitals, while Eu $p$21 appears as multiple nearly flat multiplet-derived bands between about $p$22 eV and $p$23. At low temperature, ARPES reveals anti-crossings between flat $p$24 bands and dispersive conduction bands, a kink near $p$25 meV, and a Fermi-velocity reduction by about $p$26 in the $p$27 band. Fitting the low-energy kink yields a renormalized $p$28-level energy $p$29 meV and renormalized hybridization $p$30 meV. Because P $p$31 is not resolved as a distinct low-energy quasiparticle sector, explicit $p$32 orbitals in a six-orbital model are chemically motivated rather than directly fixed by ARPES, but the paper still supports a three-sector $p$33-$p$34-$p$35 hierarchy with essential $p$36-$p$37 hybridization (2206.13030).

By contrast, the vanadate $p$38-$p$39 model functions as a cautionary reference. In $p$40VO$p$41, unrestricted Hartree–Fock on a full V $p$42 plus O $p$43 basis reproduces the observed $p$44-type alternating orbital order with $p$45-type antiferromagnetism and the complementary $p$46-type alternating orbital order with $p$47-type antiferromagnetism, but only when explicit oxygen states, weak self-doping, and the nominally empty $p$48 sector are retained. The paper’s conclusion is that a $p$49-only truncation is not accurate enough. For six-orbital $p$50-$p$51-$p$52 construction, the implication is straightforward: orbitals should be excluded because they are demonstrably inactive, not because ionic counting suggests they are empty (Rościszewski et al., 2018).

6. Computational strategies, observables, and limitations

Several complementary computational strategies appear in this literature. The $p$53- versus $p$54 comparison uses single-site DMFT with rotationally invariant Kanamori interactions, full SU(2)-symmetric spin-flip and pair-hopping terms, CT-HYB continuous-time quantum Monte Carlo as impurity solver, $p$55, and Anisimov double counting for the $p$56 model (Parragh et al., 2013). Orbital-resolved DFT$p$57 for $p$58 uses PBEsol, orthogonalized atomic orbitals as projectors, occupation-matrix eigenstates as the corrected local orbitals, and comparison to Wannier-function projectors in order to diagnose projector mismatch and spurious Hubbard forces (Warda et al., 22 Aug 2025). The vanadate work uses unrestricted Hartree–Fock on finite clusters with Slater–Koster hopping, Harrison rescaling under distortion, explicit oxygen interactions, and a self-doping parameter to shift the total electron count away from the ideal ionic value (Rościszewski et al., 2018).

The key observables are likewise heterogeneous. In DMFT-based orbital models, orbital polarization $p$59, effective crystal-field splitting $p$60, and Fermi-surface topology are primary diagnostics (Parragh et al., 2013). In orbital-resolved DFT$p$61, the decisive quantities are occupation-matrix eigenvalues, orbital-specific $p$62 values, and structural distortions such as octahedral tilt and metal or oxygen off-centering (Warda et al., 22 Aug 2025). In pyrochlore iridates, the relevant constraints include the Curie–Weiss law

$p$63

insulating power-law resistivity

$p$64

negative magnetoresistance, and the observed strengthening of $p$65-$p$66–basal O-$p$67 hybridization with $p$68 substitution (Kumar et al., 2020). In EuNi$p$69P$p$70, hybridization gaps, kink structure, non-monotonic temperature dependence across $p$71 K, and spectral-weight transfer are the empirical signatures that any effective model must reproduce (2206.13030).

The limitations are equally clear. No cited work presents a universally accepted six-orbital $p$72-$p$73-$p$74 Hamiltonian with fully determined hopping integrals, Coulomb tensors, and crystal-field parameters for all material classes. The $p$75-versus-$p$76 study does not contain $p$77 orbitals and neglects $p$78 and $p$79, although it explicitly notes that this omission may matter (Parragh et al., 2013). The monouranate work is formulated in DFT$p$80, not as a many-body low-energy Hamiltonian, and its best-performing setup is not literally a six-orbital basis (Warda et al., 22 Aug 2025). The pyrochlore iridate study is primarily experimental and does not extract microscopic hopping amplitudes, SOC strengths, or exchange constants (Kumar et al., 2020). EuNi$p$81P$p$82 establishes the necessity of $p$83-$p$84 hybridization and correlation, but leaves the $p$85 sector mostly implicit and does not determine a complete tight-binding parametrization (2206.13030). The vanadate study shows how much can be learned from an explicit metal–ligand model, but it is a full $p$86-$p$87 reference framework treated at unrestricted Hartree–Fock level rather than a finished six-orbital $p$88-$p$89-$p$90 reduction (Rościszewski et al., 2018).

Taken together, these results define the six-orbital $p$91-$p$92-$p$93 model less as a fixed textbook Hamiltonian than as a construction principle. The correct reduced manifold must separate localized and delocalized channels, retain explicit ligand states where charge transfer and bond selectivity are active, track correlated-shell occupancies rather than nominal valences, and include self-energy renormalization of level splittings. When those conditions are met, a six-orbital $p$94-$p$95-$p$96 model becomes an economical representation of mixed-valent, Hund-coupled, or hybridization-driven low-energy physics; when they are not, the reduction can misidentify the active orbitals and thereby build the wrong many-body regime directly into the model [(Parragh et al., 2013); (Warda et al., 22 Aug 2025)].