Functionals¶

XC functionals are resolved through libxc at runtime. Any functional name libxc accepts works: short names ("LDA", "PBE", "B3LYP", "PBE0"), libxc XC_… identifiers, weighted-sum custom strings ("0.2*HF + 0.8*GGA_X_PW91, GGA_C_PW91" for PW1PW), and vibe-qc’s own aliases.

For open-shell DFT you choose between UKS (unrestricted) and ROKS (spin-pure restricted open-shell). ROKS supports LDA, GGA, meta-GGA, global-hybrid, and range-separated-hybrid functionals; see the dedicated ROHF and ROKS page for the UKS-vs-ROKS decision and the invocation.

⚠️ B3LYP convention, vibe-qc ships the ORCA definition¶

The b3lyp keyword resolves to libxc id 475 (HYB_GGA_XC_B3LYP5, VWN5 local correlation, the ORCA / ADF convention), as it has since v0.8.0. v0.12.0 adds explicit flavor spellings (b3lyp5, b3lypg) and the cross-code audit below, but the default is unchanged.

“B3LYP” is a fixed hybrid recipe (Stephens-Devlin-Chabalowski- Frisch 1994); codes differ only in which Vosko-Wilk-Nusair parametrisation fills the 0.19-weighted LSDA-correlation slot (Hertwig & Koch, Chem. Phys. Lett. 268, 345 (1997)). The field is split into two camps:

Camp	Codes	vibe-qc spelling
VWN5 (Ceperley-Alder fit)	ORCA, TURBOMOLE, ADF, CRYSTAL	`b3lyp` (= `b3lyp5`)
VWN-RPA (“VWN(III)” in Gaussian’s naming)	Gaussian, libxc, PySCF (`b3lyp` and `b3lypg`, ≥ 2.3), Psi4 (≥ 1.2), NWChem, Q-Chem	`b3lyp/g` (= `b3lypg`)

vibe-qc follows the ORCA camp: it is internally consistent with lda / svwn (also VWN5), and it is the flavor CRYSTAL, vibe-qc’s primary periodic parity reference, computes.

Flavor gap: ~6.8 mHa on H₂/STO-3G, ~10-15 mHa per heavy atom (it is exactly 0.19·(E_c[VWN-RPA] − E_c[VWN5])). Measured cross-code on H₂/STO-3G at 1.4 bohr: b3lyp = −1.1586001 Ha, matching CRYSTAL14’s B3LYP keyword to 1e-9 Ha and ORCA’s bare B3LYP to ~1 µHa; b3lyp/g = −1.1654009 Ha, matching PySCF 2.13’s b3lyp to <1e-9 Ha and ORCA’s B3LYP/G to ~1 µHa.

Practical guidance:

For Gaussian / PySCF / Psi4 parity, pass functional="b3lyp/g" (or the PySCF-style spelling "b3lypg"; functional="402" and functional="XC_HYB_GGA_XC_B3LYP" resolve to the same functional). PySCF evaluates the VWN-RPA flavor for both xc="b3lyp" and xc="b3lypg".
For ORCA / TURBOMOLE / CRYSTAL / CP2K parity, the default is already right; functional="b3lyp5" is an explicit spelling of the same functional for unambiguous cross-code tables.
When publishing, say which B3LYP you used (cite Hertwig & Koch 1997, the auto-citation block does this for you, see § Citations).
There is deliberately no b3lyp3 alias: libxc’s LDA_C_VWN_3 is the Ceperley-Alder fit III, numerically identical to VWN5 on closed-shell densities, not Gaussian’s “VWN(III)” (the RPA fit), so the obvious name would select the wrong flavor exactly when users reach for it.

Recipe table¶

vibe-qc resolves these short names in xc.cpp::resolve_alias() (plus everything libxc accepts directly):

Name	Type	HF-exchange	libxc id	Notes
`lda` / `svwn`	LDA	0 %	1 + 7	Slater exchange + VWN5 correlation. Fast sanity check.
`pbe`	GGA	0 %	101 + 130	Workhorse for solid state.
`blyp`	GGA	0 %	106 + 131	Classic molecular GGA.
`b3lyp` (= `b3lyp5`)	hybrid	20 %	475	de-facto for organics; VWN5 / ORCA convention, see the flavor box above.
`b3lyp/g` (= `b3lypg`)	hybrid	20 %	402	Gaussian-compatible B3LYP (VWN-RPA), what Gaussian / PySCF / Psi4 mean by the bare name.
`pbe0` / `pbeh`	hybrid	25 %	406	PBE + 25% Fock exchange.
`pw1pw`	hybrid	20 %	weighted-sum	Bredow’s 1-parameter periodic hybrid; cite Bredow & Gerson PRB 61, 5194 (2000).
`tpss`	meta-GGA	0 %	202 + 231	Non-empirical τ-dependent MGGA; Tao-Perdew-Staroverov-Scuseria PRL 91, 146401 (2003).
`tpssh`	hybrid meta-GGA	10 %	457	10 % global hybrid of TPSS; Staroverov-Scuseria-Tao-Perdew JCP 119, 12129 (2003).
`m06-l` (`m06l`)	meta-GGA	0 %	203 + 233	Pure Minnesota meta-GGA; Zhao-Truhlar JCP 125, 194101 (2006).
`m06-2x` (`m062x`)	hybrid meta-GGA	54 %	450 + 236	54 % global hybrid for thermochemistry + non-covalent interactions; Zhao-Truhlar TCA 120, 215 (2008).
`scan`	meta-GGA	0 %	263 + 267	Strongly-constrained meta-GGA; Sun-Ruzsinszky-Perdew PRL 115, 036402 (2015). Grid-sensitive, use a fine grid (prefer `r2scan`).
`r2scan`	meta-GGA	0 %	497 + 498	Re-regularized SCAN, smooth, grid-friendly; Furness et al JPCL 11, 8208 (2020). Recommended default of the family.
`r2scan01`	meta-GGA	0 %	645 + 642	r²SCAN with the fourth-order gradient expansion restored (larger η); Furness et al JCP 156, 034109 (2022). Aimed at slowly-varying densities where r²SCAN slightly underbinds.
`r2scan0`	hybrid meta-GGA	25 %	660	25 % global hybrid of r²SCAN (PBE0-like).
`r2scanh`	hybrid meta-GGA	10 %	659	10 % global hybrid of r²SCAN (TPSSh-like).
`wb97x`	range-sep hybrid	15.77 %→100 %	464	Long-range-corrected hybrid GGA; Chai-Head-Gordon 2008. ω = 0.3. Direct SCF only (see § Range-separated hybrids).
`wb97x-d`	range-sep hybrid + D	22.2 %→100 %	471	ωB97X-D, Chai-Head-Gordon 2008. ω = 0.2 + intrinsic CHG dispersion. Use `vibeqc.run_wb97x_d` for the full XC + dispersion total.
`hse06`	screened RSH	25 %→0 %	428	Heyd-Scuseria-Ernzerhof screened hybrid; 25 % HF short-range, 0 % long-range, ω = 0.11. The solid-state band-gap workhorse.

The full libxc functional set is also reachable directly. Use the libxc identifier if your shorthand isn’t in the table: functional="XC_HYB_GGA_XC_M06_2X", functional="XC_GGA_X_RPBE,XC_GGA_C_PBE", etc.

PW1PW: Bredow’s periodic hybrid¶

pw1pw is the first non-libxc-built-in custom hybrid in vibe-qc. The functional is:

PW1PW = 0.20 × HF exchange  +  0.80 × GGA_X_PW91 exchange  +  GGA_C_PW91 correlation

Implemented via libxc’s custom-string path: vibeqc.vibeqc_xc_to_pyscf translates "pw1pw" to ".2*HF + .8*GGA_X_PW91, GGA_C_PW91" for the PySCF-grid V_xc evaluation in periodic UKS / RKS dispatch.

import vibeqc as vq

result = vq.run_periodic_job(
    mgo,
    basis="pob-tzvp",
    method="RKS",
    functional="pw1pw",
    kmesh=(4, 4, 4),
    output="mgo-pw1pw-444",
)

Cite: Bredow, T.; Gerson, A. R. Phys. Rev. B 61, 5194 (2000). PW1PW is the strict-comparison reference for the Peintinger 2013 SI basis- set paper’s HF + PW1PW totals on ~60 compounds.

Defining custom functionals at runtime (`define_functional`)¶

Added in v0.11.3: you can register a CRYSTAL-style one-parameter hybrid (or any weighted-sum functional) from the Python input without a C++ edit. This is the same mechanism that ships PW1PW, PBEh-3c, and the double-hybrid SCF pieces internally.

import vibeqc as vq

vq.define_functional(
    "my-hybrid",
    [("GGA_X_PW91", 0.80), ("GGA_C_PW91", 1.00)],
    hf_exchange_fraction=0.20,
)

# From this point on, "my-hybrid" resolves exactly like a built-in:
f = vq.Functional("my-hybrid")
assert f.hf_exchange_fraction == 0.20
assert f.is_hybrid

# Use it in any driver:
result = vq.run_rks(mol, basis, vq.RKSOptions(functional="my-hybrid"))
result = vq.run_periodic_job(system, basis, method="RKS",
                             functional="my-hybrid", kpoints=[8,8,8])

components is a list of (libxc_name, weight) pairs. Any libxc name that xc_functional_get_number recognises works, "GGA_X_PBE", "GGA_C_LYP", "MGGA_C_BC95", etc. Weights scale each component’s contribution to the XC energy; they need not sum to 1.0.

hf_exchange_fraction is the global HF-exchange fraction for hand-mixed hybrids (PW1PW-style). Set it to 0.0 for a pure-DFT custom mix; non-zero makes the functional a hybrid and routes it through the exact-exchange code path automatically.

Thread-safe. Call define_functional once at startup, before any SCF job that references the alias. Registration is case-insensitive; “MY-HYBRID” and “my-hybrid” resolve to the same alias.

A full example is in examples/molecular/define_functional_demo.py.

Choosing a functional (decision tree)¶

Periodic ionic insulator (oxides, halides, sulphides): pw1pw or b3lyp or pbe0 for hybrids; pbe for pure GGA. Cite the published reference for the system class.
Periodic semiconductor / metal: pbe (most common). Hybrids hit the dense-Lpq RAM ceiling fast, see multi_k_scf.md § Open issues.
Molecular organic geometry optimisation: r²SCAN-3c, the turnkey “Swiss army knife” composite (see composites.md), or b3lyp with D3(BJ) (see molecules.md) / pbe0 for more reactive systems.
Molecular thermochemistry: the r²SCAN-3c / ωB97X-3c / HSE-3c composite methods are runnable today (composites.md); pbe0 + def2-TZVP + D3(BJ) is the conventional mid-tier workhorse.
Open-shell radicals: prefer uks driver over rks. Same functional table works.

⚠️ Plain B3LYP for noncovalent interactions¶

Plain B3LYP (no dispersion correction) systematically under-binds noncovalent interactions by 20-50% on the S22 set. Always pair B3LYP with a dispersion correction when noncovalent interactions matter:

result = vq.run_job(
    mol,
    basis="def2-tzvp",
    method="rks",
    functional="b3lyp",
    dispersion="d3bj",  # or "d4"
)

D3(BJ) is the standard. D4 (Caldeweyher-Bannwarth-Grimme, 2019) ships alongside D3(BJ): use dispersion="d4" to opt in. There are two D4 backends, the optional dftd4 package (production default, bit-exact to upstream) and an in-tree native MPL-2.0 implementation (compute_d4(mol, func, backend="native")) that is parity-validated for H-Ne (correlated CPKS/PBE38 reference C6 within a few percent of dftd4); use the dftd4 backend for elements outside that set. Both are documented in molecules.md § Dispersion.

Meta-GGA functionals (TPSS / TPSSh, M06-L / M06-2X, SCAN / r²SCAN)¶

vibe-qc evaluates the kinetic-energy density τ(r) = ½ Σ_i |∇ψ_i(r)|² on the DFT grid (using the AO gradients that already power the GGA path), so the τ-dependent meta-GGA family is now available in molecular RKS + UKS + ROKS:

opts = vq.RKSOptions()
opts.functional = "r2scan"    # or tpss / tpssh / m06-l / m06-2x /
                              # scan / r2scan01 / r2scan0 / r2scanh
result = vq.run_rks(mol, basis, opts)

SCAN vs r²SCAN. SCAN (scan) satisfies all 17 exact constraints a semilocal functional can, but its sharp enhancement factor is notoriously grid-sensitive, on vibe-qc’s default medium grid the H₂O/def2-SVP energy sits ~1 mHa off the fine-grid limit. For SCAN, set a fine grid (opts.grid.n_radial = 120, opts.grid.lebedev_order = 41 or finer). r²SCAN (r2scan) is the re-regularized SCAN, same constraint satisfaction, a smooth enhancement factor, grid-stable to ~µHa on the default grid. It is the recommended default of the family; r2scan0 (25 % HF) and r2scanh (10 % HF) are its global hybrids. r²SCAN01 (r2scan01) re-regularizes r²SCAN with a larger η so the fourth-order term of the slowly-varying gradient expansion (dropped by r²SCAN) is restored; it targets slowly-varying densities such as bulk metals and bonding regions where r²SCAN slightly underbinds.

Coverage:

Molecular RKS / UKS / ROKS, closed-shell + open-shell SCF live; parity to PySCF.dft within grid-quadrature noise (~µHa on def2-SVP / medium grid for closed-shell H₂O; ~10-50 µHa on STO-3G / grids.level=5 for O₂ triplet UKS). ROKS retains PySCF-parity coverage for the integer-occupation convention and has dedicated fractional frontier-shell coverage for degenerate OH(2Pi).
Periodic SCF, not yet. cpp/src/periodic_xc.cpp will reject meta-GGA functionals at the Functional::eval_* call with a clear error pointing here. Periodic τ-grid + V_τ is a follow-up workstream.
Analytic gradients, available for molecular RKS + UKS (vibeqc.compute_gradient_rks / compute_gradient_uks). The XC Pulay force includes the τ term; finite-difference validated to ~3e-9 (TPSS/TPSSh) / ~1e-7 (M06-2X) on H₂/STO-3G. Geometry optimisation with meta-GGAs works end-to-end.
Analytic Hessian / CPKS, not yet. The meta-GGA second-derivative kernel (eval_unpolarised_fxc) raises on MGGA; finite-difference Hessians work as a workaround.
Laplacian-dependent meta-GGAs (TB09, …), not supported. vibe-qc populates τ but not ∇²ρ on the grid; libxc functionals with XC_FLAGS_NEEDS_LAPLACIAN are refused at Functional construction with a clear error.

Range-separated hybrids (ωB97X, ωB97X-D)¶

A range-separated (CAM) hybrid makes the exact-exchange admixture position-dependent:

EXX(r₁₂) = cam_alpha + cam_beta · erf(ω · r₁₂)

so the exchange operator is cam_alpha·(1/r₁₂) + cam_beta·(erf(ω·r₁₂)/r₁₂). vibe-qc reads the (ω, α, β) triple from libxc at construction, Functional("wb97x").is_range_separated / rsh_omega / cam_alpha / cam_beta, and the SCF driver builds the second, erf-attenuated exchange matrix K_erf(ω) alongside the ordinary K.

wb97x (Chai-Head-Gordon 2008) is the first range-separated hybrid in vibe-qc: a long-range-corrected functional, 15.77 % HF at short range ramping to 100 % HF at long range, ω = 0.3 bohr⁻¹.

opts = vq.RKSOptions()
opts.functional = "wb97x"
result = vq.run_rks(mol, basis, opts)

ωB97X-D, range-separated hybrid + dispersion¶

wb97x-d (Chai-Head-Gordon 2008) is the dispersion-corrected re-parameterisation of ωB97X (ω = 0.2 bohr⁻¹, 22.2 % HF short-range → 100 % long-range). It has two pieces: the range-separated-hybrid XC functional (libxc XC_HYB_GGA_XC_WB97X_D), and an intrinsic empirical “-D” dispersion correction of the older “DFT-D2” family. The dispersion is geometry-only, no SCF coupling, so the complete ωB97X-D energy is E_SCF + E_disp. Use the dispatcher :func:vibeqc.run_wb97x_d for the full total:

result = vq.run_wb97x_d(mol, basis)          # closed- or open-shell
print(result.e_total)        # E_SCF + E_dispersion
print(result.scf.energy, result.dispersion.energy)

The CHG dispersion is E_disp = −Σ_{A<B} C6_AB/R^6 · 1/(1 + 6·(R/R0_AB)^-12) with the Grimme-2006 D2 atomic C6 / vdW-radius tables (H-Xe); a heavier element raises a clear error. vibeqc.compute_chg_dispersion(mol) exposes the standalone term. The bare functional="wb97x-d" alias through run_rks / run_uks gives the XC/SCF part only (the range-separated hybrid), the same alias-vs-dispatcher split as b2plyp / run_b2plyp; use run_wb97x_d for the published total.

Coverage and caveats:

Molecular RKS / UKS / ROKS energies, live; parity to PySCF.dft at grid-quadrature precision (~µHa, def2-SVP; ROKS coverage includes the ωB97X exchange split and the ωB97M-V meta-GGA/VV10 combination).
Direct SCF only, the erf-attenuated K is built by the direct-SCF Fock kernel. density_fit=true with a range-separated functional is rejected with a clear error (the RI path has no erf-attenuated 3-centre integrals yet). The molecular drivers select the direct builder automatically for RSH functionals, so no user action is needed, just leave density_fit at its default.
Hard open-shell radicals, ωB97X’s 100 % long-range HF makes open-shell SCF stiffer. O₂, CH₃, and atomic radicals converge with the defaults; an orbital-near-degenerate case such as the OH ²Π radical can plateau near the gradient tolerance with plain DIIS, set opts.level_shift = 0.5 to converge it cleanly (standard QC practice for stiff open-shell SCF).
Newton / TRAH second-order acceleration is disabled for RSH functionals (their orbital-Hessian exchange response is not yet range-separation aware); DIIS + SOSCF + the quadratic fallback carry convergence.
Analytic gradients for RSH functionals are not yet available (finite-difference works). ωB97X-V and periodic RSH are queued, see below.

HSE06, screened range-separated hybrid¶

hse06 (Heyd, Scuseria, Ernzerhof; the Krukau-Vydrov-Izmaylov- Scuseria 2006 re-parameterisation, libxc XC_HYB_GGA_XC_HSE06) is a screened RSH: 25 % HF at short range tapering to 0 % at long range, the opposite ramp direction to ωB97X. In vibe-qc’s erf-form CAM representation this is cam_alpha = +0.25, cam_beta = −0.25 (EXX(r→0) = 0.25, EXX(r→∞) = 0), so it runs through the same erf-attenuated-K machinery as ωB97X, just with a negative long-range coefficient. No separate erfc kernel is needed.

opts = vq.RKSOptions()
opts.functional = "hse06"
result = vq.run_rks(mol, basis, opts)

HSE06 is primarily a solid-state band-gap functional. Molecular HSE06 is available as above, and periodic HSE06 ships now on the routes that build a genuine erfc short-range exchange kernel: the Ewald family (jk_method="slab_ewald_2d" / AUTO on dim=2 slabs, plus the Γ and multi-k EWALD_3D drivers) and jk_method="bipole". The periodic assembly is K_HF = 0.25 · K_erfc(ω = 0.11), a pure short-range direct lattice sum with no Madelung/exxdiv seam (the erfc kernel has no G → 0 divergence; this matches CRYSTAL’s SR-exchange treatment). Routes whose exact exchange is full-range only (GDF at Γ and multi-k, GPW/GAPW, COSX) fail closed on any range-separated functional rather than silently running HSE06 as PBE0, which is what they did before 2026-07-12. Long-range-corrected RSH (ωB97X, CAM-B3LYP, LC-ωPBE) fail closed on every periodic route: their full-range exchange arm needs an exxdiv treatment that is not yet validated. Analytic periodic gradients for screened hybrids are not available (the gradient entry points fail closed too); use finite differences.

Recently landed, and still-queued refinements¶

Several families below landed after the v0.8.0 cut; each bullet marks what ships now and what is still queued. Track the remainder at docs/roadmap.md § XC functional library expansion:

Range-separated hybrids, molecular wb97x, wb97x-d (via vibeqc.run_wb97x_d), hse06, and now wb97x-v and wb97m-v ship now (see § Range-separated hybrids above). The VV10 non-local correlation that ωB97X-V / ωB97M-V depend on landed (vibeqc.compute_vv10; xc="vv10" is the standalone functional), so these are the complete functionals, evaluated self-consistently on the SCF grid (worked walk-through: Modern functionals: VV10 and the ωB97X-V / ωB97M-V wave). Still queued: RSH analytic gradients, RSH + density fitting, and periodic RSH.
r²SCAN family, molecular scan / r2scan / r2scan0 / r2scanh ship now (see § Meta-GGA functionals above). Periodic SCAN / r²SCAN also ship (the periodic meta-GGA XC kernel landed in v0.12.x); the r²SCAN gradient (cell-opt Pulay τ term) is queued.
Composite “3c” methods, r²SCAN-3c, ωB97X-3c, and HSE-3c are runnable today (turnkey: functional + matching basis + gCP + D4 in one call); B3LYP-3c / B97-3c / PBEh-3c are PENDING_GCP_DATA (the framework is wired, the per-basis gCP tables are pending). See composites.md for the recipe table, status flags, and worked examples.
Double hybrids, B2PLYP (Grimme 2006), DSD-PBEP86 (Kozuch-Martin 2011), revDSD-PBEP86-D4 (Santra-Sylvetsky-Martin 2019), and PWPB95 (Goerigk-Grimme 2011) ship now via vibeqc.run_b2plyp / vibeqc.run_dsd_pbep86 / vibeqc.run_revdsd_pbep86 / vibeqc.run_pwpb95, with vibeqc.run_double_hybrid(name, …) as the generic dispatcher. PWPB95 is vibe-qc’s first meta-GGA double hybrid and its first spin-opposite-scaled one (the same-spin MP2 coefficient is zero). The SCS-MP2 / SOS-MP2 machinery the double-hybrid line rides is also live; see the MP2 and double hybrids page. ωB97M(2) (range-separated meta-GGA double hybrid) remains blocked, but no longer on VV10 (that landed): unlike the other double hybrids it is not a composition of stock libxc components, its SCF base is a re-optimised B97M power-series functional that libxc does not ship, so it needs a dedicated from-scratch meta-GGA kernel. xc="wb97m(2)" raises a message saying so. D4 dispersion ships via the optional dftd4 package, pass dispersion="d4" to run_b2plyp / run_dsd_pbep86 / run_revdsd_pbep86 / run_pwpb95 for the published X-D4 total (for revDSD-PBEP86, D4 is the native parametrisation), or call vibeqc.compute_d4 directly for arbitrary functionals. D3(BJ) is also wired through the molecular runner via the existing vibeqc.compute_d3bj machinery.

Programmatic access¶

from vibeqc import Functional

fn = Functional("PBE0")
print(fn.hf_exchange_fraction())    # 0.25
print(fn.kind())                    # XCKind.GGA   (well, GGA-hybrid)

# Custom libxc composition strings (comma-separated; X then C):
fn = Functional("XC_LDA_X,XC_LDA_C_VWN")            # explicit Slater + VWN
fn = Functional("XC_GGA_X_PBE,XC_GGA_C_PBE")        # PBE from separate parts

# Weighted-sum custom strings (PW1PW pattern):
fn = Functional("0.2*HF + 0.8*GGA_X_PW91, GGA_C_PW91")

The Functional ctor docstring lists every known short alias the resolver accepts.

Citations¶

Every run_job call writes a {stem}.bibtex and {stem}.references sibling alongside {stem}.out, pre-assembled with all the per-functional papers your job needs to cite. The list below is the human-readable equivalent of what the runtime emits; look at the auto-generated files first.

B3LYP (all flavor spellings, b3lyp, b3lyp5, b3lyp/g, b3lypg): Becke, J. Chem. Phys. 98, 5648 (1993); Lee, Yang, Parr, Phys. Rev. B 37, 785 (1988); Stephens, Devlin, Chabalowski, Frisch, J. Phys. Chem. 98, 11623 (1994); Vosko, Wilk, Nusair, Can. J. Phys. 58, 1200 (1980) (both the VWN5 fit used by the default b3lyp and the RPA fit used by b3lyp/g are from this paper); Hertwig, Koch, Chem. Phys. Lett. 268, 345 (1997) (which flavor is which, cite it so your Methods section pins the variant).
PBE: Perdew, Burke, Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
PBE0: PBE 1996 above plus Adamo, Barone, J. Chem. Phys. 110, 6158 (1999).
PW91 / PW1PW: Perdew, Chevary, Vosko, Jackson, Pederson, Singh, Fiolhais, Phys. Rev. B 46, 6671 (1992); Bredow, Gerson, Phys. Rev. B 61, 5194 (2000) for the PW1PW hybrid mixing.
B2PLYP (and other double hybrids): Grimme, J. Chem. Phys. 124, 034108 (2006), plus the B3LYP components above.
TPSS / TPSSh: Tao, Perdew, Staroverov, Scuseria, Phys. Rev. Lett. 91, 146401 (2003); Staroverov, Scuseria, Tao, Perdew, J. Chem. Phys. 119, 12129 (2003) for the 10 % global-hybrid (TPSSh) variant.
M06-L / M06-2X: Zhao, Truhlar, J. Chem. Phys. 125, 194101 (2006) (M06-L, pure meta-GGA); Zhao, Truhlar, Theor. Chem. Acc. 120, 215 (2008) (Minnesota M06 family, including the 54 %-hybrid M06-2X).
SCAN: Sun, Ruzsinszky, Perdew, Phys. Rev. Lett. 115, 036402 (2015).
ωB97X: Chai, Head-Gordon, J. Chem. Phys. 128, 084106 (2008).
ωB97X-D (and its CHG / DFT-D2-type dispersion): Chai, Head-Gordon, Phys. Chem. Chem. Phys. 10, 6615 (2008); the D2 atomic C6 / vdW-radius set is Grimme, J. Comput. Chem. 27, 1787 (2006).
HSE06: Heyd, Scuseria, Ernzerhof, J. Chem. Phys. 118, 8207 (2003); erratum ibid. 124, 219906 (2006); Krukau, Vydrov, Izmaylov, Scuseria, J. Chem. Phys. 125, 224106 (2006) (the “06” re-parameterisation).
r²SCAN (and r2scan0 / r2scanh): Furness, Kaplan, Ning, Perdew, Sun, J. Phys. Chem. Lett. 11, 8208 (2020); erratum ibid. 11, 9248 (2020).
r²SCAN01: Furness, Kaplan, Ning, Perdew, Sun, J. Chem. Phys. 156, 034109 (2022).
D3(BJ) / D4 dispersion: Grimme, Antony, Ehrlich, Krieg, J. Chem. Phys. 132, 154104 (2010); Grimme, Ehrlich, Goerigk, J. Comput. Chem. 32, 1456 (2011) (BJ damping extension); Caldeweyher, Bannwarth, Grimme, J. Chem. Phys. 150, 154122 (2019).
libxc itself: Lehtola, Steigemann, Oliveira, Marques, SoftwareX 7, 1 (2018).

Pre-v0.8.x workflows hand-curated this list from docs/citing.md. The runtime database now does it for you, see the citations user guide for the full schema, the auto-citations tutorial for the end-to-end manuscript workflow, and AGENTS.md § “Citation database ownership” for the rule on keeping the database in sync when adding a new functional.