Modern functionals: VV10 and the ωB97X-V / ωB97M-V wave

vibe-qc’s range-separated hybrids spent a release without their intended correlation partner: the “-V” members (ωB97X-V, ωB97M-V) are defined with the VV10 nonlocal correlation term, and without it they cannot be run as their authors meant. That gap is now closed. With the VV10 kernel in the C++ core, those functionals evaluate completely and self-consistently, and the same wave adds the r2SCAN01 meta-GGA and the revDSD-PBEP86-D4 double hybrid. This page is the worked companion to the functionals user guide; it shows how to invoke each new functional and what it is for. For the older, conventional range-separated hybrid ωB97X (which carries no VV10 term), start with Range-separated hybrids: ωB97X.

Status: preview

VV10 and the -V range-separated hybrids are production for molecular RKS and UKS energies, but their analytic nuclear gradients are incomplete: the gradient kernel omits the long-range exact-exchange term and the VV10 nonlocal term (GitLab #571), so compute_gradient_rks and compute_gradient_uks refuse them unless you pass allow_incomplete=True, and run_job(optimize=True) falls back to finite differences of the energy. Periodic use is not available. r2SCAN01 has an analytic gradient and is not affected. See Methods: theory and status.

Every snippet below uses water at a fixed geometry so you can paste and run them directly:

import vibeqc as vq

mol = vq.Molecule(
    [
        vq.Atom(8, [0.0, 0.00, 0.00]),
        vq.Atom(1, [0.0, 1.43, -0.98]),
        vq.Atom(1, [0.0, -1.43, -0.98]),
    ]
)

VV10: the nonlocal correlation piece

VV10 (Vydrov and Van Voorhis 2010) adds a genuinely nonlocal correlation term: a double integral over all pairs of grid points that recovers long-range dispersion from the density alone, with no empirical pairwise C6 table. It is the ingredient that completes the “-V” functionals below. vibe-qc ships it both as a standalone functional and as a kernel you can call on any grid:

basis = vq.BasisSet(mol, "cc-pvdz")

opts = vq.RKSOptions()
opts.functional = "vv10"          # rPW86 exchange + PBE correlation + the VV10 nonlocal term
result = vq.run_rks(mol, basis, opts)
print(f"E(VV10) = {result.energy:.6f} Ha")
# E(VV10) = -76.504441 Ha

The nonlocal potential is folded into the SCF exchange-correlation build, so both the energy and its potential are exact (not a post-hoc add-on). The C++ kernel is exposed directly as vibeqc.compute_vv10(coords, weights, rho, sigma, b, C) for evaluation on an arbitrary density, with the published parameters b = 5.9, C = 0.0093. vibe-qc’s kernel matches PySCF’s _vv10nlc on a shared water / cc-pVDZ grid to within 1e-8 Ha (tests/test_vv10.py).

Inside an SCF the nonlocal double integral runs on its own, coarser grid: GridOptions.vv10_grid_factor (default 3.0) divides the radial and angular resolution of the XC grid for the VV10 term only, the same practice Vydrov and Van Voorhis used in their implementation paper (a coarse SG-1 grid for the nonlocal term under a (75, 302) grid for the semilocal part, because the nonlocal term is far less sensitive to grid fineness). Set opts.grid.vv10_grid_factor = 1.0 to evaluate VV10 on the full XC grid when you need the dense reference.

ωB97X-V and ωB97M-V: the complete range-separated hybrids

These are the functionals that VV10 unlocks. ωB97X-V is a range-separated GGA hybrid plus VV10; ωB97M-V is a range-separated meta-GGA hybrid plus VV10, and is among the best-performing functionals on broad main-group benchmarks. Both compose three advanced pieces at once: the erf-attenuated long-range exact exchange, the semilocal (meta-)GGA, and the VV10 nonlocal term. You select them by name, exactly like any other functional:

opts = vq.RKSOptions()
opts.functional = "wb97m-v"       # range-separated meta-GGA + VV10 in one functional
result = vq.run_rks(mol, basis, opts)
print(f"E(ωB97M-V) = {result.energy:.6f} Ha")
# E(ωB97M-V) = -76.383379 Ha    (ωB97X-V gives -76.385779 Ha)

vibe-qc reproduces PySCF’s dft.RKS(xc="wb97m-v", nlc="VV10") to within 5e-5 Ha at the same grid (tests/test_wb97m_v.py); ωB97X-V matches its PySCF reference to the same tolerance (tests/test_vv10.py). The Functional resolver carries the recipe, so you can confirm what you are running:

fn = vq.Functional("wb97m-v")
print(fn.is_range_separated)   # True
print(fn.rsh_omega)            # 0.3  (bohr^-1)
print(fn.needs_vv10)           # True
print(fn.vv10_b, fn.vv10_C)    # 6.0 0.01  (the -V functionals re-fit b and C)

Important

The “-V” range-separated hybrids run through direct SCF only. As with plain ωB97X, the erf-attenuated exchange has no 3-centre density-fitting path yet, so the molecular driver selects the direct Fock builder automatically; leave density_fit at its default. They are molecular RKS / UKS only (periodic RSH and periodic VV10 are queued), and have no analytic gradient yet, finite-difference geometry optimization works via run_job(..., optimize=True).

r2SCAN01: a re-regularized meta-GGA

r2SCAN01 (Furness et al. 2022) targets a known weakness of r²SCAN: it restores the fourth-order term of the slowly-varying gradient expansion that r²SCAN drops, by re-regularizing with a larger η. The practical effect is better behaviour for slowly-varying densities, bulk metals and bonding regions, where r²SCAN slightly underbinds. It is a pure meta-GGA (no exact exchange), so it is cheap, and it is τ-only (laplacian-free), which fits vibe-qc’s meta-GGA grid:

basis_svp = vq.BasisSet(mol, "def2-svp")

opts = vq.RKSOptions()
opts.functional = "r2scan01"
result = vq.run_rks(mol, basis_svp, opts)
print(f"E(r2SCAN01) = {result.energy:.6f} Ha")

For this water geometry in def2-SVP, vibe-qc returns E = -76.3151 Ha (-76.315067 in a run on a vibe-qc 0.17.10 development build), matching PySCF’s MGGA_X/C_R2SCAN01 to 3.9 µHa (the asserted reference in tests/test_xc.py). Unlike the range-separated hybrids above, r2SCAN01 has analytic molecular gradients (the Pulay force includes the τ term), so native geometry optimization works.

revDSD-PBEP86-D4: a re-parametrized double hybrid

A double hybrid runs a hybrid-DFT SCF step and adds a scaled MP2 correlation correction. revDSD-PBEP86-D4 (Santra, Sylvetsky and Martin 2019) is a careful re-fit of DSD-PBEP86 with two distinguishing features: an asymmetric MP2 scaling (different coefficients for opposite-spin and same-spin pairs), and D4 as its native dispersion, the functional coefficients were fit jointly with the D4 damping, so D4 is part of the definition rather than an afterthought. Use the dedicated dispatcher:

basis = vq.BasisSet(mol, "cc-pvdz")
result = vq.run_revdsd_pbep86(mol, basis)

print(f"E(hybrid SCF step)   = {result.rks.energy:.6f} Ha")
print(f"E(MP2 correction)    = {result.mp2.e_correlation:.6f} Ha")   # 0.5922*E_os + 0.0636*E_ss
print(f"E(revDSD-PBEP86)     = {result.e_total:.6f} Ha")
# E(revDSD-PBEP86)     = -76.271466 Ha

The SCF base is 0.69 HF + 0.31 PBE exchange + 0.4210 P86 correlation; the MP2 correction uses c_os = 0.5922 and c_ss = 0.0636. The dispatcher returns a DoubleHybridResult carrying .rks, .mp2, and .e_total, and matches a PySCF reference to within 5e-5 Ha (tests/test_revdsd_pbep86.py). To obtain the published revDSD-PBEP86-D4 total, add the native dispersion:

result = vq.run_revdsd_pbep86(mol, basis, dispersion="d4")

Note

dispersion="d4" needs the optional dftd4 package (pip install dftd4); without it, drop the keyword for the SCF + MP2 total. Note also that the older revDSD-PBEP86-D3BJ uses a different coefficient fit and is not reachable through this alias, the revdsd-pbep86 name is the D4-native parametrization.

Theory

Range separation, restated. The -V hybrids build on the erf-split exchange of the previous tutorial: a fraction \(\alpha\) of exact exchange at short range that rises to \(\alpha+\beta\) at long range, with a semilocal (meta-)GGA for the rest (Mardirossian, Narbe and Head-Gordon, Martin (2014). *ωB97X-V: A 10-parameter, range-separated hybrid, generalized gradient approximation density functional with nonlocal correlation, designed by a survival-of-the-fittest strategy.* Physical Chemistry Chemical Physics, **16**(21), 9904–9924. doi:[10.1039/C3CP54374A](https://doi.org/10.1039/C3CP54374A), Mardirossian, Narbe and Head-Gordon, Martin (2016). *ωB97M-V: A combinatorially optimized, range-separated hybrid, meta-GGA density functional with VV10 nonlocal correlation.* Journal of Chemical Physics, **144**(21), 214110. doi:[10.1063/1.4952647](https://doi.org/10.1063/1.4952647)). For ωB97X-V the split is \(\alpha = 0.167\), \(\beta = 0.833\) and for ωB97M-V \(\alpha = 0.15\), \(\beta = 0.85\), both with \(\omega = 0.3\ \text{bohr}^{-1}\): as in ωB97X the long-range limit is 100 percent exact exchange, which puts the \(-1/r\) tail in the exchange potential. What these functionals add is a nonlocal correlation term.

The VV10 nonlocal kernel. Semilocal correlation cannot describe the long-range dispersion that arises from correlated charge fluctuations on distant fragments. VV10 (Vydrov, Oleg A. and Van Voorhis, Troy (2010). *Nonlocal van der Waals density functional: The simpler the better.* Journal of Chemical Physics, **133**(24), 244103. doi:[10.1063/1.3521275](https://doi.org/10.1063/1.3521275)) adds it as a double integral over the electron density,

\[ E_c^\text{nl} = \int \rho(\mathbf r)\Bigl[\beta + \tfrac12\!\int \rho(\mathbf r')\,\Phi(\mathbf r, \mathbf r')\,d\mathbf r'\Bigr] d\mathbf r , \qquad \Phi = -\frac{3}{2}\,\frac{1}{g\,g'\,(g + g')}, \quad g = \omega_0(\mathbf r)\,R^2 + \kappa(\mathbf r), \]

with \(R = |\mathbf r - \mathbf r'|\), \(g'\) the same expression at \(\mathbf r'\), and

\[ \omega_0 = \sqrt{\omega_g^2 + \tfrac13\omega_p^2}, \quad \omega_g^2 = C\,\Bigl|\frac{\nabla\rho}{\rho}\Bigr|^4, \quad \omega_p^2 = 4\pi\rho, \quad \kappa = b\,\frac{3\pi}{2}\Bigl(\frac{\rho}{9\pi}\Bigr)^{1/6}, \quad \beta = \frac{1}{32}\Bigl(\frac{3}{b^2}\Bigr)^{3/4}. \]

Here \(\rho\) is the electron density, \(\omega_p\) the local plasma frequency, and the constant \(\beta\) makes \(E_c^\text{nl}\) vanish for the uniform electron gas, so the nonlocal term is a pure correction. The two empirical parameters do distinct jobs: \(C\) enters \(\omega_g\) and so sets the size of the asymptotic \(C_6\) dispersion coefficient, while \(b\) controls \(\kappa\) and thus the short-range damping, that is, how the dispersion switches off where the densities overlap. The standalone vv10 functional (rPW86 exchange plus PBE correlation plus VV10) uses \(b = 5.9\), \(C = 0.0093\); the -V functionals use the pair \(b = 6.0\), \(C = 0.01\) chosen by their authors (Functional.vv10_b, vv10_C).

Why ωB97X-V and ωB97M-V need it. The empirical coefficients of the semilocal exchange and correlation and the exact-exchange fractions of these functionals were optimised on a large training set together with the VV10 term already present. The semilocal part therefore does not describe noncovalent interactions on its own, and running it without VV10 evaluates a different, unbalanced functional. Cost is the price: the double integral runs over pairs of grid points, \(O(N_\text{grid}^2)\), which is why vibe-qc evaluates it on a coarser grid (GridOptions.vv10_grid_factor, default 3.0) than the semilocal part. The nonlocal potential is included self-consistently, not added afterwards. Two limits remain, both stated in the status note above: no complete analytic gradient, and no density-fitted exchange for the erf-attenuated part.

What ships, and what is still queued

The table is the snapshot for this functional wave:

Functional

xc= string

Status

VV10 (standalone)

vv10

✅ shipped, molecular RKS / UKS, self-consistent

ωB97X-V

wb97x-v

✅ shipped, molecular, direct SCF, FD gradient

ωB97M-V

wb97m-v

✅ shipped, molecular, direct SCF, FD gradient

r2SCAN01

r2scan01

✅ shipped, molecular, analytic gradient

revDSD-PBEP86(-D4)

revdsd-pbep86

✅ shipped via run_revdsd_pbep86, D4 native

ωB97M(2)

wb97m(2)

⏳ gated, needs a from-scratch B97M meta-GGA base (not in libxc)

Periodic RSH / periodic VV10

(n/a)

⏳ queued

RSH analytic gradients, RSH + density fitting

(n/a)

⏳ queued

xc="wb97m(2)" raises a clear message explaining why it is held back: it is the one member of this family that is not a composition of stock libxc parts (its SCF base is a re-optimised B97M power series that libxc does not ship).

References

  • VV10. O. A. Vydrov, T. Van Voorhis, J. Chem. Phys. 133, 244103 (2010).

  • ωB97X-V. N. Mardirossian, M. Head-Gordon, Phys. Chem. Chem. Phys. 16, 9904 (2014).

  • ωB97M-V. N. Mardirossian, M. Head-Gordon, J. Chem. Phys. 144, 214110 (2016).

  • r2SCAN01. J. W. Furness, A. D. Kaplan, J. Ning, J. P. Perdew, J. Sun, J. Chem. Phys. 156, 034109 (2022).

  • revDSD-PBEP86. G. Santra, N. Sylvetsky, J. M. L. Martin, J. Phys. Chem. A 123, 5129 (2019).

  • D4 dispersion. E. Caldeweyher, C. Bannwarth, S. Grimme, J. Chem. Phys. 150, 154122 (2019).

Each functional’s route is registered in the bundled citation database, so a job that uses one auto-emits the right reference in its .bibtex and .references siblings; the entries above are the authoritative form.

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