From Exchange Models to Full XC

Exchange-only CIDER

The CIDER23X and CIDER24X files represent an exchange functional. A common use is a surrogate-hybrid construction

\[E_{\mathrm{xc}} = a E_{\mathrm{x}}^{\mathrm{CIDER}} + (1-a)E_{\mathrm{x}}^{\mathrm{PBE}} + E_{\mathrm{c}}^{\mathrm{PBE}},\]

with \(a=0.25\) for the PBE0-surrogate form used in the corresponding work. The mixing fraction and external semilocal terms are part of the functional definition. They must be supplied explicitly when an exchange-only model is initialized. [1][2]

CIDER26XC functional form

The CIDER26XC models return the full exchange-correlation energy. In the notation of the CIDER26XC work, the energy density contains an additive PBE baseline and learned exchange and correlation corrections,

\[e_{\mathrm{xc}}^{\mathrm{CIDER}} = \sum_\sigma e_{\mathrm{x}}^{\mathrm{PBE}}[n_\sigma] + e_{\mathrm{c}}^{\mathrm{PBE}}[n_\uparrow,n_\downarrow] + \sum_\sigma e_{\mathrm{x}}^{\mathrm{LDA}}[n_\sigma] f_{\mathrm{x}}(\mathbf{X}^{\mathrm{x}}_\sigma) + e_{\mathrm{c}}^{\mathrm{PBE}} f_{\mathrm{c}}\!\left(\tfrac{1}{2}\sum_\sigma \mathbf{X}^{\mathrm{c}}_\sigma\right).\]

Exchange is evaluated separately for the two spin channels. Correlation uses a spin-coupled PBE multiplier and the spin-averaged CIDER feature vector. The exchange descriptors are scale invariant. Correlation additionally receives density information to learn its nontrivial behavior under uniform coordinate scaling; see Uniform Scaling.

The YAML model contains its additive PBE contributions. The corresponding GPAW construction uses xmix=1.0, xkernel=None, and ckernel=None. CiderPress validates this composition so that the stored full-XC form is evaluated as trained. See Choosing a CIDER Functional for the complete call.

CIDER26XC feature vectors

The exchange feature vector \(\mathbf{X}^{\mathrm{x}}_\sigma\) contains the scale-invariant semilocal descriptors and the three version-j nonlocal density features used by the CIDER23X nonlocal meta-GGA models (Semilocal Features (SL) and Nonlocal Density Features (NLDF)). CIDER26XC represents the kinetic-energy information with the bounded indicator

\[t = \frac{\tau-\tau_0}{\tau+\tau_0},\]

where \(\tau\) is the kinetic-energy density and \(\tau_0\) its uniform electron gas value as defined in Semilocal Features (SL). The CIDER23X models use the iso-orbital indicator \(\alpha\) of Sun et al.[3] for this role. \(\alpha\) vanishes for single-orbital densities, which is convenient for nonempirical functional design, but \(t\) proved better behaved numerically within the CIDER26XC models and is bounded to \([-1,1]\). Combined with the reduced-gradient descriptor, the two carry the same iso-orbital information.

The correlation feature vector \(\mathbf{X}^{\mathrm{c}}_\sigma\) carries one additional descriptor, the density itself, written \(X_6 = n\) in the CIDER26XC work. In the serialized correlation feature list it follows the semilocal descriptors and precedes the nonlocal density features. The scale-invariant features enforce the exchange uniform-coordinate-scaling constraint. The density descriptor lets the correlation model represent its different uniform-coordinate-scaling behavior.

Before entering the Gaussian-process kernel, every feature is mapped onto a finite interval by the bounded transforms stored in the serialized model (implemented in ciderpress.dft.transform_data; the \(t\) indicator is handled by the SLTMap transform).

Self-consistent training

For an exchange-only target, exact exchange can be evaluated on a chosen density matrix. For a reaction-energy-trained full-XC functional, the density is part of the prediction: once the model is placed in an SCF loop, orbital relaxation changes both the features and the remaining Kohn–Sham energy contributions.

CIDER26XC iterates the training density to account for orbital relaxation. At iteration \(k\), descriptors and residual labels are constructed from the current densities, the model is refit, and self-consistent calculations with that model generate the next densities. The process is converged when fixed-density and self-consistent errors agree at the required resolution. Abdallah et al.[4]

CiderPress provides the descriptor, model, kernel, and mapping primitives used by this process. Dataset construction and orchestration are supplied by the surrounding training workflow; Training and Mapping Workflow describes the data expected by CiderPress.

Optional long-range correction

CIDER26XCCHEMD4 was trained with a D4 term removed from each target and adds the same term when the model is evaluated. D4 depends on the nuclear geometry and does not enter the electronic SCF potential. The PySCF interface includes the correction exactly once in the total energy and adds its analytical nuclear derivative to molecular gradients.

CIDER26XCCHEM uses its learned electronic XC contribution. CIDER26XCCHEMD4 combines that learned contribution with the D4 term described above, which supplies an explicit long-range asymptote.

[1]

Kyle Bystrom and Boris Kozinsky. Nonlocal machine-learned exchange functional for molecules and solids. Phys. Rev. B, 110:075130, Aug 2024. URL: https://link.aps.org/doi/10.1103/PhysRevB.110.075130, doi:10.1103/PhysRevB.110.075130.

[2]

Kyle Bystrom, Stefano Falletta, and Boris Kozinsky. Training machine-learned density functionals on band gaps. Journal of Chemical Theory and Computation, 20(17):7516–7532, 2024. PMID: 39178337. URL: https://doi.org/10.1021/acs.jctc.4c00999, doi:10.1021/acs.jctc.4c00999.

[3]

Jianwei Sun, Bing Xiao, Yuan Fang, Robin Haunschild, Pan Hao, Adrienn Ruzsinszky, Gábor I. Csonka, Gustavo E. Scuseria, and John P. Perdew. Density functionals that recognize covalent, metallic, and weak bonds. Phys. Rev. Lett., 111(10):106401, 2013. doi:10.1103/PhysRevLett.111.106401.

[4]

Mohamed Samy Abdallah, Zhuotao Jin, Boris Kozinsky, and Kyle Bystrom. Machine-learned exchange-correlation functionals in the cider framework and application to chemistry and surface science. Manuscript forthcoming; public identifier pending, 2026.