Electronic Features in CiderPress

CIDER models predict a grid-resolved exchange or exchange-correlation energy density from an electronic feature vector. Density-based models use functionals of \(n(\mathbf r)\); generalized Kohn–Sham models may also depend on the one-particle density matrix \(n_1(\mathbf r,\mathbf r')\):

\[E_{\mathrm{xc}} = \int d^3\mathbf r\, e_{\mathrm{xc}}\!\left(\mathbf x[n,n_1](\mathbf r)\right).\]

The feature representation determines what spatial information is available to the regression, which exact scaling behavior can be imposed, which backends can evaluate the model, and which derivatives are available. The serialized feature representation is part of the functional definition.

Feature families

Electronic information exposed by each feature family

Family

Inputs

Information represented

Packaged use

SL

Density, gradient, and optionally kinetic-energy density

Local GGA or meta-GGA environment

Present in every packaged family

NLDF

Density integrated through density-dependent real-space kernels

Rotationally invariant finite-neighborhood density shape

CIDER23X and CIDER26XC

SDMX

Smoothed one-particle density matrix

Approximate exchange-hole and orbital information

CIDER24X

Every packaged model begins with a semilocal block. Its density ingredients also enter the energy baselines, kernel length scales, normalization factors, and low-density regularization.

Representation in the code

For each grid point, CiderPress evaluates raw features \(\mathbf X_0\), applies the model’s physical normalizers \(\mathcal N\), and then applies the bounded feature maps \(\mathcal T\) used by the regression:

\[\mathbf X_0(\mathbf r) \xrightarrow{\mathcal N} \widetilde{\mathbf X}_0(\mathbf r) \xrightarrow{\mathcal T} \mathbf X_1(\mathbf r) \xrightarrow{f_{\mathrm{ML}}} e_{\mathrm{xc}}(\mathbf r).\]

The corresponding objects are:

Feature representation objects

Stage

Class

Role

Feature declaration

FeatureSettings

Stores the ordered semilocal, NLDF, or SDMX settings used by a model

Physical normalization

FeatNormalizerList

Converts the raw feature powers to the representation expected by the energy form

Bounded feature map

FeatureList

Maps normalized features to the coordinates of one regression kernel

Mapped evaluator

MappedXC or MappedXC2

Evaluates the mapped exchange or full-XC model and its derivatives

SemilocalSettings, NLDFSettingsVJ, and SDMXSettings define the individual feature blocks. Their numerical plans are implemented by SemilocalPlan, NLDFSplinePlan, and SDMXPlan.

The settings, normalizers, and mapped kernels are serialized together in a CIDER model file. The PySCF and GPAW interfaces reconstruct them when the functional is loaded.

The equations and parameter conventions for each block are given below. Uniform-scaling constraints are derived in Uniform Scaling, and the exchange and correlation energy forms used by CIDER26XC are defined in From Exchange Models to Full XC. Backend support for the packaged families is listed in Choosing a CIDER Functional.

Feature definitions