Numerical Evaluation of NLDF Features

The packaged CIDER23X and CIDER26XC models use the Version-J algorithm. Its feature definition and exponent parameterization are given in Nonlocal Density Features (NLDF). The numerical construction follows the kernel expansion introduced for CIDER23X, which adapts the convolution method of Román-Pérez and Soler to density-dependent CIDER kernels. [1][2]

Kernel expansion

Suppressing the constant normalization of the feature, a Version-J NLDF is

\[G_i(\mathbf r) = \int \mathrm d^3\mathbf r'\, \Phi\!\left(a_0(\mathbf r'),a_i(\mathbf r), |\mathbf r-\mathbf r'|\right)n(\mathbf r'),\]

where

\[\Phi(a,b,R)=\exp[-(a+b)R^2].\]

The density-dependent exponents are represented on a finite set of positive values \(\{q_\alpha\}\). Source and target interpolation functions give

\[\Phi(a,b,R) \simeq \sum_{\alpha\beta} p^a_\alpha(a)\,\Phi_{\alpha\beta}(R)\,p^b_\beta(b), \qquad \Phi_{\alpha\beta}(R) = \exp[-(q_\alpha+q_\beta)R^2].\]

This is Equation 61 of the CIDER23X paper. It converts the feature into three operations:

\[\begin{split}\theta_\alpha(\mathbf r') &= p^a_\alpha(a_0(\mathbf r'))n(\mathbf r'), \\ F_\beta(\mathbf r) &= \sum_\alpha \int \mathrm d^3\mathbf r'\, \Phi_{\alpha\beta}(|\mathbf r-\mathbf r'|) \theta_\alpha(\mathbf r'), \\ G_i(\mathbf r) &\simeq \sum_\beta p^b_\beta(a_i(\mathbf r))F_\beta(\mathbf r).\end{split}\]

Only the middle line is spatially nonlocal, and every kernel in that line is independent of the density. CiderPress therefore evaluates a finite set of ordinary convolutions and performs local interpolation before and after them.

The potential applies the transpose of the same discrete operations. It first propagates \(\partial E_{\mathrm{xc}}/\partial G_i\) through the target coefficients, applies the adjoint convolution, and then differentiates the source coefficients and source density. Derivatives of \(p_\alpha(a)\) supply the response of the semilocal exponents \(a_0[n]\) and \(a_i[n]\).

Exponent coefficients and spatial convolutions

Exponent interpolation and spatial convolution are separate numerical choices. Their implementation objects are:

Numerical representations used in the NLDF evaluator

Operation	Implementation	Representation
Exponent coefficients	NLDFGaussianPlan	Projects \(\exp(-aR^2)\) onto the Gaussian functions at \(q_\alpha\) using their overlap matrix, as in CIDER23X Equations 63–65.
Exponent coefficients	NLDFSplinePlan	Tabulates the projection coefficients as functions of the exponent coordinate and evaluates them with cubic splines.
Molecular convolution	PyscfNLDFGenerator	Expands the spatial source and convolved fields in atom-centered Gaussian auxiliary bases.
Periodic convolution	LibCiderPW	Applies the fixed kernels on the distributed reciprocal-space grid.

The current PySCF default uses NLDFSplinePlan for the exponent coefficients. Its spatial convolution still uses the atom-centered Gaussian auxiliary bases described below. GPAW also uses NLDFSplinePlan, with an even-tempered exponent grid \(q_\alpha=q_0\lambda^\alpha\).

Molecular auxiliary-basis algorithm

The molecular path implements the CIDER23X construction of Equations 67–78. For each source channel, it performs the following sequence:

1. CiderGrids records the atomic radial and angular structure of PySCF’s partitioned quadrature grid.

2. LCAONLDFGenerator forms \(\theta_\alpha\) on that grid, resolves each atomic contribution into spherical harmonics, and projects the radial components onto an even-tempered Gaussian auxiliary basis.

3. ConvolutionCollection applies analytically evaluated Gaussian convolution integrals to obtain the coefficients of \(F_\beta\) in a second auxiliary basis.

4. LCAOInterpolatorDirect evaluates the convolved fields on the molecular quadrature grid. The target coefficients then produce \(G_i\).

The LCAONLDFGenerator.get_potential call in ciderpress/dft/lcao_nldf_generator.py reverses this sequence using the stored forward intermediates. Analytical nuclear gradients also differentiate the atomic partition, auxiliary projection, and grid interpolation; see PySCF Numerical and Derivative Implementation.

Periodic FFT and PAW algorithm

For a periodic cell, the Fourier transform of a fixed Gaussian kernel is

\[\widetilde\Phi_{\alpha\beta}(\mathbf k) = \left(\frac{\pi}{q_\alpha+q_\beta}\right)^{3/2} \exp\!\left[-\frac{|\mathbf k|^2} {4(q_\alpha+q_\beta)}\right].\]

ciderpress.gpaw.cider_fft constructs the source interpolation coefficients and passes the source channels to LibCiderPW. The compiled evaluator transforms them to reciprocal space, multiplies by \(\widetilde\Phi_{\alpha\beta}\), and transforms the convolved fields back to the real-space grid. Target interpolation produces the NLDFs. The reverse calls supply the smooth-grid potential and the reciprocal-kernel contribution to stress.

The smooth density in PAW lacks the all-electron source inside augmentation spheres. PASDW augments the source before the FFT so that the convolved field has the correct exterior behavior, then projects that field onto atomic support grids for the all-electron-minus-pseudo energy and potential correction. This is the construction in CIDER23X Equations 82–98. [1] Its equations and implementation objects are documented in GPAW FFT, PAW, and Radial Implementation.

Numerical parameters

Parameters that define the NLDF discretization

Path	Parameter	Effect
GPAW	qmax	Largest interpolation exponent, in inverse bohr squared.
GPAW	lambd	Ratio between adjacent exponents. With an automatically selected Nalpha, a smaller ratio gives a denser exponent grid.
GPAW	Nalpha	Number of exponent channels. The value is inferred from qmax, lambd, and the model exponents when omitted.
PySCF	alpha_min, alpha_max, aux_lambd	Exponent range and spacing used when constructing the molecular NLDF generator.
PySCF	lmax, aug_beta	Angular truncation and spacing of the atom-centered auxiliary basis.

The host quadrature or real-space grid adds another discretization to these kernel and auxiliary representations. PAW calculations also contain the finite PASDW projection bases described on the GPAW implementation page.

The features described above are labeled the Verion-J NLDF features. Version-I and Version-K definitions are listed in Nonlocal Density Features (NLDF). They are experimental features and are not used in the packaged Version-J functionals described here.

[1] (1,2)

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]

Guillermo Román-Pérez and José M Soler. Efficient Implementation of a van der Waals Density Functional: Application to Double-Wall Carbon Nanotubes. Phys. Rev. Lett., 103(9):096102, August 2009. doi:10.1103/PhysRevLett.103.096102.