Nonlocal Density Features (NLDF)

Nonlocal density features (NLDFs) describe the density in a neighborhood of each integration point. A fixed-kernel convolution has the form

\[G[n](\mathbf{r}) = \int \mathrm{d}^3\mathbf{r}'\, k(\mathbf{r}-\mathbf{r}') n(\mathbf{r}')\]

An exponentially decaying fixed kernel introduces a dimensional length scale that conflicts with uniform coordinate scaling. CIDER instead uses density-dependent kernel exponents. For a scaled density \(n_\lambda(\mathbf{r})=\lambda^3 n(\lambda\mathbf{r})\), if \(k[n_\lambda](\mathbf{r}-\mathbf{r}')=\lambda^j k[n](\lambda(\mathbf{r}-\mathbf{r}'))\), then

\[G[n_\lambda](\mathbf{r}) = \lambda^j G[n](\lambda\mathbf{r})\]

The resulting power law can be combined with a normalizer to produce the scaling behavior required by the functional. Typically, this means constructing a scale-invariant normalized feature, as explained in Uniform Scaling.

Three NLDF versions are implemented in the molecular feature framework: i, j, and k. Version i and j features can be evaluated together through the combined ij setting. The packaged CIDER23X and CIDER26XC models use version j. GPAW supports version j. The other versions and alternative kernel specifications are model-development interfaces.

Version J

Version J is the NLDF implemented in Bystrom and Kozinsky[1]. In the raw form computed by the settings layer, it reads

\[G_i[n](\mathbf{r}) = \int \text{d}^3\mathbf{r}' \exp(-(a_0[n](\mathbf{r}')+a_i[n](\mathbf{r}))|\mathbf{r}-\mathbf{r}'|^2) n(\mathbf{r}')\]

Equation 18 of Bystrom and Kozinsky[1] includes a constant prefactor determined by the two exponent parameters. CiderPress represents the convolution and its normalization in separate serialized objects: NLDFSettingsVJ defines the raw feature, and the model’s normalizer defines its uniform-electron-gas value.

Equations 20 and 22 of Bystrom and Kozinsky[1] write the exponent with a constant term \(B_i\) and one modulation term \(C_i\), using the reduced gradient for a GGA functional and the kinetic-energy density for a meta-GGA functional:

\[a_i[n](\mathbf{r}) = \pi\left(\frac{n}{2}\right)^{2/3} \left[B_i + C_i\left(\frac{|\nabla n|^2}{8n\tau_0}\right)\right]\]

\[a_i[n](\mathbf{r}) = \pi\left(\frac{n}{2}\right)^{2/3} \left[B_i + C_i\left(\frac{\tau}{\tau_0}-1\right)\right]\]

Here \(\tau\) is the kinetic-energy density and \(\tau_0\) its uniform electron gas value, both defined in Semilocal Features (SL). The settings layer generalizes these to a single expression that carries both modulations,

\[a_i[n](\mathbf{r}) = \pi\left(\frac{n}{2}\right)^{2/3} \left[B_i + D_i\left(\frac{|\nabla n|^2}{8n\tau_0}\right) + C_i\left(\frac{\tau}{\tau_0}-1\right)\right]\]

with \(D_i\) the gradient coefficient. The packaged CIDER23X and CIDER26XC meta-GGA models set \(D_i=0\), recovering Equation 22.

In the serialized settings the parameter array is [a0, grad_mul, tau_mul], and tau_mul is ignored for a GGA exponent. Only the constant term maps directly: \(B_i\) is a0, while \(D_i\) and \(C_i\) are grad_mul and tau_mul multiplied by \(1.2\,(6\pi^2)^{2/3}/\pi \approx 5.803\), the conversion applied in get_cider_exponent().

Using different parameters for each \(i\) produces several feature length scales in one auxiliary expansion. The source exponent \(a_0\) is shared by all features. Both the GGA and meta-GGA constructions obey

\[a_i[n_\lambda](\mathbf{r}) = \lambda^2 a_i[n](\lambda \mathbf{r})\]

This exponent scaling makes \(G_i[n](\mathbf{r})\) scale invariant:

\[G_i[n_\lambda](\mathbf{r}) = G_i[n](\lambda\mathbf{r})\]

The settings format can multiply the source density by another semilocal quantity \(b(\mathbf{r}')\). The multiplier contributes its uniform scaling power to the raw feature. This option is experimental and is not used by the packaged functionals.

The settings layer defines the squared-exponential specification se and the se_ar2, se_a2r4, and se_erf_rinv variants. Rational-kernel variants use the vdw_param setting. The packaged models select se. Kernel strings, their ordered parameters, and their normalization form part of the serialized model; see NLDFSettingsVJ.

Version I

Version I uses the source exponent \(a_0[n](\mathbf r')\) and omits the target exponent \(a_i[n](\mathbf r)\). Several integration kernels are available, giving the general form

\[G_*[n](\mathbf{r}) = \int \text{d}^3\mathbf{r}' k_*(a_0[n](\mathbf{r}'), |\mathbf{r}-\mathbf{r}'|) n(\mathbf{r}')\]

The \(*\) symbol stands for the kernel specification. With \(R=|\mathbf r-\mathbf r'|\), the scalar kernels implemented by NLDFSettingsVI are:

Version-I scalar kernels
Setting	\(k_*(a,R)\)
`se`	\(\exp(-aR^2)\)
`se_r2`	\(R^2\exp(-aR^2)\)
`se_apr2`	\(aR^2\exp(-aR^2)\)
`se_ap`	\(a\exp(-aR^2)\)
`se_ap2r2`	\(a^2R^2\exp(-aR^2)\)
`se_lapl`	\((4a^2R^2-6a)\exp(-aR^2)\)

The se_lapl kernel is the three-dimensional Laplacian of se with respect to the displacement vector. Earlier development versions used the coefficient -2a for this experimental feature. No packaged functional uses Version I or se_lapl; the Version-I specification may change in a future release.

Version I also provides vector features of the form

\[\mathbf{g}_*[n](\mathbf{r}) = \int \mathrm{d}^3\mathbf{r}'\, (\mathbf{r}'-\mathbf{r}) k_*(a_0[n](\mathbf{r}'), |\mathbf{r}-\mathbf{r}'|) n(\mathbf{r}')\]

The implemented vector kernels are:

se_grad uses \(k_*(a,R)=a\exp(-aR^2)\).
se_rvec uses \(k_*(a,R)=\exp(-aR^2)\).

To construct rotationally invariant descriptors, the vector integrals \(\mathbf{g}_*[n](\mathbf{r})\) must be dotted with the density gradient or with another vector integral. For example, the se_grad vector can be dotted with itself to form the scalar \(G=\mathbf{g}_*\cdot\mathbf{g}_*\).

Version I also accepts a semilocal source multiplier \(b(\mathbf r')\).

Version-I variants are experimental model-construction options. A custom combination defines a distinct descriptor and requires independent forward/adjoint, scaling, and low-density validation.

Version K

Version K is a modification of Version J meant to remove the need for the squared-exponential kernel to depend on the density at \(\mathbf{r}'\). However, without the dependence on \(a_0[n](\mathbf{r}')\), the NLDFs can have large contributions from the core electrons in the valence region. Version K therefore multiplies the density by a function that decays when \(a_i(\mathbf{r})\ll a_0(\mathbf{r}')\):

\[G_i[n](\mathbf{r}) = \int \mathrm{d}^3\mathbf{r}'\, \exp\!\left(-a_i[n](\mathbf{r})|\mathbf{r}-\mathbf{r}'|^2\right) \exp\!\left(-\frac{3a_0[n](\mathbf{r}')}{2a_i[n](\mathbf{r})}\right) n(\mathbf{r}')\]