added k and r space radial basis

python/rascaline/rascaline/utils/__init__.py

@@ -3,7 +3,12 @@
 import metatensor
 
 from .power_spectrum import PowerSpectrum  # noqa
-from .splines import RadialIntegralFromFunction, RadialIntegralSplinerBase  # noqa
+from .splines import (  # noqa
+    KSpaceSpliner,
+    RadialIntegralFromFunction,
+    RadialIntegralSplinerBase,
+    RealSpaceSpliner,
+)
 
 
 # path that can be used with cmake to access the rascaline library and headers

python/rascaline/rascaline/utils/atomic_density.py

@@ -0,0 +1,19 @@
+from abc import ABC
+
+
+class AtomicDensityBase(ABC):
+    ...
+
+
+class GaussianDensity(AtomicDensityBase):
+    def __init__(self, atomic_gaussian_width: float):
+        self.atomic_gaussian_width = atomic_gaussian_width
+
+
+class DeltaDensity(AtomicDensityBase):
+    ...
+
+
+class LODEDensity(AtomicDensityBase):
+    def __init__(self, potential_exponent: int):
+        self.potential_exponent = potential_exponent

python/rascaline/rascaline/utils/radial_basis.py

@@ -0,0 +1,171 @@
+from abc import ABC, abstractmethod
+from typing import Union
+
+import numpy as np
+
+
+try:
+    from scipy.integrate import quad
+
+    HAS_SCIPY = True
+except ImportError:
+    HAS_SCIPY = False
+
+
+class RadialBasisBase(ABC):
+    """
+    Base class for evaluating the radial basis.
+
+    :parameter orthonormalization_cutoff: Provide value if the radial integral should be
+        orthonormalized. If :py:obj:`None` no orthonormalization will be performed.
+    """
+
+    def __init__(self, orthonormalization_cutoff: float):
+        self.orthonormalization_cutoff = orthonormalization_cutoff
+
+    @abstractmethod
+    def compute(
+        self, n: float, ell: float, integrand_positions: Union[float, np.ndarray]
+    ) -> Union[float, np.ndarray]:
+        """Method calculating the radial basis.
+
+        Explicitly implemented in child classes."""
+        ...
+
+    def compute_derivative(
+        self, n: float, ell: float, integrand_positions: np.ndarray
+    ) -> np.ndarray:
+        """Derivative of the radial basis.
+
+        Used for radial integrals with delta like atomic densities."""
+        displacement = 1e-6
+        mean_abs_positions = np.abs(integrand_positions).mean()
+
+        if mean_abs_positions < 1.0:
+            raise ValueError(
+                "Numerically derivative of the radial integral can not be performed "
+                "since positions are too small. Mean of the absolute positions is "
+                f"{mean_abs_positions:.1e} but should be at least 1."
+            )
+
+        radial_basis_pos = self.compute(n, ell, integrand_positions + displacement / 2)
+        radial_basis_neg = self.compute(n, ell, integrand_positions - displacement / 2)
+
+        return (radial_basis_pos - radial_basis_neg) / displacement
+
+    def compute_gram_matrix(
+        self,
+        max_radial: int,
+        max_angular: int,
+    ) -> np.ndarray:
+        """Orthornomalize the basis.
+
+        :returns: orthornomalization matrix of shape (max_angular + 1, max_radial,
+            max_radial)
+        """
+
+        if not HAS_SCIPY:
+            raise ValueError("Orthornomalization requires scipy!")
+
+        # Gram matrix (also called overlap matrix or inner product matrix)
+        gram_matrix = np.zeros((max_angular + 1, max_radial, max_radial))
+
+        def integrand(
+            integrand_positions: np.ndarray,
+            n1: int,
+            n2: int,
+            ell: int,
+        ) -> np.ndarray:
+            return (
+                integrand_positions**2
+                * self.compute(n1, ell, integrand_positions)
+                * self.compute(n2, ell, integrand_positions)
+            )
+
+        for ell in range(max_angular + 1):
+            for n1 in range(max_radial):
+                for n2 in range(max_radial):
+                    gram_matrix[ell, n1, n2] = quad(
+                        func=integrand,
+                        a=0,
+                        b=self.orthonormalization_cutoff,
+                        args=(n1, n2, ell),
+                    )[0]
+
+        return gram_matrix
+
+    def compute_orthonormalization_matrix(
+        self,
+        max_radial: int,
+        max_angular: int,
+    ) -> np.ndarray:
+        """Compute orthonormalization matrix
+
+        :returns: orthornomalization matrix of shape (max_angular + 1, max_radial,
+            max_radial)
+        """
+
+        gram_matrix = self.compute_gram_matrix(max_radial, max_angular)
+
+        # Get the normalization constants from the diagonal entries
+        normalizations = np.zeros((max_angular + 1, max_radial))
+
+        for ell in range(max_angular + 1):
+            for n in range(max_radial):
+                normalizations[ell, n] = 1 / np.sqrt(gram_matrix[ell, n, n])
+
+                # Rescale orthonormalization matrix to be defined
+                # in terms of the normalized (but not yet orthonormalized)
+                # basis functions
+                gram_matrix[ell, n, :] *= normalizations[ell, n]
+                gram_matrix[ell, :, n] *= normalizations[ell, n]
+
+        orthonormalization_matrix = np.zeros_like(gram_matrix)
+        for ell in range(max_angular + 1):
+            eigvals, eigvecs = np.linalg.eigh(gram_matrix[ell])
+            orthonormalization_matrix[ell] = (
+                eigvecs @ np.diag(np.sqrt(1.0 / eigvals)) @ eigvecs.T
+            )
+
+        # Rescale the orthonormalization matrix so that it
+        # works with respect to the primitive (not yet normalized)
+        # radial basis functions
+        for ell in range(max_angular + 1):
+            for n in range(max_radial):
+                orthonormalization_matrix[ell, :, n] *= normalizations[ell, n]
+
+        return orthonormalization_matrix
+
+
+class GTOBasis(RadialBasisBase):
+    """Primitive (not normolized nor orthonormlized) GTO radial basis."""
+
+    def __init__(self, max_radial, cutoff):
+        super().__init__(orthonormalization_cutoff=np.inf)
+        self.max_radial = max_radial
+        self.cutoff = cutoff
+        self.sigmas = np.ones(self.max_radial, dtype=float)
+
+        for i in range(1, self.max_radial):
+            self.sigmas[i] = np.sqrt(i)
+        self.sigmas *= self.cutoff / self.max_radial
+
+    def compute(
+        self, n: float, ell: float, integrand_positions: Union[float, np.ndarray]
+    ) -> Union[float, np.ndarray]:
+        return integrand_positions**n * np.exp(
+            -0.5 * (integrand_positions / self.sigmas[n]) ** 2
+        )
+
+    def compute_derivative(
+        self, n: float, ell: float, integrand_positions: Union[float, np.ndarray]
+    ) -> Union[float, np.ndarray]:
+        return n * integrand_positions ** (n - 1) * self.compute(
+            n, ell, integrand_positions
+        ) - integrand_positions / self.sigmas[n] ** 2 * self.compute(
+            n, ell, integrand_positions
+        )
+
+
+class MonomialBasis(RadialBasisBase):
+    ...