SOAP and LODE radial integrals¶
On this page, we describe the exact mathematical expression that are implemented in the radial integral and the splined radial integral classes i.e. Splined radial integrals. Note that this page assumes knowledge of spherical expansion & friends and currently serves as a reference page for the developers to support the implementation.
Preliminaries¶
In this subsection, we briefly provide all the preliminary knowledge that is needed to
understand what the radial integral class is doing. The actual explanation for what is
computed in the radial integral class can be found in the next subsection (1.2). The
spherical expansion coefficients
the atomic density function
as implemented in Atomic density, often chosen to be a Gaussian or Delta function, that defined the type of density under consideration. For a given central atom in the system, the total density function around is then defined as .the radial basis functions
as implementated Radial basis functions, on which the density is projected. To be more precise, the actual basis functions are of the formwhere
are the real spherical harmonics evaluated at the point , i.e. at the spherical angles that determine the orientation of the unit vector .
The spherical expansion coefficient
In practice, the atom centered density
Thus, instead of having to compute integrals for arbitrary densities
which are completely specified by
the density function
the radial basis
the position of the neighbor atom
relative to the center atom
The radial integral class¶
In the previous subsection, we have explained how the computation of the spherical expansion coefficients can be reduced to integrals of the form
If the atomic density is spherically symmetric, i.e. if
The key point is that the dependence on the vectorial position
Delta Densities¶
Here, we consider the especially simple special case where the atomic density function
Thus, in this particularly simple case, the radial integral is simply the radial basis
function evaluated at the pair distance
Gaussian Densities¶
Here, we consider another popular use case, where the atomic density function is a Gaussian. In featomic, we use the convention
The prefactor was chosen such that the “L2-norm” of the Gaussian
but does not affect the following calculations in any way. With these conventions, it can be shown that the integral has the desired form
with
where
Derivation of the General Case¶
We now derive an explicit formula for radial integral that works for any density. Let
and thus we have the desired form
where
Derivation of the explicit radial integral for Gaussian densities¶
Denoting by
where the first factor no longer depends on the integration variable
Analytical Expressions for the GTO Basis¶
While the above integrals are hard to compute in general, the GTO basis is one of the few sets of basis functions for which many of the integrals can be evaluated analytically. This is also useful to test the correctness of more numerical implementations.
The primitive basis functions are defined as
In this form, the basis functions are not yet orthonormal, which requires an extra linear transformation. Since this transformation can also be applied after computing the integrals, we simply evaluate the radial integral with respect to these primitive basis functions.
Real Space Integral for Gaussian Densities¶
We now evaluate
the result of which can be conveniently expressed using
where