Background / Formulation of the Model
The purpose of this section is to give a brief summary of the mathematics behind the linear ACE models.
Invariant Properties
To explain the main ideas in the simples non-trivial setting, we consider systems of indistinguishable particles. A configuration is an mset $R := \{ \bm r_j \}_j \subset \mathbb{R}^3$ with arbitary numbers of particles and we wish to develop representation of properties
\[ \varphi\big(R) \in \mathbb{R}\]
which are invariant under permutations (already implicit in the fact that $R$ is an mset) and under isometries $O(3)$. To make this explicit we can write this as
\[\varphi\big( \{ Q \bm r_{\sigma j} \}_j \big) = \varphi\big( \{ \bm r_{j} \}_j \big) \qquad \forall Q \in O(3), \quad \sigma \text{ a permutation}.\]
To that end we proceed in three steps:
Density Projection / Atomic Base
We define the "atomic density"
\[\rho({\bm r}) := \sum_j \delta({\bm r} - {\bm r}_j)\]
Then we choose a one-particle basis
\[\phi_v({\bm r}) = \phi_{nlm}({\bm r}) = R_n(r) Y_l^m(\hat{\bm r})\]
and project $\rho$` onto that basis,
\[A_{v} = A_{nlm} = \langle \phi_{nlm}, \rho \rangle = \sum_j \phi_{nlm}({\bm r}_j).\]
Density correlations
Next, we form the $N$-correlations of the density, $\rho^{\otimes N}$ and project them onto the tensor project basis,
\[ {\bm A}_{{\bm nlm}} = \Big\langle \otimes_{t = 1}^N \phi_{n_t l_t m_t}, \rho^{\otimes N} \Big\rangle = \prod_{t = 1}^N A_{n_t l_t m_t}.\]
The reason to introduce these is that in the next step, the symmetrisation step the density project would loose all angular information while the $N$-correlations retain most (though not all) of it.
Symmetrisation
Finally, we symmetrize the $N$-correlations, by integrating over the $O(3)$-Haar measure,
\[ B_{\bm nlm} \propto \int_{O(3)} {\bm A}_{\bm nlm} \circ Q \, dQ \]
Because of properties of the spherical harmonics one can write this as
\[ {\bm B} = \mathcal{U} {\bm A},\]
where ${\bm A}$ is the vector of 1, 2, ..., N correlations (the maximal $N$ is an approximation parameter!) and $\mathcal{U}$ is a sparse matrix (the coupling coefficients).
If one symmetrised all possible $N$-correlations then this would create a spanning set, but one can easily reduce this to an actual basis. This construction then yields a basis of the space of symmetric polynomials.
Notes:
- Because of permutation symmetry only ordered ${\bm v}$ tuples are retained
General Setting
TODO: introduce the general setting with general equi-variant properties and general symmetry groups.