# Introduction to ACE Models

The purpose of this section is to give a brief summary of the mathematics behind linear ACE parameterisations of invariant atomic properties. It is not required to use the `ACE1pack`

and `ACE1`

packages and can be skipped on a first reading.

### Invariant Properties

To explain the main ideas in the simplest 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

### Linear Dependence

The construction described above introduces a lot of linear dependence which is removed in the ACE basis construction in a mixed symbolic / numerical procedure. In the end we no longer index the symmetrized basis functions as $B_{\bm nlm}$ but as $B_{{\bm nl}i}$ with $i$ indexing the linearly independent basis functions from the ${\bm nl}$ block.