Introduction to ACE Models

This section summarises the mathematics behind linear ACE parameterisations of invariant atomic properties. It is not required to use the ACEpotentials package 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,

\[ {\bm A}_{\mathbf{nlm}} = \Big\langle \otimes_{t = 1}^N \phi_{n_t l_t m_t}, \rho^{\otimes N} \Big\rangle = \prod_{t = 1}^N A_{n_t l_t m_t}.\]

The reason to introduce these is that in the next step, the symmetrisation step, the density project would lose all angular information while the $N$-correlations retain most (though not all) of it.

Symmetrisation

Finally, we symmetrise the $N$-correlations, by integrating over the $O(3)$-Haar measure,

\[ B_{\mathbf{nlm}} \propto \int_{O(3)} {\bm A}_{\mathbf{nlm}} \circ Q \, dQ \]

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 $B_{\mathbf{nl}i}$ with $i$ indexing the linearly independent basis functions from the $\mathbf{nl}$ block.