Additional symmetries for matrix-valued ACE bases about ace.jl

[MatthiasSachs]

I think, one way to approach this issue would be to

1. generate a span of function that satisfy the required symmetry property by applying the symmetrization operation

   $$ \overline{B}_k(\{r_i\}, \{r_j\}) = B_k(\{r_i\}, \{r_j\}) + S \circ B_k(\{-r_i\}, \{r_j\}),$$

   to an equivariant ACE basis $B_k,, k=1,\dots, N_{\rm basis}$.

2. follow the usual steps to convert this set of functions to a basis, i.e., compute the Gramian for the such obtained set of functions and then obtain coupling coefficients from the corresponding SVD.

However, I am not sure where exactly (and if at all?) in the ACE.jl code base we should include this.

[zhanglw0521]

If I understand correctly we encountered a similar (or same?) issue in ACEhamiltonians implementation, which raised from the fact that
$$H_{IJ} = H_{JI}^\ast,$$
or say,
$$H(r_{bond}, {r_{env}}) = [H(-r_{bond}, {r_{env}})]^*.$$

This was currently fixed in the ACEhamiltonians package rather than in ACE, by editing the A2Bmap of the mentioned $B$ basis. That is to say, assume $B(R) = UA(R)$, $S\circ B(R^{dual}) = \tilde{U}A(R^{dual}) = \tilde{U}PA(R)$, where $P$ are some permutation matrix, then we set
$$\bar{B} = (U+\tilde{U}P) A$$
to be the new symmetric basis.

This introduces potentially new linear dependence which can be removed by adopting another SVD.