Atomic Cluster Expansion (ACE)

Background

The energy of a system can be assumed that the total energy E of the system as a sum of atomic contributions \(E_i\)

\[ E = \sum_iE_i \]

So, it’s straightforward to write down the energy of a collection of atoms \(i= 1, . . . , N\) in a many-atom expansion:

\[\begin{split} \begin{aligned} E =\;& V_0 + \sum_i V^{(1)}(r_i) + \frac{1}{2}\sum_{ij} V^{(2)}(r_i, r_j) \\ &+ \frac{1}{3!}\sum_{ijk} V^{(3)}(r_i, r_j, r_k) \\ &+ \frac{1}{4!}\sum_{ijkl} V^{(4)}(r_i, r_j, r_k, r_l) + \cdots \end{aligned} \end{split}\]

\(r_i\) is the position of atom \(i\) and the potential \(V^{(2)}, V^{(3)},...\) are symmetric, uniquely defined, and zero if two or more indices take identical values. Commonly, \(V_0\) is a constant offset that can be set to zero and \(V^{(1)}\) is the chemical potential

So, the above equation can be written into atomic contributions for each separated energy:

\[\begin{split} \begin{aligned} E_i = &\sum_i V^{(1)}(r_i) + \frac{1}{2}\sum_{ij} V^{(2)}(r_i, r_j) \\ &+ \frac{1}{6}\sum_{ijk} V^{(3)}(r_i, r_j, r_k) \\ &+ \frac{1}{24}\sum_{ijkl} V^{(4)}(r_i, r_j, r_k, r_l) + \cdots \end{aligned} \end{split}\]

As you can see here, such atomic expansion requires higher order Tylor expansion, the convergence is slow. E,g. Bulk metal potential \(V^{(K)}\) up to \(K>15\) is required. Even with cutoff that just taking account of nearby atoms within cutoff \(r_c\), the evaluation of the leading termof order \(K + 1\) in the sum Eq. (3) scales as \(N_c^K\), where \(N_c\) corresponds to a typical number of neighbors within the cutoff sphere. e.g. for an accurate potentials one requires cutoffs that in a closed packed materials imply \(N_c ≈ 10^2 ... 10^3\). It’s challenging to sum expansions within acceptable time.

1.  ATOMIC CLUSTER EXPANSION

We first define the inner product in Hilbert space:

\[ <f|g> = \int f^\star(\sigma)g(\sigma)d\sigma \]

Next a set of orthogonal and complete basis functions \(\phi_v(r)\) with \(v = 0,1,2,... \) that depend only on a single bond \(r\) are

introduced:

\[\begin{split} \begin{aligned} &\int \phi_v(r)^\star\phi_u(r)dr = \delta_{vu}\\ &\sum_v\phi_v(r)^\star\phi_v(r') =\delta(\mathbf{r}-\mathbf{r'}) \end{aligned} \end{split}\]

[!NOTE]

Proof of completeness: Set of basis functions \(\phi_v\) is capable of describing any possible local atomic environment.

If the basis is complete: \(f(r)=\sum_v c_v\phi_v(r)\), where \(c_v\) is the coefficient for the basis function

To find a specific coefficient, we project \(f(r)\) into the basis, which is the inner product, so, \(c_v=\int \phi_v^\star(r^\prime)f(r^\prime)dr^\prime\)

The we substitute the \(c_v\) obtained by projection to the original formula: \(f(r)=\sum_v\Big[\int \phi_v^\star(r^\prime)f(r^\prime)dr^\prime\Big]\phi_v(r)=\sum_v\int[\phi_v^\star(r^\prime)\phi_v(r)]f(r^\prime)dr^\prime\)

And \(\phi_v^\star(r^\prime)\phi_v(r)=\delta(r^\prime-r)\)

Define the cluster expansion

Here we define the basis functions for the expansion of the atomic energy from the product of single-bond basis functions. By choosing \(\phi_0= 1\), a hierarchical expansion is obtained.

A cluster \(\alpha\) with \(K\) elements contains \(K\) bonds \(\alpha = (j_{1i},j_{2i},. . . , j_{Ki})\), where the order of entries in \(\alpha\) does not matter, and the vector \(v = (v_1,v_2,. ..,v_K )\) contains the list of single-bond basis functions in the cluster. Only single-bond basis functions with v > 0 are considered in ν. The cluster basis function is given by:

\[ \Phi_{\alpha v} = \phi_{v_1}(r_{j_1i})\phi_{v_2}(r_{j_2i})...\phi_{v_K}(r_{j_Ki}) \]

And the orthogonality and completeness of the one-bond basis functions can be transferred to cluster basis when \(0\le K\le N-1\), \(\alpha \) is any arbitrary cluster:

\[\begin{split} \begin{aligned} \langle \Phi_{\alpha v} \mid \Phi_{\beta \mu} \rangle &= \delta_{\alpha\beta}\,\delta_{v\mu},\\ 1 + \sum_{\gamma \subseteq \alpha}\sum_v \Phi_{\gamma v}^*(\boldsymbol{\sigma})\, \Phi_{\gamma v}(\boldsymbol{\sigma}') &= \delta(\boldsymbol{\sigma}-\boldsymbol{\sigma}'). \end{aligned} \end{split}\]

We abbreviate the left hand inner product as a kernel function: \(k(\sigma, \sigma')= 1+ \sum_{\gamma\subseteq \alpha}\sum_v \Phi^\star_{\gamma v}(\sigma)\Phi_{\gamma_v}(\sigma'),\) then the expansion of atomic energy can be written as:

\[ E_i(\sigma) =\langle k(\sigma, \sigma')\mid E_i(\sigma')\rangle = J_0+\sum_{\alpha v} J_{\alpha v}\Phi_{\alpha v} (\sigma) \]

Where the expansion coefficients \(J_{\alpha v}\) are obtained by projection:

\[ J_{\alpha v} = \langle \Phi_{\alpha v}(\sigma)\mid E_i(\sigma)\rangle \]