The error metric (objective function)

Functional form

The error metric, or objective function, \(f(\mathbf{x})\) is defined in terms of \(K\) criteria, each of which specifies the desired attributes of a bound or resonance state:

The weight \(w_k\) scales errors in this criterion, relative to other criteria.
The energy \(E_k\).
The set of possible spins \(\{J_k^\pi\}\).
The scaling factor \(h_k\) for the half-width error, relative to the energy error; this can be set to zero if there is no half-width.
The half-width \(\Gamma_k\).

The objective function is the weighted sum of the difference between the criteria and the MCAS output states.

\begin{align} f(\mathbf{x}) &= \sum_{k=1}^K w_k \cdot \left[f_E(E_{mcas}, E_k) + h_k \cdot f_\Gamma(\Gamma_{mcas}, \Gamma_K) \right] + f_J(J_{mcas}^\pi, \{J_k^\pi\}) \\ f_E(E_{mcas}, E_k) &= \lvert E_{mcas} - E_k \rvert \\ f_\Gamma(\Gamma_{mcas}, \Gamma_k) &= \lvert \Gamma_{mcas} - \Gamma_k \rvert \\ f_J(J_{mcas}^\pi, \{J_k^\pi\}) &= \begin{cases} 0 & \text{if } J_{mcas}^\pi \in \{J_k^\pi\} \\ 10^6 & \text{otherwise} \end{cases} \end{align}

Note

There is a very large penalty for states with incorrect spin, which should effectively exclude any solutions from including an incorrect spin.

Limitations of experimental data

For resonance states, available experimental data may comprise energies and widths, while for bound states only energies are available. Width measurements are limited by experimental precision, and are typically binned. Sufficiently thin resonances can occur within a single bin and are designated to have width equivalent to that of the entire bin, which represents an upper limit for \(\Gamma\) rather than being a precise value.

Note

It may also be desirable to specify an upper bound for \(E\) rather than a precise value.

Intruder states are states whose energies have not been measured in an experiment. The model may output energies that do not match any of the provided criteria; if it’s a low-spin orbit then it might be a previously-undetected state, if it’s a high-spin orbit then it’s likely a mistake in the model and should be ignored. In either case, the model should not be penalised.

Note

Clarify that the calculation/output should not be penalised. It’s not apparent from the functional form, as described above.

Selecting appropriate states for comparison

Each criterion \(k \in \{1, 2, \dots, K\}\) must be paired with a state from the MCAS output in order to evaluate the objective function \(f(\mathbf{x})\). This pairing must respect the following properties:

There should be no states below the lowest measured bound state (the ground state).
States should only be paired if they have matching spin parities. Note that this does not necessarily mean there is a unique set of paired states, because the set \(\{J_k^\pi\}\) may (a) be empty; or (b) contain multiple elements.