Figures of Merit in the Sir Program

Abstract

Given several sets of phases it would be rather time consuming to compute and interpret all the corresponding electron density maps to see which yield the correct structure. It is instead easier to compute some appropriate functions, called figures of merit (FOM); their values are expected to be extreme for the correct solution and thus allow an a-priori estimate of the goodness of each phase set. The most commonly used FOM’s are:

• 1) ABS (absolute FOM) represents a measure of the internal consistency of the employed triplet relationships in estimating the phases. It is defined as

  $$ABS = \frac{{\sum {_h{a_h}} }}{{\sum {_h < {a_h} > } }} = \frac{A}{{{A_e}}}$$

  ((1))

  for a correct structure A should be close to the theoretically estimated A_e and ABS≈1.0. In practice it has been found that often for the correct set of phases A>Ae and ABS values between 0.9 and 1.3 indicate a promising phase set. Higher values indicate an overconsistency and are typical of some troublesome structures.

• 2) R _α FOM; it is a measure of how much triplets deviate from their expected statistical behavior and is defined as

  $$ {R_\alpha } = \frac{{\sum\nolimits_h {\left| {{\alpha _h} - < {\alpha _h} > } \right|} }}{{\sum\nolimits_\alpha { < {\alpha _h} > } }} $$

  ((2))

  it should be minimum for the correct set of phases.