**Structure invariants estimate (%invariants command)**

Up to 10000 triplets relating reflections with normalized E values greater than a given threshold (strong triplets) are stored for active use in the phasing process. Also triplets (psi-zero triplets) relating two reflections with large |E| and one with |E| close to zero are generated: they are used to define a special figure of merit (PSCOMB).

Negative quartets are generated by combining the psi-zero triplets in pairs, and those with cross-magnitudes smaller than a given threshold are estimated by means of their first representation, as described by Giacovazzo (1976). These quartets are used to provide an important contribution to the CPHASE figure of merit.

Active triplets may be estimated according to Cochran’s (1955) P3 distribution; the concentration parameter of the von Mises distribution is given by:

Triplets can also be estimated according to their second representation (*i.e.* P10 formula, as described by Cascarano *et al.*, 1984). The concentration parameter of the new von Mises distribution is given by:

where q is a function (positive or negative) of all the magnitudes in the second representation of the triplet. The G values are rescaled on the C values and the triplets are ranked in decreasing order of G. The top relationships represent a better selection of triplets with phase value close to zero than that obtained when ranking according to C. These triplets will be actively used in the phase determination process.

Triplets characterized by a negative G represent a sufficiently good selection of relationships close to 180 degrees, to be used for the calculation of a powerful FOM (CPHASE). Triplets with G close to zero are expected to have values widely dispersed around 90 or 270 degrees and are used to compute an enantiomorph sensitive FOM. A similar FOM is also computed using quartets estimated with a very small concentration parameter.

As a default, triplets are estimated according to P10 formula.

The parameter C of the Cochran distribution (say P3) is suitably modified when the pseudo information in the section Normalization procedure is available. Then triplet phases are no longer expected to be around zero (see quoted references) and may lie anywhere between 0 and 2π.

**Directives in the %invariants command**

**The following directives must be added after the command %invariants in the input file to activate specific non-default procedures:
**

**gmin x**

Positive triplets with G <

*x*are not actively used.

Default value of

*x*is 0.6 (in any case 0.2 <

*x*≤ 1.0 is an accepted range of value) (see the Example 1 for its use).

**cochran **

To use the P3 formula. (By default P10 formula is used).

**emax ****x**

Maximum value of E for the weak reflections for the psi-zero search (up to a maximum of **nrpsizero **reflections). *x* > 0.3 not accepted.

Minimum value for the normalized structure factors in the sigma2 search. *x* < 1.0 not accepted.

The default value is the minimum value of |E| for the strongest reflections selected by the normalization procedure.

**npsizero**

To not calculate psi-zero triplets (default uses them).

**nrpsizero**** n**

The number of weakest reflections for the psi-zero search.

Default = number computed by normalization routine.

**nrtriplets n**

The number of strongest reflections for the sigma2 search (up to Emin = 1.0).

Default = number computed by normalization routine (nstrong).

Value less than nstrong/2 is not accepted.

**nlarge n**

To supply the number of largest reflections for the psi-zero search (default = number of strongest reflections used for sigma2 search).

nlarge > nrtriplets is not accepted.

**nquartets **

To not calculate negative quartets (default uses them).

**numk n**

The number of free-vectors used by P10 formula.

Default value is provided by EXPO2014 according to the structural complexity:

n = min (natom / msym + 30, 70)

where natom is the number of atoms in the cell and msym is the number of symmetry operators of the space group.

**Example**

**Example 1**

Use of gmin directive.

%structure mes %job MES - data from home diffractometer %data pattern mes.pow cell 8.588 9.931 11.105 90.0 93.754 90.0 content C 24 N 4 O 20 S 4 H 52 spacegroup p 21/c wavelength 1.5406 %extraction %normal %invariants gmin 0.4 %continue

**References**

Cascarano G., Giacovazzo C., Burla M.C., Nunzi A. & Polidori G. (1984). *Acta Cryst. *A**40**, 389-394.

Cochran W.(1955). *Acta Cryst. ***8**, 473-478.

Giacovazzo C. (1976).* Acta Cryst. *A**32**, 958-966.