**Fourier map calculation and crystal structure optimization (%fourier command)**

The sets of phases generated by the tangent routine are passed to the fast fourier transform routine written by L.F.Ten Eick (1977) and subsequently modified by the MULTAN team (Main, 1978; Main* et al.,* 1980).

In the default run, the best phasing set selected by CFOM is processed. When the first ranked CFOM set does not provide a reasonable structure model, see the ALLTRIALS strategy for exploring all the stored phasing sets.

The E-map, whose coefficients are the normalized experimental structure factor moduli and the phases are obtained by Direct Methods, is calculated. The map is then chemically interpreted and the structure model is recovered.

Special positions are handled, peaks very close to symmetry elements are moved onto symmetry elements, the site symmetry is defined and the crystallographic occupancy factor is evaluated.

As a next step, for improving the structure model derived from the E-map, a Fourier recycling strategy is applied and then a structure model optimization process is carried in a default way by

1. suitably weighted least squares (*w*LSQ) (Altomare *et al**.*, 2006). The procedure is automatically applied in case of inorganic compounds;

2. the resolution bias correction algorithm (RBM) (Altomare *et al.*, 2008a, 2008b; 2009, 2010a, 2010b). It represents the default choice in case of organic and metal organic compounds.

At the end of the Expo2014 run, non default model optimization strategies can be applied if the model is incomplete and/or very rough.

It is possible to graphically choose different RBM procedures for model optimization by **Refine > Resolution Bias Modification (RBM)** from the upper EXPO2014 menu.

A non standard approach (directive **edmo**) has been introduced in EXPO2014 to improve the phasing process and consequently for optimizing the electron density map. Based on real space techniques (Altomare *et al.*, 2002) originally used for phase extension and refinement of proteins (Giacovazzo & Siliqi, 1997; Burla *et al.*, 2000), it combines an Electron Density Modification (EDM) procedure with Fourier map calculations, enabling to reduce the errors on the Direct Methods phases of the strong reflections (*i.e.*, with E>1.0) and to extend the phases also to the subset of reflections with E larger than 0.8. The structure model obtained by the E-map calculated with all the phased reflections is automatically refined by the structure model optimization process. The procedure can be carried out both for X-ray and neutron data; in this last case it has to be applied only in case of crystal structures with not negative scattering factor atoms.

**Directives in the %fourier command**

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

**edmo**

To enable the Electron Density Modification (EDM) approach, followed by an automatic structure model optimization procedure. Except for the Fourier directives **set** and **recycle 0**, all the rest of the Fourier directives are not valid for the EDM procedure. The directive **recycle 0** has to be used to require the E-map calculation only (the automatic structure model optimization is not carried out).

**fragment**** string**

To supply a known structural fragment to be completed. s*tring* is the name of the file in which, for each atom, are stored: chemical element, atomic coordinates (*x*, *y*, *z*), isotropic thermal factor.

**fwhm ****D**

To change the value of *D* which is the parameter for the reflection overlapping condition: two reflections are considered to be in overlapping if 2θ1 – 2θ2 < *D**FWHM (2θ1 and 2θ2 are the peak positions of the two reflections).

Default value is approximately 0.9 (depending on the ratio between the number of observations and the number of parameters to refine).

**grid x**

To define the grid of the Fourier map. The default value of

*x*is computed by the program (approximately 1/3 Å).

**integration x n**

*x*is the radius of integration for searching peaks in the electron density map. (default value is 0.5).

*n*may be 1 for using the gaussian fitting of the Fourier peak in the map or 2 for using the peak average density. (default value is 2).

**layx**

To print the Fourier map in section of constant x. This directive can be used only if **map** directive is given.

**layy**

To print the Fourier map in section of constant y. This directive can be used only if **map** directive is given.

**layz**

To print the Fourier map in section of constant z. This directive can be used only if **map** directive is given.

**level ****n**

All numerical values in the E-map greater than *n* will be underlined with **** in the line-printer output to facilitate contouring.

The default is 100. The map is automatically scaled between ± 999 (approximately) at the grid points at which it is calculated.

Note, however, that it is printed between ± 99. (This directive can be used only if associated to **map** directive).

**limits ****x1 x2 x3**

To print the E-map from 0 to *x1* along x, from 0 to *x2* along y, from 0 to *x3* along z. (This directive can be used only if associated to the **map** directive).

**map**

To print the E-map on the line-printer. The printing of the E-map is rarely used.

The output is in sections of constant y with x across the page and z running down it on a grid of about three points per angstrom.

The maximum number of grid points across the page is 36 (i.e. about 12 angstroms in length). If more are needed, a new page is started.

**peaks ****n**

The number of peaks to be searched for in the E-map. The default is the number of peaks which

∑(occ(i)*m) = 1.3 * *n*

where occ(i) is the crystallographic site occupancy factor of the i-th peak, m is the number of symmetry operators, *n* is the number of non-hydrogen atoms in the unit cell.

**radius El x**

The value

*x*of the radius assigned to the chemical element

*El*to evaluate the connectivity.

**recycle n**

To stop the Fourier recycling procedure after the cycle number

*n*. If

*n*is equal to zero only the E-map is calculated.

**set n**

The serial number of the phasing set from which the E-map is calculated. If ‘

**set**

*n*’ is not specified, the set with the highest CFOM combined figure of merit is used, and then if the %fourier command is used again r, the set with the next highest combined CFOM will be used.

**sogf x**

The percentage of the reflections with the largest |F| structure factor moduli actively used in the Fourier recycling routine.

The reflections with the smallest |F| values corresponding to the rejected percentage are considered with zero weights in the refinement process (default value is 100.0).

**sogi x**

The percentage of the most intense groups of overlapping reflections actively used in the Fourier recycling routine (default value is 91.0).

**thre x**

*x*is the threshold for the E value of reflections used in the Electron Density Modification procedure. x should be positive (its default value is 0.8).

**thrm x**

*x*is the fraction of the largest intensity Fourier map points used in the Electron Density Modification procedure (0< x ≤ 1.0). Its default value is 0.025.

**Examples**

**Example 1**

Only the E-map corresponding to the set number 7 and the complete E-map calculation and structure model optimization procedure for set number 10 are requested by the user.

```
%structure nbpo
%job titolo nbpo.dat
%data
pattern nbpo.pow
content Nb 20 O 120 P 28
wavelength 1.0001
cell 29.8661 8.7215 8.7860 90.000 91.769 90.000
spacegroup c 2/c
synchrotron
%extraction
%normal
%invar
%phase
%fourier
set 7
recyc 0
%fourier
set 10
%continue
```

**Peak labelling procedure (%changelabel command)**

The frequent incorrect labelling (in term of atomic species) of the atomic positions, as obtained for example at the end of a Direct Methods procedure, can strongly affect the efficiency of the approaches used for crystal structure refinement. In this situation, the correct structural model is difficult to establish. The atomic label can be automatically modified using chemical information about the structure. The procedure (Altomare* et al.*, 2002) requires: 1) the prior information on the number and coordination of the heavy atoms (the procedure can only handle tetrahedral and octahedral coordinations); 2) the range of the typical distances between heavy atoms; 3) the range of the typical distances between heavy and light atoms.

The peak labelling procedure starts only from a list of atomic positions, which are supplied by an external file (they can be selected from the Fourier map of Expo2014).

The command %changelabel must follow the commands %extraction and %fragment.

**Directives in the %changelabel command**

**The following directives must be added after the command %changelabel in the input file to activate the peak labelling procedure:**

**dlight ****d tol**

*d* is the expected heavy atom (*i.e.* atomic species for which Z>10) to light atom distance;

*tol* is the tolerance around the ideal distances. Default value = 0.5.

The program uses only one value of ‘d’ for all types of light atoms.

**dweight d tol**

*d*is the expected heavy atom to heavy atom distance;

*tol*defines the tolerance around the ideal distances. Default value = 0.5.

The program uses only one value of ‘d’ for all types of heavy atoms.

**label n1 El1 n2 El2**

n is the number of atoms to relabel using the El element;

El is the atomic species to assign to the heavy atoms.

Only a maximum of two combination (n, El) can be used.

**coord xi**

x specifies the expected number of heavy atoms surrounding each i-th heavy atom (the coordination type) (i = 1,2).

**Examples**

**Example 1**

Use of the peak labelling procedure to relabel the atoms contained in the vfi.fra file, supplied by using the command %fragment. The procedure requires that the extraction process must be carried out (%extraction).

```
% window
% structure vfi
% job vfi - Synchrotron data
%data
cell 18.9752 18.9752 8.1044 90.0 90.0 120.0
space p 63
wave 1.5193
cont al 18 p 18 o 114
pattern vfi.pow
synchrotron
%extraction
%fragment vfi.fra
%changelabel
label 3 Al 3 P
coord 4 4
dwei 3.1 0.5
dlig 1.9 0.5
% continue
```

The vfi.fra file contains the chemical labels, the fractional coordinates and the isotropic thermal factors for each atom in the supplied model.

**The POLPO procedures for polyhedrally coordinated structure (%polyhedra command)**

The POLPO1 (Altomare *et al.*, 2000) and POLPO2 (Giacovazzo *et al.*, 2002) procedures can be applied for completing polyhedral (tetrahedral or octahedral) coordinated structures when the cations have been located and the coordination is *a priori* known. The procedures, which exploit the experimental information about the cation connectivity, are based on the generation of several structure models compatible with the coordination information, by using the Monte Carlo algorithm. POLPO1 is suggested when all the cations are rightly positioned and labelled. POLPO2 is able to position one or more missing cations and the surrounding anions. The POLPO procedures start only from the positions of cations, which are supplied by an external file (they can be selected from the Fourier map of Expo2014). The command %polyhedra must follow the commands %extraction and %fragment.

**Directives in the %polyhedra command**

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

**missing n1 n2 El tetrahedron (octahedron) d tol1 tol2**

*n1*and

*n2*are the two order numbers (taken from an external file or from the list of atom positions in a Fourier map) of the cations between which the missing cation(s) must be located;

*El*is the missing cation label,

*tetrahedron*(

*octahedron*) is the type of coordination of the missing cation. (The procedure can only handle octahedron and tetrahedron);

*d*is the expected cation-anion distance;

*tol1*,

*tol2*are the tolerance parameters for the tetrahedron (octahedron) distances and angles, respectively. Default values are 0.3.

**octahedron El d tol1 tol2**

*d*is the expected octahedral coordination distance around the

*El*chemical element;

*tol1*and

*tol2*are the tolerance parameters for distances and angles, respectively.

**tetrahedron El d tol1 tol2**

d is the expected tetrahedral coordination distance around the

*El*chemical element;

*tol1*and

*tol2*are the tolerance parameters for the distances and angles, respectively.

**Examples**

**Example 1**

The POLPO1 procedure is applied to locate the light atoms around the specified heavy atoms contained in the crox_cr.fra file, supplied by using the command %fragment. The POLPO1 procedure requires that the extraction process must be carried out (%extraction).

```
%structure crox
%job CROX
%data
pattern crox.pow
content cr 8 o 21
wavelength 1.39222
cell 5.447 6.5576 12.1147 106.382 95.715 77.970
spacegroup p -1
%extraction
%fragment crox_cr.fra
%polyhedra
octahedron 1 1.92 0.2 0.2
tetrahedron 2 1.75 0.2 0.2
tetrahedron 3 1.75 0.2 0.2
tetrahedron 4 1.75 0.2 0.2
%continue
```

cr .859 .211 .068 1.500 cr .280 .234 .898 1.700 cr .358 .377 .258 2.100 cr .721 .799 .528 3.200

**Example 2**

The POLPO2 procedure is applied to locate the two missing silicon cations and the polyhedral coordinated oxygen anions. The POLPO2 procedure requires that the extraction process must be carried out (%extraction).

```
%structure sapo
%job SAPO
%data
cell 21.9410 13.6912 7.1243 90.000 90.000 90.000
spacegroup p m m n
wavelength 1.54056
content si 32 o 64 n 2 c 48
pattern sapo.pow
%extraction
%fragment sapo_si.fra
%polyhedra
tetrahedron 1 1.6 0.2 0.2
tetrahedron 2 1.6 0.2 0.2
missing 1 2 Si tetrahedron 1.6 0.2 0.2
missing 1 2 Si tetrahedron 1.6 0.2 0.2
%continue
```

Si 1.17900 0.13400 0.11500 Si 0.98700 0.14200 0.11900

**Direct Methods combined with Simulated Annealing (%sdirect command)**

The crystal structure solution by powder diffraction data via Direct Methods can be very difficult if the diffraction data are of low quality and if no heavy atoms are present in the molecule. On the contrary, the use of direct space methods does not require good quality diffraction data but only the expected molecular model, and the structure solution is limited principally by the number of degrees of freedom used to describe the model.

The two approaches can be combined: the information contained in the electron density map as calculated by Direct Methods, is associate with the Simulated Annealing (SA) algorithm (direct space techniques), to be a powerful tool for crystal structure solution, especially for organic structures (Altomare *et al*., 2003, 2008c).

The combined procedure starts from a list of atomic positions (they can be selected from the Fourier map of Expo2014), supplied by an external file via the %fragment command, and used by the procedure as pivot peaks around which a flexible external model can be accommodated.

The command %sdirect must follow the commands %extraction and %fragment.

**Directives in the %sdirect command**

**The following directives must be added after the command %sdirect in the input file to activate the Direct Methods combined with SA procedure:**

**extmod string**

To supply the atomic positions of the external model. s

*tring*is the name of the file in which, for each atom, are stored: chemical element and atomic coordinates (

*x*,

*y*,

*z*).

**Example 1**

The procedure requires that the extraction process must be carried out (%extraction).

```
%structure alpha_lga
%job alpha_lga
%data
cell 10.282 8.779 7.068 90.000 90.000 90.000
spacegroup p 21 21 21
wavelength 1.54056
content o 16 c 20 n 4
pattern alpha_lga.pow
%extraction
%fragment alpha.fra
%sdirect
extmod alpha_inizmod.fra
%continue
```

**References**

Altomare A., Cascarano G., Giacovazzo C. & Viterbo D. (1991). *Acta Cryst. *A**47**, 744-748.

Altomare, A., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G. & Rizzi, R. (2000). *J. Appl. Cryst.* **33**, 1305–1310.

Altomare A., Giacovazzo C., Ianigro M., Moliterni A.G.G., Rizzi R. (2002). J. Appl. Cryst. **35**, 21-27.

Altomare A., Giacovazzo C., Ianigro M., Moliterni A.G.G., Rizzi R. (2003). J. Appl. Cryst. **36**, 230-238.

Altomare, A., Cuocci, C., Giacovazzo, C., Moliterni, A.G.G., Rizzi, R. (2006). *J. Appl. Cryst. ***39, **558-562.

Altomare, A., Cuocci, C., Giacovazzo, C., Kamel, G. S., Moliterni, A. & Rizzi, R. (2008a).* Acta Cryst.* **A64**, 326-336.

Altomare, A., Cuocci, C., Giacovazzo, C., Moliterni, A. & Rizzi, R. (2008b).* J. Appl. Cryst.* **41**, 592-599.

Altomare, A., Cuocci, C., Giacovazzo, C., Moliterni, A. & Rizzi, R. (2008c).* * *J. Appl. Cryst.* **41**, 56-51.

Altomare, A., Cuocci, C., Giacovazzo, C., Moliterni, A., Rizzi, R. (2009). *Acta Cryst.* **A65**, 183-189.

Altomare, A., Cuocci, C., Giacovazzo, C., Moliterni, A., Rizzi, R. (2010a). *J. Appl. Cryst*. **43**, 798-804.

Altomare, A., Cuocci, C., Giacovazzo, C., Moliterni, A., Rizzi, R. (2010b). *Z. Kristallogr.* **225**, 548-551.

Giacovazzo, C., Altomare, A., Cuocci, C., Moliterni,A.G.G and Rizzi, R. (2002). *J. Appl. Cryst.* **35**, 422±429.

Main P. (1978). *Acta Cryst. *A**34**, 31-38.

Main P., Fiske S.J., Hull S.E., Lessinger L., Germain G., Declercq J.P. & Woolfson M.M. (1980) – MULTAN80, a system of computer programs for the automatic solution of crystal structures from x-ray diffraction data – Univ. of York, England.

Ten Eick L.F. (1977). *Acta Cryst. *A**33**, 486-492.