# Reciprocal space

This section is primarily for people working with crystallographic data.

Gemmi supports three reflection file formats:

MTZ files – the most popular format in macromolecular crystallography,

structure factor mmCIF files – used for data archiving in the Protein Data Bank,

and small molecule structure factor CIF files (usually with extension hkl).

Reflection files store Miller indices (*hkl*) with various associated
numbers. Initially, the numbers represent observations derived from
diffraction images (reflection intensities and error estimates).
Then other quantities derived from these are added,
as well as quantities derived from a macromolecular model
that is built to explain the experimental data.

Even electron density maps, which are used in the real space, are nowadays stored mostly as map coefficients in reflection files. Molecular viewers can Fourier-transform them on the fly and show in the real space. Therefore, switching between the real and reciprocal space is also included in Gemmi, as well as calculation of structure factors from the model.

## MTZ format

MTZ format has textual headers and a binary data table, where all numbers
are stored in a 32-bit floating point format.
The headers, as well as data, are stored in class `Mtz`

.
We have two types of MTZ files:

merged – single record per (

*hkl*) reflectionunmerged – multi-record

In merged files the columns of data are grouped hierarchically
(Project -> Crystal -> Dataset -> Column).
Normally, the column name is all that is needed; the hierarchy can be ignored.
In Gemmi, the hierarchy is flattened:
we have a list of columns and a list of datasets.
Each columns is associated with one dataset, and each dataset has properties
`dataset_name`

, `project_name`

and `crystal_name`

,
which is enough to reconstruct the tree-like hierarchy if needed.

In unmerged files it is records, not columns, that are the associated with datasets. The BATCH column links records to batches (~ diffraction images). The batch header, in turn, stores dataset ID.

### Reading

In C++, the MTZ file can be read using either stand-alone functions:

```
Mtz read_mtz_file(const std::string& path)
template<typename Input> Mtz read_mtz(Input&& input, bool with_data)
```

or member functions of the Mtz class, when more control over the reading process is needed.

In Python, we have a single function for reading MTZ files:

```
>>> import gemmi
>>> mtz = gemmi.read_mtz_file('../tests/5e5z.mtz')
```

### class Mtz

The Mtz class has a number of properties read from the MTZ header (they are the same in C++ and Python):

```
>>> mtz.cell # from MTZ record CELL
<gemmi.UnitCell(9.643, 9.609, 19.029, 90, 101.224, 90)>
>>> mtz.spacegroup # from SYMINF
<gemmi.SpaceGroup("P 1 21 1")>
>>> mtz.title # from TITLE
''
>>> mtz.history # from history lines
['From cif2mtz 17/ 5/2019 12:15:14']
>>> # The properties below are also read from the MTZ file,
>>> # although they could be recalculated from the data.
>>> mtz.nreflections # from MTZ record NCOL
441
>>> # Header SORT contains 1-based column indices, 1,2,3,0,0 means H,K,L order.
>>> mtz.sort_order # from SORT, 0,0,0,0,0 = unspecified, cf. Mtz::sort()
[0, 0, 0, 0, 0]
>>> mtz.min_1_d2 # from RESO
0.0028703967109323
>>> mtz.max_1_d2 # from RESO
0.3611701726913452
```

The resolution can also be checked using functions:

```
>>> mtz.resolution_low() # sqrt(1 / min_1_d2)
18.665044863474495
>>> mtz.resolution_high() # sqrt(1 / max_1_d2)
1.6639645192598425
```

Importantly, Mtz class has a list of datasets and a list of columns.
Datasets are stored in the variable `datasets`

:

```
std::vector<Mtz::Dataset> Mtz::datasets
```

```
>>> mtz.datasets
gemmi.MtzDatasets([<gemmi.Mtz.Dataset 0 HKL_base/HKL_base/HKL_base>, <gemmi.Mtz.Dataset 1 5e5z/5e5z/1>])
```

In the MTZ file, each dataset is identified internally by an integer “dataset ID”. To get dataset with the specified ID use function:

```
Dataset& Mtz::dataset(int id)
```

```
>>> mtz.dataset(0)
<gemmi.Mtz.Dataset 0 HKL_base/HKL_base/HKL_base>
```

Dataset has a few properties that can be accessed directly:

```
struct Dataset {
int id;
std::string project_name;
std::string crystal_name;
std::string dataset_name;
UnitCell cell;
double wavelength; // 0 means not set
};
```

Python bindings provide the same properties:

```
>>> mtz.dataset(0).project_name
'HKL_base'
>>> mtz.dataset(0).crystal_name
'HKL_base'
>>> mtz.dataset(0).dataset_name
'HKL_base'
>>> mtz.dataset(0).cell
<gemmi.UnitCell(9.643, 9.609, 19.029, 90, 101.224, 90)>
>>> mtz.dataset(0).wavelength
0.0
```

Columns are stored in variable `columns`

:

```
std::vector<Mtz::Column> Mtz::columns
```

```
>>> mtz.columns[0]
<gemmi.Mtz.Column H type H>
>>> len(mtz.columns)
8
```

To get the first column with the specified label use functions:

```
>>> mtz.column_with_label('FREE')
<gemmi.Mtz.Column FREE type I>
```

If the column names are not unique, you may specify the dataset:

```
>>> mtz.column_with_label('FREE', mtz.dataset(0))
>>> # None
>>> mtz.column_with_label('FREE', mtz.dataset(1))
<gemmi.Mtz.Column FREE type I>
```

To get all columns of the specified type use:

```
>>> mtz.columns_with_type('Q')
[<gemmi.Mtz.Column SIGFP type Q>, <gemmi.Mtz.Column SIGI type Q>]
```

Different programs use different column names for the same thing.
To access the column free set flags you may use function `rfree_column`

which searches for column of type `I`

named
FREE, RFREE, FREER, FreeR_flag or R-free-flags:

```
>>> mtz.rfree_column()
<gemmi.Mtz.Column FREE type I>
```

### Column

Column has properties read from MTZ headers COLUMN and COLSRC:

```
struct Column {
int dataset_id;
char type;
std::string label;
float min_value = NAN;
float max_value = NAN;
std::string source; // from COLSRC
// (and functions and variables that help in accessing data)
};
```

Python bindings provide the same properties:

```
>>> intensity = mtz.column_with_label('I')
>>> intensity.dataset_id
1
>>> intensity.type
'J'
>>> intensity.label
'I'
>>> intensity.min_value, intensity.max_value
(-0.30090001225471497, 216.60499572753906)
>>> intensity.source
'CREATED_17/05/2019_12:15:14'
```

C++ function `Mtz::Column::size()`

is wrapped in Python as `__len__`

:

```
>>> len(intensity)
441
```

In both C++ and Python `Column`

supports the iteration protocol:

```
>>> [x for x in intensity if x > 100]
[124.25700378417969, 100.08699798583984, 216.60499572753906]
>>> numpy.nanmean(intensity)
11.505858...
```

### Data

All the data in MTZ file is stored as one array of 32-bit floating point numbers. This includes integers such as Miller indices. In MTZ class this data is stored in a C++ container:

```
std::vector<float> Mtz::data
```

In Python, this data can be conveniently accessed as NumPy array, as described in a separate section below.

The Column class from the previous section provides another way to access the data.

We can also copy selected columns into AsuData – a class that stores values together with Miller indices:

```
>>> mtz_5wkd = gemmi.read_mtz_file('../tests/5wkd_phases.mtz.gz')
>>> mtz_5wkd.get_float('FOM')
<gemmi.FloatAsuData with 367 values>
>>> mtz_5wkd.get_int('FREE')
<gemmi.IntAsuData with 367 values>
>>> mtz_5wkd.get_f_phi('FWT', 'PHWT')
<gemmi.ComplexAsuData with 367 values>
>>> mtz_5wkd.get_value_sigma('FP', 'SIGFP')
<gemmi.ValueSigmaAsuData with 367 values>
```

These functions are also available for data from SF-mmCIF. Missing values (MNF in the MTZ format) are skipped. In case of value-sigma pairs both numbers must be present.

### Batch headers

Unmerged (multi-record) MTZ files store a list of batches:

```
std::vector<Mtz::Batch> Mtz::batches
```

```
>>> # here we use mdm2_unmerged.mtz file distributed with CCP4 7.1
>>> mdm2 = gemmi.read_mtz_file(mdm2_unmerged_mtz_path)
>>> len(mdm2.batches)
60
```

Each batch has a number of properties:

```
>>> batch = mdm2.batches[0]
>>> batch.number
1
>>> batch.title
'TITLE Batch 2'
>>> batch.axes
['AXIS']
>>> batch.dataset_id # link to a dataset
2
>>> mdm2.dataset(2).wavelength
0.8726
```

Most of the batch header properties are not decoded, but they can be accessed directly if needed:

```
>>> list(batch.ints)
[185, 29, 156, 0, -1, 1, -1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0]
>>> batch.floats[36], batch.floats[37] # start and end of phi
(80.0, 80.5)
```

One peculiarity of the MTZ format is that instead of the original Miller indices it stores indices of the equivalent reflection in the ASU + the index of the symmetry operation that transforms the former to the latter. It can be useful to switch between the two:

```
>>> mdm2.switch_to_original_hkl()
True
>>> mdm2.switch_to_asu_hkl()
True
```

The return value indicates that the indices were switched.
It would be `False`

only if it was a merged MTZ file.
Keeping track what the current indices mean is up to the user.
They are “asu” after reading a file and they must be “asu” before writing
to a file.

### Data in NumPy and pandas

In Python Column has a read-only `array`

property that provides
a view of the data compatible with NumPy. It does not copy the data,
so the data is not contiguous (because it’s stored row-wise in MTZ):

```
>>> intensity.array.shape
(441,)
>>> intensity.array.strides
(32,)
```

Here is an example that uses the array property to make a plot similar to AUSPEX:

```
import sys
from matplotlib import pyplot
import gemmi
for path in sys.argv[1:]:
mtz = gemmi.read_mtz_file(path)
intensity = mtz.column_with_label('I')
sigma = mtz.column_with_label('SIGI')
if intensity is None or sigma is None:
sys.exit("Columns I and SIGI not in the file: " + path)
x = mtz.make_1_d2_array()
y = intensity.array / sigma.array
pyplot.figure(figsize=(5,5))
pyplot.hexbin(x, y, gridsize=180, bins='log', cmap='gnuplot2')
pyplot.show()
```

To test this program we run it on data from 5DEI and we get result similar to #3:

The Mtz class also has a read-only property named `array`

.
It provides access to all the MTZ data as 2D NumPy array
(C-style contiguous), without copying it:

```
>>> mtz.array.shape
(441, 8)
```

It helps to have labels on the columns. A good data structure for this is Pandas DataFrame:

```
>>> import pandas
>>> df = pandas.DataFrame(data=mtz.array, columns=mtz.column_labels())
>>> # now we can handle columns using their labels:
>>> I_over_sigma = df['I'] / df['SIGI']
```

The data in NumPy array is float32. To cast Miller indices in the DataFrame to integer type:

```
>>> df = df.astype({name: 'int32' for name in ['H', 'K', 'L']})
```

Another way to get Miller indices as a N×3 array of integers is:

```
>>> hkl = mtz.make_miller_array()
```

The same method is available also in ReflnBlock (which represents SF-mmCIF
and is described in a later section). Similarly, AsuData
has a property `miller_array`

.

There is also a standalone function `make_miller_array()`

that returns all reflections in a resolution shell.
To get all unique reflections up to the same resolution
as the example MTZ file we can do:

```
>>> dmin = mtz.resolution_high() - 1e-6 # the 1e-6 margin is for numerical errors
>>> gemmi.make_miller_array(mtz.cell, mtz.spacegroup, dmin)
array([[-5, 0, 1],
[-5, 0, 2],
[-5, 0, 3],
...,
[ 5, 2, 0],
[ 5, 2, 1],
[ 5, 2, 2]]...)
```

This function has two optional parameters (not used above): `dmax`

and `unique`

.
Setting `unique=False`

returns all equivalent reflections.
By default, only reflections from the reciprocal space ASU are returned.

Note: if you’d like to only count the reflections, use function
`count_reflections`

that takes the same parameters:

```
>>> gemmi.count_reflections(mtz.cell, mtz.spacegroup, dmin)
441
>>> gemmi.count_reflections(mtz.cell, mtz.spacegroup, 2.5, dmax=3.0)
52
>>> gemmi.count_reflections(mtz.cell, mtz.spacegroup, 2.5, dmax=3.0, unique=False)
188
```

Back to NumPy arrays. The N×3 array of Miller indices can be used with a number of vectorized functions:

```
>>> gops = mtz.spacegroup.operations()
>>> gops.centric_flag_array(hkl) # vectorized is_reflection_centric()
array([ True, True, True, ..., False, False, False])
>>> gops.epsilon_factor_array(hkl) # vectorized epsilon_factor()
array([1, 1, 1, ..., 1, 1, 1], dtype=int32)
>>> gops.epsilon_factor_without_centering_array(hkl) # epsilon_factor_without_centering()
array([1, 1, 1, ..., 1, 1, 1], dtype=int32)
>>> gops.systematic_absences(hkl) # vectorized is_systematically_absent()
array([False, False, False, ..., False, False, False])
>>> mtz.cell.calculate_d_array(hkl) # vectorized calculate_d()
array([1.91993507, 1.92853496, 1.91671381, ..., 1.76018669, 1.72347773,
1.67531162])
>>> mtz.cell.calculate_1_d2_array(hkl) # vectorized calculate_1_d2()
array([0.27128571, 0.26887162, 0.27219833, ..., 0.3227621 , 0.33665777,
0.35629424])
```

You may also check a different project (not associated with Gemmi) for working with reflection data in Python: ReciprocalSpaceship.

### Modifying

To change the data in-place, without changing the number of columns and rows, simply modify the array with data. In Python, you can use NumPy functions. As an example, let’s add 0.5 to each value in the last column (in this case SIGI):

```
>>> mtz.array[:,-1] += 0.5
```

A more general way to change the data is by calling Mtz.set_data().
In Python, this function takes as an argument a 2D NumPy array of floating
point numbers. The new data must have the same number of columns, but
the number of rows can differ.
As an example, let’s use set_data() to remove reflections with the resolution
above 2Å (i.e. *d* < 2Å).
To show that it really has an effect we print the appropriate
grid size before and after:

```
>>> mtz.get_size_for_hkl()
[12, 12, 24]
>>> mtz.set_data(mtz.array[mtz.make_d_array() >= 2.0])
>>> mtz.get_size_for_hkl()
[10, 10, 20]
```

Columns can be removed with `Mtz::remove_column(index)`

,
where index is 0-based column index:

```
>>> mtz.column_with_label('FREE').idx
3
>>> mtz.remove_column(_) # removes column 3 (FREE)
```

Columns can be added with `Mtz::add_column()`

:

```
>>> mtz.add_column('FREE', 'I', dataset_id=0, pos=3)
<gemmi.Mtz.Column FREE type I>
```

We also have two functions for copying columns. The first function is overwriting the destination column, the second one is inserting a new column:

```
Column& Mtz::replace_column(size_t dest_idx, const Column& src_col,
const std::vector<std::string>& trailing_cols={})
Column& Mtz::copy_column(int dest_idx, const Column& src_col,
const std::vector<std::string>& trailing_cols={})
```

The column can be copied from either the same or another Mtz object.
Copying a column between objects is more involved: *hkl*s are compared,
values for matching Miller indices are copied, the rest is filled with NaNs.
`dest_idx`

is position of the resulting column.
The second function can take negative `dest_idx`

– to add the copied column
at the end. Often, two or four consecutive columns are handled together
(value and sigma, absolute value and angle, the ABCD coefficients).
In such case you can use `trailing_cols`

:

```
>>> mtz.column_with_label('FP')
<gemmi.Mtz.Column FP type F>
>>> mtz.copy_column(-1, _, trailing_cols=['SIGFP'])
<gemmi.Mtz.Column FP type F>
>>> [col.label for col in mtz.columns]
['H', 'K', 'L', 'FREE', 'FP', 'SIGFP', 'I', 'SIGI', 'FP', 'SIGFP']
>>> # duplicated labels would need to be changed before saving the file
```

The call above checked that the next column has label SIGFP and copied it
as well. To copy the next column without checking the label pass the empty
string, i.e. `trailing_cols=['']`

.

To demonstrate other functions, we will create a complete `Mtz`

object
from scratch.
The code below is in Python, but all the functions and properties have
equivalents with the same names in C++.

We start from creating an object:

```
>>> mtz = gemmi.Mtz(with_base=True)
```

`with_base=True`

adds the base dataset (HKL_base)
with columns H, K and L.

Then we set its space group and unit cell:

```
>>> mtz.spacegroup = gemmi.find_spacegroup_by_name('P 21 21 2')
>>> mtz.set_cell_for_all(gemmi.UnitCell(77.7, 149.5, 62.4, 90, 90, 90))
```

The MTZ format evolved – first it stored “general” cell dimensions
(keyword CELL), then separate cell dimensions for datasets were added
(keyword DCELL). `set_cell_for_all()`

sets all of them:

```
>>> mtz.cell
<gemmi.UnitCell(77.7, 149.5, 62.4, 90, 90, 90)>
>>> mtz.datasets[0].cell
<gemmi.UnitCell(77.7, 149.5, 62.4, 90, 90, 90)>
```

Now we need to add remaining datasets and columns.

```
>>> mtz.add_dataset('synthetic')
<gemmi.Mtz.Dataset 1 synthetic/synthetic/synthetic>
```

The name passed to `add_dataset()`

is used to set `dataset_name`

,
`crystal_name`

and `project_name`

. These three names are often
kept the same, but if needed, they can be changed afterwards.
The cell dimensions of the new dataset are a copy of the “general” CELL.

Let’s add two columns:

```
>>> mtz.add_column('F', 'F')
<gemmi.Mtz.Column F type F>
>>> mtz.add_column('SIGF', 'Q')
<gemmi.Mtz.Column SIGF type Q>
```

By default, the new column is assigned to the last dataset and is placed at the end of the column list. But we can choose any dataset and position:

```
>>> mtz.add_column('FREE', 'I', dataset_id=0, pos=3)
<gemmi.Mtz.Column FREE type I>
>>> mtz.column_labels()
['H', 'K', 'L', 'FREE', 'F', 'SIGF']
```

Now it is time to add data.
We will use the `set_data()`

function that takes 2D NumPy array
of floating point numbers (even indices are converted to floats,
but they need to be converted at some point anyway – the MTZ format
stores all numbers as 32-bit floats).

```
>>> data = numpy.array([[4, 13, 8, 1, 453.9, 19.12],
... [4, 13, 9, 0, 102.0, 27.31]], numpy.float32)
>>> mtz.set_data(data)
>>> mtz
<gemmi.Mtz with 6 columns, 2 reflections>
```

In C++ the `set_data`

function takes a pointer to row-wise ordered data
and its size (columns x rows):

```
void Mtz::set_data(const float* new_data, size_t n)
```

We can also start the file history:

```
>>> mtz.history = ['This MTZ file was created today.']
```

`mtz.history`

in Python is a property that gets a copy of the C++ data.
Changing it (for example, `mtz.history.append()`

) won’t affect the original data.
Only assignment works: `mtz.history = [line1, line2, ...]`

or `mtz.history += ['appended item']`

.

To update properties `min_1_d2`

and `max_1_d2`

call `update_reso()`

:

```
>>> mtz.update_reso()
>>> mtz.min_1_d2, mtz.max_1_d2
(0.0266481872714499..., 0.03101414716625602)
```

You do not need to call `update_reso()`

before writing an MTZ file –
the values for the RESO record are re-calculated automatically when the file
is written.

### Reindexing, ASU, sorting, …

The reindexing function changes:

Miller indices of reflections (if the new indices would be fractional, the reflection is removed),

space group,

unit cell parameters (in MTZ records CELL and DCELL, and in batch headers).

Reindexing takes an operator that is applied to the Miller indices. Information about the operation is passed to a Logger.

```
>>> mtz.set_logging(sys.stdout)
>>> mtz.reindex(gemmi.Op('k,l,h'))
Real space transformation: y,z,x
Space group changed from P 21 21 2 to P 21 2 21.
```

If reindex() is called on merged data, it should be followed by a call to ensure_asu().

See also: gemmi-reindex.

A merged MTZ file can, in principle, use any of the equivalent hkl indices for each reflection, but it is preferable to always use, for a given space group, indices from the same reciprocal-space ASU. The choice of ASU is arbitrary, but fortunately both CCP4 and CCTBX use the same reciprocal-space ASU definitions. If you have different hkls in Mtz, you may switch to the usual ones with:

```
>>> mtz.ensure_asu()
```

(You may also call `mtz.ensure_asu(tnt=True)`

to use the ASU defined
in the TNT
program, but it is unlikely that you will ever need it).

When appropriate, changing hkl indices is also:

shifting phases (column type P),

changing phase probabilities – Hendrickson-Lattman coefficients (column type A),

swapping anomalous pairs, such as I(+)/I(-), F(+)/F(-) and E(+)/E(-),

changing the sign of anomalous differences (column type D).

To sort data rows by the *h,k,l* indices call `Mtz::sort()`

:

```
>>> mtz.sort() # returns False iff the data was already sorted
False
>>> mtz.sort_order # sort() always sorts by h,k,l and sets sort_order to:
[1, 2, 3, 0, 0]
```

If you’d like to use the first 5 columns for sorting (for multirecord data),
call `mtz.sort(use_first=5)`

.

“Sometimes you may want to reduce the symmetry of your space group and explicitly generate the symmetry related reflections. In most cases you will want to expand to P1 and generate data for a full hemisphere of reciprocal space.” — from documentation of the command EXPAND in Bart Hazes’ SFTOOLS.

Gemmi also has such a function:

```
>>> mtz.expand_to_p1()
```

The reflections that are added may not be in the ASU and are not sorted. You may call ensure_asu() and sort() afterwards.

### Writing

In C++, the MTZ file can be written to a file or to a memory buffer using one of the functions:

```
#include <gemmi/mtz.hpp>
void Mtz::write_to_cstream(std::FILE* stream) const
void Mtz::write_to_file(const std::string& path) const
void Mtz::write_to_string(std::string& str) const
```

In Python we have a single function for writing to a file:

```
>>> mtz.write_to_file('output.mtz')
```

Here is a complete C++ example how to create a new MTZ file:

```
#include <gemmi/mtz.hpp>
int main() {
gemmi::Mtz mtz(/*with_base=*/true);
mtz.title = "My new MTZ file";
mtz.history.push_back("Created from scratch");
mtz.spacegroup = gemmi::find_spacegroup_by_name("P 21 21 21");
mtz.set_cell_for_all(gemmi::UnitCell(77.7, 149.5, 62.4, 90, 90, 90));
mtz.add_dataset("synthetic");
mtz.datasets.back().wavelength = 0.8;
mtz.add_column("F", 'F', -1, -1, false);
mtz.add_column("SIGF", 'Q', -1, -1, false);
const float data[] = { 2, 3, 4, 200.4f, 10.5f,
2, 3, 5, 596.1f, 7.35f };
mtz.set_data(data, 2*5);
mtz.write_to_file("my_new.mtz");
}
```

## SF mmCIF

The mmCIF format is also used to store reflection data from crystallographic
experiments. Reflections and coordinates are normally stored in separate
mmCIF files. The files with reflections are called structure factor mmCIF
or shortly SF mmCIF. Such file for the 1ABC PDB entry
is usually named either `1ABC-sf.cif`

(if downloaded through RCSB website)
or `r1abcsf.ent`

(if downloaded through PDBe website or through FTP).

SF mmCIF files usually contain one block, but may have
multiple blocks, for example merged and unmerged data in separate blocks.
Merged and unmerged data is expected in mmCIF categories `_refln`

and `_diffrn_refln`

, respectively.
Usually, also the unit cell, space group, and sometimes the radiation
wavelength is recorded.

The support for SF mmCIF files in Gemmi is built on top of the generic
support for the CIF format.
We have class `ReflnBlock`

that wraps `cif::Block`

and a function `as_refln_blocks`

:

```
// in C++ it can be called:
// auto rblocks = gemmi::as_refln_blocks(gemmi::read_cif_gz(path).blocks);
std::vector<ReflnBlock> as_refln_blocks(std::vector<cif::Block>&& blocks)
```

In Python this function takes `cif.Document`

as an argument:

```
>>> doc = gemmi.cif.read('../tests/r5wkdsf.ent')
>>> doc
<gemmi.cif.Document with 1 blocks (r5wkdsf)>
>>> rblocks = gemmi.as_refln_blocks(doc)
```

Blocks are moved from the Document to the new list:

```
>>> doc
<gemmi.cif.Document with 0 blocks ()>
>>> rblocks
gemmi.ReflnBlocks([<gemmi.ReflnBlock r5wkdsf with 17 x 406 loop>])
```

When ReflnBlock is created some of the mmCIF tags are interpreted to initialize the following properties:

```
>>> rblock = rblocks[0]
>>> rblock.entry_id
'5wkd'
>>> rblock.cell
<gemmi.UnitCell(50.347, 4.777, 14.746, 90, 101.733, 90)>
>>> rblock.spacegroup
<gemmi.SpaceGroup("C 1 2 1")>
>>> rblock.wavelength
0.9791
```

To check if block has either merged or unmerged data, in C++ use function
`ok()`

, and in Python:

```
>>> bool(rblock)
True
```

Normally, one block has only one type of data, merged and unmerged. Which one is used can be checked with the function:

```
>>> rblock.is_unmerged()
False
```

But it is syntactically correct to have both types of data in two tables in one block. In such case you can switch which table is used:

```
>>> rblock.use_unmerged(True)
>>> rblock.use_unmerged(False)
```

Finally, ReflnBlock has functions for working with the data table:

```
std::vector<std::string> ReflnBlock::column_labels() const
size_t ReflnBlock::get_column_index(const std::string& tag) const
template<typename T>
std::vector<T> ReflnBlock::make_vector(const std::string& tag, T null) const
std::vector<std::array<int,3>> ReflnBlock::make_index_vector() const
std::vector<double> ReflnBlock::make_1_d2_vector() const
std::vector<double> ReflnBlock::make_d_vector() const
```

We will describe these functions while going through its Python equivalents.
`column_labels()`

returns list of tags associated with the columns,
excluding the category part. Unlike in the MTZ format, here the tags
must be unique.

```
>>> rblock.column_labels()
['crystal_id', 'wavelength_id', 'scale_group_code', 'index_h', 'index_k', 'index_l', 'status', 'pdbx_r_free_flag', 'F_meas_au', 'F_meas_sigma_au', 'F_calc_au', 'phase_calc', 'pdbx_FWT', 'pdbx_PHWT', 'pdbx_DELFWT', 'pdbx_DELPHWT', 'fom']
```

All the data is stored as strings. We can get integer or real values from the selected column in an array. In Python – in NumPy array:

```
>>> rblock.make_int_array('index_h', -1000) # 2nd arg - value for nulls
array([-26, -26, -26, ..., 25, 26, 26], dtype=int32)
>>> rblock.make_float_array('F_meas_au') # by default, null values -> NAN
array([12.66, 13.82, 24.11, ..., nan, 9.02, nan])
>>> rblock.make_float_array('F_meas_au', 0.0) # use 0.0 for nulls
array([12.66, 13.82, 24.11, ..., 0. , 9.02, 0. ])
>>> # Miller indices hkl are often used together, so we have a dedicated function
>>> rblock.make_miller_array()
array([[-26, 0, 1],
[-26, 0, 2],
[-26, 0, 3],
...,
[ 25, 1, 0],
[ 26, 0, 0],
[ 26, 0, 1]], dtype=int32)
```

We also have convenience functions that returns arrays of 1/*d*^{2}
or just *d* values:

```
>>> rblock.make_1_d2_array().round(4)
array([0.2681, 0.2677, 0.2768, ..., 0.301 , 0.2782, 0.2978])
>>> rblock.make_d_array().round(3)
array([1.931, 1.933, 1.901, ..., 1.823, 1.896, 1.832])
```

**Example 1**

The script below renders the same colorful *I*/*σ* image as in the previous
section, but it can take as an argument a file downloaded directly from
the wwPDB (for example,
`$PDB_DIR/structures/divided/structure_factors/de/r5deisf.ent.gz`

).

```
import sys
from matplotlib import pyplot
import gemmi
for path in sys.argv[1:]:
doc = gemmi.cif.read(path)
rblock = gemmi.as_refln_blocks(doc)[0]
intensity = rblock.make_float_array('intensity_meas')
sigma = rblock.make_float_array('intensity_sigma')
x = rblock.make_1_d2_array()
y = intensity / sigma
pyplot.figure(figsize=(5,5))
pyplot.hexbin(x, y, gridsize=70, bins='log', cmap='gnuplot2')
pyplot.show()
```

**Example 2**

In this example we compare columns from two files using Python pandas.

Here we use the 5WKD entry, which has only 367 measured reflections, but the same script works fine even for 1000x more reflections. You’d only need to tweak the plots to make them more readable.

We use two data files (SF-mmCIF MTZ) downloaded from the RCSB website. First, we make pandas DataFrames from both files, and then we merge them into a single DataFrame:

```
import pandas
from matplotlib import pyplot
import gemmi
MTZ_PATH = 'tests/5wkd_phases.mtz.gz'
SFCIF_PATH = 'tests/r5wkdsf.ent'
# make DataFrame from MTZ file
mtz = gemmi.read_mtz_file(MTZ_PATH)
mtz_df = pandas.DataFrame(data=mtz.array, columns=mtz.column_labels())
# (optional) store Miller indices as integers
mtz_df = mtz_df.astype({label: 'int32' for label in 'HKL'})
# make DataFrame from mmCIF file
cif_doc = gemmi.cif.read(SFCIF_PATH)
rblock = gemmi.as_refln_blocks(cif_doc)[0]
cif_df = pandas.DataFrame(data=rblock.make_miller_array(),
columns=['H','K','L'])
cif_df['F_meas_au'] = rblock.make_float_array('F_meas_au')
cif_df['d'] = rblock.make_d_array()
# merge DataFrames
df = pandas.merge(mtz_df, cif_df, on=['H', 'K', 'L'])
```

We want to compare the FP column from the MTZ file and the F_meas_au column from mmCIF. We start with plotting one against the other:

```
# plot FP from MTZ as a function of F_meas_au from mmCIF
pyplot.rc('font', size=8)
pyplot.figure(figsize=(2, 2))
pyplot.scatter(x=df['F_meas_au'], y=df['FP'],
marker=',', s=1, linewidths=0)
pyplot.xlim(xmin=0)
pyplot.ylim(ymin=0)
pyplot.show()
```

The numbers are similar, but not exactly equal.
Let us check how the difference between the two values depends on the
resolution. We will plot the difference as a function of 1/*d*.
This extremely small dataset has only three Miller indices *k* (0, 1 and 2),
so we can use it for coloring.

```
# plot the ratio FP : F_meas_au as a function of 1/d
pyplot.figure(figsize=(3, 3))
pyplot.scatter(x=1/df['d'],
y=df['FP']/df['F_meas_au'],
c=df['K'], # color by index k
marker='.', s=16, linewidths=0)
pyplot.xlim(xmin=0)
pyplot.show()
```

Apparently, some scaling has been applied. The scaling is anisotropic
and is the strongest along the *k* axis.

## MTZ <-> mmCIF

The library contains classes that facilitate conversions: MtzToCif and CifToMtz. This code is used in gemmi command-line utilities mtz2cif and cif2mtz and in WebAssembly converters.

The converters can use *spec files* for customization, see the command-line
program documentation for details.

```
>>> cif2mtz = gemmi.CifToMtz()
>>> cif2mtz.spec_lines = ['pdbx_r_free_flag FREE I 0',
... 'F_meas_au FP F 1',
... 'F_meas_sigma_au SIGFP Q 1']
>>> cif2mtz.convert_block_to_mtz(rblock)
<gemmi.Mtz with 6 columns, 406 reflections>
>>>
>>> # and convert it back
>>> cif_string = gemmi.MtzToCif().write_cif_to_string(_)
```

## SX hkl CIF

In the small molecule world reflections are also stored in separate CIF files. Similarly to SF mmCIF files, we may take cif::Block and wrap it into the ReflnBlock class:

```
>>> doc = gemmi.cif.read('../tests/2242624.hkl')
>>> gemmi.hkl_cif_as_refln_block(doc[0])
<gemmi.ReflnBlock 2242624 with 7 x 71 loop>
```

The methods of ReflnBlock are the same as in the previous section.

```
>>> print(_.make_d_array(), _.make_float_array('F_squared_meas'), sep='\n')
[1.7710345 1.03448487 0.71839568 ... 0.67356073 0.69445847 0.56034145]
[173.17 246.01 11.61 ... 84.99 46.98 16.99]
```

These hkl files in the CIF format are not to be confused with the SHELX hkl files. The latter is often included in the coordinate cif file, as a text value of _shelx_hkl_file:

```
>>> cif_doc = gemmi.cif.read('../tests/4003024.cif')
>>> hkl_str = gemmi.cif.as_string(cif_doc[0].find_value('_shelx_hkl_file'))
>>> print(hkl_str[:87])
-1 0 0 20.05 0.74
0 0 -1 20.09 0.73
0 0 1 19.76 0.73
```

## Intensity merging

TBD

## AsuData: merged reflections

AsuData is an array of symmetry-unique Miller indices and values.

In C++, in principle, the values can be of any type.

In Python, we have separate classes for different value types: IntAsuData, FloatAsuData, ComplexAsuData, ValueSigmaAsuData. The last one is for a pair of numbers, typically a float value with associated sigma.

Such an array can be created in two ways:

From reflection data read from an MTZ or SF-mmCIF file, for example:

>>> mtz.get_value_sigma('F', 'SIGF') <gemmi.ValueSigmaAsuData with ... values>Examples of functions giving other AsuData classes were given in an example above.

The choice of ASU differs between programs. By default, reflections in AsuData are moved to the ASU used in CCP4 and sorted. If you want to keep original Miller indices and the order from the file, add parameter

`as_is=True`

. You can also switch the indices and sort reflections using separate functions:>>> asu_data = mtz.get_value_sigma('F', 'SIGF', as_is=True) >>> asu_data.ensure_asu() >>> asu_data.ensure_sorted() >>> asu_data <gemmi.ValueSigmaAsuData with ... values>

From a reciprocal-space grid, which will be introduced in the next section:

>>> grid = rblock.get_f_phi_on_grid('pdbx_FWT', 'pdbx_PHWT', [54,6,18]) >>> asu_data = grid.prepare_asu_data(dmin=1.8, with_000=False, with_sys_abs=False) >>> asu_data <gemmi.ComplexAsuData with 407 values>Arguments of the

`prepare_asu_data`

function are optional. By default, the resolution is not limited, the (000) reflection is not included and systematic absences are also not included.

Each item in AsuData has two properties, hkl and value:

```
>>> asu_data[158]
<gemmi.ComplexHklValue (-6,2,5) (-1.376941204071045-0.19008657336235046j)>
>>> _.hkl, _.value
([-6, 2, 5], (-1.376941204071045-0.19008657336235046j))
```

Both Miller indices and values can be accessed as NumPy arrays:

```
>>> asu_data.miller_array
array([[-26, 0, 1],
[-26, 0, 2],
[-26, 0, 3],
...,
[ 25, 1, 0],
[ 26, 0, 0],
[ 26, 0, 1]], dtype=int32)
>>> asu_data.value_array[158:159]
array([-1.3769412-0.19008657j], dtype=complex64)
```

AsuData has several methods common with Mtz and ReflnBlock:

```
>>> asu_data.get_size_for_hkl()
[54, 6, 18]
>>> asu_data.get_size_for_hkl(sample_rate=3.0) # explained in the next section
[90, 8, 30]
>>> asu_data.data_fits_into([54,6,18])
True
>>> asu_data.make_1_d2_array()
array([0.268129 , 0.26766333, 0.27679217, ..., 0.30102324, 0.27818915,
0.2978438 ], dtype=float32)
>>> asu_data.make_d_array()
array([1.9312037, 1.9328829, 1.9007417, ..., 1.8226361, 1.8959632,
1.8323385], dtype=float32)
```

The data can be edited as two NumPy arrays and then copied into new AsuData object. In this example we exclude low-resolution data:

```
>>> d = asu_data.make_d_array()
>>> gemmi.ComplexAsuData(asu_data.unit_cell,
... asu_data.spacegroup,
... asu_data.miller_array[d<8],
... asu_data.value_array[d<8])
<gemmi.ComplexAsuData with 399 values>
```

## Resolution bins

To analyse reflections in resolution shells (bins), we need to first calculate bin boundaries. Class Binner provides four ways (Binner.Method) of determining the boundaries:

EqualCount – each bin has (almost) equal number of reflections,

Dstar – bin widths are uniform in

*d*^{–1},Dstar2 – bin widths are uniform in

*d*^{–2},Dstar3 – bin widths are uniform in

*d*^{–3}.

As a toy example, we will use a tiny MTZ file and 4 bins. The boundaries are set by one of the setup functions:

```
>>> mtz = gemmi.read_mtz_file('../tests/5wkd_phases.mtz.gz')
>>> binner = gemmi.Binner()
>>> binner.setup(4, gemmi.Binner.Method.Dstar3, mtz)
```

The MTZ file may contain multiple datasets with different unit cells. In such case we may explicitly specify the unit cell:

```
>>> binner.setup(4, gemmi.Binner.Method.Dstar3, mtz, cell=mtz.get_cell(1))
```

Alternatively, the binner can be set up with ReflnBlock or with an array
of *hkl* or *d*^{–2}:

```
>>> binner.setup(4, gemmi.Binner.Method.Dstar3, rblock)
>>> binner.setup(4, gemmi.Binner.Method.Dstar3,
... mtz.make_miller_array(), mtz.get_cell())
>>> binner.setup_from_1_d2(4, gemmi.Binner.Method.Dstar3,
... mtz.make_1_d2_array(), mtz.get_cell())
```

The unit cell and the upper bin boundaries (in Å^{–2}) are stored internally:

```
>>> binner.cell
<gemmi.UnitCell(50.347, 4.777, 14.746, 90, 101.73, 90)>
>>> binner.limits
[0.122247130469427..., 0.193954142690122..., 0.254107666379415..., inf]
>>> binner.size # number of bins
4
```

The minimum and maximum of *d*^{–2} in the setup data is also stored,
but it is not used when mapping *hkl* to bins. Reflections from outside
the resolution range are put into the first or the last bin.

```
>>> print('%g %g' % (binner.min_1_d2, binner.max_1_d2))
0.00164605 0.307803
```

Helper functions can return the limits of each bin in Å:

```
>>> print('%g - %g' % (binner.dmin_of_bin(2), binner.dmax_of_bin(2)))
1.98377 - 2.27065
```

After the limits and the unit cell are set, the binner can assign reflections to bins:

```
>>> binner.get_bin((2,1,2))
0
>>> binner.get_bin((1,2,2))
2
```

It is more efficient to get bin numbers for all reflections at once.
Similarly to the setup functions above, the argument can be Mtz,
ReflnBlock, array of Miller indices or array of *d*^{–2}:

```
>>> binner.get_bins(mtz)
array([3, 3, 3, ..., 3, 3, 3], dtype=int32)
>>> # note: there are also 0's, 1's and 2's, but they are in the middle
>>> binner.get_bins(mtz.make_miller_array())
array([3, 3, 3, ..., 3, 3, 3], dtype=int32)
>>> binner.get_bins_from_1_d2(mtz.make_1_d2_array())
array([3, 3, 3, ..., 3, 3, 3], dtype=int32)
```

These are all Binner’s functions.
Now we will show how the bin numbers can be used.
First, let use `numpy.bincount()`

to find out how many reflections
are in each bin.

```
>>> bins = binner.get_bins(mtz)
>>> counts = numpy.bincount(bins)
>>> for i in range(binner.size):
... print('bin %d d= %6.3f - %5.3f count: %d' %
... (i+1, binner.dmax_of_bin(i), binner.dmin_of_bin(i), counts[i]))
bin 1 d= 24.648 - 2.860 count: 108
bin 2 d= 2.860 - 2.271 count: 86
bin 3 d= 2.271 - 1.984 count: 93
bin 4 d= 1.984 - 0.000 count: 80
```

Now let’s calculate R_{work} and R_{free} of the whole dataset.

```
>>> # First find the needed MTZ columns and access them as NumPy arrays.
>>> fc = mtz.column_with_label('FC_ALL').array
>>> fo = mtz.column_with_label('FP').array
>>> sigma = mtz.column_with_label('SIGFP').array
>>> rfree = mtz.rfree_column().array
>>>
>>> # Then define a function to calculate R-factor.
>>> def calculate_rfactor(fo, fc):
... return numpy.nansum(numpy.abs(fo-fc)) / numpy.nansum(fo)
...
>>> # And calculate the values (the free flags are 0/1 here).
>>> print('Rwork', calculate_rfactor(fo[rfree==0], fc[rfree==0]))
Rwork 0.16427532
>>> print('Rfree', calculate_rfactor(fo[rfree==1], fc[rfree==1]))
Rfree 0.19464952
```

In the example above we use `nansum`

to handle correctly NaN values
that can be present in the data (missing observations).
Sometimes it is easier to remove these NaNs first:

```
>>> present = ~numpy.isnan(fo)
>>> fc = fc[present]
>>> fo = fo[present]
>>> sigma = sigma[present]
>>> rfree = rfree[present]
>>> bins = bins[present]
```

Before we get to calculation of R-factors in bins, let’s try to calculate
mean *F*_{obs}/σ. We can use bincount with weights for this:

```
>>> numpy.bincount(bins, weights=fo/sigma) / numpy.bincount(bins)
array([17.45109335, 14.27616408, 11.48073644, 9.47130558])
```

The bincount function here would not handle correctly NaNs, and it can perform only summations. That’s why gemmi provides similar bin-functions for the most common scenarios. For calculation of the mean (the same as above, but it skips NaNs):

```
>>> gemmi.binmean(bins, fo/sigma)
array([17.45109335, 14.27616408, 11.48073644, 9.47130558])
```

For calculation of the R-factors between two columns:

```
>>> gemmi.binrfactor(bins, obs=fo, calc=fc)
array([0.15737026, 0.20364942, 0.21249298, 0.2510972 ])
>>> # alternatively, we can use R_iso formula - mean value in denominator
>>> gemmi.binrfactor(bins, fo, fc, riso=True)
array([0.16200339, 0.20942611, 0.22133223, 0.25476191])
```

And for calculation of the Pearson’s correlations. This one returns a list of Correlation objects:

```
>>> correlations = gemmi.bincorr(bins, obs=fo, calc=fc)
>>> for n, c in enumerate(correlations):
... print('bin %d n=%3d CC=%g <Fc>/<Fo>=%g' %
... (n+1, c.n, c.coefficient(), c.mean_ratio()))
bin 1 n=108 CC=0.968352 <Fc>/<Fo>=0.942802
bin 2 n= 86 CC=0.947074 <Fc>/<Fo>=0.944833
bin 3 n= 93 CC=0.871931 <Fc>/<Fo>=0.920127
bin 4 n= 80 CC=0.895339 <Fc>/<Fo>=0.97123
```

Note that the bin numbers passed to these functions don’t need to
correspond to resolution shells.
We could use the free flag value (which is here 0 or 1)
as a bin number to calculate R_{work}
and R_{free}:

```
>>> gemmi.binrfactor(rfree, obs=fo, calc=fc)
array([0.16427531, 0.19464953])
```

And we could combine bin numbers and free flags to calculate R_{work}
and R_{free} in each shell:

```
>>> n = binner.size
>>> bins_and_rfree = bins + n * (rfree == 0)
>>> r = gemmi.binrfactor(bins_and_rfree, fo, fc)
>>> for i in range(n):
... print('bin %d Rwork=%.4f Rfree=%.4f' % (i+1, r[i], r[n+i]))
...
bin 1 Rwork=0.1545 Rfree=0.2898
bin 2 Rwork=0.2099 Rfree=0.1244
bin 3 Rwork=0.2208 Rfree=0.1349
bin 4 Rwork=0.2552 Rfree=0.1819
```

The same trick works with bincount. One can calculate % of free reflections in shells as:

```
>>> counts = numpy.bincount(bins_and_rfree)
>>> 100. * counts[n:] / counts[:n]
array([6.93069307, 3.61445783, 9.41176471, 5.26315789])
```

## Reciprocal-space grid

The reciprocal-space data can be alternatively presented on a 3D grid indexed by Miller indices. This grid is represented by C++ class ReciprocalGrid, which shares most of the properties with the real-space class Grid.

In C++, the `<gemmi/fourier.hpp>`

header defines templated function
`get_f_phi_on_grid()`

that can be used with MTZ and SF mmCIF data,
as well as with AsuData.
Here, we focus on the usage from Python.

Mtz, ReflnBlock and ComplexAsuData classes
have method `get_f_phi_on_grid`

that takes three mandatory arguments:
column names for the amplitude and phase, and the grid size.
It returns reciprocal grid of complex numbers
(`ReciprocalGrid<std::complex<float>>`

in C++,
`ReciprocalComplexGrid`

in Python).
Symmetry-related reflections are filled automatically (with phase shift).
All the missing values are set to 0:

```
>>> grid = rblock.get_f_phi_on_grid('pdbx_FWT', 'pdbx_PHWT', [54,6,18])
>>> grid
<gemmi.ReciprocalComplexGrid(54, 6, 18)>
```

If we’d ever need to put data from a single column on a grid, we can use analogical function:

```
>>> rblock.get_value_on_grid('F_meas_au', [54,6,18])
<gemmi.ReciprocalFloatGrid(54, 6, 18)>
```

The grids in examples above have capacity to store reflections
with -27<*h*<27, -3<*k*<3 and -9<*l*<9.
Reflections outside of this range would be silently ignored.
To check if the size is big enough you can call:

```
>>> rblock.data_fits_into([54,6,18])
True
>>> rblock.data_fits_into([52,6,18])
False
```

To access the data you can use either the NumPy interface (`grid.array`

),
or getter and setter:

```
>>> grid.get_value(7, 1, -5)
(31.747737884521484-57.06287384033203j)
>>> grid.set_value(7, 1, -5, 12+34j)
>>> grid.get_value(7, 1, -5)
(12+34j)
```

The functions above throw IndexError if the reflection is outside of the grid. If you prefer zero instead of the error, use:

```
>>> grid.get_value_or_zero(20, 30, 40)
0j
```

We can also iterate over points of the grid.

```
>>> for point in grid:
... pass
>>> point.value # point is the last point from the iteration
(-178.310546875+99.20561218261719j)
```

Miller indices and resolution corresponding to the point can be obtained with:

```
>>> grid.to_hkl(point)
[-1, -1, -1]
>>> grid.calculate_1_d2(point)
0.0496019399792323
>>> grid.calculate_d(point)
4.490044743125198
```

### Grid size

To get an appropriate size, we can use method `get_size_for_hkl`

that has two optional parameters: `min_size`

and `sample_rate`

.
`min_size`

sets explicitly the minimal size of the grid:

```
>>> rblock.get_size_for_hkl()
[54, 6, 18]
>>> rblock.get_size_for_hkl(min_size=[64,8,0])
[64, 8, 18]
```

The actual size can be increased to make room for all reflections, to obey restrictions imposed by the spacegroup, and to make the size FFT-friendly (currently this means factors 2, 3 and 5).

`sample_rate`

sets the minimal grid size in relation to *d*_{min}.
It has the same meaning as the keyword SAMPLE in the CCP4 FFT program.
For example, `sample_rate=3`

requests grid size that corresponds
to real-space sampling *d*_{min}/3.
(N.B. 3 here is equivalent to Clipper oversampling parameter equal 1.5).

```
>>> rblock.get_size_for_hkl(sample_rate=3.0)
[90, 8, 30]
```

### Array layout

The `get_f_phi_on_grid`

function has also two optional arguments:
`half_l`

and `axis_order`

.
The `half_l`

flag is used to shrink the size of the grid in the memory.
When set, the grid does not include data with negative index *l*.
If the data is Hermitian, i.e. if it is a Fourier transform of
the real data (electron density), the (*h* *k* *l*) reflection
with negative *l* can be restored as a complex conjugate of its Friedel mate
(*-h* *-k* *-l*).

```
>>> rblock.get_f_phi_on_grid('pdbx_FWT', 'pdbx_PHWT', [54,6,18], half_l=True)
<gemmi.ReciprocalComplexGrid(54, 6, 10)>
```

`axis_order`

can take one of the two values : `AxisOrder.XYZ`

(the default)
or `AxisOrder.ZYX`

. With the former – the *h* direction is along the first
(fast) axis of the grid. The latter results in *l* along the fast axis.
(The fast axis is first, which is a Fortran convention. This convention
affected the design of the CCP4 format, which in turn affected the design of
the grid classes in Gemmi.)

```
>>> rblock.get_f_phi_on_grid('pdbx_FWT', 'pdbx_PHWT', [54,6,18], order=gemmi.AxisOrder.ZYX)
<gemmi.ReciprocalComplexGrid(18, 6, 54)>
```

### Example

As an example, we will use numpy.fft to calculate electron density map from map coefficients. Gemmi can calculate it internally, as described in the next section, but it is instructive to do it with a general-purpose tool.

```
>>> size = rblock.get_size_for_hkl(sample_rate=2.6)
>>> full = rblock.get_f_phi_on_grid('pdbx_FWT', 'pdbx_PHWT', size)
>>> complex_map = numpy.fft.ifftn(full.array.conj())
>>> scale_factor = complex_map.size / full.unit_cell.volume
>>> real_map = numpy.real(complex_map) * scale_factor
>>> round(real_map[1][2][3], 5)
-0.40554
>>> round(real_map.std(), 5)
0.66338
```

Above, we could use `fftn(array)`

instead of `ifftn(array.conj())`

,
but the inverse FFT is more appropriate here (or so I think).

Since the data is Hermitian, using complex-to-complex FFT is equivalent to using complex-to-real FFT on a half of the data:

```
>>> half = rblock.get_f_phi_on_grid('pdbx_FWT', 'pdbx_PHWT', size, half_l=True)
>>> real_map = numpy.fft.irfftn(half.array.conj()) * scale_factor
>>> round(real_map[1][2][3], 5)
-0.40554
>>> round(real_map.std(), 5)
0.66338
```

## FFT

Internally, Gemmi is using the PocketFFT library to transform between real space and reciprocal space. This library was picked after evaluation and benchmarking of many FFT libraries.

In C++, the relevant functions are in the `<gemmi/fourier.hpp>`

header,
and the gemmi-sf2map program may serve as a code example.
Like in the previous section, here we will cover only the Python interface.

Instead of using numpy.fft as in the example above,
we can use `gemmi.transform_f_phi_grid_to_map()`

and we expect to get the same result (wrapped in a Grid class):

```
>>> gemmi.transform_f_phi_grid_to_map(half)
<gemmi.FloatGrid(72, 8, 24)>
>>> round(_.get_value(1, 2, 3), 5)
-0.40554
```

Mtz, ReflnBlock and ComplexAsuData classes have method `transform_f_phi_to_map`

that combines `get_f_phi_on_grid()`

with `transform_f_phi_grid_to_map()`

.

`transform_f_phi_to_map`

takes column names for amplitude and
phase (in degrees), and optional parameters `min_size`

, `exact_size`

and
`sample_rate`

(normally only one of them is given):

```
>>> rblock.transform_f_phi_to_map('pdbx_FWT', 'pdbx_PHWT', sample_rate=4)
<gemmi.FloatGrid(120, 12, 36)>
>>> rblock.transform_f_phi_to_map('pdbx_FWT', 'pdbx_PHWT', min_size=[127,16,31])
<gemmi.FloatGrid(128, 16, 32)>
>>> rblock.transform_f_phi_to_map('pdbx_FWT', 'pdbx_PHWT', exact_size=[70,8,24])
<gemmi.FloatGrid(70, 8, 24)>
```

The grid can be accessed as NumPy 3D array:

```
>>> round(_.array.std(), 5)
0.66338
```

and it can be stored in a CCP4 map format:

```
>>> ccp4 = gemmi.Ccp4Map()
>>> ccp4.grid = rblock.transform_f_phi_to_map('pdbx_FWT', 'pdbx_PHWT', sample_rate=2.6)
>>> ccp4.update_ccp4_header()
>>> ccp4.write_ccp4_map('5wkd.ccp4')
```

To transform the electron density back to reciprocal space coefficients
use function `transform_map_to_f_phi`

:

```
>>> ccp4.grid
<gemmi.FloatGrid(72, 8, 24)>
>>> gemmi.transform_map_to_f_phi(ccp4.grid)
<gemmi.ReciprocalComplexGrid(72, 8, 24)>
```

Now you can access hkl reflections using `Grid.get_value()`

:

```
>>> _.get_value(23, -1, -3)
(18.4402...+26.1892...j)
```

`transform_map_to_f_phi`

has one optional flag: `half_l`

.
It has the same meaning as in the function `get_f_phi_on_grid`

.
When you set `half_l=True`

you cannot access directly a data point
with negative Miller index l, but you can use its Friedel mate:

```
>>> gemmi.transform_map_to_f_phi(ccp4.grid, half_l=True)
<gemmi.ReciprocalComplexGrid(72, 8, 13)>
>>> _.get_value(-23, 1, 3).conjugate() # value for (23, -1, -3)
(18.4402...+26.1892...j)
```

Then again, you can use `transform_f_phi_grid_to_map()`

to transform it back to the direct space, and so on…

### Example

This script converts CCP4 map to MTZ file with map coefficients.

```
import sys
import gemmi
RESOLUTION_LIMIT = 1.5 # set 0 for no limit
if len(sys.argv) != 3:
sys.exit('Usage: map2mtz.py input.ccp4 output.mtz')
m = gemmi.read_ccp4_map(sys.argv[1], setup=True)
sf = gemmi.transform_map_to_f_phi(m.grid, half_l=True)
data = sf.prepare_asu_data(dmin=RESOLUTION_LIMIT)
mtz = gemmi.Mtz(with_base=True)
mtz.spacegroup = sf.spacegroup
mtz.set_cell_for_all(sf.unit_cell)
mtz.add_dataset('unknown')
mtz.add_column('FWT', 'F')
mtz.add_column('PHWT', 'P')
mtz.set_data(data)
mtz.write_to_file(sys.argv[2])
```

## Scattering factors

### X-ray form factors

The X-ray scattering contribution from an isolated atom is described by the atomic form factor. Nowadays, two form factor parametrizations are commonly used: one from the International Tables of Crystallography Vol. C (4 Gaussians + constant factor), and the other from Waasmaier & Kirfel (1995), Acta Cryst. A51, 416 (5 Gaussians). Other parametrizations exist (for example, with only two Gaussians), but are not widely used.

Currently, Gemmi includes only the ITC parametrization.

In C++, the form factor coefficients are listed in the `it92.hpp`

header.
In Python, these coefficients can be accessed as a property of an element
(only coefficients for neutral atoms):

```
>>> gemmi.Element('Fe').it92
<gemmi.IT92Coef object at 0x...>
>>> gemmi.Element('Es').it92 # -> None (no parametrization for Einsteinium)
```

or by using the function IT92_get_exact() that takes an element and a charge as arguments and returns None if this exact atom or ion is absent in the table.

The charges listed in a coordinates file might not be reliable in determining
the scattering of an atom, so by default, we use coefficients for neutral atoms
regardless of the formal charge. To use coefficients for ions, you need to first
change `IT92::ignore_charge`

:

```
>>> gemmi.IT92_set_ignore_charge(False)
>>> gemmi.IT92_get_exact(gemmi.Element('Mg'), +2) # for Mg2+
<gemmi.IT92Coef object at 0x...>
```

You can get the coefficients as numbers:

```
>>> fe_coef = gemmi.Element('Fe').it92
>>> fe_coef.a
[11.7695, 7.3573, 3.5222, 2.3045]
>>> fe_coef.b
[4.7611, 0.3072, 15.3535, 76.8805]
>>> fe_coef.c
1.0369
```

The *a*’s and *c* should sum up to the number of electrons (*Z* - *charge*),
but the numbers from the Tables may differ slightly:

```
>>> gemmi.Element('Fe').atomic_number
26
>>> sum(fe_coef.a) + fe_coef.c
25.9904...
```

Some programs (Refmac) normalize the coefficients, which means multiplying them by a factor between 0.99995 and 1.00113.

```
>>> # let's store the original values first
>>> orig_coefs = {i : gemmi.Element(i).it92.get_coefs() for i in range(1, 99)}
>>> # now we normalize the values
>>> gemmi.IT92_normalize()
>>> # and we can see that the values has changed
>>> fe_coef.a
[11.7738..., 7.3600..., 3.5235..., 2.30535...]
```

Now let’s use function `set_coefs()`

to change the coefficients back
to the original values (for neutral atoms):

```
>>> for i in range(1, 99):
... gemmi.Element(i).it92.set_coefs(orig_coefs[i])
```

Macromolecular models may have unknown atoms (UNK) with element specified as X. By default, we use oxygen’s coefficients for X, but you may change it (as well as coefficients of any other atom):

```
>>> c_coefs = gemmi.Element('C').it92.get_coefs()
>>> gemmi.Element('X').it92.set_coefs(c_coefs)
```

The coefficients can be used to directly calculate the sum of Gaussians – the structure factor contribution:

```
>>> fe_coef.calculate_sf(stol2=0.4) # argument: (sin(theta)/lambda)^2
9.303603172302246
```

The large number of reflections in macromolecular crystallography makes direct calculation of structure factors inefficient. Instead, we can calculate electron density on a grid and use FFT to obtain structure factors. In the simplest case, the atom’s contribution to the electron density at a grid point can be calculated as:

```
>>> # arguments are distance^2 and isotropic ADP
>>> fe_coef.calculate_density_iso(r2=2.3, B=50)
0.5279340744018555
```

The C++ interface provides more functions to calculate the electron density. We have separate functions to work with isotropic and anisotropic ADPs. And for efficiency, the computations are split into two stages: the first one is performed once per atom, the second one – for each nearby grid point.

In the usual scenario, we add *f’* (the real component of anomalous
scattering – see the next section) to the constant coefficient *c*.

### Anomalous scattering

The anomalous dispersion is wavelength dependent.
Gemmi provides function `cromer_liberman`

that calculates
real and imaginary components *f’* and *f”*
for isolated atoms from Z=3 to Z=92.

As the name suggests, we use the Cromer-Liberman algorithm.
This algorithm, as is noted on the
pyFprime website,
“fails in computing *f’* for wavelengths < 0.16 Å (> 77.48 keV)
for the heaviest elements (Au-Cf)
and fails to correctly compute *f’*, *f”* and *μ*
for wavelengths > 2.67 Å (< 4.64 keV) for very heavy elements (Am-Cf).”

The implementation is contained in a single C++ header file
`fprime.hpp`

with no dependencies.
All the data is embedded in the code.
The binary size after compilation is about 100kB.

Reportedly, the data tables synthesised by C.T. Chantler are more accurate. Consider using them instead. They are available from the NIST website and from the XrayDB project.

The code in `fprime.hpp`

is based on the X-ray spectroscopy project
Larch
and should give the same results as the
f1f2_cl
function there. The Fortran code in Larch is, in turn, based on the
Brennan and Cowan
routines. Which, in turn, were based on the
original program
from Don Cromer. Along the way, the code was extensively modified.
Importantly, the Jensen correction has been removed (as is recommended
in the chapter 4.2.6 of ITvC)
and the Kissel and Pratt (1990)
correction has been added.
Therefore, it gives different results than the
crossec program,
which was contributed to CCP4 directly by Don Cromer in the 1990’s.

The `cromer_liberman`

function is available in both C++ and Python:

```
>>> gemmi.Element('Se').atomic_number
34
>>> gemmi.cromer_liberman(z=_, energy=10332.0) # energy in eV
(-1.41862..., 0.72389...)
>>> # use gemmi.hc to convert wavelength [A] to energy [eV]
>>> gemmi.cromer_liberman(z=34, energy=gemmi.hc/0.71073)
(-0.09201..., 2.23336...)
```

The same values can be printed from the command line program gemmi-fprime.

### Electron form factors

The electron form factors are parametrized as 5 Gaussians,
using coefficients from International Tables for Crystallography
Volume C, edition 2011, table 4.3.2.2 (pp. 282-283),
“Elastic atomic scattering factors of electrons for neutral atoms
and *s* up to 2.0 A^{–1}”.
The same parametrization is used in cctbx and in CCP4.

In gemmi, were refer to these coefficients as C4322 (after the volume and
table number in the International Tables).
For C++, the form factor coefficients are tabulated
in the `c4322.hpp`

header.
In Python, they can be accessed as a property of an element:

```
>>> fe_coef = gemmi.Element('Fe').c4322
>>> fe_coef.a
[0.3946, 1.2725, 1.7031, 2.314, 1.4795]
>>> fe_coef.b
[0.2717, 2.0443, 7.6007, 29.9714, 86.2265]
```

and, similarly to the X-ray coefficients, they can be used to calculate structure factors and density:

```
>>> fe_coef.calculate_sf(stol2=0.4) # argument: (sin(theta)/lambda)^2
0.9971507787704468
>>> fe_coef.calculate_density_iso(r2=2.3, B=50)
0.13794779777526855
```

Unlike for X-ray form factors, we do not add anomalous scattering here.

### Neutron scattering

For neutrons, we use bound coherent scattering lengths from NCNR, which are based on Neutron News, Vol. 3, No. 3, 1992.

In C++, this data is in the `neutron92.hpp`

header.
In Python, it can be accessed as a property of an element:

```
>>> h_neut = gemmi.Element('H').neutron92
>>> h_neut.get_coefs() # bound coherent scat. length in fm (10^-15 m)
[-3.739]
>>> h_neut.calculate_sf(0) # the argument is ignored
-3.739
>>> h_neut.calculate_density_iso(r2=1.3, B=25)
-0.17104358254308388
```

We don’t store data for individual isotopes, except for deuterium.

```
>>> gemmi.Element('D').neutron92.get_coefs()
[6.671]
```

### Direct summation

Gemmi has functions to calculate structure factor by direct summation of structure factors from individual atoms. This route is not commonly used in macromolecular crystallography and it was implemented primarily to check the accuracy of FFT-based computations. Therefore, the efficiency was not a priority here. In particular, space group specific optimizations, described by Bourhis et al (2014), are not included.

In Python classes StructureFactorCalculatorX and StructureFactorCalculatorE perform direct summation using X-ray and electron form factors, respectively. Class StructureFactorCalculatorN does calculations for neutrons. The C++ interface is similar, although it uses a single templated class StructureFactorCalculator. These classes are constructed with UnitCell as a parameter (we don’t pass SpaceGroup because UnitCell already contains a list of symmetry operations).

```
>>> st = gemmi.read_structure('../tests/4oz7.pdb')
>>> calc_e = gemmi.StructureFactorCalculatorE(st.cell)
```

Now we can compute structure factors from Model for any (hkl):

```
>>> calc_e.calculate_sf_from_model(st[0], (3,4,5))
(54.50873...+53.39498...j)
```

Similarly, for small molecules:

```
>>> small = gemmi.read_small_structure('../tests/1011031.cif')
>>> small.change_occupancies_to_crystallographic()
>>> calc_x = gemmi.StructureFactorCalculatorX(small.cell)
>>> calc_x.calculate_sf_from_small_structure(small, (0,2,4))
(17.849694728851315-6.557871454633539e-15j)
```

For each atom, the Debye-Waller factor (used in the structure factor calculation) is obtained using either isotropic or anisotropic ADPs (B-factors). If anisotropic ADPs are non-zero, isotropic ADP is ignored.

#### Addends

StructureFactorCalculator contains also addends – per-element values
that are to be added to the form factor coefficient *c*
(which is the same as adding it to the total atomic contribution
to the structure factor):

```
>>> calc_x = gemmi.StructureFactorCalculatorX(st.cell)
>>> calc_x.addends
<gemmi.Addends object at 0x...>
```

When calculating X-ray structure factors, one may want to include *f’*
(real part of the anomalous scattering).
To do this, *f’* needs to be set for each element present in the system:
This is done with function `set_addend()`

, which sets angle-independent
value that will be added to the value calculated from the form factors.

```
>>> energy = gemmi.hc / 0.8 # for wavelength 0.8A
>>> for symbol in ['C', 'N', 'O', 'S', 'Cu']:
... el = gemmi.Element(symbol)
... fp, _ = gemmi.cromer_liberman(z=el.atomic_number, energy=energy)
... calc_x.addends.set(el, fp)
...
>>> calc_x.addends.get(gemmi.Element('Cu'))
0.265503853559494
```

Alternatively, you could call a single function (here we first need to remove the previous values, so it makes two functions):

```
>>> calc_x.addends.clear()
>>> calc_x.addends.add_cl_fprime(energy)
```

which calculates *f’* for all elements handled by the Cromer-Liberman
algorithm (*Z* from 3 to 92). Although it seems wasteful, it takes
well below 1ms.

Structure factors calculated at this point incorporate the addends:

```
>>> calc_x.calculate_sf_from_model(st[0], (3,4,5))
(182.3655...+269.0002...j)
```

Addends can also be employed to calculate the electron scattering from X-ray form factors, according to the Mott–Bethe formula:

```
>>> calc_x.addends.clear()
>>> calc_x.addends.subtract_z()
>>> calc_x.mott_bethe_factor() * calc_x.calculate_sf_from_model(st[0], (3,4,5))
(54.065...+52.968...j)
```

The next section gives slightly more details on the Mott-Bethe formula.

### Density for FFT

To use FFT to calculate structure factors, we first need to calculate density of the scatterer (usually electrons) on a grid. For this we use

in C++ – class DensityCalculator templated with a form factor table,

in Python – classes DensityCalculatorX (corresponding to X-ray form factors), DensityCalculatorE (electron form factors) and DensityCalculatorN (neutrons).

DensityCalculator contains a grid. The size of the grid is determined
from two parameters that we need to set: `d_min`

which corresponds to
our resolution limit, and `rate`

– the oversampling rate (1.5 by default).

```
>>> dencalc = gemmi.DensityCalculatorE()
>>> dencalc.d_min = 2.5 # 2.5A
>>> dencalc.rate = 1.5 # we could skip it, this is the default value
```

These two parameters are only used to determine the spacing of the grid. In this case, about 2.5Å / (2 * 1.5) = 0.83Å.

As with StructureFactorCalculator, here we also have addends:

```
>>> dencalc.addends
<gemmi.Addends object at 0x...>
```

To create a grid and calculate the density we use two function calls. Almost all the work is in the latter:

```
>>> dencalc.grid.setup_from(st)
>>> dencalc.put_model_density_on_grid(st[0])
```

Calling `put_model_density_on_grid`

is equivalent to these three functions:

```
>>> dencalc.initialize_grid()
>>> dencalc.add_model_density_to_grid(st[0])
>>> dencalc.grid.symmetrize_sum()
```

`initialize_grid()`

, in this case, uses `d_min`

and `rate`

to determine required grid spacing and uses this spacing to setup the grid.
If `d_min`

is not set, but the grid size is already set,
initialize_grid() only zeros all the grid values.

At this point, we have a grid with density:

```
>>> dencalc.grid
<gemmi.FloatGrid(48, 48, 50)>
```

which can be transformed (as described above) into a structure factor grid:

```
>>> sf_grid = gemmi.transform_map_to_f_phi(dencalc.grid)
>>> sf_grid
<gemmi.ReciprocalComplexGrid(48, 48, 50)>
>>> sf_grid.get_value(3, 4, 5)
(54.5276...+53.4189...j)
```

In addition to `d_min`

and `rate`

, which govern the grid density,
DensityCalculator has two more parameters that affect the accuracy
of the calculated structure factors:

`cutoff`

(default: 1e-5) – density cut-off in the same unit as the map. It is used to determine the atomic radius in which the density is calculated (density at the radius distance should be approximately`cutoff`

). A smaller cutoff means more accurate but slower calculations.`blur`

(default: 0) – Gaussian dampening (blurring) factor – artificial temperature factor*B*_{extra}added to all atomic B-factors (the structure factors must be later corrected to cancel it out).

Choosing these parameters is a trade-off between efficiency and accuracy.
*B*_{extra} is the most interesting one.
It is discussed in the ITfC vol B,
section 1.3.4.4.5 by G. Bricogne, and further in papers by
J. Navaza (2002) and by
P. Afonine and A. Urzhumtsev (2003),
but no formula for the optimal value exists.
The value of *B*_{extra} that
gives the most accurate results depends on the resolution, oversampling,
atomic radius cut-off, and on the distribution of B-factors
(in particular, on the minimal B-factor in the model).
Additionally, increasing the dampening makes the computations slower
(because it increases the atomic radius).

*B*_{extra} can be set explicitly (it can be negative):

```
>>> dencalc.blur = 10
```

or using the formula from Refmac (which is a function of the grid spacing
and *B*_{min}). If the formula results in a negative number,
Refmac sets *B*_{extra}to 0, and Servalcat, which employs the same
formula, uses the negative value:

```
>>> dencalc.set_refmac_compatible_blur(st[0], allow_negative=False)
>>> dencalc.blur
31.01648695048263
```

The sfcalc program can be used to test different choices
of *B*_{extra}.

If the density was blurred, you need to calculate it by either
adding option `unblur`

to `prepare_asu_data()`

:

```
>>> asu_data = grid.prepare_asu_data(dmin=dencalc.d_min, unblur=dencalc.blur)
```

or multiplying individual structure factor by
`dencalc.reciprocal_space_multiplier(inv_d2)`

.

#### Mott-Bethe formula

(It’s a niche topic that most of the readers should skip.)

To calculate *f*_{e} according to the Mott-Bethe formula
we first employ addends to calculate *f*_{x}–*Z*:

```
>>> dc = gemmi.DensityCalculatorX()
>>> dc.d_min = 2.5
>>> dc.addends.subtract_z()
>>> dc.grid.setup_from(st)
>>> dc.set_refmac_compatible_blur(st[0])
>>> dc.put_model_density_on_grid(st[0])
>>> grid = gemmi.transform_map_to_f_phi(dc.grid)
```

Then we multiply it by –1/(2*π*^{2}*a*_{0}*d*^{2})
to get *f*_{e}.

We either multiply individual values by `mott_bethe_factor()`

(which includes `reciprocal_space_multiplier()`

):

```
>>> dc.mott_bethe_factor([3,4,5]) * grid.get_value(3,4,5)
(54.063...+52.970...j)
```

or we call `prepare_asu_data()`

with `mott_bethe=True`

:

```
>>> asu_data = grid.prepare_asu_data(dmin=2.5, mott_bethe=True, unblur=dencalc.blur)
>>> asu_data.value_array[numpy.all(asu_data.miller_array == [3,4,5], axis=1)]
array([54.063...+52.970...j], dtype=complex64)
```

That is all.
If you would like to separate positions of hydrogen nuclei
and electron clouds then, assuming that the model
has positions for electrons, call `subtract_z()`

as:

```
>>> dc.addends.subtract_z(except_hydrogen=True)
```

change hydrogen coordinates to proton positions, and subtract Z=1 by adding c=-1:

```
>>> for cra in st[0].all():
... if cra.atom.is_hydrogen():
... dc.add_c_contribution_to_grid(cra.atom, -1)
```

### Scaling and bulk solvent correction

Anisotropic scaling of calculated structure factors **F** and bulk solvent
correction are usually handled together because the bulk solvent parameters
are optimized along with the scaling parameters.
In Gemmi, both are implemented in one class: Scaling.

The bulk solvent correction is optional and used only with crystallographic
data. The solvent (not modeled as individual atoms) occupies a significant
volume of a macromolecular crystal, so accounting for it makes a difference.
If we have a model with atomic coordinates, we can create a mask,
as described in the section about solvent masking,
to model bulk solvent as a flat scatterer.
Then, we Fourier-transform the mask and obtain **F**_{mask}.

The solvent correction is parametrized as:

F_{sol}=k_{sol}exp(–B_{sol}s^{2}/4)F_{mask}

*k*_{sol} and *B*_{sol} are refined together with the overall
anisotropic scaling parameters, *k*_{ov} and **B**_{ov}:

F_{tot}=k_{ov}exp(–s^{T}B_{ov}s/ 4) (F_{cryst}+F_{sol})

**B**_{ov} is a crystallographic anisotropic tensor that obeys
the symmetry; effectively, from a single fittable parameter for a cubic
crystal to 6 parameters for a primitive crystal.
All the parameters (*k*_{ov}, **B**_{ov},
*k*_{sol} and *B*_{sol}), or only selected ones,
are optimized to make the total calculated amplitude
*F*_{tot} = |**F**_{tot}|
match the diffraction data *F*_{obs}.
But according to what criterion?

First, there is least-squares scaling, which minimizes
∑(*F*_{tot} – *F*_{obs})^{2}.
While this target function has fallen out of favor in macromolecular
crystallography (because the errors are not really Gaussian-distributed),
it is still routinely used for the calculation of statistics such as R-factors.

The workhorse algorithm for non-linear least squares optimization
is the Levenberg-Marquardt method. Unlike other popular algorithms,
such as BFGS and conjugate gradient methods, L-M works only for least squares.
This is because while the other methods estimate the Hessian from the result of
the previous optimization step (BFGS) or ignore the second derivatives (CG),
L-M approximates the Hessian as a squared Jacobian (**J**^{T} **J**),
which holds only for quadratic forms. However, for these forms L-M tends
to converge faster than the other methods.

Here is a full example. We read *F*_{obs} from an mtz file,
then we read a corresponding model and use it to calculate
**F**_{cryst} and **F**_{mask}.
These functions were described in the previous sections.
Then we instantiate the Scaling class, pass the data points to it,
and finally we perform the least squares scaling.

**Note: functions in class Scaling are likely to change soon**.

```
>>> # read Fobs
>>> mtz = gemmi.read_mtz_file('../tests/5e5z.mtz')
>>> mtz.update_reso()
>>> f_obs = mtz.get_value_sigma('FP', 'SIGFP')
>>> # calculate Fcryst
>>> st = gemmi.read_structure('../tests/5e5z.pdb')
>>> d_min = mtz.resolution_high() - 1e-9
>>> dc = gemmi.DensityCalculatorX()
>>> dc.d_min = d_min
>>> dc.set_refmac_compatible_blur(st[0])
>>> dc.grid.setup_from(st)
>>> dc.put_model_density_on_grid(st[0])
>>> grid = gemmi.transform_map_to_f_phi(dc.grid, half_l=True)
>>> f_cryst = grid.prepare_asu_data(dmin=d_min, unblur=dc.blur)
>>> # calculate Fmask
>>> mask_grid = gemmi.FloatGrid()
>>> mask_grid.setup_from(st, spacing=min(0.6, d_min / 2 - 1e-9))
>>> masker = gemmi.SolventMasker(gemmi.AtomicRadiiSet.Refmac)
>>> masker.put_mask_on_float_grid(mask_grid, st[0])
>>> fmask_gr = gemmi.transform_map_to_f_phi(mask_grid, half_l=True)
>>> f_mask = fmask_gr.prepare_asu_data(dmin=d_min)
>>> # now we can start with Scaling
>>> scaling = gemmi.Scaling(st.cell, st.find_spacegroup())
>>> scaling.use_solvent = True
>>> scaling.prepare_points(f_cryst, f_obs, f_mask)
>>> scaling.fit_isotropic_b_approximately()
>>> wssr = scaling.fit_parameters()
>>> print(f'RMSE: {(wssr / len(f_obs)) ** 0.5:.4f}')
RMSE: 6.4176
>>> print(f'R-factor: {scaling.calculate_r_factor():.2%}')
R-factor: 17.73%
>>> # get scaled Ftotal
>>> f_tot = f_cryst.copy()
>>> scaling.scale_data(f_tot, f_mask)
```

Least squares are sensitive to outliers. To make the scaling less sensitive, we must change the target function. The absolute differences in R-factor are less affected by outliers than the squared differences. If we choose a target that’s more similar to the R-factor, we’ll get more robust scaling, and also a lower R-factor.

This requires a different optimization algorithm. Actually, it’s also possible
to *robustify* least squares and keep using (modified) Levenberg-Marquardt,
as was reviewed
for the bundle adjustment procedure, but I haven’t found any research indicating
these methods are preferable to simply using another algorithm.

TBC