# 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*) reflection - unmerged – 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')
```

### Metadata¶

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
>>> mtz.min_1_d2 # from RESO
0.0028703967109323008
>>> mtz.max_1_d2 # from RESO
0.36117017269134527
```

The resolution can also be checked using functions:

```
>>> mtz.resolution_low() # sqrt(1 / min_1_d2)
18.665044863474492
>>> 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
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 number;
std::string project_name;
std::string crystal_name;
std::string dataset_name;
UnitCell cell;
double wavelength = 0.;
};
```

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

### 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_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
0.8726000000000002
```

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

```
>>> 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[36:38] # start and end of phi
[80.0, 80.5]
```

One pecularity 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.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:

Another way to access all the MTZ data in Python is through the buffer protocol: For example, to view the data as 2D NumPy array (C-style contiguous) without copying do:

```
>>> import numpy
>>> all_data = numpy.array(mtz, copy=False)
```

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

```
>>> import pandas
>>> df = pandas.DataFrame(data=all_data, 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`

.
And finally, gemmi has 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]]...)
```

We also have 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 Mtz object we can use function set_data(). In Python, we pass to this function a 2D NumPy array of floating point numbers.

In the previous section, we got such an array (`all_data`

)
with the original data.
Now, as an example, let us remove reflections with the resolution
above 2Å (i.e. *d* < 2Å) and copy the result back into `mtz`

.
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(all_data[mtz.make_d_array() >= 2.0])
>>> mtz.get_size_for_hkl()
[10, 10, 20]
```

The metadata in an `Mtz`

object can also be modified.
To illustrate it, 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`

automatically added 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.float)
>>> 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 = ...`

or `mtz.history += ...`

.

To update properties `min_1_d2`

and `max_1_d2`

call `update_reso()`

:

```
>>> mtz.update_reso()
>>> mtz.min_1_d2, mtz.max_1_d2
(0.02664818727144997, 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.

### Writing¶

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

```
// In exactly one compilation unit define this before including one of
// mtz.hpp, to_mmcif.hpp, to_pdb.hpp.
#define GEMMI_WRITE_IMPLEMENTATION
#include <gemmi/mtz.hpp>
void Mtz::write_to_stream(std::FILE* stream) const
void Mtz::write_to_file(const std::string& path) const
```

and in Python using:

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

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

```
#define GEMMI_WRITE_IMPLEMENTATION
#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');
mtz.add_column("SIGF", 'Q');
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
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 gemmi
import numpy
import pandas
from matplotlib import pyplot
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_data = numpy.array(mtz, copy=False)
mtz_df = pandas.DataFrame(data=mtz_data, 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
```

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)
```

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.

## 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
```

## 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 both MTZ and SF mmCIF data.
Here, we focus on the usage from Python.

Both Mtz and ReflnBlock classes (as well as the AsuData class introduced below)
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 buffer protocol (in the same way as in the Grid class), 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)
0.04960193997923228
>>> grid.calculate_d(point)
4.490044743125198
```

### AsuData¶

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: Int8AsuData, 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 (which was read from an MTZ or SF-mmCIF file):

>>> asu_data = mtz.get_value_sigma('F', 'SIGF') >>> asu_data.ensure_asu() >>> asu_data <gemmi.ValueSigmaAsuData with ... values>The choice of ASU differs between programs, so if this matters, calling

`ensure_asu()`

will switch to the ASU used in CCP4.

- From a reciprocal-space grid:

>>> asu_data = grid.prepare_asu_data(dmin=1.8, with_000=False, with_sys_abs=False) >>> asu_data <gemmi.ComplexAsuData with 407 values>

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

```
>>> asu_data[158]
<gemmi.ComplexHklValue (-6,2,5) (-1.37694,-0.190087)>
>>> _.hkl, _.value
([-6, 2, 5], (-1.376941204071045-0.19008657336235046j))
```

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.

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)
```

We can also go back from AsuData to ReciprocalComplexGrid:

```
>>> asu_data.get_f_phi_on_grid([54,6,18])
<gemmi.ReciprocalComplexGrid(54, 6, 18)>
```

and we can use other method common for 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()
>>> new_data = gemmi.ComplexAsuData(asu_data.unit_cell,
... asu_data.spacegroup,
... asu_data.miller_array[d<8],
... asu_data.value_array[d<8])
>>> new_data
<gemmi.ComplexAsuData with 399 values>
>>> hkl_grid = new_data.get_f_phi_on_grid([54, 6, 18])
>>> gemmi.transform_f_phi_grid_to_map(hkl_grid)
<gemmi.FloatGrid(54, 6, 18)>
```

The last two calls could be replaced with:

```
>>> new_data.transform_f_phi_to_map(exact_size=[54, 6, 18])
<gemmi.FloatGrid(54, 6, 18)>
```

### 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 explicitely 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)
>>> array = numpy.array(full, copy=False)
>>> complex_map = numpy.fft.ifftn(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 equivant 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)
>>> array = numpy.array(half, copy=False)
>>> real_map = numpy.fft.irfftn(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
```

Both Mtz and ReflnBlock 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:

```
>>> array = numpy.array(_, copy=False)
>>> 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(2, True)
>>> 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])
m.setup()
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 (one Gaussian more). Other parametrizations exist (for example, with only two Gaussians), but are not widely used.

Currently, Gemmi includes only the ITC parametrization. The other ones will be added if needed. Furthermore, we ignore charges of atoms. For example, the ITC provides separate form factors for Cu, Cu1+ and Cu2+, but we use only the first one. Again, if it turns out to be useful, we will add also the charged parametrizations.

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

header.
In Python, they can be accessed as a property of an element (this may change).

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

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
```

Fun fact: for neutral atoms the *a*’s and *c* should sum up to *Z*,
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 them.
To change the coefficients use function `set_coefs()`

.
Here, to make it simple, we change only c:

```
>>> new_c = 26 - sum(fe_coef.a)
>>> fe_coef.set_coefs(fe_coef.a + fe_coef.b + [new_c])
```

or you can used them to directly calculate the sum of Gaussians – the structure factor contribution:

```
>>> fe_coef.calculate_sf(stol2=0.4) # argument: (sin(theta)/lambda)^2
9.31320248504031
```

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)
0.5281308588290554
```

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*.

We may also want to add a Gaussian dampening
*B*_{extra} to ADPs.
It is a trick that improves the accuracy. *B*_{extra} added
in the real space is then cancelled out in the reciprocal space by re-scaling
the structure factors.

### Anomalous scattering¶

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

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-Libermann 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.

Admittedly, 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_libermann`

function is available in both C++ and Python:

```
>>> gemmi.Element('Se').atomic_number
34
>>> gemmi.cromer_libermann(z=_, energy=10332.0) # energy in eV
(-1.41862..., 0.72389...)
>>> # use gemmi.hc to convert wavelength [A] to energy [eV]
>>> gemmi.cromer_libermann(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
0.9971508092073326
>>> fe_coef.calculate_density_iso(r2=2.3, B=50)
0.13794779230350634
```

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

### 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. If you would like to use it in performance sensitive contact us.

In Python classes StructureFactorCalculatorX and StructureFactorCalculatorE perform direct summation using X-ray and electron form factors, respectively. 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))
(54.50873699946013+53.39498671218216j)
```

#### 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_libermann(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-Libermann
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))
(182.3655966448982+269.0002625524398j)
```

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))
(54.065658093072436+52.968332363622224j)
```

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) and DensityCalculatorE (electron form factors).

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`

– 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
```

All these two parameters mean is that the spacing between grid planes needs to be at least 2.5Å / (2 * 1.5) = 0.67Å.

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.set_grid_cell_and_spacegroup(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()
```

Now we have density calculated on the grid:

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

which we can transform (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)
(54.52764892578125+53.418941497802734j)
```

In addition to `d_min`

and `rate`

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

`r_cut`

(default: 1e-5) – density cut-off. Atomic radius in which we calculate the density is picked in such a way that, approximately, we include only points with values greater than`r_cut`

(note: this variable may change in the future).`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,
chapter 1.3 by G. Bricogne, section 1.3.4.4.5, and further in papers by
J. Navaza (2002) and by
P. Afonine and A. Urzhumtsev (2003),
but without a formula for optimal value.
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 atomic radius).

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

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

or using the formula from Refmac:

```
>>> dencalc.set_refmac_compatible_blur(st[0])
>>> 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¶

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.set_grid_cell_and_spacegroup(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)
(54.063699106275386+52.97058061981685j)
```

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([54.063698+52.97058j], 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 and fitting of the bulk solvent parameters are usually
performed together. Both are implemented in the same class (Scaling),
but it doesn’t prevent you from scaling *F*s without any bulk solvent,
or from calculating the bulk solvent correction without anisotropic scaling.

The bulk solvent occupies significant volume of a macromolecular crystal;
adding its contribution significantly changes structure factors.
Usually, the bulk solvent is modelled as a flat scatterer.
First, we create a mask of the bulk solvent, as described in
a section about solvent masking.
Then, we Fourier-transform the mask and obtain *F*_{mask}.
Usually, the solvent correction has two parameters:

*F*_{sol} = *k*_{sol} exp(-*B*_{sol} *s*^{2}/4) *F*_{mask},

and we also have overall anisotropic scaling parameters
(*k*_{ov} and **B**_{ov}).
All the parameters, i.e. *k*_{ov}, **B**_{ov},
*k*_{sol} and *B*_{sol}, or only selected ones,
are optimized to make the total calculated structure factors *F*_{calc}
match the diffraction data *F*_{obs}.

TBC

class Scaling

```
>>> grid = gemmi.FloatGrid()
>>> grid.setup_from(st, spacing=1.0)
>>> masker = gemmi.SolventMasker(gemmi.AtomicRadiiSet.Cctbx)
>>> masker.put_mask_on_float_grid(grid, st[0])
>>> gemmi.transform_map_to_f_phi(grid, half_l=False).prepare_asu_data(dmin=2.1)
<gemmi.ComplexAsuData with 1848 values>
```