Integrating atomic basins


Critic2 provides several methods to integrate the attractor (Bader) basins associated to the maxima of a field. In QTAIM theory, this field is the electron density, the attractors are (usually) nuclei and the basins are the atomic regions. The integrated properties are atomic properties (e.g. atomic charges, volumes, moments, etc.). The attractor basins are defined by a zero-flux condition of the electron density: no gradient paths cross the boundary between attractor regions. This makes the basins local to each attractor, but their definition is a relatively complex algorithmic problem.

The simplest way of integrating an attractor basin is bisection. A number of points distributed in a small sphere around the atom are chosen, each of them determining a ray. On each ray, a process of bisection is started. A point belongs to the basin if the gradient path traced upwards ends up at the position of the attractor we are considering. If the end-point is a different attractor, then the point is not in the basin. By using bisection, it is possible to determine the basin limit (called the interatomic surface, IAS). The bisection algorithm is implemented in critic2, and can be accessed with the INTEGRALS keyword. Bisection works best with analytical fields such as molecular wavefunctions or WIEN2k, but can be used with any field.

The qtree algorithm is based on the recursive subdivision of the irreducible Wigner-Seitz (IWS). In qtree, the smallest symmetry-irreducible portion of space is considered and a tetrahedral mesh of points is superimposed on it. The gradient path is traced from all those points and the points are assigned to different atoms (the points are “colored”). The integration is performed by quadrature over the points belonging to a given basin. The qtree algorithm is accessed through the QTREE keyword, and is suited for small crystals and analytical fields (e.g. WIEN2k or elk).

Lastly, integration algorithms based on grid discretization are very popular nowadays thanks to the widespread use of pseudopotential/plane-waves DFT methods. Critic2 provides the integration method of Yu and Trinkle (YT). The algorithm is based on the assignment of integration weights to each point in the numerical grid by evaluating the flow of the gradient using the neighboring points. This algorithm is extremely efficient and robust and is strongly recommended in the case of fields on a grid. The associated keyword is YT.

An alternative to the YT method is the method proposed by Henkelman et al. (Comput. Mater. Sci. 36, 254-360 (2006), J. Comput. Chem. 28, 899-908 (2007), J. Phys.: Condens. Matter 21, 084204 (2009)), which is implemented in critic2 through the keyword BADER.

The field that determines the basins being calculated is always the reference field (see REFERENCE). In general, it is necessary to define one or more properties to be integrated inside the basins that use other scalar fields. For instance, to calculate the electron population inside an ELF basin, the ELF would be the reference field and the electron density would be an integrable property.

List of Properties Integrated in the Attractor Basins (INTEGRABLE)

INTEGRABLE "expr.s" 
INTEGRABLE ... [NAME name.s]

Critic2 uses an internal list of all properties that will be integrated in the attraction basins. This list can be modified by the user with the INTEGRABLE keyword. This keyword has a syntax similar to the list of properties calculated at the critical points, POINTPROP.

A single INTEGRABLE command assigns a new quantity to be integrated in the atomic basins. The new integrable property is related to field id.s (given as field number or identifier). This quantity can be the field value itself (F), its valence component (if the field is core-augmented, FVAL), the gradient norm (GMOD), the Laplacian (LAP), or the valence-component of the Laplacian (LAPVAL). If no keyword is given after id.s, F is used by default.

With the MULTIPOLES (or MULTIPOLE) keyword, the multipole moments of the field are calculated up to an l equal to lmax.i (default: 5). This keyword only applies to the BADER and YT integration methods. For the others, it is equivalent to the field value (same as F). The units for all calculated multipoles are atomic units.

The keyword DELOC activates the calculation of the delocalization indices (DIs) using field id.s. The calculation of DIs requires that the field is a Quantum ESPRESSO pwc file, which contains the Bloch states. There are two modes of operation with DELOC:

  • Calculate the DIs using maximally localized Wannier functions (MLWF). This requires wannier90 checkpoint files to be provided with the pwc file, and is the default behavior in this case. More details are given below. The use of Wannier functions can be forced using the WANNIER keyword in INTEGRABLE DELOC.

  • Calculate the DIs using Bloch states. This is much less efficient than using Wannier functions but it is simpler because it does not require an additional wannier90 calculation and allows calculating DIs in metals. More details are given below. The use of Bloch states can be forced using the PSINK keyword in INTEGRABLE DELOC.

Also, see the example for the calculation of DIs in solids.

Both ways of calculating the DIs generate by default two checkpoint files containing the (complex) overlap matrices (sij) or the matrix of atomic integrals of the exchange-correlation density (fa). The delocalization indices can be calculated from these files bypassing having to load the pwc and wannier90 checkpoint files with the DELOC_SIJCHK and DELOC_FACHK keywords. This is useful because the pwc files can be quite large, so it may be interesting to delete them after the sij or fa matrices are computed. Both keywords accept a string pointing to the corresponding file. The DI calculation from these matrices will typically take only a few seconds.

In addition, it is possible to define an integrable property using an expression involving more than one field (expr.s). For instance, if the spin-up density is in field 1 and the spin-down density is in field 2, the atomic moments can be obtained using:

LOAD AS "$1+$2"

Note that the quotation marks are required.

The additional keyword NAME can be used with any of the options above to change the name of the integrable property, for easy identification in the output.

The keyword CLEAR resets the list to its initial state (volume, electron population, and Laplacian in crystals; electron population and Laplacian in molecules). Using the INTEGRABLE keyword will print a report on the list of integrable properties.

The default integrable properties are:

  • Volume (1), in crystals only.

  • Pop (fval): the value of the reference field is integrated. If the reference field is the density, then this is the number of electrons in the basin. If core augmentation is active for this field, only the valence contribution is integrated.

  • Lap (lap(fval)): the Laplacian of the reference field. The integrated Laplacian has been traditionally used as a check of the quality of the integration because the exact integral is zero regardless of the basin (because of the divergence theorem). However, it is difficult to obtain a zero in the Laplacian integral in critic2 in some cases because of numerical inaccuracies:

    • In fields based on a grid, the numerical interpolation gives a noisy Laplacian.

    • In FPLAPW fields (WIEN2k and elk), the discontinuity at the muffin surface introduce a non-zero contribution to the integral.

    If f is a core-augmented field, only the valence Laplacian is integrated.

Integrating Delocalization Indices in a Solid With Maximally Localized Wannier Functions

The keyword DELOC activates the calculation of localization and delocalization indices (DIs) in a crystal using the procedure described in the literature. DIs can be calculated only if the loaded field contains information about individual Kohn-Sham states and the orbital rotation that leads to the maximally localized Wannier functions (MLWF). This is done by using a field loaded from a .pwc file (generated by the pw2critic.x utility in Quantum ESPRESSO) together with a checkpoint (chk) file from wannier90. The former contains the electronic wavefunctions and the latter the orbital rotation. For maximum consistency, the pwc file can also be used to provide the structural information for the run via the CRYSTAL keyword.

In addition to these data, the calculation of DIs has a few requirements: the grid must be consistent with that of the reference field, and the DIs can be calculated using YT or BADER only.

A typical delocalization index calculation comprises the following steps:

  • Run a PAW calculation, then obtain an all-electron density using pp.x with plot_num=21. This creates a cube file (rhoae.cube) that gives the Bader basins for the calculation (the pseudo-valence density is not valid for this purpose).

  • Run an SCF calculation with norm-conserving pseudopotentials and the same ecutrho as the calculation in the first step, so the two grids have the same size.

  • Use the open_grid.x utility in Quantum ESPRESSO to unpack the symmetry of the k-point grid, in preparation for the wannier90 run (this would normally be accomplished by a non-SCF calculation, but with open_grid.x it is easier and much faster).

  • Use pw2critic.x on the output of open_grid.x to generate the pwc file.

  • Run wannier90 on the result of open_grid.x to generate the chk file.

  • Load the all-electron density and the pwc and chk files as two fields in critic2. Set the former as the reference density and the latter as INTEGRABLE DELOC. If the system is spin-polarized, two checkpoint files will be necessary, one for each spin component.

  • Run YT or BADER. YT is usually more accurate but takes longer than BADER.

Additional options for the DI calculation follow. The NOU option disables the use of the U rotation matrices to calculate the MLWFs. This makes critic2 calculate the DI using Wannier functions obtained by using a straight Wannier transformation from the Bloch states. This is naturally much slower than the maximally localized version, since overlaps cannot be discarded, and should be used only if wannier90 failed to converge for the particular case under study.

By default, two checkpoint files are generated during a DI calculation run. These files have the same name as the pwc file but with -sij and -fa suffixes. The former checkpoint file contains the atomic overlap matrices, and the latter, the exchange-correlation density (\(F_{AB}\)) integrals required for the DI calculation. The presence of any of these two files makes critic2 read the information from the files and bypass the corresponding calculations, which are quite time consuming in general. The keywords NOSIJCHK and NOFACHK deactivate reading and writing these checkpoint files. You can calculate the DIs from these files directly without the pwc file using the DELOC_SIJCHK and DELOC_FACHK options to INTEGRABLE.

By default, the overlap between two MLWFs whose centers are a certain distance away are discarded. The WANCUT keyword controls this distance: overlaps are discarded if the centers are wancut.r times the sum of their spreads away. By default, wancut.r = 4.0. A very large wancut.r will prevent critic2 from discarding any overlaps. The appropriateness of the chosen WANCUT can be checked a posteriori by comparing the integrated electron population obtained by sum of the localization and delocalization indices to the value obtained from a straight integration of the electron density.

See the example for some sample input files and more details.

Integrating Delocalization Indices in a Solid With Bloch States

The DIs can also be calculated without doing the transformation to Wannier functions, using the Bloch states provided by the calculation. This is typically far less efficient than using Wannier functions but has two advantages:

  • It is simpler because it does not require the additional wannier90 step.

  • Using Bloch states, DIs can be calculated in metals (the Wannier transformation is ill-defined if some bands are partially occupied).

The calculation of DIs using Bloch states requires a .pwc file only, generated by the pw2critic.x utility in Quantum ESPRESSO. The pwc file contains electronic wavefunctions and the crystal structure. For maximum consistency, the pwc file should be used to provide the structural information for the run via the CRYSTAL keyword.

In addition to these data, the calculation of DIs with Bloch states has a few requirements: the grid must be consistent with that of the reference field, and the DIs can be calculated using YT or BADER only. A typical delocalization index with Bloch states calculation comprises the following steps:

  • Run a PAW calculation, then obtain an all-electron density using pp.x with plot_num=21. This creates a cube file (rhoae.cube) that gives the Bader basins for the calculation (the pseudo-valence density is not valid for this purpose).

  • Run an SCF calculation with norm-conserving pseudopotentials and the same ecutrho as the calculation in the first step, so the two grids have the same size. This calculation must be run without k-point symmetry, i.e., with nosym=.true, and noinv=.true..

  • Use pw2critic.x to generate the pwc file.

  • Load the all-electron density and the pwc files as two fields in critic2. Set the former as the reference density and the latter as INTEGRABLE DELOC.

  • Run YT or BADER. YT is usually more accurate but takes longer than BADER.

By default, two checkpoint files are generated during a DI calculation run. These files have the same name as the pwc file but with -sij and -fa suffixes. The former checkpoint file contains the atomic overlap matrices, and the latter, the exchange-correlation density (\(F_{AB}\)) integrals required for the DI calculation. The presence of any of these two files makes critic2 read the information from the files and bypass the corresponding calculations, which are quite time consuming in general. The keywords NOSIJCHK and NOFACHK deactivate reading and writing these checkpoint files. You can calculate the DIs from these files directly without the pwc file using the DELOC_SIJCHK and DELOC_FACHK options to INTEGRABLE.

The keywords NOU and WANCUT have no effect on the calculation of DIs using Bloch states.

See the example for some sample input files and more details.


INTEGRALS {GAULEG ntheta.i nphi.i|LEBEDEV nleb.i}
          [CP ncp.i] [RWINT] [VERBOSE]

The BISECTION keyword integrates the attractor basins using bisection. If the Gauss-Legendre quadrature is used (GAULEG keyword), ntheta.i and nphi.i are the number of \(\theta\) (polar angle) and \(\phi\) (azimuthal angle) points.

In the case of a Lebedev-Laikov quadrature, selected via the LEBEDEV keyword, only the total number of points in the spherical quadrature is needed. The actual value of nleb.i is the smallest number larger than the one given by the user that is included in the list: 6, 14, 26, 38, 50, 74, 86, 110, 146, 170, 194, 230, 266, 302, 350, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810.

By using the CP keyword, a single non-equivalent CP (ncp.i) is integrated. Otherwise, all the CPs of the correct type (found using AUTO) are integrated. If RWINT is present, read (if they exist) and write the .int files containing the interatomic surface limit for the rays associated to the chosen quadrature method.

Defaults: ntheta.i = nphi.i = 50, nleb.i = 4802.

                 LEBEDEV nleb.i} [CP ncp.i] [NR npts.i]
                [R0 r0.r] [REND rend.r]

The SPHEREINTEGRALS keyword integrates the volume, field and Laplacian of the reference field in successive spheres centered around each of the attractor CPs. The meaning of the GAULEG and LEBEDEV keywords is the same as in INTEGRALS.

A total number of npts.i spheres are integrated per nucleus. The grid is logarithmic, so that the region near the nucleus has a higher population of points. The grid starts at the radius r0.r and ends at rend.r (bohr in crystals, angstrom in molecules). If rend.r < 0 then the final radius is taken as half the nearest neighbor distance for each atom times abs(rend.r).

Default: npts.i = 100. In GAULEG, ntheta.i = 20 and nphi.i = 20. In LEBEDEV, nleb.i = 770. r0.r = 1d-3 bohr. rend.r = rnn/2 for each CP. id.i = 0 (all attractors).

Qtree (QTREE)

General Syntax

The QTREE integration method integrates QTAIM atomic properties by discretization of the smallest part of the crystal that reproduces the whole system by symmetry. QTREE is specific for periodic crystals, and its use is recommended with fields not given on a grid. For fields on a grid, either BADER or YT are better alternatives. QTREE is based on a hierarchical subdivision of the irreducible part of the WS cell, employing a tetrahedral grid. The integration region is selected so as to maximize the use of symmetry, and partitioned into tetrahedra. The syntax of QTREE consists of the maximum subdivision level (maxlevel.i), perhaps followed by the pre-splitting level (plevel.i), and then a series of optional keywords that control the behavior and the options for the integration.

QTREE [maxlevel.i [plevel.i]] [MINL minl.i] [GRADIENT_MODE gmode.i]
      [QTREE_ODE_MODE omode.i] [STEPSIZE step.r] [ODE_ABSERR abserr.r]
      [INTEG_MODE level.i imode.i] [INTEG_SCHEME ischeme.i] [KEASTNUM k.i]
      [PLOT_MODE plmode.i] [PROP_MODE prmode.i] [MPSTEP inistep.i]
      [QTREEFAC f.r] [CUB_ABS abs.r] [CUB_REL rel.r] [CUB_MPTS mpts.i]
      [SPHFACTOR {ncp.i fac.r|at.s fac.r}] [SPHINTFACTOR atom.i fac.r]
      [DOCONTACTS] [WS_ORIGIN x.r y.r z.r] [WS_SCALE scale.r]
      [COLOR_ALLOCATE {0|1}] [SETSPH_LVL lvl.i] [VCUTOFF vcutoff.r]

In QTREE, the tetrahedra that comprise the IWS enter a recursive subdivision process in which each of the tetrahedra is divided in 8 at each level, up to a level given by the user. This subdivision level is controlled by the maxlevel.i argument given after the QTREE keyword (default, 6). Every tetrahedron vertex is assigned to a non-equivalent atom in the unit cell by tracing a gradient path and identifying its endpoint. Once all the points in the tetrahedral mesh are assigned, the tetrahedra are integrated and the properties assigned to the corresponding atoms. The space near the atoms is integrated using a beta-sphere, which improves the accuracy of the integration.

In the simplest (and most common) approach, qtree can be executed using:

QTREE [maxlevel.i [plevel.i]]

where maxlevel.i is the level of subdivision. The optional plevel.i value corresponds to the pre-splitting level of the tetrahedra. The initial tetrahedra list is split into smaller tetrahedra plevel.i times. This can be useful in cases where a very high accuracy (and therefore a very high level) is required, but there is not enough memory available to advance to higher maxlevel.i. However, using a relatively high plevel.i incurs an overhead, because the atom assigning procedure is not as efficient when smaller tetrahedra are used.

Steps of the QTREE Algorithm

  1. The WS cell is constructed and split into tetrahedra, all of which have in common, at least, the origin of the WS cell. Then, the site symmetry of the origin is calculated and the tetrahedra that are unique under the operations of this group are found. This is what we call the irreducible Wigner-Seitz cell (IWS). Note, however, that it is only “irreducible” in the local site symmetry of the origin, not in the full set of space group operations. The IWS is the region that is integrated in later steps of QTREE. We will refer to a IWS tetrahedra as an IWST.

    It is possible, through the WS_ORIGIN keyword, to shift the origin of the WS cell away from the (0 0 0) position. Because the symmetry of the WS cell is determined by the site symmetry of the origin, the number and shape of IWST change depending on the origin chosen. A general position (with no symmetry) will make the IWS exactly equal to the WS.

    Also, for large systems, the user can choose to shrink the size of the original WS cell in order to integrate a smaller region, using the WS_SCALE keyword (most likely in combination with WS_ORIGIN, to move the integration region around). If a value is given to WS_SCALE (for instance, rws.r), then all the vectors connecting the origin of the WS cell with the vertex are shrunk by a factor rws.r, and the volume of the integration region is decreased by a factor equal to the cube of rws.r. The IWS is calculated using the smaller WS cell and integrated in the same way. Note that this integration region is non-periodic: it does not fill the volume of the solid and it does not integrate to the total number of electrons per cell.

  2. Non-overlapping spheres are chosen centered on each of the atoms in the cell, the so-called beta-spheres. Atoms equivalent by symmetry share the same beta-sphere radius (\(\beta_i\) for atom \(i\)). The beta-sphere has two roles in QTREE:

    • The atomic properties are integrated inside the beta-spheres using a 2D cubature. The cubature can be a product of two 1D Gauss-Legendre quadratures or a Lebedev quadrature of the sphere. Both methods, and the number of nodes can be selected using the INT_SPHEREQUAD_* keywords shown below. The radial quadrature can be any of the available in critic2, and is controlled by the INT_RADQUAD_* options. The default values, however, are usually fine, integrating the beta-spheres in a matter of seconds with a precision that is orders of magnitude better than the overall QTREE performance.

      This beta-sphere integration removes the error of the finite-elements integration of a region where the integrated scalar fields present the steeper variations in value. By removing the high-error regions from the grid integration, the accuracy of QTREE is enhanced. In particular, this increase in precision outweighs the loss of precision caused by creating an additional interface between the grid and the sphere.

    • The space inside the beta-sphere of an atom is assumed to be inside the basin of that atom. The terminus of any gradient path that reaches the interior of the beta-sphere \(i\) is assumed to be the atom \(i\). It is known that most of the steps in the integration of the gradient of the electron density are spent in the close vicinity of the terminus. Therefore, this modification saves precious function evaluations.

    The default beta-sphere radius is set to \(0.80\) times half the nearest-neighbor distance. Both i) and ii) above assume that the beta-sphere is completely contained inside the basin of the atom. This may turn out not to be true for the default beta-sphere radius (specially for cations in ionic systems). In these cases, the keyword SPHFACTOR is used. An example of the use of this keyword is:

    SPHFACTOR 1 0.70   # Make b_1 = 0.70 * rNN2(1) (atom type 1)

    If SPHFACTOR < 0, use the method by Rodriguez et al. to determine the beta-sphere radii: a collection of points around the atom are selected and the angle between the gradient and the radial direction is determined. If all the angles are less than 45 degrees, the sphere is accepted. In solids, this strategy usually yields spheres that are too large.

    In the case that any atomic SPHFACTOR is zero (the default value for all atoms), then a pre-computation at a lower level is done to ensure that all beta-spheres lie within the desired basins. There are two methods for this, and the method can be chosen using the AUTOSPH keyword.

    Method number one involves using a reduced version of QTREE. The pre-computation usually takes no longer than some minutes (and usually only few seconds) and the spheres are guaranteed to be inside the basins. The keyword SETSPH_LVL controls the level of the pre-computation, that must not be higher than 7. The default value is 6 or maxlevel.i, whichever is smaller.

    The second method (default) traces gradient path on a coarse sphere around each nucleus, and reduces the sphere until all of the points are inside the basin. NOCHECKBETA is used in this case.

    An additional factor the user can define is the SPHINTFACTOR. It is possible to consider the sphere where GPs terminate different from the one that is integrated. If SPHINTFACTOR is defined, for example as:


    then the sphere associated to atom 1 where the integration is done has a radius which is 0.75 times that of the sphere where GP terminate.

    The CHECKBETA and NOCHECKBETA keywords activate and deactivate the check that ensures that the beta-spheres is completely contained inside the basin.

    If a beta-sphere is not strictly contained in the basin, QTREE detects it and stops immediately (specifically, QTREE checks that every tetrahedron that is partly contained in a beta-sphere has vertex termini that are all assigned to the same atom as the atom that owns the beta-sphere).

    For a new system, it is always a good idea to start with a low-level QTREE (say, level 4) to check if the default beta-spheres are adequate. If one of the beta-spheres is too large, the error message looks like:

    An undecided tetrahedron is overlapping with a beta-sphere.
    Make beta-spheres smaller for this system.
    terms:            1          -1           2           1

    which indicates that there is a tetrahedron that is partly contained in the sphere of the first atom (terminus -1) and that has a vertex corresponding to atom 2. Modifying the sphfactor solves this problem:

    SPHFACTOR 1 0.70

    At lower levels, QTREE is reasonably fast, so a trial-and-error selection of beta-spheres is acceptable. Note that the beta-spheres used in QTREE have no relation to the ones reported after an AUTO calculation.

  3. If the cell is periodic (which means that WS_SCALE was not set), the contacts between the faces of the IWST are found. These contacts are used in a later step to copy the termini information between tetrahedron faces.

    The determination of the tetrahedra contacts in a periodic integration region is deactivated if the NOCONTACTS keyword is used. The DOCONTACTS keyword does the opposite thing: it activate the calculation of contacts. By default, the contacts are not calculated.

  4. A grid is built for each of the IWST. For each of the grid points, the termini of the gradient paths starting at them is calculated. The positions of the grid points are determined by subdividing the IWST maxlevel.i times. In each subdivision step, a parent tetrahedron is divided in 8 smaller tetrahedron (all with the same volume, \(V/8\)) by splitting each edge of the parent tetrahedron in two. There are two possible ways of doing this, the choice being irrelevant to the performance of QTREE.

    For a given IWST, the size of the grid is given by \(\begin{equation} S = n*(n+1)*(n+2)/6 \end{equation}\) where \(n = 2^l + 1\) and \(ll\) is the maxlevel.i, the approximate scaling being as \(8^l\). The termini information on the grid is saved to the array trm of type integer*1, with size \(n_t * S\), where \(n_t\) is the number of IWST (in fact, the integer type is that which is the result of selected_int_kind(2)).

    The subdivision level is the main input parameter for QTREE, controlling the accuracy (and cost!) of the integration. For small-medium sized systems \(4-5\) are low cost integrations (seconds), \(6-7\) are medium cost (minutes) and \(8-9\) are the slowest and most accurate (hours). The level is input in the call to the QTREE integration:

    QTREE 6   # maxlevel.i is 6

    By default, the integration level is 5.

    In addition to trm, more work space can be allocated if the integration is restricted to the volume and charge or to the volume, charge and Laplacian. The number and type of properties to integrate is controlled by the PROP_MODE keyword. The following values are allowed:

    • 0: Only volume is integrated. This amounts to canceling the finite elements integration of tetrahedra and is equivalent to INTEG_MODE 0 (see below).

    • 1: Only charge and volume. If the integration uses the value of the density at the grid points (INTEG_MODE 11, see below) In addition to trm, another real*8 array, fgr, is allocated (strictly it is selected_real_kind(14), not real*8). In fgr, the value of the density at the grid points is stored.

    • 2: Charge, volume and Laplacian. In a similar way to 1, if the information on the grid points is used during the integration (INTEG_MODE 11), an additional real*8 array, lapgr, is allocated. It contains the value of the Laplacian of the electron density at the grid points.

    • 3: All the integrable properties calculated by the module. The number of properties varies with the interface being used. No fgr or lapgr are allocated, as the grid points need to be recomputed during integration.

    The default value for PROP_MODE is 2.

    The termini of the grid points contained in a beta-sphere is marked previous to the beginning of the subdivision.

  5. Each tetrahedron is subdivided recursively up to a level maxlevel.i, and integrated at the same time. The IWST integration is relatively independent of one another, so for the moment we will focus on just one IWST, which we will call the base tetrahedron.

    The result of the integration of the different IWST can be used for other IWST:

    • When the integration ends, the termini of the four faces of a base tetrahedron are copied to its neighbors’ trm, according to the contacts determined previously.

    • Depending of the method chosen (see GRADIENT_MODE below), the gradient path integration may be aware of the neighboring grid points, that may belong to other IWST. In particular, the gradient mode number 3 integrates a gradient path following grid points. When the endpoint is reached, all the grid points that have been traversed by the path are assigned the same common terminus. Therefore, there is the possibility that gradient paths starting inside a given base tetrahedron write the trm of other IWST.

  6. A tetrahedra stack is built and initialized with one element: the base tetrahedron. An iterator works on the stack, performing at each step the following tasks:

    • Pop a tetrahedron from the stack.

    • The termini of the vertex of the tetrahedron are calculated, if they are not already known.

    • If all the termini of the tetrahedron correspond to the same atom, the tetrahedron is “painted”. This means that all the grid points that are in the interior or border of the tetrahedron are assigned the same color as its vertex, thereby saving the tracing of the gradient paths.

      “Painting” can be dangerous whenever a (curved) IAS crosses the face of the tetrahedron being painted. To this end, a minimum level is defined, using the keyword QTREE_MINL. If the subdivision level of the tetrahedron is lower or equal than minl.i, the tetrahedron is not painted. (Note: the base tetrahedron corresponds to level 0).

      Furthermore, if all the termini correspond to the same atom and are located outside of the beta-sphere region, the tetrahedron is integrated and does not enter another subdivision process. Once more, this only happens to tetrahedra with a level of subdivision strictly greater than minl.i. The “inner integration method” is the quadrature method used to integrate these tetrahedra.

      If all the termini are located inside the beta-sphere region, the tetrahedron does not subdivide, but the properties are not integrated, because this region corresponds to the sphere integration addressed in point 2. If the tetrahedron is across the border of a beta-sphere, it is divided further.

    • If the tetrahedron is at subdivision level equal to maxlevel.i, then it does not subdivide, it is integrated and the properties are assigned to the atoms. There are several possibilities depending of the nature of its vertex’ termini:

      • If all the termini correspond to the same atom, and the tetrahedron is completely inside or outside of this atom’s sphere, it corresponds to the case in 6.3.

      • If it is completely inside an atom basin, but on the border of a beta-sphere, the part of the tetrahedron that is outside of the sphere is integrated and assigned to the atom. Another integration method is required for this, different from the “inner integration”. We will refer to this method as “border, same-color integration”.

      • If the tetrahedron has termini corresponding to different atoms, its properties are integrated and split into contribution to atoms, according to the number of termini each atom has. These tetrahedra are located on the IAS, and require a third class of integration, “border, diff-color integration”.

    • A tetrahedron that has not been integrated continues the subdivision. In this step, 8 new tetrahedra are pushed into the stack. The edges of the parent tetrahedron are split in two. By construction, the newly generated points also correspond to grid points.

      The subdivision scheme is:

      • 1, 1-2, 1-3, 1-4
      • 2, 1-2, 2-3, 2-4
      • 3, 1-3, 2-3, 3-4
      • 4, 1-4, 2-4, 3-4
      • 2-3, 1-2, 1-3, 1-4
      • 1-4, 1-2, 2-3, 2-4
      • 1-4, 1-3, 2-3, 3-4
      • 2-3, 1-4, 2-4, 3-4

      where “a” represents a vertex of the parent tetrahedron and “a-b” the midpoint of both vertex. Each of the 8 child tetrahedra enclose the same volume, equal to \(V/8^l\), where \(l\) is the subdivision level and \(V\) is the volume of the base tetrahedron.

    • When the stack is empty, the work on the base tetrahedron is finished.

  7. “Inner integration”. The inner integration is a quadrature that is applied to tetrahedra that are completely contained in the non-beta-sphere region of a basin. It can apply to a tetrahedron of any level, as long as this level is greater than minl.i. The integrated properties are assigned to a single atom.

    In the current implementation of QTREE, several integration methods are possible, and are controlled by the INTEG_MODE keyword. The possible values of INTEG_MODE are:

    • 11 : use the information of the density, Laplacian and properties at the vertex of the tetrahedron to integrate. The integral is approximated by a quadrature of four terms, each corresponding to a volume that is 1/4 of the volume of the tetrahedron and multiplied by the value of the properties at the vertex. This integration method is useful if only charge or charge and Laplacian are being integrated, because the information gained during the gradient path tracing, and saved in the fgr and lapgr arrays, is used. It is not very accurate for large tetrahedra.

    • 12 : use the CUBPACK routines. CUBPACK provides an adaptive tetrahedron integration method based on recursive subdivision and an integration rule with 43 nodes (degree 8), that is equivalent to the DCUTET library by Bernsten et al. The integration rule is fully symmetric under the Th group operations. The error estimation is compared to the error requested by the user, that is controlled using the CUB_ABS (absolute error), CUB_REL (relative error) and CUB_MPTS (maximum number of function evaluations) keywords. If CUB_MPTS is exceeded, an error message is output, but the QTREE integration continues.

      Note that, no matter how low the error requirements are, the CUBPACK integration spends, at least, 43 function evaluations per tetrahedron, so it is quite expensive if compared to other integration modes. This should be used for large tetrahedra (see below) or for really accurate calculations.

    • 1…10 : use a non-adaptive rule from the KEAST library (Keast et al., 1986), the number corresponding to:

      • 1 – order = 1, degree = 0
      • 2 – order = 4, degree = 1
      • 3 – order = 5, degree = 2
      • 4 – order = 10, degree = 3
      • 5 – order = 11, degree = 4
      • 6 – order = 14, degree = 4
      • 7 – order = 15, degree = 5
      • 8 – order = 24, degree = 6
      • 9 – order = 31, degree = 7
      • 10 – order = 45, degree = 8

      In particular, the first KEAST rule uses the barycenter of the tetrahedron.

    The syntax of the INTEG_MODE keyword is:

    INTEG_MODE lvl.i mode.i

    where mode.i is one of the modes above and lvl.i is the level to which it applies. This means that, if a tetrahedron of a given level is integrated, the value of INTEG_MODE(level) is checked to decide on the method.

    Another INTEG_MODE value is possible:

    • -1 : do not integrate and force the tetrahedron into the subdivision process. This value of INTEG_MODE can be combined with a positive value at higher levels, amounting to a recursive integration in the style of CUBPACK. Of course, -1 is not an acceptable value of INTEG_MODE for the last level, maxlevel.i.

    As setting these INTEG_MODE by hand could be confusing, QTREE provides sets of INTEG_MODE values, which we will call “integration schemes”. An integration scheme is a full set of INTEG_MODEs for all levels. Integration schemes are selected with the INTEG_SCHEME keyword, that can have the following values:

    • 0: do not integrate, only calculate volume and plot (see below). This is equivalent to setting PROP_MODE to 0.

    • 1: subdivide each tetrahedron up to the highest level and then integrate using the vertex information. This is most useful if PROP_MODE is 1 (only charge and volume) or 2 (charge, volume and Laplacian) because the information of the gradient path tracing (fgr and lapgr arrays) are used:
      INTEG_MODE = -1 -1 ... -1 11
      !            ^^           ^^
      !        QTREE_MINL    `maxlevel.i`
    • 2: subdivide each tetrahedron up to the highest level and then integrate using the barycenter.
      INTEG_MODE = -1 -1 ... -1 1.
    • 3: barycentric integration at all levels of subdivision. Less accurate but faster than 2.
      INTEG_MODE = 1 1 ... 1 1.
    • 4: one of the Keast rules (given by the KEASTNUM keyword) is used at all levels. If KEASTNUM is n,
      INTEG_MODE = n n ... n n.
    • 5: CUBPACK, at all levels. Reserve this one for special occasions.
      INTEG_MODE = 12 12 ... 12 12.
    • 6: this scheme and the next are (poor) attempts at trying an adaptive integration scheme. They are not more reliable or efficient than, for instance, scheme 2. Integration scheme 6 calculates levels 4, 5, and 6 using CUBPACK, and the rest with subdivision up to the highest level and vertex-based integration.
      INTEG_MODE = 12 12 12 -1 ... -1 11.
    • 7: same as 6 but the final integration uses only the barycenter.
      INTEG_MODE = 12 12 12 -1 ... -1 1.
    • -1: let the user enter the INTEG_MODEs by hand.

    The default integration scheme is 2, suitable for low and medium-accuracy calculations.

  8. “Border, same-color integration”. This integration method applies to tetrahedra that have reached the maximum subdivision level and sit on the interface between a beta-sphere and the atomic basin. Some of the vertex are inside the sphere and some are outside. The out-of-sphere part is integrated and added to the atomic properties, while the in-the-sphere part is ignored because it has already been integrated.

    The integration works by assuming that the sphere radius is much larger than the tetrahedron characteristic lengths and, therefore, that the sphere surface can be considered a plane that intersects the tetrahedron. The intersection points of the sphere with the tetrahedron edges are easily calculated and, for simplicity, we refer to them as the “middle” of the edges. There are three cases:

    • One vertex is outside, three inside. The tetrahedron formed by the vertex outside and the three middle points of the edges that stem from it form a tetrahedron by itself, which is integrated and added to the atom properties.

    • Three vertex are outside, one inside. The difference between the whole tetrahedron integration and the small tetrahedron inside the sphere is added to the atom properties. The small tetrahedron is formed by the vertex that is inside the sphere and the three edges connected to it.

    • Two vertex are inside, two outside. The region outside of the sphere is a “triangular prism”, that is split in three tetrahedra and integrated.

    Note that the INTEG_MODE of the maximum subdivision level (maxlevel.i) applies to all the sub-integrations of the border, same-color integration.

  9. “Border, diff-color integration”. As in the case of “border, same-color integration”, this method only applies to tetrahedra which are at their maximum subdivision level. In this case, the termini of the vertex corresponding to, at least, two different atoms.

    In the current implementation of QTREE, the tetrahedron is integrated as a whole. Then, the properties are equally assigned to each of the termini atoms. For instance, if the termini are (1 1 1 3), the properties of the tetrahedron are integrated, then 3/4 of them assigned to atom 1 and 1/4 to atom 3.

  10. Gradient path tracing. The gradient path start always at grid points, and are traced using one of three methods, controlled by the GRADIENT_MODE keyword, that can assume the following values:

    • 1: “full gradient”. This method is plain ODE integration. It is carried out ignoring the grid information. The gradient path is terminated whenever it enters a beta-sphere region.

    • 2: “color gradient”. At each point of the gradient path, the neighboring grid points are checked. If all of them correspond to the same atom, then the terminus of the gradient path is assigned to that atom.

      In a tetrahedral mesh, the meaning of “neighboring grid points” is not as clear as in a cubic mesh. For a given point x, the neighbors are calculated by first converting x to convex coordinates, that range from 0 to \(2^l\), restricted to \(x_1 + x_2 + x_3 \leq 2^l\), where \(l\) is the subdivision levels. The neighboring points are \((x_1\pm 1, x_2\pm 1, x_3\pm 1)\) If any of these neighbors are not valid points in the tetrahedron, they are discarded and not checked. This is the default, except in the grid module.

    • 3: “qtree gradient”. This method behaves much like the “full gradient”, but whenever the gradient path steps near a grid point, it is projected to it. When a projection occurs, the grid point is pushed onto a stack. At the end of the gradient path, when the terminus is known, all the grid points in the stack are popped and assigned the terminus.

      The projection regions are spheres located around each grid point, whose radius is controlled by the QTREEFAC keyword. The radius of these spheres is \(m_l/2^{l_{\rm max}}/{\rm qtreefac}\), where \(m_l\) is the smallest edge length of all IWST and \(l_{\rm max}\) is the maximum subdivision level (maxlevel.i). Note that QTREEFAC equal to 1 is the maximum value allowed, and corresponds to touching spheres along at least one tetrahedron edge. By default, QTREEFAC is 2. Lower levels of QTREEFAC tend to give errors when assigning the grid points that lie on the IAS of two atoms (although only there). With higher levels, the time saving diminishes, and “qtree gradient” is equivalent to “full gradient”.

      Additionally, the projection can be started only after a certain number of initial steps. The MPSTEP keyword controls this value. The default MPSTEP is 0.

    • -1, -2, -3: these correspond to the same as their positive values, but each gradient path terminus is compared to their “full gradient” version, using the best available ODE integration method. Information about the results of the comparison are output to stdout, and a .tess file is generated (difftermxx.tess, where xx is the subdivision level) containing the position of the points where both termini differ.

    If the integration region is not periodic, then methods “color” and “qtree” are not defined. There are two possible options, controlled by the “KILLEXT” and “NOKILLEXT” keywords. If KILLEXT is active (the default behavior), the gradient path tracing is killed whenever it leaves the integration region, independently of the GRADIENT_MODE being used. The terminus is then assigned to an “unknown” state, and the tetrahedra it generates are not integrated. If NOKILLEXT is active, the gradient path is continued as a “full gradient”, until the terminus is found.

    The default is KILLEXT because, if the integration region is not periodic, the integral over atoms that are partially contained in it is most likely not meaningful to the user.

    The ODE integration method can be chosen using the QTREE_ODE_MODE keyword, that can assume the following values:

    • 1 : Euler method, fixed step, 1st order.

    • 2 : Heun method, fixed step, 2nd order.

    • 3 : Kutta method, fixed step, 3rd order.

    • 4 : Runge-Kutta method, fixed step, 4th order.

    • 5 : Euler-Heun embedded method, adaptive step. 1st order with 2nd order error estimation. 2 evaluations per step.

    • 6 : Bogacki-Shampine embedded method, adaptive step. 3rd order with 5th order error estimation. The FSAL (first step also last) allows only 4 evaluations per step. Local extrapolation.

    • 7 : Runge-Kutta Cash-Karp embedded method, adaptive step. 4th order with 5th order error estimation. 6 evaluations per step.

    • 8 : Dormand-Prince 4-5 embedded method, adaptive step. 4th order with 5th order error estimation. 6 evaluation per step, with FSAL. Local extrapolation.

    For embedded methods (4-8), the absolute error requested from the method can be set using the ODE_ABSERR keyword. The default of this variable is chosen so that reasonable stepsizes are kept. This default is 1d-3 for Euler-Heun and 1d-4 for the rest. Typically, methods with greater accuracy (7 and 8) save evaluations by increasing stepsize to values much larger than their lower accuracy counterparts.

    The step size of the fixed step methods (1-4) is controlled with the STEPSIZE keyword (bohr). In the variable step methods (5-8) the value of STEPSIZE is the length of the starting step.

    The default QTREE_ODE_MODE is Dormand-Prince (8).

  11. When the integration of the base tetrahedron is finished, the termini of the grid points located at each of its four faces are copied to the corresponding neighboring IWST, if DOCONTACTS is active.

Once the integration of the IWST is completed, the atomic properties are summed. The final result is output, together with an analysis of the contribution of each subdivision level to the total integrated properties.

It is possible to plot the basins obtained by QTREE using the PLOT_MODE keyword. It can assume the values:

  • 0: no plotting is done.

  • 1: a single tess file is written containing a description of the unit cell CPs, the IWS, and balls corresponding to all the grid points that have been sampled.

  • 2: same as 1 but only the balls that are either on the face of an IWST or close to a IAS are output.

  • 3: the full WS cell

  • 4: a file for the full WS cell and several files, containing a description of each of the integrated basins. Note that the basins need not be connected.

  • 5: same as 4 but only balls belonging to faces of IWST and IAS are output.

The default value is 0. If PLOT_MODE is > 0, then the sticks that form the tetrahedra are written to .stick files. The PLOTSTICKS and NOPLOTSTICKS keywords control this behavior.

Additional Considerations

  • The integration of the volume is not done using the beta-sphere / basin separation because the volume of each tetrahedron is exactly known. The integrated cell volume for a periodic integration region will always be exact (if it is not, then it is an error). The integrated cell charge, on the contrary, is a measure of how well the tetrahedra are being integrated, but not of how well the IAS is being determined.

  • For very high levels of QTREE (say 10-11, depending on the amount of memory your computer has), memory usage may be a problem. The COLOR_ALLOCATE keyword controls the amount of memory allocated for the color and property arrays. The syntax is:


    Using a zero value, the color vector (and possibly the properties vectors, depending on PROP_MODE) is allocated only for the current IWST. This saves memory but makes the computation slower, especially if the GRADIENT_MODE is 2 or 3. In addition, setting COLOR_ALLOCATE to 0 deactivates the passing of colors through the contacts (DOCONTACTS and NOCONTACTS keywords) and the plotting (sets PLOT_MODE to 0). If COLOR_ALLOCATE is 1, the color (and optionally the properties) of all the IWST are saved. By default, COLOR_ALLOCATE is 1 if maxlevel.i is less than 9 and 0 if the maximum level is higher.

Yu and Trinkle method (YT)

YT [NNM] [NOATOMS] [WCUBE] [BASINS [OBJ|PLY|OFF] [ibasin.i]] [RATOM ratom.r]
   [DISCARD expr.s] [JSON file.json]

The Yu and Trinkle (YT) method calculates the attraction basins of the reference fields and computes QTAIM integrals in them. The reference field must be defined on a grid. Hence YT will not work directly with wien2k, elk, aiPI,… densities, although those can be transformed into a grid by appropriate use of the LOAD keyword. The algorithm proceeds by running over grid nodes in decreasing order of density. If a point has no neighboring points with higher density, then it’s a local maximum. If it does, but all of them belong to the same basin then that point belongs to the interior of that basin as well. Otherwise, it is sitting on top of the interatomic surface. The actual fraction of a grid point on an IAS belonging to a particular basin is calculated by evaluating the trajectory flow to neighboring points.

The YT algorithm is described in J. Chem. Phys. 134 (2011) 064111 which should be consulted for further details. Please, cite this reference if you use this keyword in your work.

The located maxima in the field are identified by default with the closest nucleus. If non-nuclear maxima are expected, use the NNM keyword to assign only maxima that are only within 1 bohr of the closest atom. This distance can be changed using the RATOM keyword (ratom.r in bohr (crystals) or angstrom (molecules)), which also controls the distance below which two maxima are considered the same maximum. Changing the default ratom.r using the RATOM keyword automatically activates the detection of NNM. The NOATOMS option is appropriate for scalar fields where the maxima are not expected to be at the atomic positions (or at least not all of them). If NOATOMS is used, all the maxima found are given as NNM. This is useful for fields such as the ELF, the Laplacian, etc.

The WCUBE option makes critic2 write cube files for the integration weights of each attractor. In YT, these weights are values zero (outside the basin), one (inside the basin), or some intermediate value near the atomic basin boundary. The generated cube files have names <root>_wcube_xx.cube, where xx is the attractor number.

Use the BASINS option to write a graphical representation of the calculated basins. The format can be chosen using the OBJ, PLY, and OFF keywords (default: OBJ). If an integer is given after the format selector (ibasin.i), then plot only the basin for that attractor. Otherwise, plot all of them. The basin surfaces are colored by the value of the reference field, in the default gnuplot scale.

Any maxima that is not assigned to an existing atom or non-nuclear critical point is automatically added to the critical point list. It is possible to get more information about these maxima by using the CPREPORT keyword.

In some cases, particularly if there is a vacuum region in your system (for instance, if your system is a molecule or a surface), multiple spurious maxima will appear due to numerical noise in the grid values. The number of spurious attractors will increase the computational cost and serve little purpose, as the vacuum region will integrate to zero anyway. In these cases, the DISCARD keyword can be used to make critic2 ignore any attractor that matches the expression expr.s when it is evaluated at that point. For instance, if the electron density is given by field $rho, and we want to discard low-density critical points, we could use:

DISCARD "$rho < 1e-7"

The arithmetic expression may include any field, not just the reference field.

By using the JSON keyword a JavaScript Object Notation (JSON) file is created containing the molecular or crystal structure, information about the reference field and the results of the integration.

Not all the properties defined by the INTEGRABLE keyword are integrated inside the basins. Only the subset of those properties that are grids, have F or FVAL as the integrand and are congruent with the reference grid are considered. This limitation can be circumvented by using LOAD AS to define grid from fields or expressions that are not given on a grid. In addition, no core is used even if the CORE keyword is active. The volume is always integrated. A xyz file (<root> is always written, containing the unit cell description (with border, see WRITE) and the position of the maxima, labeled as XX.

Note that in the output (“List of basins and local properties”), “Charge” refers not to the integrated electron density (because critic2 does not know whether a given field is an electron density or not) but to the value of the integral of the reference field in its own basins (which may not make much sense if you are integrating, for instance, the ELF or the Laplacian). Loading a second field and using INTEGRABLE and the field number is the way to go in such cases.

Usage of the YT algorithm for grid fields is strongly recommended, as it is much more efficient, robust and accurate than the alternatives.

Henkelman et al. method (BADER)

The algorithm by Henkelman et al. is implemented in critic2, and can be used with the BADER keyword:

      [DISCARD expr.s] [JSON file.json]

The BADER algorithm uses the reference field to calculate the QTAIM basins. This field must be defined on a grid. BADER assigns grid nodes to basins using the near-grid method incrementally described in Comput. Mater. Sci. 36, 354-360 (2006), J. Comput. Chem. 28, 899-908 (2007), and J. Phys.: Condens. Matter 21, 084204 (2009). Please, cite these references if you use this method. The same method is implemented in the bader program available from the Henkelman research group’s website.

The output and the optional keywords have the same meaning as in YT. Using BADER as an alternative to YT is recommended in very large grids because of its more efficient memory usage, but in general it gives less accurate integrations (at least in my experience). The weight cubes written by WCUBE contain zeros for the grid points outside the basin, and ones inside.