Plotting gradient paths

Overview

Gradient paths are the solution of the differential equation \(x^\prime = {\mathbf \nabla}f({\mathbf x})\) where \(f({\mathbf x})\) is a scalar field. They play an important role in QTAIM theory because gradient paths of the electron density cannot cross the boundary between atomic regions. In consequence, a gradient path plot is a simple way to investigate the shape and properties of a basin. Gradient paths originate at maxima (if the field is the density they are usually the nuclei) and end at the minima (the crystal voids) or at infinity in case of a gas-phase molecule.

There are two keywords for making gradient path representations in critic2: GRDVEC (2D gradient paths plus a contour plot) and FLUXPRINT (3D).

Gradient Path Representations in a Plane (GRDVEC)

GRDVEC
   {FILES|ROOT|ONAME} rootname.s
   PLANE x0.r y0.r z0.r x1.r y1.r z1.r x2.r y2.r z2.r
   SCALE sx.r sy.r
   EXTENDX zx0.r zx1.r
   EXTENDY zy0.r zy1.r
   OUTCP sx.r sy.r
   HMAX hmax.r
   ORIG x.r y.r z.r atr.i up.i down.i
   CP cp.i up.i down.i
   CPALL
   BCPALL up.i down.i
   RBCPALL bup.i bdown.i rup.i rdown.i
   CHECK
        x.r y.r z.r
        ...
   ENDCHECK/END
   CONTOUR {F,GX,GY,GZ,GMOD,HXX,HXY,HXZ,HYY,HYZ,HZZ,LAP} 
     nptsu.i nptsv.i {LIN niso.i [cini.r cend.r]|
     LOG niso.i [zmin.r zmax.r]|ATAN niso.i [zmin.r zmax.r]|
     BADER|i1.i i2.i...}
ENDGRDVEC/END

GRDVEC makes a plot in a plane. The plot contains the gradient paths originating from a set of points, critical or otherwise, inside that plane, in addition to a contour representation of a scalar field. The GRDVEC syntax consists of a GRDVEC…ENDGRDVEC environment that accepts a set of input lines (in any order) that control the characteristics of the plot. The syntax of GRDVEC originated from (and is similar to) Bader’s AIMPAC suite of programs.

By using the FILES keyword (equivalently, ROOT or ONAME), the user sets the root name of the output files containing the information for the plot (default: <root>). These files include:

  • <root>.grd : gradient path data.

  • <root>.dat : values of the reference field on the plane.

  • <root>.iso, <root>.neg.iso : positive (green) and negative (blue) contour lines.

  • <root>.gnu : gnuplot script file that generates the merged gradient/contour plot.

  • <root>-label.gnu : gnuplot script file loaded in <root>.gnu containing the information for the position of the CPs in the plot plane.

PLANE specifies the plane for the plot using three points: x0 is the origin, x1 the end of the x-axis and x2 the end of the y-axis. By default, these points are in crystallographic coordinates in a crystal, and in molecular Cartesian coordinates in a molecule (default units: angstrom). The two axes of the plane can be scaled using the SCALE keyword. If sx.r (sy.r) is given, the total length of the x-axis (y-axis) is scaled by sx.r (sy.r). If EXTENDX is used, extend the x-axis by zx0.r (initial point of the x-axis) and zx1.r (end point). The keyword EXTENDY performs the equivalent operation on the y-axis. The units for EXTENDX and EXTENDY are bohr (crystals) or angstrom (molecules) unless changed by the UNITS keyword.

The plot plane may contain regions that are traversed by gradient lines originating at critical points located inside the plane but outside the plot region. If this is the case, the OUTCP option allows the user to extend the plane for the CP labels. The sx.r and sy.r in OUTCP are scale parameters for the plane, same as in SCALE, but only apply to CP labels. The x-axis extends \((s_x-1)\times l_x\) in each direction, where \(s_x\) is sx.r and \(l_x\) is the length of the x-axis. The sy.r variable works the same way. The plane determined by the vectors given in PLANE acts as a clipping plane while the scaled plane determines the gradient path origins.

With HMAX, you can set the maximum distance from a CP to the plane to be included in the plot (units: bohr in crystals, angstrom in molecules). Default: 1d-4 bohr.

The ORIG keyword adds a source of gradient lines to the plot. Its coordinates are x.r, y.r and z.r. Crystallographic coordinates are used in a crystal, and molecular Cartesian coordinates in a molecule (default units: angstrom). atr.i is 1 if the point is to be treated as a ncp or ccp (the up and down trajectories start from points located on a sphere centered on the origin) and it is 0 if the point is to be treated as a bcp or ccp (a circle is built around the CP in the plane determined by two eigenvectors whose eigenvalues have equal sign. The remaining eigenvector determines a unique direction). up.i and down.i are the number of gradient paths to be started in the upwards and downwards direction, respectively.

The CP keyword accepts a critical point identifier from the complete CP list (or the complete atom list). The number of upwards and downwards gradient paths must be given. A special case is the CPALL keyword, which adds as origins every critical point in the CP list on the selected plane. The default number of gradient paths is 36 down for ncps and 36 up for ccps, and 2 up and 2 down for bcps and rcps. The BCPALL keyword is similar to CPALL, except that only the bond critical points are included as origins. If BCPALL is used, the user must supply the number of gradient lines in the upwards and downwards directions. In a similar way, RBCPALL includes bond and ring critical points, and the user must give the number of upwards and downwards gradient paths for bonds (bup.i, bown.i) and rings (rup.i, rdown.i).

The CHECK environment allows the user to enter the crystallographic coordinates of a CP of the scalar field to add it as an origin. If the point given is not a CP or if it lies outside the selected plane, it is excluded from the list of points that are sources of gradient paths. The valid CPs in the CHECK list are identified and an adequate number of gradient paths are started according to its character: for a ncp and ccp, 36 upwards or downwards and for a bcp or rcp, 2 upwards and 2 downwards.

The CONTOUR keyword makes critic2 generate a plot in which the gradient paths calculated in GRDVEC are merged with a contour plot, in the spirit of the CONTOUR option to PLANE. The scalar field for the contour plot can be selected with F (current reference field), GX, GY,… The syntax for the derivatives is the same as in PLANE. After this, the user must specify the number of points in each direction of the plane (nptsu.i and nptsv.i) and the contour values and type of mapping. The contour distribution can be: logarithmic (LOG, with niso.i contours), arctangent (ATAN, with niso.i contours), same as in the aimpac program (BADER, {1,2,4,8}x10^{-3,-2,-1,0,1}), linear (LIN, niso.i contours from r0.r to r1.r), or the user can specify the contour values manually (no keyword). In LOG and ATAN, the default contours range from the minimum to the maximum value of the field in the plot. These quantities can be changed by passing the optional zmin.r and zmax.r parameters to LOG/ATAN.

Note that GRDVEC is able to handle non-orthogonal axes for the plot plane. If the two plane axes determined in the PLANE keyword are non-orthogonal, the final graph will correctly reflect the actual appearance of the plane by conserving the original angle between the x- and y- axis. Also, note that at most 2 gradient lines may be traced from bcps and rcps, either upwards or downwards. Thus, for example, BCPALL 2 2 is equivalent to BCPALL 2 100 or BCPALL 100 100.

Three-Dimensional Gradient Path Representations (FLUXPRINT)

FLUXPRINT
  POINT {1|-1|0} x.r y.r z.r [step.r epsi.r]
  NCP cp.i ntheta.i nphi.i [step.r epsi.r]
     [LVEC x.i y.i z.i]
  BCP cp.i 1 [step.r epsi.r] [LVEC x.i y.i z.i]
  BCP cp.i {0|-1} n.i [step.r epsi.r]
     [LVEC x.i y.i z.i]
     [BRAINDEAD|QUOTIENT|DYNAMICAL]
  RCP cp.i -1 [step.r epsi.r] [LVEC x.i y.i z.i]
  RCP cp.i {0|1} n.i [step.r epsi.r]
     [LVEC x.i y.i z.i]
     [BRAINDEAD|QUOTIENT|DYNAMICAL]
  CCP cp.i ntheta.i nphi.i [step.r epsi.r]
     [LVEC x.i y.i z.i]
  GRAPH igraph.i [step.r epsi.r]
  COLOR r.i g.i b.i
  TEXT|TESSEL|TESS|OBJ|PLY|OFF|CML
  SHELLS ishl.i
  NOSYM
ENDFLUXPRINT/END

The FLUXPRINT keyword prints three-dimensional gradient paths. There are several plotting commands:

  • POINT: plot a gradient path starting at point (x.r y.r z.r) in crystallographic coordinates (crystals) or in molecular Cartesian coordinates (molecules, default unit: angstrom). step.r is the maximum step (Cartesian coordinates) for the gradient path tracing algorithm. If step.r > 0, use it also as the initial step. If step.r < 0, use a small step as initial the initial step (1d-3). epsi.r is the gradient norm stop criterion. The default values are 0.1 for step and 1e-9 for epsi.r. The {1|-1|0} field controls the direction of the path. An ascending gradient path is obtained with 1 while -1 issues a descending path. 0 = -1 + 1 makes FLUXPRINT represent both ascending and descending paths.

  • NCP: print gradient paths starting from a (small) sphere centered on the nuclear CP identified by cp.i (this identifier comes from the complete CP list). The number of points is controlled by ntheta.i (number of points sampling the azimuthal angle) and nphi.i (number of points sampling the polar angle). cp.i specifies a ncp in the main cell up to a lattice translation. The LVEC optional keyword allows the user to enter a lattice vector to displace the represented gradient paths from their initial position, which is given by the complete CP list written by AUTO.

  • BCP: print gradient paths starting at the vicinity of a bond CP, identified by cp.i. If the gradient path is ascending (1 in the fourth field), the (unique) bond path associated to the bcp is represented. If -1 is given instead, the IAS associated to the bcp is sampled starting from a small circle surrounding the bcp, with n.i points on it. With a 0 value, both tasks are performed.

    The three keywords BRAINDEAD, QUOTIENT and DYNAMICAL establish the method employed in generating the starting angular grid. With BRAINDEAD, critic2 uses a uniform angular grid. Using QUOTIENT, the uniform grid is remapped by \(x^{l_1/l_2}\) where \(l_1\) and \(l_2\) are the two negative eigenvalues at the bcp. This way, the points get accumulated around the bcp with lowest eigenvalue (highest if absolute value is taken). DYNAMICAL uses a linearized model of the interatomic surface and predicts the initial angles critic2 has to take in order to generate a uniform distribution of points a given distance away. This distance is calculated as 90% of the distance to the nearest ccp found in a coarse exploration of the IAS omega-limits. Unfortunately, this algorithm works only in cases where the bcp has significant but not too large ellipticty. Also, there is no gain in using this method in cases where the number of ccps is different than 4.

    By default, BRAINDEAD is used. H1 is experimental.

  • RCP: print gradient paths starting at the neighbourhood of a ring CP. The situation is analogous to that of the bcps.

  • CCP: print gradient paths starting at the vicinity of a cage CP. Again, the situation is symmetric to the ncp case.

  • GRAPH: represent the complete graph in the unit cell. This means:

    • All the bond paths for which both ncps and the bcp lay inside the main unit cell.

    • All the ring paths for which both ccps and the rcp lay inside the main unit cell.

    The critical points on the boundary of the main cell are also represented.

    The igraph.i value represents the amount of information that is to be printed. It is the sum of these two values, each representing an element to plot:

    • 1 : print ring paths associated to the rcps.

    • 2 : print bond paths associated to the bcps.

Several parameters of the plot generated by FLUXPRINT can be changed. The color to be applied to the paths resulting from FLUXPRINT commands can be changed using the COLOR keyword (three integers from 0 to 255 for the red, green, and blue components). By default, NCP uses the color corresponding to the originating atom, and a gold color is used otherwise. The format of the output file can be controlled with the following keywords: TEXT (plain text file), TESS or TESSEL (tessel), OBJ (Wavefront obj), PLY (ply format), OFF (Geomview’s off), and CML (Chemical Markup Language). The CML format, which can be read with the avogadro program, is used by default. The color specifications are not passed to the CML format. To keep the number of points manageable, critic2 writes one gradient path point every certain distance along the path.

Finally, the SHELLS keyword applies only to graph and graphcp. It represents the number of unit cell shells where the graph is going to be plotted. Thus, 0 represents the main unit cell; 1, the main unit cell and its 26 neighbours; and so on. By default, SHELLS adopts the -1 value, which is equivalent to 0 for the graph keyword and means that the partial graph generated in GRAPHCP is not replicated by symmetry.

The NOSYM keyword instructs critic2 to write the complete list of CPs in the unit cell, together with the identity matrix as the only operation in the space group, to the output.