Defining the gridΒΆ

  • grid: specifies an input section with the specifications of the grid of points.

It is used for the solution of the vibrational problem.


  npoints 501
  unit angstroms
  range 1.48 , 2.65
  type 0


  • npoints: is the number of grid points Np.

Typical runs use 100 to 500 points.

npoints 301
  • units: is optional and specifies the unit of measure of the grid specifications; possible values are angstroms (default)
    or bohrs.
  • range: specifies the range of the grid in terms of rmin and rmax the lower and upper bond lengths.

rmin should be strictly greater than zero and rmax strictly greater than rmin. As elsewhere in the program, the value may be separated by a space or a comma.

  • type: Grid type

Is an integer number \ge 0 which specifies the type of grid.

Duo support not only uniformely spaced grids (default), which correspond to type 0, but also various kind on non-uniformely spaced ones, which are particularly useful for near-dissociation, very weakly bound states. Example:

type 0

(Default) In the case of uniformely-spaced grids the mesh points rj, j=0, Np-1 are given by r_j = r_\mathrm{min} + \Delta j \qquad \mbox{where} \qquad \Delta = \frac{ r_\mathrm{max}-r_\mathrm{min}} {N_p - 1}

Non-uniformely spaced grids are based on a change of variables from r to z=f(r); it is then the transformed variable z that is uniformely sampled. The transformed variables z are parametrised by two parameters, r_e and \alpha, which have to be specified for the grid types > 0 (see below).

Transformed variable currently implemented are (Phys. Rev. A, 78:052510, 2008):

\quad z=\exp( -e^{-\alpha (r-r_e)} )

\quad z=1 - \left(1+ e^{\alpha (r-r_e)} \right)^{-1}

\quad z= \arctan( \alpha (r-r_e) )

\quad z= (y-1)/(y+1) with y=(r/r_e)^\alpha

All the transformed grids have the property of decreasing the density of points for large r, so that one does not waste too many points in regions where the potential is almost constant and the corresponding vibrational wave function slowly varying.

  • re: (alias ref) Reference bond length used for type > 0 (see above).
  • alpha: Parameter \alpha for type >0 (see above).