Subject: Accuracy of Levedev grids

From: Christoph van Wüllen <[email-masked]>
Date: Fri, 3 Mar 2000 10:13:32 +0100

Accuracy of Lebedev grids
=========================

Few days ago it was discussed here where to obtain Lebedev grids.

To check the accuracy of Lebedev grids for numerical quadrature on a sphere,
I evaluated the integral of normalized spherical harmonics Y_ml(th, ph)
over the whole sphere, giving the Integral I_ml. For angular momentum l>0,
the result should be zero, and I define the error of the grid

  Err(l) = Max(m=-l to m=l) |I_ml|

The error is zero for l=0 (the weights are normalized) and l uneven (due
to symmetry). The following table gives the errors for even l=2..64 for
the lebedev grids with 6, 14, 38, 50, 86, 110, 194, 302 and 434 points
(taken from my DFT program) and for the lebedev grids with 590 and 770
points (taken from http://www.fos.su.se/~mhl/science/software.html).
The first column is a 3200-point reference grid which integrates exactly
all Y_ml up to l=79. This was used to check the accuracy of the Y_ml, for
which I used code from the NumericalRecipes book.

It can be seen up to which l the Levedev grids integrate the Y_ml exactly.
More important is that the higher grids contain parameters which are not
given very accurately, especially the accuracy of the 590/770 grid suffers
>from too few digits given.

- Are these grid (freely) available with more accurate parameters?
- If anyone wants to check his favourite angular integration grid, I can
   send him the code to test it, it is just a small program.

TABLE: ACCURACY OF LEBEDEV GRIDS WITH 6/14/38/50/110/194/302/434/590/770
        ANGULAR INTEGRATION POINTS. 3200=REFERENCE GRID ACCURATE TO L=79

  L/N  3200     6    14    38    50    86   110   194   302   434   590   770
   2  -13.9 -15.4 -15.2 -12.3 -15.7 -12.3 -12.3 -11.9 -13.2 -15.3  -9.5  -9.5
   4  -14.2   1.1 -14.9 -11.5 -15.1 -11.1 -11.6 -11.6 -11.6 -12.3  -9.0  -8.3
   6  -14.2   0.8   0.9 -11.8 -15.1 -11.0 -11.1 -11.3 -11.3 -12.4  -8.7  -8.1
   8  -14.1   1.2   1.0 -11.3 -14.5 -10.9 -11.2 -10.8 -11.0 -12.4  -8.5  -8.0
  10  -14.0   1.1   0.8   1.1 -14.4 -10.7 -11.0 -10.9 -11.1 -12.4  -8.5  -8.1
  12  -14.4   1.4   1.1   0.9   1.1 -10.5 -10.9 -10.9 -10.9 -12.0  -8.3  -7.6
  14  -14.2   1.2   1.2   0.6   0.8 -11.6 -10.8 -10.4 -11.0 -11.9  -8.2  -7.9
  16  -14.0   1.5   1.2   0.7   0.9   1.1 -10.6 -10.8 -10.6 -11.7  -8.4  -7.5
  18  -13.8   1.3   0.8   1.0   0.4   1.0   1.1 -10.1 -10.6 -11.3  -8.0  -7.6
  20  -13.5   1.5   1.4   1.2   0.9   0.5   1.0 -10.6 -10.5 -10.8  -7.9  -7.2
  22  -13.4   1.4   1.1   1.0   1.1   0.8   0.7 -10.4 -10.6 -10.4  -8.0  -7.2
  24  -14.1   1.6   1.2   0.9   1.2   0.6   0.8   1.1 -10.8 -10.3  -7.8  -7.1
  26  -13.6   1.5   1.5   0.6   0.9   0.8  -0.1   1.0 -10.4  -9.7  -7.8  -7.2
  28  -14.0   1.7   1.4   1.4   1.1   0.8   0.7   1.0 -10.3  -9.4  -7.6  -7.1
  30  -13.8   1.6   1.3   1.1   1.1   1.0   0.8   0.6   1.1  -8.5  -7.7  -7.2
  32  -13.9   1.7   1.5   1.3   1.2   1.2   1.1   0.6   1.0  -7.5  -7.5  -7.0
  34  -13.9   1.6   1.5   1.1   1.2   1.1   1.0   0.2   0.9  -6.2  -7.6  -7.0
  36  -13.4   1.7   1.5   1.1   1.2   1.2   1.3   0.6   0.9   1.1  -7.5  -6.7
  38  -13.4   1.7   1.4   1.5   1.1   1.0   1.1   0.5   0.3   1.0  -7.5  -6.8
  40  -13.5   1.8   1.7   1.4   1.3   0.8   1.0   1.0   0.3   1.0  -7.4  -6.7
  42  -13.4   1.7   1.5   1.2   1.1   0.9   0.9   1.0   0.2   0.9   1.1  -6.9
  44  -13.5   1.8   1.5   1.3   1.4   1.1   1.1   1.0   0.5   0.9   1.0  -6.7
  46  -13.7   1.8   1.7   1.4   1.2   1.3   1.1   1.2   0.1   0.4   0.9  -6.7
  48  -13.7   1.9   1.6   1.5   1.3   1.4   1.2   1.2   0.8   0.4   1.0   1.1
  50  -13.4   1.8   1.5   1.3   1.5   1.0   1.1   1.1   0.8   0.1   0.8   1.0
  52  -13.6   1.9   1.7   1.4   1.3   1.3   1.2   1.0   1.1   0.3   0.8   1.0
  54  -13.2   1.8   1.6   1.4   1.0   1.2   1.4   0.9   0.9  -0.2   0.5   0.9
  56  -13.5   1.9   1.7   1.6   1.5   1.0   1.2   1.0   1.1   0.6   0.5   1.0
  58  -13.1   1.9   1.7   1.4   1.2   1.3   1.2   0.8   1.1   0.6   0.2   0.7
  60  -13.3   1.9   1.8   1.5   1.6   1.4   0.9   1.0   1.3   0.9   0.2   0.7
  62  -13.6   1.9   1.7   1.4   1.4   1.1   1.3   1.1   1.1   1.0  -0.3   0.5
  64  -13.3   2.0   1.7   1.5   1.5   1.4   1.3   1.1   1.0   1.0   0.4   0.6
-- 
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Christoph van Wullen       | Fon (University):  +49 234 32 26485
Theoretical Chemistry      | Fax (University):  +49 234 32 14109
Ruhr-Universitaet          | Fon/Fax (private): +49 234 33 22 75
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