7.3. Details on the numerical integration grids¶

As in all other popular grid schemes, our grids are constructed from assembling a set of atomic grids into a molecular one, using Becke’s approach. Each individual atomic grid is build based on optimized parameters for that atom, and are composed of an angular and a radial part, that are defined separately.

The whole scheme was updated from ORCA 5.0, but we tried to keep things as close as possible to the previous one. First, the overall construction of these grids needs to be explained.

7.3.1. The angular grid scheme¶

Instead of using a single angular grid throughout the whole atom, most schemes apply a so-called grid pruning in order to reduce the number of grid points outside of the most important regions, as we do in ORCA. In the current scheme, we split the atomic grids into five regions, using Lebedev grids with the following number of points on each of those:

Table 7.1 Different angular grid schemes used in ORCA. The numbers indicate the Lebedev grids used.¶
AngularGrid	Region 1	Region 2	Region 3	Region 4	Region 5
1	14	26	50	50	26
2	14	26	50	110	50
3	26	50	110	194	110
4	26	110	194	302	194
5	26	194	302	434	302
6	50	302	434	590	434
7	110	434	590	770	590

The ideal cutoffs between those regions were subjected to optimization, and are defined for all atoms. Whenever we refer to a AngularGrid flag in ORCA, one of these schemes is chosen.

7.3.2. The radial grid scheme¶

The number of radial points (\(n_r\)) for a given atom is simply defined using the equation first defined by Krack and Köster:

\[n_r = (15 \times \varepsilon - 40) + b \times ROW\]

where \(\varepsilon\) is called the IntAcc of that grid in ORCA, b is any number and ROW refers to the row of the periodic table for that atom. In its original formulation, \(b\) was set to \(5\), but here it as now optimized and varies slightly depending on the AngularGrid schemes shown above.

One important thing to note is that each increase of IntAcc by \(1\), adds \(15\) radial points to the atomic grids, as in the previous grid scheme. These IntAcc values were optimized for each angular grid:

Table 7.2 Optmimized IntAcc parameters for the exchange-correlation and COSX grids.¶
AngularGrid	XC	COSX
1	4.004	3.816
2	4.004	4.020
3	4.159	4.338
4	4.388	4.871
5	4.629	4.871
6	4.959	4.871
7	4.959	4.871

After defining the number of radial points \(n_r\), the actual radial grid is then defined from a Gauss-Chebyschev quadrature using the so-called M3 mapping from Treutler and Ahlrich:

\[r = \frac{\xi}{ln 2} ln \frac{2}{1-x}\]

where \(-1 \le x \le 1\), and the center of the grid (\(x=0\)) coincides with the value of \(\xi\). These \(\xi\) parameters were also optimized for each atom type.

7.3.3. The DEFGRIDs¶

With all that in mind, we can now present how the DEFGRIDs are built in terms of their AngularGrid scheme and IntAccs, which define the angular and radial parts of the atomic grids. More details can be found in Ref. [383].

Table 7.3 Angular grid schemes used in different part of ORCA. The XC and COSX grids are separated by a slash, and multiple COSX grid schemes are separated by a comma.¶
Grid Name	SCF	TD-DFT	CP-SCF	MP2 grad	MP2 2ndder
DEFGRID1	3 / 1, 1, 2	1 / 1	1 / 1	3	1
DEFGRID2	4 / 1, 2, 3	1 / 1	1 / 1	4	4
DEFGRID3	6 / 2, 3, 4	3 / 2	3 / 2	5	4

OBS.: The IntAccs for TD-DFT and the CP-SCF are 3.467 for the XC and 3.067 for the COSX instead of the default. These numbers can be smaller here and we exploit this to increase the overall speed.

From the Table 7.3 one can see, for instance, that the default SCF XC grid now is defined from AngularGrid 4 (with no extra final integration in the end). The default COSX uses a 1,2,3 grid scheme, with the COSX third grid being used to update the energy after the SCF converges and for the gradients.

7.3.4. Other details and options¶

The new adaptive pruning. The current pruning scheme uses lower grids close to the nucleus, and far away from the bonding region. However, if the basis set has polarized functions close to the nuclei, or diffuse Gaussians, this might not be sufficient.

To improve the grids for these problems, we now use by default an adaptive pruning scheme, that detects core-polarization, diffuse functions and steep basis set orbitals by analyzing the expectation value of the position operator, \(\langle \hat r \rangle\), and fixes the grid accordingly. This can increase the grids in these cases by 10-20%, but gives significantly better results. To use the non-adaptive scheme, just set %METHOD GRIDPRUNING OLDPRUNING END. For a completely unpruned grid, set GRIDPRUNING to UNPRUNED.

A simpler Gauss-Legendre angular grid. By setting AngularGrid to \(0\), instead of using the Lebedev grids, a Gauss-Legendre angular grid will be built, as suggested by Treutler and Ahlrich [851]. The number of \(\theta\) points is defined as \(0.4 n_r\) by default and the number of \(\phi\) points is chosen as to avoid crowding close to the poles.

These grids are in general less efficient than the Lebedev’s, but are useful if one needs to construct extremely large grids for specific applications.

The SpecialGrid Option. Sometimes, you might want to increase the integration accuracy for some atoms that need special care, while it is not necessary to enlarge the grid generally. ORCA provides you with a basic mechanism to increase the radial integration accuracy for a few atoms while maintaining the chosen grid for all others.

%METHOD
   # a maximum of 64 assignments can be made
   #  in = 0 : no changes are made
   #  in > 0 : the grid will be changed for all atoms with
   #           atomic number=in to IntAcc=an
   #  in < 0 : only the specific n'th atom will have its
   #           IntAcc value changed to an
   SpecialGridAtoms  i1, i2, i3,...,in;
   SpecialGridIntAcc a1,a2,a3,...,an;
END

OBS.: Starting from ORCA 5.0 it is not necessary to use this option anymore unless you have very specific reasons. The basis set is considered during the grid construction and that is automatically extended if needed.

7.3.5. SCF grid keyword list¶

A complete description of the SCF grid options is given below. There are keywords specific to the XC integration, COSX integration and a general part that applies to all:

%METHOD
# XC grids
AngularGrid    1    #Lebedev50
               2    #Lebedev110
               3    #Lebedev194
               4    #Lebedev302
               5    #Lebedev434 
               6    #Lebedev590 
               7    #Lebedev770 
               0    #SimpleGrid 
IntAcc         5.0  # determines no. of radial points

# COSX grids
AngularGridX   1,1,2,4,5    # the first three are used in the SCF
                            # the 4th in the MP2 gradient and
                            # the 5th for MP2 second derivatives
IntAccX        3.56,-1,4.5  # if a -1 is given, the default IntAcc is used.

# General
NThetaMax      0.4          # only for AngularGrid=0, multiplier for nr
GridPruning    Unpruned     # no Pruning
               OldPruning   # the old pruning
               Adaptive     # default (and recommended)
HGridReduced   true         # Reduce grids for H and He by one
                            # unit (default and recommended)
BFCut          1e-10        # basis fcn. cut. Is adjusted according to
                            # convergence tolerances
WeightCut      1e-14        # grid weight cut. default: 1e-14
END

7.3.6. Changing TD-DFT, CP-SCF and Hessian grids¶

TD-DFT. The grids used in CIS or TD-DFT can be changed in their respective block:

%TDDFT  # or %CIS, they are equivalent
  # XC grids
  IntAccXC 3.467
  GridXC   1

  #COSX grids
  IntAccX  3.076
  GridX    1
END

CP-SCF. The CP-SCF grids are changed in the %METHOD block:

%METHOD 
  # XC grids
  Z_IntAccXC  3.467
  Z_GridXC    1

  #COSX grids
  Z_IntAccX   3.076
  Z_GridX     1
  Z_GridX_RHS 2
END

OBS.: The Z_Grid_RHS is only used in MP2 and the number here has a different meaning. It refers to which of the COSX grids used in the SCF will be chosen, rather than an AngularGrid scheme. The default is to use the second COSX grid.

Hessian. The XC grids used to compute the DFT terms in the Hessian are automatically chosen to be one unit higher than the SCF grids. Because of the second derivative terms, we found that it is better to have a slightly higher XC grid here. The COSX grid can be changed freely:

%FREQ 
  HessGridX = 2,2,2,2
END

These four numbers refer to the possible usages of COSX in the Hessian, as explained in Sec. Frequency calculations - numerical and analytical

Non-local functionals (VV10 and alike). The default non-local grid is defined by AngularGrid 2, and is not recommended to be changed. In any case, these can be altered by using:

%METHOD
  # non-local grids
  VdwAngularGrid    2        # same scheme as the SCF ones
  VdWIntAcc         5.0      # determines no. of radial points
  VdwGridPruning    Adaptive # default
  VdwDistTCut       10       # cutoff distance between grid points, in angstroem
END

7.3.7. When should I change from the default grids?¶

In general, the errors from the default grids are rather small and reasonable for most applications. After benchmarking against the GMTNK55 test set with the default !DEFGRID2, we found an error of about \(0.01 \pm 0.03\) kcal/mol from DFT (compared to a reference grid), and \(0.05 \pm 0.10\) kcal/mol for the COSX (compared to the analytical integration). We also benchmarked geometries, excitation energies and frequencies, and all errors are systematically low.

However, there might still be cases where an improved grid is needed:

If you need to be confident that your energy error is below 0.2 kcal/mol;
When dealing with anions with large negative charges (\(<-3\));
For very subtle intermolecular interactions;
When dealing with weird electronic structures;
With large conjugated systems - graphene-like structures and large polyaromatics.

When needed, the !DEFGRID3 is very large and conservative - it was built to cover almost all these cases. In contrast, !DEFGRID1 will yield grids of the size close to previous ORCA versions defaults, but still with increased accuracy.

AngularGrid	Region 1	Region 2	Region 3	Region 4	Region 5
1	14	26	50	50	26
2	14	26	50	110	50
3	26	50	110	194	110
4	26	110	194	302	194
5	26	194	302	434	302
6	50	302	434	590	434
7	110	434	590	770	590

AngularGrid	Region 1	Region 2	Region 3	Region 4	Region 5
1	14	26	50	50	26
2	14	26	50	110	50
3	26	50	110	194	110
4	26	110	194	302	194
5	26	194	302	434	302
6	50	302	434	590	434
7	110	434	590	770	590

AngularGrid	Region 1	Region 2	Region 3	Region 4	Region 5
1	14	26	50	50	26
2	14	26	50	110	50
3	26	50	110	194	110
4	26	110	194	302	194
5	26	194	302	434	302
6	50	302	434	590	434
7	110	434	590	770	590