IGS | IntGrid | NumAcc

[-](igs|intgrid|numacc)   ±N

This keyword (Integration Grid Selection) specifies the integration grid to be used for numerical integrations. Both "pruned" and "unpruned" grids can be specified. Pruned grids are grids that have been optimized to use the minimal number of points required to achieve a given level of accuracy. Pruned grids are used by default when available (currently defined for H through Kr).

Specific grids may be selected by giving an integer value N as the argument to the option. N may have one of these forms:

• A small positive integer (0 ≤ N < 1000), which requests a pre-defined pruned grid, as shown in the following table:

  N	(Nr, NΩ)	L	Description	Ref.
  1	(50,194)	23	50-point radial quadrature and   194-point Lebedev angular quadrature. See the 2, Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512. 5(a) Lebedev, V.I. USSR Comp. Math. and Math. Phys. 1975, 15(1), 44-51. Zh. vychisl. Mat. mat. Fiz. 1975, 15(1), 48-54. (b) Lebedev, V.I. USSR Comp. Math. and Math. Phys. 1976, 16(2), 10-24. Zh. vychisl. Mat. mat. Fiz. 1976, 16(2), 293-306.
  2	(75,302)	29	75-point radial quadrature and   302-point Lebedev angular quadrature. See the Gaussian integral option: FineGrid.	2, Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512. 6Lebedev, V.I. Siberian. Math. J. 1977, 18(1), 99-107. Sibirskii Matematicheskii Zhurnal 1977, 18(1), 132-142.
  3	(99,590)	41	99-point radial quadrature and   590-point Lebedev angular quadrature. See the Gaussian integral option: UltraFine.	2, Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512. 7Lebedev, V. I.; Skorokhodov, A. L. Russian Acad. Sci. Dokl. Math. 1992, 45, 587-592.
  4	(110,770)	47	110-point radial quadrature and   770-point Lebedev angular quadrature	2, Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512. 7Lebedev, V. I.; Skorokhodov, A. L. Russian Acad. Sci. Dokl. Math. 1992, 45, 587-592.
  5	(150,974)	53	150-point radial quadrature and   974-point Lebedev angular quadrature. See the Gaussian integral option: SuperFineGrid.	2, Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512. 7Lebedev, V. I.; Skorokhodov, A. L. Russian Acad. Sci. Dokl. Math. 1992, 45, 587-592.
  6	(185,1202)	59	185-point radial quadrature and   1202-point Lebedev angular quadrature	2, Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512. 8Lebedev, V. I. Russian Acad. Sci. Dokl. Math. 1995, 50, 283-286.
  7	(225,5810)	131	225-point radial quadrature and   5810-point Lebedev angular quadrature	2, Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512. 9Lebedev, V.I.; Laikov, D.N. Dokl. Math. 1999, 59, 477-481.
  N	(Nr, Nθ, Nφ)	L	Description	Ref.
  8	(50,15,30)	29	50-point radial quadrature and   450-point spherical product quadrature [Nφ ≡ L + 1 = 30, Nθ = Nφ/2 = (L + 1)/2 = 15, NΩ = Nθ⋅Nφ = (L + 1)2/2 = 450]	2Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512.
  9	(75,25,50)	49	75-point radial quadrature and   1250-point spherical product quadrature [Nφ ≡ L + 1 = 50, Nθ = Nφ/2 = (L + 1)/2 = 25, NΩ = Nθ⋅Nφ = (L + 1)2/2 = 1250]	2Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512.
  10	(99,48,96)	95	99-point radial quadrature and   4608-point spherical product quadrature [Nφ ≡ L + 1 = 96, Nθ = Nφ/2 = (L + 1)/2 = 48, NΩ = Nθ⋅Nφ = (L + 1)2/2 = 4608]	2 Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512.
  N	(Nr, NΩ)	L	Description	Ref.
  11	(23,170) (26,170)	21	either 23-point MultiExp radial quadrature for H to F   or 26-point MultiExp radial quadrature for Na to Cl combined   with 170-point Lebedev angular quadrature   (reported as SG-0 grid)	3, Gill, P. M. W.; Chien, S.-H. J. Comput. Chem. 2003, 24, 732-740. 4Chien, S.-H.; Gill, P. M. W. J. Comput. Chem. 2006, 27, 730-739.

• A large positive integer of the form mmmlll, which requests an unpruned grid with mmm radial shells around each atom, and a Lebedev angular grid in each shell, which integrates exactly all spherical harmonics of degree L = lll or less. The total number of integration points per atom is thus ≈ mmm⋅(lll + 1)²/3. For example, to specify the (99,590) unpruned grid, use -intgrid 99041. The valid values of lll are 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131.
• A large negative integer of the form −mmmlll, which requests an unpruned grid with mmm radial shells around each atom, and a spherical product grid having (lll + 1)/2 θ points and (lll + 1) φ points in each shell. Again, L = lll is the highest order of spherical harmonics that are integrated exactly by this rule. The total number of integration points per atom is therefore mmm⋅(lll + 1)²/2. For example, this form is used to specify the (96,32,64) grid commonly cited in benchmark DFT calculations: -intgrid −96063.

Only one radial quadrature scheme was available in PAMoC versions prior to 2012, namely the N_r-point Euler-Maclaurin quadrature formula which, in this context, corresponds to a (N_r + 2)-point semi-open extended trapezoidal rule combined with Handy's transformation of the radial coordinate.[1Murray, C. W.; Handy, N. C.; Laming, G. J.
Mol. Phys. 1993, 78, 997-1014., 2Gill, P. M. W.; Johnson, B. G.; Pople, J. A.
Chem. Phys. Lett. 1993, 209, 506-512.]
In the current version of PAMoC, the choice of the radial integration rule is controlled by keyword radrule, which allows the user to choose among several quadrature schemes and has a default value different from Euler-Mclaurin.
Then, the integration grid selection keyword   (igs|intgrid|numacc)   is used mainly to set the number of radial and angular points together with the pruning scheme (i.e. pruned/unpruned grid). Only in the case of N = 11, it completely define the grid, irrespectively of the value of keyword radrule.

• bck
• hrf
• stk
• stw
• arc and sur
• radrule

1. "Quadrature schemes for integrals of density functional theory"
   Murray, C. W.; Handy, N. C.; Laming, G. J. Mol. Phys. 1993, 78, 997-1014.
2. "A standard grid for density functional calculations"
   Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1993, 209, 506-512.
3. "Radial Quadrature for Multiexponential Integrands"
   Gill, P. M. W.; Chien, S.-H. J. Comput. Chem. 2003, 24, 732-740.
4. "SG-0: A Small Standard Grid for DFT Quadrature on Large Systems"
   Chien, S.-H.; Gill, P. M. W. J. Comput. Chem. 2006, 27, 730-739.
5. (a) "Values of the nodes and weights of ninth to seventeenth order Gauss-Markov quadrature formulae invariant under the octahedron group with inversion"
   Lebedev, V.I. USSR Comp. Math. and Math. Phys. 1975, 15(1), 44-51. Zh. vychisl. Mat. mat. Fiz. 1975, 15(1), 48-54.
   (b) "Quadratures on a sphere"
   Lebedev, V.I. USSR Comp. Math. and Math. Phys. 1976, 16(2), 10-24. Zh. vychisl. Mat. mat. Fiz. 1976, 16(2), 293-306.
6. "Spherical quadrature formulas exact to orders 25-29"
   Lebedev, V.I. Siberian. Math. J. 1977, 18(1), 99-107. Sibirskii Matematicheskii Zhurnal 1977, 18(1), 132-142.
7. "Quadrature formulas of orders 41, 47, and 53 for the sphere"
   Lebedev, V. I.; Skorokhodov, A. L. Russian Acad. Sci. Dokl. Math. 1992, 45, 587-592.
8. "A quadrature formula for the sphere of 59th algebraic order of accuracy"
   Lebedev, V. I. Russian Acad. Sci. Dokl. Math. 1995, 50, 283-286.
9. "A quadrature formula for the sphere of the 131-st algebraic order of accuracy"
   Lebedev, V.I.; Laikov, D.N. Dokl. Math. 1999, 59, 477-481.