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Ints

Last edited by Andrew Wildman
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Details

The INTS section handles the specification of parameters relevant to the evaluation / contraction of the operator integrals. The INTS section is optional, and the default parameters are often sufficient. The defaults are often overly conservative for some applications, and the performance of e.g. the direct electron repulsion integral (ERI) contraction can be drastically altered by a proper toggling of these parameters.

Keywords

Keyword Type Description Default
ALG String ERI contraction algorithm DIRECT
SCHWARTZ Double precision float Criterion for ERI schwartz screening 10^{-12}
RI String Method for the resolution of identity approximation (RI) FALSE
CDRI_THRESHOLD Double precision float Criterion for inclusion in Cholesky decomposition based RI 10^{-4}
FINITE_NUCLEI String Toggles use of Gaussian charge distribution instead of point charges to represent nuclei DEFAULT

The ALG Keyword

ChronusQ currently supports two types of ERI contraction algorithms:

  • DIRECT: Direct ERI contraction
    • This is the default in ChronusQ due to the negligible memory requirement and efficiency through screening. However, if the memory required to store the ERI tensor in core is readily available, the INCORE integral contraction will often be much more efficient
  • INCORE: In-Core ERI contraction
    • The integral contraction is performed in-core using optimized level-3 BLAS operations. The option is wildly memory intensive, but if the memory can be afforded, the option is highly efficient. If this option is chosen and there is not enough memory specified, the job will end and the amount of memory required for the ERI storage will be written to the log file.

The SCHWARTZ Keyword

An upper bound to the magnitude of elements within the ERI tensor is given by:1

|(\mu\nu|\lambda\kappa)| \leq \sqrt{(\mu\nu|\mu\nu)}\sqrt{(\lambda\kappa|\lambda\kappa)}

This can be used to prescreen the evaluation of (\mu\nu|\kappa\lambda) if the left hand side of the equation is below a certain value. The SCHWARTZ keyword gives the user control over this cutoff value.

The RI Keyword

Because the ERI tensor is rank 4, adding more basis functions will cause the number of significant ERIs to scale as \mathcal{O}(n^4) where n is the number of basis functions. For large systems or large basis sets, this may be computationally intractable. In these cases, one may use an auxiliary basis fit to reproduce the four index ERIs from three and two index integrals, reducing the scaling to \mathcal{O}(n^3):2,3

(\mu\nu|\lambda\kappa) = \sum_{tu} (\mu\nu|u)(t|u)^{-1}(u|\lambda\kappa)

Where t and u are auxiliary basis functions. There are two main ways to generate this auxiliary basis. The first method is to use a predetermined auxiliary basis.2 The second method is to generate this basis on the fly with a Cholesky decomposition.3 Both approaches are available in ChronusQ, toggled by the RI keyword. The options are:

  • FALSE (Default)
    • Do not do any RI approximation method
  • AUXBASIS
  • CHOLESKY
    • Form an auxiliary basis on the fly from a Cholesky decomposition. The criterion for vectors that are kept by this method is given by the CDRI_THRESHOLD keyword.

The FINITE_NUCLEI Keyword

In some relativistic methods, (e.g. scalar-only relativistic corrections or the exact two component method) using point nuclei can introduce singularities in the working equations. In these cases, it is recommended to use finite, sharply peaked charge distributions. ChronusQ will use a Gaussian to approximate the charge distribution for atom X according to the following formula:4

\rho_X = Z_X \left ( \frac{\zeta}{\pi} \right)^{\frac{3}{2}}  e^{ - \zeta R^2 }

Where the exponent is given by

\zeta = \frac{3}{2} ( 0.52917721092\mathrm{e}^{-5} \times [ 0.8636 A_X^{\frac{1}{3}} + 0.57 ] )^{-2}

and A_X is the atomic mass of atom X.

The options for FINITE_NUCLEI are

  • DEFAULT (Default)
    • Chooses either point or finite nuclear charge distributions based on the Hamiltionian
  • TRUE
    • Turns on finite nuclei
  • FALSE
    • Turns off finite nuclei

Example

Auxiliary basis RI with all ERIs stored

[INTS]
alg = incore
ri = auxbasis

Cholesky decomposition RI with ERIs formed on the fly

[INTS]
alg = direct
ri = cholesky

References

  1. Häser, M., & Ahlrichs, R. (1989). Improvements on the direct SCF method. Journal of Computational Chemistry, 10(1), 104-111. ↩

  2. Kendall, R. A., & Früchtl, H. A. (1997). The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development. Theoretical Chemistry Accounts, 97(1-4), 158-163. ↩

  3. Koch, H., Sánchez de Merás, A., & Pedersen, T. B. (2003). Reduced scaling in electronic structure calculations using Cholesky decompositions. The Journal of chemical physics, 118(21), 9481-9484. ↩

  4. Visscher, L., & Dyall, K. G. (1997). Dirac–Fock atomic electronic structure calculations using different nuclear charge distributions. Atomic Data and Nuclear Data Tables, 67(2), 207-224. ↩