F Generation of auxiliary basis

This Appendix is adopted from the Ph.D. Thesis by Alston Misquitta [66].

The following procedure can be applied to construct an auxiliary basis set for each atom in the dimer under consideration. Denoted by M is the decontracted basis set used in calculations for given dimer. The auxiliary basis, denoted by X, is developed as follows:

  1. Construct an auxiliary basis as the tensor product of M with itself. That is,
    X = M ⊗ M.

    If GM(lii) and GM(ljj) are two basis functions of M centered at the same point, where li and lj are the angular quantum numbers and αi and αj are the exponents, then the product is a basis function belonging to X centered at the same point and given by GX(lkk) where lk = li + lj and αk = αi + αj. The resulting basis X is a (large) basis including high symmetry functions compared to the original basis. All the products involving basis functions from different centers are rejected.

  2. Within each angular symmetry of X, a reduction is performed in the number of basis functions as follows. Given an ϵ, if there are n basis functions for which log αk1, log αk2,...,log αkn are in an ϵ neighborhood, then these n functions are replaced by one function with the exponent β = (αk1αk2...αkn)1∕n. Perform this reduction for the whole basis set.
  3. If necessary, the resulting basis can be reduced even further with the previous step repeated on the reduced basis with a different value of ϵ.
  4. Further pruning of the reduced basis X can be done as follows:
    1. Reject all functions of g-symmetry and higher. This is necessary if cadpac is used to obtain integrals.
    2. The ‘large’ exponents, particularly the ones of high symmetries, can generally be removed. The criterion for this removal is best found by trial and error and by monitoring the constraint conditions.

We have found the optimal values of ϵ used in the pruning process to be between 0.3 and 0.5. This procedure typically results in an auxiliary basis X that is 2 to 3 times larger than the basis used to obtain molecular orbitals and eigenvalues. While in general the auxiliary basis for a molecule should be centered also on sites between pairs of atoms in addition to atomic sites, it is our experience that the use of only atomic centers is adequate. This is also true for the optimized auxiliary basis sets [14].