Self-Consistent Field Calculation of Nuclear Magnetic Resonance Chemical Shielding Constants Using Gauge-Including Atomic Orbitals and Approximate Two-Electron Integrals

The chain-of-spheres
method (COS) for approximating two-electron
integrals is applied to Hartree–Fock and density functional
theory calculations of nuclear magnetic resonance chemical shielding
tensors, based on gauge-including atomic orbitals. The accuracy of
the approximation is compared to that of the resolution of the identity
(RI) approach, using a benchmark test set of 15 small molecules. Reasonable
auxiliary basis sets and grid sizes are selected on the basis of a
careful investigation of how approximating each of the two-electron
terms in the self-consistent field (SCF) and coupled perturbed SCF
equations affects the calculated shielding constants. It is found
that the errors are linearly additive but can have either sign. The
mean absolute relative error due to applying the RI/COS approximations
with the chosen settings to all two-electron terms is on the order
of 0.01% and therefore negligible compared to the errors due to basis
set incompleteness (∼1%) and the method used (10–50%).
Several larger organic systems are used to assess the efficiency of
the RI approximation for both Coulomb- and exchange-type integrals
(RIJK) as well as a combination of RI for Coulomb and COS for exchange
contributions (RIJCOSX). The RIJK approximation is more efficient
for small molecules, while for systems of over 100 electrons and 1000
basis functions, the RIJCOSX approximation is superior.