Abstract
The use of London atomic orbitals (LAOs) in a nonperturbative manner enables the determination of gauge-origin invariant energies and properties for molecular species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calculations for molecular systems is the evaluation of molecular integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiencies of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP), and Rys quadrature schemes are compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and weights, enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalized algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta; thus, a simple mixed scheme is put forward that selects the most efficient approach to calculate the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm.
Highlights
The impact of the use of London atomic orbitals (LAOs) on the complexity of three LAO-electron repulsion integrals (ERIs) algorithms was discussed in detail
For the MD method, the introduction of LAOs leads to a significant increase in complexity and has a negative impact on its overall efficiency, for integrals over higher angular momenta basis functions
For the HGP method, the introduction of LAOs does not significantly complicate the underlying algorithm and the efficiency of the HGP scheme for contracted basis functions remains a significant advantage in the generalized form
Summary
There has been a great deal of interest in the non–perturbative calculation of molecular energies and properties in the presence of arbitrarily strong magnetic fields.[1,2,3,4,5,6,7,8,9,10,11,12] Investigations have included the implementation of electronic structure calculations at the Hartree-Fock,[1,8] configuration– interaction,[4] coupled-cluster,[9] coupled-cluster equation of motion[12] and current densityfunctional[6,10,11] levels of theory. A number of advantageous features for integral schemes to perform non-perturbative calculations with arbitrary strength magnetic fields can be identified The first of these is that the algorithmic complexity should not be significantly increased over the corresponding zerofield scheme. A generalized Rys quadrature has been explored previously, its use was limited to low field strengths owing to the fact that the required quadrature roots and weights were being approximated by a 2D interpolation scheme. Whilst adequate for this purpose, the application of this approach to arbitrary field strengths is problematic as it would necessitate the storage of very large interpolation grids. This mixed approach provides an an effective approach to minimizing the computational cost of the LAO-ERI evaluation
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