Gaussian Expansions of Orbitals

Abstract

Using numerical calculations and analytic theory, we examine the convergence behavior of Gaussian expansions of several model orbitals. By following the approach of Kutzelnigg, we find that the errors in the energies of the optimal n-term even-tempered Gaussian expansions of s-type, p-type, and d-type exponential orbitals are ε n s ∼ exp(−π(3n) 1/2 ), ε n p ∼ exp(−π(5n) 1/2 ), and ε n d ∼ exp(−π(7n) 1/2 ), respectively. We show that such "root-exponential" convergence patterns are a consequence of the orbital cusps at r = 0, rather than the over-rapid decay of Gaussians at large r. We find that even-tempered expansions of the cuspless Lorentzian orbital also exhibit root-exponential convergence but that this is a consequence of its fat tail.

Uncertainty estimates for theoretical atomic and molecular data

“…Studies [69,70] suggest that while the FD approach for transition dipoles shows improved convergence behavior compared to the EV approach, perhaps more so than for Spherically averaged polarizability of water in its equilibrium geometry computed using sum-over-states formula (1) [73]; the experimental value is corrected for vibrational effects [75].…”

“…the diagonal dipole moments, there are technical issues with their use that still need to be overcome [70,71].…”

Atomic orbital basis sets

“…Klopper and Kutzelnigg109–111 showed that the energy of a hydrogen atom converges exponentially with the square‐root of the number of Gaussian functions used for expanding the 1s‐orbital, as shown in Eq. (12): …”

Uniform electron gases. II. The generalized local density approximation in one dimension

“…Given that the fitting errors (Table III) in the gLDA1 functional can be of the order of 0.1 mE h , we aimed to obtain the energies of the n-boxium and n-hookium to within 0.1 mE h of their complete basis set (CBS) limits. This is easily achieved for the HF, LDA1 and gLDA1 energies, because they converge exponentially [90][91][92] with the size M of the one-electron basis, but it is less straightforward for traditional post-HF energies.…”