Importance profiles. Visualization of basis set superposition errors

Recent developments in fully numerical methods promise interesting opportunities for new, compact atomic orbital (AO) basis sets that maximize the overlap to fully numerical reference wave functions, following the pioneering work of Richardson and coworkers from the early 1960s. Motivated by this technique, we suggest a way to visualize the importance of AO basis functions in polyatomic calculations, employing fully numerical calculations at the complete basis set (CBS) limit: the importance of a normalized AO basis function $|\alpha\rangle$ centered on some nucleus can be visualized by projecting $|\alpha\rangle$ on the set of numerically represented occupied orbitals $|\psi_{i}\rangle$ as $I(\alpha)=\sum_{i}\langle\alpha|\psi_{i}\rangle\langle\psi_{i}|\alpha\rangle$. Choosing $\alpha$ to be a continuous parameter describing the orbital basis, such as the exponent of a Gaussian-type orbital (GTO) or Slater-type orbital (STO) basis function, one is then able to visualize the importance of various functions on various centers in various molecules. The proposed visualization $I(\alpha)$ has the important property $0\leq I(\alpha)\leq1$ which allows unambiguous interpretation. We exemplify the method with importance profiles computed for atoms from the first three rows in a set of chemically diverse diatomic molecules. We find that the method offers a good way to visualize basis set superposition errors: the non-orthonormality of AO basis functions on different atomic centers is unambiguously revealed by the importance profiles computed for the ghost atom in an atomic calculation performed in the numerical basis set for a diatomic molecule.Comment: 10 pages, 3 figure

Basis set generation and optimization in the NISQ era with Quiqbox.jl

In the noisy intermediate-scale quantum era, ab initio computation of the electronic structure problem has become one of the major benchmarks for identifying the boundary between classical and quantum computational power. The single-particle basis set plays a key role in the electronic structure methods implemented on both classical and quantum devices. To investigate the consequences of the single-particle basis set, we propose a framework for more customizable basis set generation and basis set optimization. This framework allows configurations of composite Gaussian-type basis functions to go beyond typical Gaussian-type basis set frameworks such as the atomic orbitals and floating basis sets. Such basis set generations set the stage for more flexible variational optimization of basis set parameters. To realize this framework, we have developed an open-source electronic structure package named ``Quiqbox'' in the Julia programming language. Both the Hartree--Fock procedure and Gaussian-based electronic integral computations are implemented in this package. We compare Quiqbox with the basis set optimization package DiffiQult and find faster convergence of the basis set optimization with lower run time. We also demonstrate the additional customizability Quiqbox brings for more systematic basis set research with an example of constructing and optimizing delocalized orbitals.Comment: 15 pages, 7 figures, 5 tables, 1 listin