Application of Löwdin's canonical orthogonalization method to the Slater-type orbital configuration-interaction basis set

Abstract

We apply Löwdin's canonical orthogonalization method to investigate the linearly dependent problem arising from the variational calculation of atomic systems using Slater‐type orbital configuration‐interaction (STO‐CI) basis functions. With a specific arithmetic precision used in numerical computations, the nonorthogonal STO‐CI basis is easily linearly dependent when the number of basis functions is sufficiently large. We show that Löwdin's canonical orthogonalization method can successfully overcome such problem and simultaneously reduce the dimension of basis set. This is illustrated first through an S‐wave model He atom, and then the real two‐electron atoms in both the ground and excited states. In all of these calculations, the variational bound state energies of the two‐electron systems are obtained in reasonably high accuracy using over‐redundant STO‐CI bases, however, without using extended high‐precision technique. © 2015 Wiley Periodicals, Inc.

Linear dependence in Hylleraas configuration‐interaction calculations of He atom

“…In Equation 1, the inter‐electronic coordinate takes into account the Kato cusp condition explicitly if the power k is larger than 0. For k = 0, the Hy‐CI basis set reduces to our previous STO‐CI representation of the two‐electron wave functions . The introduce of nonindependent variable r 12 leads the numerical computation of Hamiltonian and overlap matrices much more complicated than the STO‐CI ones.…”

“…The Löwdin's canonical orthogonalization method plays a vital role in transforming the generalized eigenvalue problem to a standard one. Its implementation has been given elsewhere and here we only present a brief introduction of it. By introducing a non‐singular linear transformation matrix A , the original basis set Φ is transformed to an orthonormal basis set Φ ′, …”

“…In our earlier work, we have successfully applied Löwdin's canonical orthogonalization method to investigate the linearly dependent problems arising from the variational configuration‐interaction (CI) calculations of two‐electron atoms based on product Slater‐type orbitals (STO). Although the powerfulness of this method in improving the convergence of CI calculations has been established, the obtained energies are not satisfied yet.…”

A Program for Single-center Expansion in Laguerre-type Orbitals for the Hydrogen Molecular Ion

“…As a result of overlap integrals among these basis functions, however, problems associated with linear dependence [2,3] occasionally prevent accurate computation. This may be a severe impediment when extending the basis toward completeness.…”

“…The biggest problem, common to all exponential-type orbitals (ETOs), is in evaluating multi-center Coulomb integrals. For this reason, LTOs are applied only to single-center expansion, and actual single-center expansion calculations have been limited to small systems such as the He atom [2,3], H 2 + [9,10], or H 2 [11]. The hydrogen molecular ion (H 2 + ) is the simplest molecule and has been studied from the early days of quantum chemistry, beginning with the pioneering works of Burrau [12] and Hylleraas [13].…”