Variational Wavefunctions for H2+

Abstract

The usual approximations to the molecular orbitals of H2+ are shown to have an improper exponential dependence on the radial elliptic coordinate λ. However, the exact λ dependence is easily incorporated into simple trial functions and leads to substantially improved numerical accuracy for both the 1sσg and 2pσu states.

Non-Born-Oppenheimer corrections in an exactly solvable model of the hydrogen ion molecule

“…Within the frozen nuclei approximation [14] the electronic problem is separable in spheroidal prolate coordinates, but the solution cannot be given in an analytic closed form [15]. Approximate analytic solutions [14,16] or numeric ones [15,17] exist to any degree of accuracy for the ground and first excited states.…”

Molecular orbital studies on small molecules using H2+‐type elliptical basis orbitals. Application to H2+, H2, He2++ and H3+

“…For this reason, we choose to call Equation (6) the "boundary condition" value of a. Indeed, the importance of this factor has been recently established by Weinhold and Chinen [7], who report a variational calculation on the zX: …”

Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional

“…For one-electron systems, this formula can be rewritten with the one-electron densities, n 1 ( r ) = ψ 1 2 ( r ), where ψ 1 is the only occupied one-electron orbital. Albeit the concept of enhancement factors is introduced only for semilocal functionals, formally, effective position-dependent (fully spin-polarized) exact exchange enhancement factors ( F X,pol λ ) can be defined, as it was done by Perdew et al, for a given system dividing e X,pol λ by the local density approximation (LDA) exchange energy density (e X LDA ) to simply visualize if the exact exchange energy densities locally satisfy the suggested tighter lower bound: Generally, the spin-unpolarized exact exchange enhancement factor can be obtained from the fully spin-polarized one, by using the following transformation: where s is the reduced gradient. Since our question here is whether the suggested nonconventional exact exchange energy density is locally compatible with the semilocal SCAN or revSCAN exchange, we plot the position-dependent effective exact exchange enhancement factors, with respect to the reduced gradient, and apply a formally analogue expression to transform the effective spin-polarized exact exchange enhancement factors into spin-unpolarized ones to facilitate the comparison with the SCAN or revSCAN semilocal (spin-unpolarized) exchange enhancement factors. We consider simple prototypical one-electron systems to map how the suggested exact exchange energy densities behave in one-electron atomic regions, bonding regions, and for delocalized densities. Our first model is the H atom with the 1s orbital: Then, we model the hydrogen molecular ion (H 2 + ) at various bond lengths using a simple approximate wave function form in prolate spheroidal coordinates to facilitate the calculations: where the coordinates σ = ( d 1 + d 2 )/ R and τ = ( d 1 – d 2 )/ R can be obtained from the distances to the nuclei ( d 1 , d 2 ) and from the bond length ( R ). The corresponding parameter values can be found in Table .…”

“…Then, we model the hydrogen molecular ion (H 2 + ) at various bond lengths using a simple approximate wave function form in prolate spheroidal coordinates to facilitate the calculations: where the coordinates σ = ( d 1 + d 2 )/ R and τ = ( d 1 – d 2 )/ R can be obtained from the distances to the nuclei ( d 1 , d 2 ) and from the bond length ( R ). The corresponding parameter values can be found in Table .…”