The ground state of the ${\rm H}_3^+$ molecular ion: physics behind

TL;DR

This paper investigates the physical mechanisms binding the ${ m H}_3^+$ molecular ion using variational methods, identifying key interactions and superpositions that yield highly accurate ground state energies close to the best known values.

Contribution

It introduces a set of physically motivated variational trial functions for ${ m H}_3^+$ and explores superpositions of mechanisms to achieve highly precise energy estimates.

Findings

Superpositions of mechanisms improve energy accuracy.

The best superposition reproduces two-three significant digits of the exact energy.

Variational energies agree with the most accurate results within 2-4 significant digits.

Abstract

Five physics mechanisms of interaction leading to the binding of the ${\rm H}_3^+$ molecular ion are identified. They are realized in a form of variational trial functions and their respective total energies are calculated. Each of them provides subsequently the most accurate approximation for the Born-Oppenheimer (BO) ground state energy among (two-three-seven)-parametric trial functions being correspondingly, H$_2$-molecule plus proton (two variational parameters), H$_2^+$-ion plus H-atom (three variational parameters) and generalized Guillemin-Zener (seven variational parameters). These trial functions are chosen following a criterion of physical adequacy. They include the electronic correlation in the exponential form $\sim\exp{(\gamma r_{12})}$, where $\gamma$ is a variational parameter. Superpositions of two different mechanisms of binding are investigated and a particular one, which is a generalized Guillemin-Zener plus H$_2$-molecule plus proton (ten variational parameters), provides the total energy at the equilibrium of $E=-1.3432$\ a.u. The superposition of three mechanisms: generalized Guillemin-Zener plus (H$_2$ -molecule plus proton) plus (H$_2^+$ -ion plus H) (fourteen parameters) leads to the total energy which deviates from the best known BO energy to $\sim 0.0004$\ a.u., {\it it reproduces two-three significant digits in exact, non-BO total energy}. In general, our variational energy agrees in two-three-four significant digits with the most accurate results available at present as well as major expectation values.