Hydrogen-molecule ion

Among the few systems that can be solved exactly are the particle in a onedimensional box, the hydrogen atom, and the hydrogen molecule ion Hj. Although of limited interest chemically, these systems are part of the foundation of the quantum mechanics we wish to apply to atomic and molecular theory. They also serve as benchmarks for the approximate methods we will use to treat larger systems. 

The hydrogen molecule ion is best set up in confocal elliptical coordinates with the two protons at the foci of the ellipse and one electron moving in their combined potential field. Solution follows in mueh the same way as it did for the hydrogen atom but with considerably more algebraic detail (Pauling and Wilson, 1935 Grivet, 2002). The solution is exact for this system (Hanna, 1981). 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

Approximate Theoretical. The simplest molecular orbital problem is that of the hydrogen molecule ion (Pig KJ-3), is a preliminary example of all molecular orbital problems to come, w hich, although they may be very complicated, are elaborations on this simple example. 

A very important difference between H2 and molecular orbital calculations is electron correlation. Election correlation is the term used to describe interactions between elections in the same molecule. In the hydrogen molecule ion, there is only one election, so there can be no election correlation. The designators given to the calculations in Table 10-1 indicate first an electron correlation method and second a basis set, for example, MP2/6-31 G(d,p) designates a Moeller-Plesset electron coiTclation extension beyond the Hartiee-Fock limit canied out with a 6-31G(d,p) basis set. 

This section briefly considers the proton H+, the hydride ion H, the hydrogen molecule ion H2, the triatomic 2-electron species H3+ and the recently established cluster species +... 

The species H2 and H3+ are important as model systems for chemical bonding theory. The hydrogen molecule ion H2+ comprises 2 protons and 1 electron and is extremely unstable even in a low-pressure gas discharge system the energy of dissociation and the intemuclear distance (with the corresponding values for H2 in parentheses) are ... 

The triatomic hydrogen molecule ion H3+ was first detected by J. J. Thomson in gas discharges and later fully characterized by mass spectrometry its relative atomic mass, 3.0235, clearly distinguishes it from HD (3.0219) and from tritium... 

Why is the hydrogen molecule ion H2 stable, and what should its bond length be ... 

The traditional place to begin a quantum-mechanical study of molecules is with the hydrogen molecule ion H2+. Apart from being a prototype molecule, it reminds us that molecules consist of nuclei and electrons. We often have to be aware of the nuclear motion in order to understand the electronic ones. The two are linked. 

Figure 3.2 Potential energy curve for hydrogen molecule-ion...

In the case of the hydrogen molecule-ion H2" ", we defined certain integrals Saa, Taa, Tab, Labra- The electronic part of the energy appropriate to the Heitler-London (singlet) ground-state wavefunction, after doing the integrations... 

In Chapter 3, I showed you how to write a simple LCAO wavefunction for the electronic ground state of the hydrogen molecule-ion, H2 ... 

In order to calculate the total probability (which comes to 1), we have to integrate over both space dr and spin ds. In the case of the hydrogen molecule-ion, we would write LCAO wavefunctions... 

In Chapter 4,1 discussed the concept of an idealized dihydrogen molecule where the electrons did not repel each other. After making the Bom-Oppenheimer approximation, we found that the electronic Schrddinger equation separated into two independent equations, one for either electron. These equations are the ones appropriate to the hydrogen molecule ion. 

In our study of the hydrogen molecule-ion in Chapter 3, we considered the electron density map shown in Figure 18.8. It is obvious by inspection that the... 

Molecular structure theory is a fast-moving subject, and a lot has happened since the First Edition was published in 1995. Chapters 3 (The Hydrogen Molecule-ion) and 4 (The Hydrogen Molecule) are pretty much as they were in the First Edition, but 1 have made changes to just about everything else in order to reflect current trends and the recent literature. I have also taken account of the many comments from friends and colleagues who read the First Edition. 

The argon-hydrogen and krypton-hydrogen systems are distinguished by the fact that the reaction occurs with comparable cross-sections via both hydrogen molecule ion and rare gas ion reactants— namely,... 

THE APPLICATION OF THE QUANTUM MECHANICS TO THE STRUCTURE OF THE HYDROGEN MOLECULE AND HYDROGEN MOLECULE-ION AND TO RELATED PROBLEMS... 

The problem has already been solved for the normal state of the hydrogen molecule-ion (ZA = ZB = 1) by the use of numerical methods. A rather complete account of these calculations of Burrau (30) will be given here, since the journal in which they were published is often not available. 

Fia. 4. The Electronic Energy of the Hydrogen Molecule-ion in the Normal State as a Function of the Distance Between the Two Nuclei (Burrau)... 

Although no new numerical information regarding the hydrogen molecule-ion can be obtained by treating the wave equation by perturbation methods, nevertheless it is of value to do this. For perturbation methods can be applied to many systems for which the wave equation can not be accurately solved, and it is desirable to have some idea of the accuracy of the treatment. This can be gained from a comparison of the results of the perturbation method of the hydrogen molecule-ion and of Bureau s accurate numerical solution. The perturbation treatment assists, more-... 

Curve 1 represents the total energy of the hydrogen molecule-ion as calculated by the first-order perturbation theory curve 2, the naive potential function obtained on neglecting the resonance phenomenon curve 3, the potential function for the antisymmetric eigenfunction, leading to elastic collision. 

The contour lines represent points of relative density 1.0, 0.9, 0.8,..0.1 for a hydrogen atom. This figure, with the added proton 1.06 A from the atom, gives the electron distribution the hydrogen molecule-ion would have (in the zeroth approximation) if the resonance phenomenon did not occur it is to be compared with figure 6 to show the effect of resonance. 

The above perturbation treatment of the hydrogen molecule-ion has not before been published. 

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