The hydrogen molecule ion, Hj

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. 

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. 

These considerations suggest that it might be useful to approximate molecular orbitals as sums of atomic orbitals. Thus for the hydrogen molecule ion hJ, a one-electron system, one could consider a wave function - based on the exact orbitals for the separated atoms - of the form, in... 

In this section we make the first chemical application of the idea of resonance, in connection with the structure of the simplest of all molecules, the hydrogen molecule-ion, Hj, and the simplest of all chemical bonds, the one-electron bond, which involves one electron shared by two atoms. 

It will be recalled that the approach of molecular orbital (MO) theory starts, on the other hand, from an independent-particle model (IPM) in which both electrons occupy the same bonding MO , 1 = Xa + Xb, similar to the one used [4] for the hydrogen molecule ion, Hj. The bonding MO is in fact the approximate wavefunction for a single electron in the field of the two nuclei and allocating two electrons to this same MO, with opposite spins, yields the 2-electron wavefunction... 

The LCAO method is a simple and qualitatively useful approximation. It is based on the very reasonable idea that as the electron moves around in the nuclear framework it will at any given time be close to one nucleus and relatively far from others, and that when near a given nucleus it will behave more or less as though it were in an atomic orbital belonging to that nucleus. To develop this idea more concretely we shall use the hydrogen molecule ion Hj This is a prototype for homonuclear diatomic molecules just as the hydrogen atom is for atoms in general. 

Characteristically a covalent bond between two atoms is an electron-pair bond two electrons of opposite spin are in states described by wave functions that have the same shape. But a bond formed by only one electron has most of the same physical characteristics. Look first, therefore, at a one-electron bond, in particular at the simplest instance of it, the hydrogen-molecule ion Hj ... 

Since the hydrogen molecule ion hJ contains only one electron, the Schrbdinger equation can be solved exactly (once we have accepted the Born-Oppenheimer approximation). In the... 

In Chapter 2, the development of atomic orbitals for many-electron atoms was built upon an understanding of the orbitals obtained from the exact solution for the one-electron hydrogen atom. Similarly, it is useful to examine the electronic structure for the simplest molecular species—the hydrogen molecule ion (Hj)—to begin our discussion of molecular orbitals. Like the H atom, the ion contains only one electron. The difference is the presence of two nuclei instead of one. The potential energy of the system is a sum of the Coulomb attraction of the electron to each of the two nuclei and Coulomb repulsion between the positively charged nuclei. 

Figure 3.14 Geometry of the hydrogen molecule ion, HJ. eonsisting of two protons (p ) and one electron (e ).

The Schrodinger equation for the hydrogen molecule ion, hJ, can he solved in the Born-Oppenheimer approximation without additional approximations. The solutions are molecular orhitals that represent motion of the electron around both nuclei. 

The hydrogen molecule ion, hJ, is the simplest possible molecule. It consists of two nuclei and a single electron, as depicted in Figure 20.2. It is highly reactive, but it is chemically bonded and has been observed spectroscopically in the gas phase. We apply the Born-Oppenheimer approximation and place our coordinate system with the nuclei on the z axis and the origin of coordinates midway between the nuclei. One nucleus is at position A and the other nucleus is at position B. The Born-Oppenheimer Hamiltonian for the hydrogen molecule ion is... 

A single substance can produce several values of Bo at which resonance occurs, because different nuclear spin states will be found in different molecules in the sample and because the coupling constants at different nuclei can be different from each other. In the hydrogen molecule ion, hJ, the electron couples equally with the two protons. The molecule could be in a state with both proton spins up, in either of two states with one proton spin up and one down, or in a state with both proton spins down. Since the sum of the Mj values can equal 1, 0, or -1, we obtain a spectrum with three lines, where each line is produced by a different set of molecules. The states are nearly equally populated and the middle line is twice as intense as the other two, because there are two states with one spin up and one spin down. 

For the 1-electron bond of the hydrogen molecule ion Hj, with Is atomic orbitals, the bonding molecular orbital wave-function and corresponding valence-bond stmctures are / =ls -t-lSg =ct1s and (H-H) s(H IT ) <—> (H H). In the Linnett valence-bond structures (1) and (2) for BjHg and, the bridging B-H bonds and the C-C 7i-bonds are 1-electron bonds. For each of these bonds, the a and b atomic orbitals are a pair of boron sp and hydrogen Is orbitals, and a pair of 2p i-orbitals located on adjacent carbon atoms. 

In order to answer these questions, accurate experimental and theoretical results were needed for representative molecular systems. Theoreticians, for obvious reasons, have favored very simple systems, such as the hydrogen molecular ion (Hj) for their calculations. However, with only one electron, this system did not provide a proper test case for the molecular quantum mechanical methods due to the absence of the electron correlation. Therefore, the two-electron hydrogen molecule has served as the system on which the fundamental laws of quantum mechanics have been first tested. 

Let s consider the simplest possible molecule, Hj (the hydrogen molecule ion), which has only one electron. In Figure 3.11, Ha identifies the position of nucleus Ha, and Hb that of nucleus B. The distance between the two nuclei is Rab- The distance between the electron and each nucleus is tac and respectively. 

Most molecules and molecular ions are held together by forces in which electron-electron correlations and exchange interactions play a decisive role. An exception is the hydrogen positive ion Hj, which contains only one electron but is a very stable system. It provides us with an interesting example of how a shallow well can occur near the dissociative limit under conditions reminiscent of those involved in the formation of negative ions. 

We start with diatomic molecules, the simplest of which is Hj, the hydrogen molecule ion, consisting of two protons and one electron. Just as the one-electron H atom serves as a st u ting point in the discussion of many-electron atoms, the one-electron H2 ion furnishes many ideas useful for discussing many-electron diatomic molecules. The electronic SchrOdinger equation for is separable, and we can get exact solutions for the eigenfunctions and eigenvalues. 

We shall now use quantum mechanics to describe chemical bonds and begin with the simplest of all molecules, hJ. The hydrogen molecule ion has not been found in solids or melts, but is easily formed by electric discharge through hydrogen gas. It is also one of the most common molecules in interstellar space. The properties are well known from experimental studies, the equilibrium bond distance is Re = 106.0 pm and the dissociation energy De = 269 kJ mol Comparison with the H2 molecule. Re = 74.1 pm and De = 455 kJ mol shows that the one-electron bond in the ion is 43% longer and 41% weaker than the two-electron bond in the neutral molecule. 

The hydrogen molecular ion has long been a test bed for quantum theoretical methods, and continues so to be. We now present first the simplest quantitative treatment, leading to calculations of the electronic energy as a function of the intemuclear distance, R. The Hamiltonian for the hJ molecule may be written in the form... 

In this section we have concentrated on calculations for Hj only, which have particular relevance to the fine and hyperftne constants determined from Jefferts experiments. Many other papers deal with calculations of the vibration-rotation level energies, for which there is much less experimental data. There are also many papers dealing with the heteronuclear molecule, HD+, which is really a special case because the Bom-Oppenheimer approximation collapses, particularly for the highest vibrational levels of the ground electronic state. Even the homonuclear species Hj and Dj exhibit some fascinating and imusual effects in their near-dissociation vibration rotation levels. Finally we note that in order to match the accuracy of the experimental measurements for all the hydrogen molecular ion isotopomers, it is necessary to include radiative and relativistic effects. 

The hydrogen molecule, like the He atom, poses a real problem the Hamiltonian operator contains a term representing the repulsion between the electrons, and the presence of this term makes an exact solution of the Schrodinger equation impossible. In the case of the helium atom we turned to the hydrogen atom for guidance in the choice of approximate wavefunctions. In the case of the hydrogen molecule we turn to the hJ ion and assume that the wavefunction may be approximated by the product of two molecular orbitals... 

We know that the hydrogen molecule is a stable species. Our simple molecular orbital method predicts that Hj and He2 also possess some stability, because both have bond orders of 5. Indeed, their existence has been confirmed by experiment. It turns out that H2 is somewhat more stable than HeJ, because there is only one electron in the hydrogen molecular ion and therefore it has no electron-electron repulsion. Furthermore, Hj also has less nuclear repulsion than HeJ. Our prediction about He2 is that it would have no stability, but in 1993 He2 gas was found to exist The molecule is extranely unstable and has only a transient existence under specially created conditions. 

Several elementary processes lead to this dissociation. One is the ejection of an electron by the a particle, the hydrogen molecule becoming an hJ ion which dissociates into H-I-H+ ... 

The most simple molecule is the positive ion of the hydrogen molecule (Hj ) Its model, when we neglect the angular momentum s, is distinct from the two-centre problem only in the respect that the internuclear distance is not fixed, but vibrates around an equilibrium configuration which is also determined by the electronic configuration. Instead of each term in the two-centre problem a function of the distance r between the nuclei appears, of which the minimum determines the equilibrium configuration and the value of each of its terms transforms in the value of the terms of the separated systems. For small r the functions behave like 1/r, their distances like the distances between the terms in the case of a united nucleus. 

Typical Cl processes in which neutral sample molecules (M) react with NH to give either (a) a protonated ion [M + HJ or (b) an adduct ion [M + NHJ+ the quasi-molecular ions are respectively 1 and 18 mass units greater than the true mass (M). In process (c), reagent ions (CjHf) abstract hydrogen, giving a quasi-molecular ion that is 1 mass unit less than M. 

Now, we consider H, atoms produced from hydrogen molecules adsorbed on zinc oxide under the influence of electron (ion) impact. We suppose that in this case the energy released in interaction of an electron (ion) with an adsorbed molecule is enough to break any bond between hydrogen atoms. As a consequence, Hj atoms bounce apart over the surface. Hydrogen atoms produced in this case are similar to H atoms adsorbed on the oxide surface from the gas phase at small surface coverages. In other words, they can be chemisorbed as charged particles and thus may influence electric conductivity of zinc oxide. This conclusion is consistent with the experimental results. 

An interesting question is how H3 is formed on the emitter surface and whether H3 molecules can exist on the surface. This question can be investigated with a measurement of the appearance energy of Hj ions. Jason etal.264 find Hj in field ionization of condensed layers of hydrogen, and measure the appearance energy to be 12.7 eV. This value is 2.9 eV smaller than that of H2. Ernst Block conclude265 from a similar measurement in field ionization mass spectrometry of hydrogen that an H 3 ion is formed at the moment when a chemisorbed H atom combines... 

An intramolecular isotope effect, /hj/ d2 > of 1.6 has been observed for the loss of a hydrogen molecule from the metastable (C6D5PC6HS)+ ion formed following El of triphenyl phosphine [918]. 

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