One-electron diatomic ion

In this appendix we collect together tables of matrix elements of r, r2, r3, z, z2, rz, and r(r2 — z2) in the scaled hydrogenic basis and perturbation corrections to the ground-state energy for the LoSurdo-Stark, Zeeman, screened Coulomb, and charmonium Hamiltonians and the one-electron diatomic ion. 

For the one-electron diatomic ion, H2", then the Schrodinger equation, subject to the Bom-Oppenheimer approximation, takes the form... 

Perturbation Corrections to the Wave Function up to Fifth Order for One-Electron Diatomic Molecular Ions at Large Separation"... 

Clay, M. (1979). An algebraic approach to the calculation of the long-range intermolecular forces of a one-electron diatomic molecular ion. M. Math. Thesis, Univ. of Waterloo, Waterloo, Ontario. 

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. 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

Chlorine as a free element is diatomic (Cl2) however, as an ion, it will gain one electron to become isoelectronic with argon. The electronic configuration of the chloride ion is s22s22p63s23p6. The compound thus formed, aluminum chloride, has the formula A1C13. 

This chapter consists of the application of the symmetry concepts of Chapter 2 to the construction of molecular orbitals for a range of diatomic molecules. The principles of molecular orbital theory are developed in the discussion of the bonding of the simplest molecular species, the one-electron dihydrogen molecule-ion, H2+, and the simplest molecule, the two-electron dihydrogen molecule. Valence bond theory is introduced and compared with molecular orbital theory. The photo-electron spectrum of the dihydrogen molecule is described and interpreted. 

Similar excited states have been observed for diatomic molecules of the alkali metals. They may be interpreted as involving a molecule-ion, such as Li, with a one-electron bond, plus a loosely-bound outer electron. The internuclear distances are about 0.3 A greater than for the corresponding normal states 2 2.94 A lor Lij (2.672 A for Lit), 3.41 A for Na (3.079 A for Nat), and 4.24 A for K (3.923 A for Kt). The values of the bond energies for the one-electron bonds, as indicated by the vibrational levels, are about 60 percent of those for the corresponding electron-pair bonds. 

The first values necessary are some estimates of the ionic radii of O and BF4. For the latter we may use the value obtained thermochemically by Yauimirskii, 218 pm. An educated guess hus to be made for 02. since if we arc attempting to make it for the first tune (as was assumed ahove), we will not have any experimental data available for this species. However, we note that the CN ion, a diatomic ion which should be similar in size, has a thermocheinical radius of 177 ppm. Furthermore, an estimate based on covalent and van dec Waals radii (see Chapter 8) gives a similar value. Because OJ has lost one electron and is positively charged, it will probably be somewhat smaller than this. We can thus take 177 pm as a conservative estimate if the cation is smaller than this, the compound will be more stable than our prediction and even more likely to exist. Adding the radii we obtain an estimate of 395 pm for the interionic distance. 

Montgomerie [44] and Leach and Moss [45]. We will make use of the analyses given in these reviews. Not surprisingly, the most accurate treatments have been for one-electron molecules, the principal example being II2 and its isotopomers. Spectacular breakdown of the Bom-Oppenheimer approximation in diatomic molecules is observed for the HD+ ion, which we will describe in some detail. 

The 0 + (vanadyl) ion is one of the most stable oxo-metal species known, and probably the most stable diatomic ion. Most of these properties stem from the ground-state electronic configuration of the vanadium atom, [Ar]3d, which shows similarities to the Cu + d system. The ion also lends itself to study by ESR because of the isotopic purity of the V isotope, its high nuclear spin, / = 7/2, and the single unpaired outer electron. 

Next, we will apply the simple MO model to diatomic ions, starting with He2 +. Is this ion expected to be stable Examination of Fig. 14.30 shows that when He is combined with He+ to form He2+, two electrons are lowered in energy and only one is raised in energy. That is, the bond order is (2 - 1 )/2 = j, and He2+ is predicted to be stable. Experiments have shown that He2+ does indeed exist with a bond energy of 250 kj/mol. 

The simplest of all molecules is the ion H J since it possesses two nuclei and but one electron. If orbitals can be found to describe the states of the one electron in a two-centre field of this kind, it should be possible to develop an aufbau theory of diatomic molecules, exactly as in the one-centre instance (p. 57). And there is no reason why this should not be extended to a many-centre example. The one-electron orbitals, which extend over all nuclei, are called molecular orbitals or in the case of a crystal, crystal or Bloch orbitals (after Bloch who first used them). In the aufbau approach, the available... 

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