Chain of hydrogen atoms

Band Theory. The Linear Chain of Hydrogen Atoms... 

The Jk-th wave function of the electrons in a chain of hydrogen atoms results in a similar way. From every atom we obtain a contribution 2 cosnka, i.e. the Is function %n of the n-th atom of the chain takes the place of A0. All atoms have the same function x, referred to the local coordinate system of the atom, and the index n designates the position of the atom in the chain. The k-th wave function is composed of contributions of all atoms ... 

The model of the chain of hydrogen atoms with a completely delocalized (metallic) type of bonding is outlined in the preceding section. Intuitively, a chemist will find this model rather unreal, as he or she expects the atoms to combine in pairs to give H2 molecules. In other words, the chain of equidistant H atoms is expected to be unstable, so it undergoes a distortion in such a way that the atoms approach each other in pairs. This process is called Peierls distortion (or strong electron-phonon coupling) in solid-state physics ... 

Let us consider once more the chain of hydrogen atoms, but this time we put it together starting from H2 molecules. In the beginning the chain then consists of H atoms, and electron pairs occur between every other pair of atoms. Nevertheless, let us still assume equidistant H atoms. The orbitals of the H2 molecules interact with one another to give a... 

The one-dimensional chain of hydrogen atoms is merely a model. Flowever, compounds do exist to which the same kind of considerations are applicable and have been confirmed experimentally. These include polyene chains such as poly acetylene. The p orbitals of the C atoms take the place of the lx functions of the H atoms they form one bonding and one antibonding n band. Due to the Peierls distortion the polyacetylene chain is only stable with alternate short and long C-C bonds, that is, in the sense of the valence bond formula with alternate single and double bonds ... 

In deriving these formulae it is assumed that the wave functions fa are orthogonal to one another, as for instance are Wannier functions. If they are atomic functions falling off as e ar, a formula such as (9) must be modified by a term in the denominator to take account of the non-orthogonality. Reviews of the appropriate formulae are given in textbooks see e.g. Callaway (1964) and Wohlfarth (1953), who considered a linear chain of hydrogen atoms. 

At ultrahigh pressure and low temperature, such as 250 GPa and 77 K, solid molecular hydrogen transforms to a metallic phase, in which the atoms are held together by the metallic bond, which arises from a band-overlap mechanism. Under such extreme conditions, the H2 molecules are converted into a linear chain of hydrogen atoms (or a three-dimensional network). This polymeric H structure with a partially filled band (conduction band) is expected to exhibit metallic behavior. Schematically, the band-overlap mechanism may be represented in the following manner ... 

Figure 1 The band structure of a chain of hydrogen atoms spaced 3, 2, and 1 A apart. The energy of an isolated H atom is -13.6 eV.

Let s see how this works on a chain of hydrogen atoms (or a polyene). The original chain has one orbital per unit cell, 85a, and an associated simple band. We prepare it for deformation by doubling the unit cell, 85b. The band is typically folded. The Fermi level is halfway up the band the band has room for two electrons per orbital, but for H or CH we have one electron per orbital. 

The qualitative features of paraffin pyrolysis, on the other hand, are reasonably well understood. The decomposition is, in general, initiated by rupture of C-C bonds, carried by chains of hydrogen atoms, methyl radicals, and to some extent ethyl radicals, and terminated by assorted radical recombinations. Product inhibition occurs through the reaction... 

What remains are calculations for certain model systems fermion gases in different dimensions and with different types of interactions and the linear chain of hydrogen atoms. These have the advantage that the calculations can be carried out without approximations. They can therefore serve as benchmarks for further work. 

The first explicit use of a gap equation in the type of problems discussed here was in a calculation of total and orbital energies for a linear chain of hydrogen atoms.30 Berggren and Martino arrived at their gap equation in a way other than the one described here. They studied the HN chain in the AMO approximation [cf. (3.1)], which means that their spm orbitals belong to the Fukutome class ASDW. The pairing partners i//M were taken from the unoccupied part of the band, the lower half of which is doubly filled in RHF. The pairing was vertical in the sense of a band... 

Fig. 2.9 Semiempirical (extended Huckel theory) band structure E ip k)) for a onedimensional chain of hydrogen atoms with H-H distances of 3 A and 2 A (in grey) and 1 A (in black). The shape of the extended wave function has been iconized for three different k points, r (left), k = n/2a (middle), and X (right).

For chemists, it is probably not a big surprise that a one-dimensional chain of hydrogen atoms does not exist and that it will immediately decompose into isolated H2 molecules. The Peierls distortion has important consequences for one-dimensional systems, such as polyacetylene with C-C bond-length alterations (instead of equal C-C distances) [74], infinite molecules with platinum-platinum bonding such as Krogmann s salts K2[Pt(CN)4]Xo,3 3H2O with X = Cl or Br [75], or other one-dimensional systems [76], and it also affects three-dimensional systems, in particular elemental structures (see Section 3.4). From a group-theoretical point of view, Peierls distortions are characterized by a loss of translational symmetry in the above example, the nonequidistant chain of H atoms is less symmetrical (in terms of translational symmetry) than the equidistant one. 

Fig. 2.19 Semiempirical (extended Huckel theory) band structure, density-of-states, and crystal orbital overlap population for a one-dimensional chain of hydrogen atoms spaced at 2.0 A. Due to a Gaussian smoothing, DOS and COOP plots appear slightly...

Our discussion so far has focused on polyacetylene and related examples. The broad results however, are transfciable to many other systems. Algebraically, for example, our discussion applies equally well to the case of a one-dimensional chain of hydrogen atoms bearing l.v orbitals. Recall the one-to-one correspondence between the orbitals of finite molecules and their polyene analogs. With one electron pet-atom, Figure 13.3 indicates a half-filled band foi the geometry 13.34. This will distort (in a Peierls fashion) to a solid composed of H2 molecules (13.35) as chemical... 

Finally, there is the important domain of intermediate density, which is really the only experimentally accessible one lattices with r.>1.3 are unstable relative to the molecular solid, while those with ra<1.0 are likely to remain experimentally inaccessible in the foreseeable future. In this domain, neither the van der Waals nor the electron gas model is appropriate. There are no calculations in this regime to which comparison can be made, although results from calculations on one-dimensional chains of hydrogen atoms [35] suggest that the dimensional scaling values may be somewhat low here. 

Let us take our beloved chain of hydrogen atoms in the lx state, to which we already owe so much (Fig. 9.14). 

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