Hydrogen molecular ions

Table 3.2 Historically significant calculations for the electronic ground state of the hydrogen molecular ion...

The hydrogen molecular ion is rapidly (within a day at a standard gas density n of 104 cm-3) converted to H3 via the well-studied reaction ... 

In quantum mechanics, as we have already seen, one can approximately describe the hydrogen molecular ion as consisting of Ha+ and Hb, or Hb+ and Ha. Some combination of wave functions representing these two configurations is needed as an approximation of the actual state of affairs. The state of H2+ can then be thought of as a resonance hybrid of the two. 

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. 

Fig. 7.1. Three regimes of interaction in the hydrogen molecular ion. (a) At large distances, R>16 a.u., the. system can be considered as a neutral hydrogen atom plus a proton. The polarization of the hydrogen atom due to the field of the proton generates a van der Waals force, (b) At intermediate distances, 16>/ >4 a.u. the electron can tunnel to the vicinity of another proton, and vice versa. A resonance force is generated, which is either attractive or repulsive, (c) At short distances, R<4 a.u., proton-proton repulsion becomes important. (Reproduced from Chen, 1991c, with permission.)...

In the following, we present a treatment of the hydrogen molecular ion problem using a time-dependent Schrddinger equation ... 

Fig. 7.4. Wavefunctions of the hydrogen molecular ion. (a) The exact wavefunctions of the hydrogen molecular ion. The two lowest states are shown. The two exact solutions can be considered as symmetric and antisymmetric linear combinations of the solutions of the left-hand-side and right-hand-side problems, (b) and (c), defined by potential curves in Fig. 7.3. For brevity, the normalization constant is omitted. (Reproduced from Chen, 1991c, with permission.)...

In this subsection, we show that by evaluating the modified Bardeen integral, Eq. (7.14), with the distortion of the hydrogen wavefunction from another proton considered, as shown by Holstein (1955), an accurate analytic expression for the exact potential of the hydrogen molecular ion is obtained. 

For R>7 bohr, the difference between the exact solution and the leading terms (Equations (7.33) and (7.34)) is less than 1 meV. Therefore, the interpretation of the resonance energy in terms of tunneling is verified quantitatively in the case of the hydrogen molecular ion. Furthermore, the comparison with the soluble case of the hydrogen molecular ion is also a verification of the accuracy of the perturbation theory presented in Chapter 2. 

As shown in Fig. 7.1, as the proton-proton separation becomes even smaller, the picture of the resonance becomes obscured, and the proton-proton repulsion is no longer screened by the electron. Slater (1963) showed that the Morse function can match the exact potential curve for the hydrogen molecular ion very precisely ... 

Second, the sensitivity of AFM. In a typical AFM (Binnig et al., 1986), the force sensitivity is about 0.01 nN. In the range of 4-10 a.u., the resonance force in the hydrogen molecular ion is 4 nN to 0.01 nN. Therefore, the resonance force (attractive atomic force) of a single chemical bond, extended over a distance of 3 A, can be detected. On the other hand, the van der Waals force of a pair of neutral atoms, when it is distinguishable from the total force. 

To make a quantitative treatment, we define a system including a tip and a sample, as shown in Fig. 7.6. Independent electron approximation is applied. The Schrbdinger equation is identical to Eq. (7.6), with the potential surface shown in Fig. 7.6. Similar to the treatment of hydrogen molecular ion, a separation surface is drawn between the tip and the sample. The exact position of the surface is not important. Define two subsystems, the sample S and the tip T, with potential surfaces Hs and Ut, respectively, as shown in Fig. 7.6 (c) and... 

This effect can be illustrated by Fig. 14.2. The effective range of local modification of the sample states is determined by the effective lateral dimension 4ff of the tip wavefunction, which also determines the lateral resolution. In analogy with the analytic result for the hydrogen molecular ion problem, the local modification makes the amplitude of the sample wavefunction increase by a factor exp( — Vi) 1.213, which is equivalent to inducing a localized state of radius r 4tf/2 superimposed on the unperturbed state of the solid surface. The local density of that state is about (4/e — 1) 0.47 times the local electron density of the original stale in the middle of the gap. This superimposed local state cannot be formed by Bloch states with the same energy eigenvalue. Because of dispersion (that is, the finite value of dEldk and... 

The factor 0.47 is the relative intensity of the tip-induced local states, estimated from an analogy to the hydrogen molecular ion problem. For example, for (E/r- ,) = 4 eV and kf = 1/4 , when atomic resolution is achieved, that is, r < 2 A, the energy resolution is about 0.2 eV. Even for the reconstructed surfaces, such as Si(lll) 7X7, when the adatoms are barely resolved, that is, t aA, the energy resolution is 0.12 eV. 

Landau, L. D., and Lifshitz, L. M. (1977). Quantum Mechanics, third edition, Pergamon Press, Oxford. The treatment of the hydrogen molecular ion is presented as a problem on page 312. 

See Contact stress Hexagonal symmetry 132 Hohenberg-Kohn theorem 113 Hydrogen molecular ion 173 history 173 repulsive force 185 resonance interaction 177 van der Waals force 175 Hydrogen on silicon 336 Image force 56—59, 72, 93 concept 56 effect on tunneling 74 field emission, in 56 jellium model, in 93 observability by STM 72... 

Underpotential deposition 339 van der Waals force 56, 175 in the hydrogen molecular ion problem 175 tunneling, in 186... 

Almost all studies of quantum mechanical problems involve some attention to many-body effects. The simplest such cases are solving the Schrodinger equation for helium or hydrogen molecular ions, or the Born— Oppenheimer approximation. There is a wealth of experience tackling such problems and experimental observations of the relevant energy levels provides a convenient and accurate method of checking the correctness of these many-body calculations. 

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