Exact Theoretical Energies

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. 

Clearly, the successful reproduction of the experimental result is, in part, related to the high quality of the potential energy surface. A more direct evaluation of the accuracy of transition-state theory can be obtained via a comparison to other (more exact) theoretical approaches to the calculation of the rate constant, all using the same potential energy surface. Table 6.3 shows such a comparison. We observe that transition-state theory does overestimate the rate constant but the agreement is quite reasonable, especially when the simplicity of the calculation is taken into account. 

Such UPS data offer a unique opportunity to compare experimental data and MO calculations on the nature of the molecular orbitals, including ground state and both singlet and triplet excited states. Although the ionization potentials determined from MO calculations were not exact, relative energies were in good agreement with the theoretical values (Sections III and VIII). 

The principle message of this paper is that few-body atomic systems such as helium and lithium can be solved essentially exactly for all practical purposes in the nonrelativistic limit, and there is a systematic procedure for calculating the relativistic and other higher-order QED corrections as perturbations. The solution of the problem of calculating Bethe logarithms means that the theoretical energy levels are complete up to and including terms of order Ry. 

It has been shown by de Boer P], Pitzer [ °], and Guggenheim that the classical law of corresponding states can be given an exact theoretical basis provided that (1) the total potential energy arising from the intermolecular forces can be written as a sum of pair potential functions, p(ri ), where ra is the distance between the centers of molecules i andj, and (2) that 9 itself is of the form... 

To conclude this subsection we observe that the utilization of the standard Fermi-Golden rule rate expression, which leads to Eq. 1, is subject to a number of constraints. In particular, when the adiabatic electronic energy gap between P BH and P B "H is too small ( kT at room T) its use would be questionable. Nevertheless, even in this case an exact theoretical treatment of the problem is unlikely to reveal dispersive kinetics at room T from the Q distribution. 

The calculation of the surface energy of metals has been along two rather different lines. The first has been that of Skapski, outlined in Section III-IB. In its simplest form, the procedure involves simply prorating the surface energy to the energy of vaporization on the basis of the ratio of the number of nearest neighbors for a surface atom to that for an interior atom. The effect is to bypass the theoretical question of the exact calculation of the cohesional forces of a metal and, of course, to ignore the matter of surface distortion. 

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