Approximate 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. 

Theoretical energy curves for one-electron bonds between two atoms are calculated for bond orbitals formed by hybridization of 2s and 2p orbitals, 35 and 3 orbitals, and 35, 3p, and 3d orbitals, the same radial part being used for the orbitals in a set. It is found that for s-p hybridization the bond energy is closely proportional to S3, with 5 the magnitude of the angular part of the bond orbital in the bond direction. This relation is less satisfactorily approximated in the case of s-p-d hybridization. 

The picture has not been confirmed experimentally because time scales of less than 10 ps are not accessible at present and there are difficulties envisaged In reducing this limit below 1 ps. However, as a theoretical model It fits much of the experimental data and Is of much greater value than that which uses the continuous slowing down approximation whereby energy Is assumed to be deposited continuously along the track. 

Fig. 5.1 Theoretical energy curves (a-J, f) for the hydrogen molecule. Hj. compared with the experimental curve (e). Curves o d show successive approximations m the wave function as discussed in the text. Curve /is the repulsive interaction of two electrons of like spin.

The average emf associated with this process is approximately 2.3 V and the theoretical energy density is 2.6 k Wh/kg at 350°C. 

Approximate theoretical estimates of the mean pairing energies for the relevant ions of the first transition series have been made from spectroscopic data. In Table 20-1 these energies, along with A0 values for some complexes... 

The theoretical energy required for the process is that for establishing the desired concentration of H+ and OH -ions in the outer phases of the membrane from their concentration in the membrane which is approximately 10 mol/L. The free energy of this process is ... 

The approximate SAP energies, (T ) for Zn + calculated for all four types of pair functions when using Fq. (51) are compared in Table 9.2 with the accurate values EpEiy[(T ) For each type the correction (50) consists of just one term defined by the relevant leading AEC obtained from our theoretical formulas (43)-(46). 

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