How Do We Determine Molecular 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. 

Experimental. The vibrational spectrum of an ideal harmonic oscillator would consist of one line at frequency v corresponding to A = hv, where A is the distance between levels on the vertical energy axis in Fig. 10-la. In the harmonic oscillator, AE is the same for a transition from one energy level to an adjacent level. A selection rule An = 1, where n is the vibrational quantum number, requires that the transition be to an adjacent level. 

What is the energy difference in electron volts and kilojoules per mole between the ground state of TI, and its lowest vibrationally excited state The vibrational spectrum of TI, has lines at 2191.2064. 1941, 1821. 170h. 1591, 1479. 1368. 1257,1145, 103.1. 918. 800, 677, 548, 411. 265. 117 cm . 

What is tlie sum of all the energy gaps as determined from the vibrational speetrum 

The second correction is much larger. The residual energy that the molecule ion has in the ground state above the T),- at the e(]nilibrintn bond length is the zero point energy. /PH. 

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