Electron integrated overall energies

So, STOs give "better" overall energies and properties that depend on the shape of the wavefunction near the nuclei (e.g., Fermi contact ESR hyperfine constants) but they are more difficult to use (two-electron integrals are more difficult to evaluate especially the 4-center variety which have to be integrated numerically). GTOs on the other hand are easier to use (more easily integrable) but improperly describe the wavefunction near the nuclear centers because of the so-called cusp condition (they have zero slope at R = 0, whereas Is STOs have non-zero slopes there). 

The first summation in Equation 9-34 is over all the orbital energies of the occupied molecular orbitals, (again, the factor 2 is needed because there are two electrons in each molecular orbital). The terms Jy and Ky are determined by operating the Coulomb operator (Equation 9-31) and the exchange operator (Equation 9-32) on/integrating overall space. The last summation term in Equation 9-34 refers to the inter-nuclear repuslion potential for a particular nuclear configuration. 

Measurement of integrated absorption requires a knowledge of the absorption line profile. At 2000-3000 K, the overall line width is about 10-2 nm which is extremely narrow when compared to absorption bands observed for samples in solution. This is to be expected, since changes in molecular electronic energy are accompanied by rotational and vibrational changes, and in solution collisions with solvent molecules cause the individual bands to coalesce to form band-envelopes (p. 365). The overall width of an atomic absorption line is determined by ... 

The overall current of electron transfer is then obtained by integrating Eqns. 8-7 and 8-8 with respect to electron energy e as shown in Eqn. 8-9 ... 

The overall transfer currents of redox electrons can be obtained by integrating these equations with respect to electron energy e from the lower edge of the conduction band to infinity for the conduction band mechanism and firom minus infinity to the upper edge of the valence band for the valence band mechanism. 

We have m x m equations because each of the m spatial MO s i// we used (recall that there is one HF equation for each ip, Eqs. 5.47) is expanded with m basis functions. The Roothaan-Hall equations connect the basis functions (p (contained in the integrals F and S, Eqs. 5.55, above), the coefficients c, and the MO energy levels . Given a basis set energy levels e. The overall electron distribution in the molecule can be calculated from the total wavefunction P, which... 

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