Simple perturbation theory

In situations where maximum overlap criteria are important, it is often possible to construct simple perturbation theory arguments to account for broad geometric trends. Burdett has illustrated in a very neat fashion how the geometries of binary metal carbonyls M(CO) may be rationalized using such an angular overlap model (55). The same problem has also been tackled by Elian and Hoffmann using more complete extended Hiickel calculations (78). 

Lo et a/,102 have calculated spin-orbit coupling constants for first- and second-row atoms and for the first transition series, results agreeing with the work of Blume and Watson. Karayanis103 has extended the calculation to triply ionized rare earths. However, with very heavy atoms relativistic effects on the part of the wavefunction near the nucleus become severe, leading to a breakdown of the conditions under which simple perturbation theory ought to be applied. Lewis and co-workers104 have used relativistic self-consistent Dirac-Slater and Dirac-Fock wavefunctions to evaluate spin orbit coupling... 

The subject of multiphoton excitation spectroscopy began in 1931 when Goppert-Mayer [450] wrote a theoretical paper in which she calculated the transition rate for an atom in the presence of two photons rather than just one. At the time, the process seemed rather exotic, and it was reassuring that the calculated rate was so low as to guarantee that it could not readily be observed in the laboratory with conventional sources. This conclusion was reassuring because it implies that a simple perturbative theory (one photon per transition is the weak-field limit) is adequate for most purposes. 

For the numerical implementation of these formulae one can either use a perturbation expansion for the functions coupled Hartree-Fock (CHF) equations for each order.124 This procedure seems to be more advantageous for numerical calculations, but the resulting expressions are quite complicated. For this reason we do not reproduce them here, but refer to the original paper.124 It seems to be an acceptable compromise to use the second order CHF equations for the dynamic as and (Js, but use simple perturbation theory for the dynamic ys using as unperturbed wave functions the results of the solutions of the first and second order CHF equations. One should point out, however, that in our calculations124 the second numerical derivatives... 

An adiabatic mode analysis of measured vibrational spectra is possible with a simple perturbation theory approach that was already published in the sixties [44]. The basic equation of vibrational spectroscopy (compare with Eq. 19) can be written in matrix form according to Eq. (74)... 

Figs. 8 and 27. Again, attempts to predict the shifts when replacing sulfur by pyrrole type nitrogen cannot be based on electronegativity considerations and simple perturbation theory ). Consider f.i. the third and the fourth level of 38, la2 and 2b, respec-... 

The interrelationships between the classical, thermally activated electron transfer processes and optical electron transfer spectroscopy are clear from this brief list. Most of the points noted on this list need some modification in the application to real systems. Most of these modifications can be based on simple perturbation theory or elaborate computational arguments. The physical models on which the above statements are based and their necessary modifications are dealt with below. 

Although the potentials used for the early quantum calculations described above are now known to be quite inadequate, these calculations did show that simple perturbation theory ideas based on long-range multipole moments are far from adequate for most aspects of rotationally inelastic HF-HF collisions. 

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