HIGHER-ORDER CALCULATIONS

MOLECULAR orbital calculations of the type described in these Notes are often called zero—order calculations because of the many assumptions involved. Higher order approaches are available in considerable profusion. Unfortunately, most of these are not very convenient for use by organic chemists. It is not difficult to include overlap (i.e., take Sy 0) and make corrections in Coulomb integrals for nonself-consistent fields, in resonance integrals for bond lengths, and in the o-bond framework for angle strain. All of this may or may not constitute a first-order approach, depending upon one s point of view. 

There are some fundamental approximations in the simple LCAO method that are harder to evaluate. One is the validity of the linear combination of atomic orbitals as an approximation to molecular orbitals. Another is the assumption of localized o bonds. A proper treatment probably should take account of the so-called cr—ir interactions. Beyond these rather basic assumptions is the bothersome business of dealing explicitly with interelectronic repulsions. These repulsions are expected to be functions of molecular geometry as well as the degree of self-consistency of the molecular field. Thus, cyclobutadiene must have considerably greater interelectronic repulsion than butadiene, with the same number of tr electrons. 

The usual procedures for calculating interelectronic repulsions in molecules are complicated. Space does not permit discussion of more than the elements of one, perhaps typical, approach, which is of interest here because it starts with our regular LCAO molecular orbitals calculated as desired with or without overlap. 

The steps involved are, first, calculation of the one-electron molecular orbital energies for the field of the nuclei and (T—bond electrons. Usually much more detailed account is taken of molecular geometry than is done in the simple MO theory. The repulsions between the electrons in the same and different molecular orbitals are then calculated for particular electronic configurations (such as the lowest state). The usual MO coefficients are used to determine the fraction of the time a given electron spends in a particular orbital. The exclusion principle is employed to reject all terms that amount to having two electrons with the same spin in a given atomic orbital. 

The result is to have the total ir-electron energy (attraction and repulsion) of a configuration set up on the basis of one-electron molecular orbitals that were obtained without consideration of interelectronic repulsion. It would, of course, be only a coincidence if this energy were to represent the minimum possible calculated energy. The energies of a number of excited configurations with one, two, or several electrons in normally unoccupied molecular orbitals are also calculated. These excited states may have more or less interelectronic repulsion than the presumed lowest state. 

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Cl results can vary a little bit from one software program to another for open-shell molecules. This is because of the HF reference state being used. Some programs, such as Gaussian, use a UHF reference state. Other programs, such as MOLPRO and MOLCAS, use a ROHF reference state. The difference in results is generally fairly small and becomes smaller with higher-order calculations. In the limit of a full Cl, there is no difference. 

We finally note that this analysis resulted in an explicit identification of the renormalization factors Zv, Zn, Z of polymer theory with renormalization factors of field theory. We can thus take over higher order results for the renormalization factors and the resulting RG flow, established in field theory. This is a most useful result since a good representation of the RG mapping is crucial in the application of the theory, but higher order calculations are most complicated. 

Analytical basis set for higher-order calculations of transition amplitudes... 

The calculation of the temperature-dependent values of the thermal function is performed using a variety of procedures. Some of these procedures were developed internally while others were developed at other locations. A general description of the traditional equations used in the JANAF project follows. Numerous higher order calculational schemes are used whenever the data are sufficient. These procedures are detailed on the appropriate tables. 

The efficient way of calculating the fifth-order energy terms would be by combining the CC scheme with Wigner s 2n + 1 rule. The 2n + 1 rule in MBPT calculations was effectively exploited in fourth-, fifth-, and higher-order calculations by Bartlett and co-workers.I6-32 39 It is particularly suitable in fifth-order calculations. This method is effectively based on Eq. (24) according to which we can express the fifth-order energy as... 

Something similar happens if the s, (5-expansion is applied to study models of m-vector magnets with long-range-correlated quenched disorder [65,71] also in the case of magnets, as well as for polymers the first order e, (5-expansion leads to a controversial phase diagram. In order to obtain a clear picture and more reliable information, one should proceed to higher order calculations. 

As we have already seen for the leptonic sector and, as we shall see later for the hadronic sector, the SM in lowest order provides a wonderfully successful description of a huge range of experimental phenomena. But to test the theory more profoimdly requires the comparison of precise higher order calculations with precise experimental data. In the on shell renormalization scheme, all calculations are expressed in terms of the parameters Mz and The former is known to great accuracy and we would clearly like to have an equally precise value for the latter. Unfortunately, this is precluded by the limited precision in our measurements of Mw-Thus, taking e.g. the UA2 value (Table 5.1) Mw = (80.2 1.5) GeV/c and (Table 8.3) Mz = (91.177 0.006) GeV/c, we obtain the rather imprecise estimate Sw = 0.226 0.029. 

It therefore seems advisable for the present to adopt a procedure similar to the one used in comparing the SM with experiment in lowest order (as was done even before the discovery of the W and Z°). Thus we fit as many higher order calculations, containing the now unknown parameter s, to experiment and examine the compatibifity of the various determinations. This programme is complicated by the fact that the higher order formulae involve the unknown top and Higgs masses as is explained in Section 7.8. 

The treatment of other atoms is similar to that of helium. In zero order we neglect electron-electron repulsions, and in higher-order calculations these repulsions are treated with the same approximation methods as used with the helium atom. From this point on in the chapter we will discuss only the results of such calculations. 

Although only higher-order calculations will predict the difference between T - and S simple LCAO-MO theory does predict an increase in the acidity of 2-naphthol upon excitation. The calculated electron densities of the ground and the first excited state are shown in Fig. 11 (Sandorfy,... 

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