Other Simple Variation Treatments

The energy of the bond for this function too is essentially resonance energy. Dickinson2 introduced an additional term, dependent on two additional parameters, in order to take polarization into account. He wrote for the (not yet normalized) variation function 

It was found by Guillemin and Zener3 that another variation function containing only two parameters provides a very good value for the energy, within 0.01 v.e. of the correct value, the equilibrium separation of the nuclei being 1.06 A, as for all functions discussed except the simple one of the preceding section. This function is 

1 The introduction of such a function to take care of polarization was firet made (for the hydrogen molecule) by N. Rosen, Phys. Rev. 38, 2099 (1931). 

A still more simple variation function giving better results has been recently found by James.1 This function is 

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In conclusion, we note that in all the examples discussed in the present chapter the use of charge, current and spin densities has allowed us to obtain results that, besides exposing the physics, are of completely general form they do not depend either on the use of simple orbital-type wavefunctions (see e.g. Slichter, 1964, Chap. 7) or on the availability (hypothetical) of complete sets of exact unperturbed functions. In other words, the treatment of a vast range of properties, using variational approximations of any form whatever, has been reduced in a unified way to the calculation of a small number of density functions. 

In the unrestricted treatment, the eigenvalue problem formulated by Pople and Nesbet (25) resembles closely that of closed-shell treatments.-On the other hand, the variation method in restricted open-shell treatments leads to two systems of SCF equations which have to be connected in one eigenvalue problem (26). This task is not a simple one the solution was done in different ways by Longuet-Higgins and Pople (27), Lefebvre (28), Roothaan (29), McWeeny (30), Huzinaga (31,32), Birss and Fraga (33), and Dewar with co-workers (34). 

There are various other ways of examining the variate in question in this case. Let us first examine simple one-way ANOVA of the variate by sex as in Table 16.16. In neither of the two cases was there any indication of significant treatment differences at any reasonable level. Because the two sexes did not show any pretreatment differences based on the two-factor analysis of the covariate, let us combine the two sexes and analyze the data by one-way ANOVA as in Table 16.17. In this case, because of the increased sample sizes for combining the two sexes, there was indication of some treatment differences (p = 0.0454). Unfortunately, this analysis assumes that because there were no pretreatment differences between the two sexes, that pattern will hold during the posttreatment period. That often may not be the case because of biological reasons. 

The constructions of different approximations will be done in the sections that follow on the basis of the variational principle for molecular electronic energy in the SLG-based approximation. We shall demonstrate that this treatment leads to a mechanistic model which can in a sense be considered a generic or deductive form of MM. It means that although the simple balls-and-springs model can hardly be justified from any general point of view, it does not mean that any other mechanistic model cannot be justified at all. And that is what we shall provide. 

A variety of diverse synthetic methods have been empioyed for the preparation of cyclopropane (1 j. Schlatter and Demjanov and Dojarenko pyrolyzed cyclopropyltrimethylammonium hydroxide at 320°C using platinized asbestos as the catalyst. About equal amounts of cyclopropene (1) and cyclopropyidimethylamine are formed, contaminated with some dimethyl ether and ethylene. Treatment with dilute hydrochloric acid removed the amine from the gas stream and 1 was separated from the other products by gas chromatography. Alder-Rickert cleavage of the Diels-Alder adduct formed from cycloheptatriene and dimethyl acetylenedicarboxylate resulted only in the formation of a polymer and trace amounts of 1. A simple approach by Closs and Krantz based on the synthesis of 1-methylcyclopropene involved the addition of allyl chioride to a suspension of sodium amide in mineral oil at 80°C. Under the conditions employed, 1 could readily escape from the reaction mixture. Though a number of variations were tried, the yield of 1 never exceeded 10%. 

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