Elliptical-type basis functions

Elliptical-type basis functions have been shown to be useful in calculations for diatomic molecules by a number of authors. They are most often defined in terms of the elliptical coordinates A, and as follows... 

Comparison of seif-consistent field calculations using elliptical-type basis functions with fully numerical studies of diatomic molecules. ... 

Elliptical-type basis functions can provide a very accurate representation of orbitals in diatomic molecules. In Table III, we compare the results of calculations employing basis sets of elliptical-type functions with fully numerical studies. It can be seen that the two approaches yield results of comparable accuracy. It should be emphasized that the basis set approach, in contrast to the fully numerical technique, affords a compact representation of the wavefunction. 

The use of different types of basis functions in molecular calculations continues. The Os or Hulthdn-type function has been used in calculations on Hs and Ha,33 and is more effective than the Is-type function. Hj-type elliptical orbitals have also been employed in variational calculations on Hi, Ha, He 4, and H.4.34 The H orbitals were also used as basis orbitals for SCF calculations. A four-function basis set with two (non-linear) variational parameters yields 99% of the Hartree-Fock energy. 

For H2 and D2 the procedure is much the same but there is more experimental information with which we can compare. Explicitly electron-correlated functions of the James-Coolidge type were chosen as basis functions. They were defined in terms of elliptical coordinates as ... 

In the expansion [Eq. (12)], the products x aXvb are formally included where Xu is on center a and x on center b. It is convenient to represent the 2p orbitals as linear combinations of 2pir, 2p r, and 2pa, where the notation of Roothaan24 is used. There are 15 unique two-center density functions involving 2s and 2p-type orbitals, and these can be written in terms of sin m and cos m and some functions in the con-focal elliptical coordinates and 77. For the case studied here, m is 0, 1, or 2 so that only five basis functions span the cylindrical point groups. One question, then, is Are all 15 functions necessary, or can the chaige density analyst get by with as few as five functions to effectively represent these two-center bond-type density functions This depends on... 

Contours like this are qualitatively the same sort of thing we obtain from sedimentation-diffusion experiments as shown in Figure 2.9. Therefore let us consider the relationship between the two types of data. In general, exactly the same factors affect both the intrinsic viscosity and the friction factor ratio, but the functional dependencies are somewhat different. Figure 4.13b shows how a contour of f/f0 selected from a sedimentation-diffusion study and an intrinsic viscosity contour selected on the basis of viscosity experiments might overlap. In this case the solvation-ellipticity combination is characterized unambiguously a/b = 2.5 and (ml b/m2) = 1.0. Figure 4.13b shows the complementarity of viscosity and sedimentation-diffusion data. 

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