Orbit size, optimum

The observations that there is an "optimum" orbit size and that peaks split for orbits not too much larger than the optimum orbit suggest that the optimum orbit occurs because of special circumstances. One possible circumstance is a coincidence of frequencies for ions with low and high z-mode amplitudes so that if there are mass discriminating differences in the way the ions populate the trap or in the way ions are excited, then systematic mass measurement errors can be expected. Excitation of the cyclotron mode does produce a spread in cyclotron radii, and mass discriminating z-mode excitation is discussed elsewhere in this chapter. Thus, frequency variations that cause systematic mass errors are due in part to trap field inhomogeneities. These effects are evident at low ion populations and may be due in part to excitation induced ion cloud deformation which increases with ion number. 

By choosing the contracting scheme, it is possible to determine the contracted gaussian functions which best approximate Slater functions,88 Hartree-Fock atomic orbitals,89-90 etc. It should be noted that the size and balance (revealed, for example, by the optimum ratio of the number of s-type and p-type functions) of the basis are definite factors which control the quality of the final result (e.g., ref. 90). 

Given a density functional, DFT calculations also require the choice of an atomic orbital basis set. The larger the basis set the closer calculations approach the complete basis set limit and the smaller the basis set error. At the same time, computational demands increase rapidly with increasing basis set size. The optimum choice of basis set is that which provides the optimum compromise of accuracy and computational effort. 

It has been shown earlier (Chapter 2) that it is possible to use the vaiiation principle to derive a single differential equation whose solutions are the opti-iimm orbitals of a single-determinant wavefunction. We now wish to carry this derivation forward in order to be able to have a practical method of obtaining (approximations to) these optimum orbitals. In particular, since the solution of a highly non-linear differential equation by numerical methods for molecules of arbitrary geometry is out of the question, it is desirable to transform the differential equation into an algebraic equation which will he more amenable to a systematic method of solution independently of the size and shape of the molecule, radical, ion or group of molecules. 

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