Errors Not Yet Corrected

The question, what conditions are to be fulfilled by a density matrix to be the image of a wave function, that is, to describe a real physical system is opened till today. The contracted Schrodinger-equations derived for different order reduced density matrices by H. Nakatsui [1] give opportunity to determine density matrices by a non-variational way. The equations contain density matrices of different order, and the relationships needed for the exact solutions are not yet known in spite of the intensive research activity [2,3]. Recently perturbation theory corrections were published for correcting the error of the energy obtained by minimizing the density matrix directly applying the known conditions of N-representability [4], and... 

An interesting problem is the precise calculation and measurement of the Lamb shift 6 which we describe here, commenting on the main points of interest. First, there is a disparity - not yet accounted for - both between the at present most precisely known theoretical values of S, as well as between experiment and theory. Another important point is the opportunity provided to obtain information on the structure and properties of corrections which are not given directly by QED. In contrast to the anomalous magnetic moment, the Lamb shift characterizes the properties of bound electrons, i.e. it takes account of not only the QED effects but the effects arising from the nuclear structure. If the corrections independent of QED are far beyond the error limits of measurements for an anomalous magnetic moment, the corrections... 

This relationship is clearly related to equation (15.18) in particular. The term in log is of note. In the original reference [3] very limited justification for its inclusion was made. Furthermore, the values of quoted in Reference [3] were obtained from then unpublished work, and effectively have the nature of an empirical correction, which in itself is not a major problem in the utility of the model. However, when the work was published the values of the parameter were quite different [12], yet this error in the Wilke-Chang equation has not been corrected in the literature, to the authors knowledge. 

For heavy elements, all of the above non-relativistic methods become increasingly in error with increasing nuclear charge. Dirac 47) developed a relativistic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-velocity effects, an effect named after Darwin, and the very important interaction that arises between the magnetic moments of spin and orbital motion of the electron (called spin-orbit interaction). A completely correct form of the relativistic Hamiltonian for a many-electron atom has not yet been found. However, excellent results can be obtained by simply adding an electrostatic interaction potential of the form used in the non-relativistic method. This relativistic Hamiltonian has the form... 

Inspection of Table IV shows that the methodology is correctly predicting the order of stability of a series of tautomers but that the magnitudes of the values obtained by theory and experiment do not yet match closely. The reason for this is probably a complex mixture of inadequacy in theory and experimental difficulties indicated by the range of error associated with the experimental values. 

If we restrict ourselves to H-H and C-H interactions, for which the pseudodipolar terms can be neglected, then the nuyor systematic errors are due to molecular vibrations. As discussed above, they are estimated to amount to about 3 percent of the directly bonded C-H distance relative to more remote nuclear distances. In general, accurate corrections have not yet been undertaken. 

A variation calculation yields an upper bound to the true Born-Opper-heimer potential curve. Error limit calculations have not yet proved useful. Since the experimental separated and united atom energies are known, the error at the endpoints is known. An adjustment procedure which corrects at both the separated and united atom ends has been proposed. This procedure has not been fully evaluated. 

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