Large-Z approximation

In the large-Z approximation, we restrict the basis to those configurations that belong to a particular value of 1ZV, i.e. those configurations that correspond... 

Since only Coulomb potentials are involved, the matrix T v, v turns out to be energy independent. Its elements are pure numbers that depend only on N, the number of electrons, and are independent of the nuclear charge Z. The roots lK of the energy-independent interelectron repulsion matrix T v, v are also pure numbers (Table 1). In the large-Z approximation, the generalized Sturmian secular equation (41) reduces to the requirement ... 

Fig. 1 The ground state of the carbon-like isoelectronic series, calculated in the large-Z approximation. The energies divided by Z2 are shown as functions of Z. Experimental values are indicates by dots, while the energies calculated from (51) are shown as curves. The lower (solid) curve, which approaches the experimental values with increasing Z, has been corrected for relativistic effects. The upper (dashed) curve is uncorrected...

The sample size n is sufficiently large to meet the requirement for the Z approximation. 

The work of Kato11 guarantees convergence of equation (43) for sufficiently large Z. It is clear therefore that equation (43) must be approximately summed to all orders in 1/Z, in the limit of a large N value, by the TF energy of an atomic... 

As the samples are large according to the definition given earlier, the test of the two proportions using the Z approximation is appropriate. For a two-sided test of size 0.05 the critical region is defined by Z < -1.96 or Z > 1.96. The value of the test statistic is calculated as ... 

The two methods described earlier, the Z approximation and the y- test of homogeneity, are appropriate when the sample sizes are large enough. There are times, however, when the sample sizes in each group are not large enough or the proportion of events is low such that up < 5. In such cases another analysis method, one that does not require any approximation, is appropriate. 

In NONMEM, the default covariance matrix is a function of the Hessian (denoted as the R-matrix within NONMEM, which is not the same R-matrix within the MIXED procedure in SAS) and the cross-product gradient (denoted as the S-matrix within NONMEM) of the —2LL function. The standard errors are computed as the square root of the diagonals of this matrix. An approximate asymptotic (1 a) 100% confidence interval (Cl) can then be generated using a Z-distribution (large sample) approximation... 

Figure 6. The (Dirac) Hartree-Fock energy — hf and the first order Breit correction are given for helium-like ions. —grows approximately as and grows approximately as for large Z.

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