Relativistic Troullier-Martins scheme

For atoms Equation (4.27) reduces to two coupled radial equations for the large and small components of the orbitals, 

For valence states for which the associated screened PP has a depth of no more than 100 Hartree, can then be extracted from (4.52), (4.53), using a weakly relativistic 

Prototype LDA and GGA results obtained with these PPs are given in Tables 4.4 and 4.7. As Table 4.4 demonstrates, the PP calculations reproduce the AE results for both Cu2 (Mayer etal. 1996) and Au2 (Liu and van Wiillen 2000) very accurately, both on the LDA and on the GGA level. Similar agreement is found for the transition-metal compounds listed in Table 4.7, for which, however, only nonrelativistic AE reference values (Castro and Salahub 1994 Engel etal. 2001b) are available. 

Nevertheless, the relativistic corrections are not negligible even for these 3d elements. In fact, in the case of FeO relativity reduces the excitation energy from the 5 A ground state to the first excited state (5 E) from the nonrelativistic value of 0.4 eV to 0.2 eV. On the other hand, the comparison of the LDA results with experiment clearly shows the need for nonlocal corrections. The GGA results are consistently closer to the experimental data, in particular for Re. The GGA values for e are nonetheless not completely satisfying, which underlines the importance of the truly nonlocal contributions to Exc. 

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Among the various schemes for the construction of V] presently the Troullier-Martins (TM) form (Troullier and Martins 1991) seems to be most widely used. However, in contrast to the original approach (Bachelet et al. 1982 Hamann et al. 1979) the TM scheme has been formulated for nonrelativistic situations. A consistent relativistic extension (Engel et al. 2001b) is presented in the next section. 

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