Infinite-Order Regular Approximation

The quasi-relativistic model obtained by using the ZORA ansatz in combination with a fully variational derivation is the infinite-order regular approximation (lORA) previously derived by Sadlej and Snijders [46] and by Dyall and Lenthe [47]. The lORA method has recently been implemented by Klop-per et al. [48]. The ZORA model can be obtained from the lORA equation by omitting the relativistic correction term to the metric. However, the indirect renormalization contribution is as significant as the relativistic interaction operator in the Hamiltonian. This is the reason why ZORA overestimates the... 

The equations derived in Sections 2, 3, and 4 have been implemented as described in the previous Section and the methods have been applied on one-electron atoms. The energies for the lowest l.v states obtained using the infinite-order regular approximation (lORA) Hamiltonian [52] are given in Table I. A few Dirac energies are for comparison listed in Table 2. 

The operator in the second bracket is the ZORA Hamiltonian, and it is sandwiched by normalization operators. If we expand these operators as we did above, we get the FORA Hamiltonian as the first term. The higher terms differ, however, because the final energy in the previous series must be the Dirac energy, whereas here it is the energy for the approximate Hamiltonian. Inclusion of the normalization terms corresponds to a resummation of certain parts of the ZORA perturbation series to infinite order, and the name coined by Dyall and van Lenthe (1999) is lORA—infinite-order regular approximation. 

There are also ab initio DF [173] and the infinite-order regular approximation with modified metric method (lORAmm/HF) [174] theoretical studies of the electronic stmctures of MO4 (M = Os and Hs). These works, however, revealed some deficiency of the calculations that resulted in the prediction of a wrong trend in properties from Os to Hs, as compared to the more accurate calculations [170] and the experiment (see below, as well as discussions in [169]). 

The second term can be thought of as an effective kinetic energy operator that goes to the non-relativistic one when V 0. Proper renormalization gives the Infinite Order Regular Approximation (lORA) [17], often approximated by scaled ZORA [16], which improves on ZORA. 

K. G. DyaU, E. van Lenthe. Relativistic regular approximations revisited An infinite-order relativistic approximation. /. Chem. Phys., Ill (1999) 1366-1372. 

---