Non-relativistic calculations

The most difficult part of relativistic calculations is that a large amount of CPU time is necessary. This makes the problem more difficult because even non-relativistic calculations on elements with many electrons are CPU-intensive. The following lists relativistic calculations in order of increasing reliability and thus increasing CPU time requirements ... 

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

Table 1 Xenon, comparison of orbital energies for numerical Dirac and ZORA and non relativistic calculations with basis set ZORA calculations in different Coulomb matrix approximations in the UGBS basis set...

Non-relativistic calculation of spin-spin couplings are based on the four interactions between magnetic nuclei and electrons described by Ramsey, eqs. (la-b)... 

We note from Figure 1 that scalar relativistic calculations (7) are entirely unable to reproduce the experimental trend for X = Br and I. Indeed, scalar relativistic and non-relativistic calculations gave almost identical results in this case (7). However, with the inclusion of spin-orbit/Fermi contact operators, the experimental trend is reproduced nicely (9). 

The measured hyperfine splittings of 2 3Pi level were in reasonable agreement with the relativistic calculations of [96], and also with non-relativistic calculations corrected for relativistic and QED effects [114,115], The results for the hyperfine corrected 21S o — 23Pl interval in 14N5+ are compared with theory in table 3. QED corrections make up 3.5% of the measured interval. The experiment is hence sensitive to these corrections at the level of 20 ppm, the highest precision for a Lamb shift in any multiply-charged ion. 

For d = 3, the solution of Eq. (7) is the familiar Coulomb wave function in 3 dimensions. For d 3 this is not true. This represents an additional problem since the non-relativistic calculations have to be done without an explicit knowledge of the wave function. Fortunately, cancelation of all divergences can be ensured on the operator level using the Schrodinger equation in d-dimensions. Once divergences are canceled, the limit d —> 3 can be taken and a non-trivial (but now finite) matrix elements can be easily computed. 

A important relativistic effect is that 5f orbitals of actinides are larger and their electrons more weakly bound than predicted by non-relativistic calculations, hence the 5f electrons are more chemically available . This leads to ... 

Malli and Pyper (14), reproduced with permission. Basis set same as the relativistic basis set, except that non-relativistic calculations were performed using the appropriate increased value of c (velocity of light). 

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