Nonrelativistic atomic calculations

Figure 13 Nonrelativistic 6s and 6p radial wave functions (solid) versus relativistic 6s 1/2 (dotted), 6p /2 (dashed-dotted), 6/23/2 (dashed) radial wave functions of the thallium atom calculated at the Hartree-Fock and Dirac-Fock levels, respectively.

Table 5.6. Comparison of LDA [36], Cl (estimated from nonrelativistic Cl-calculations for the three innermost electrons and the experimental ionisation potentials of all other electrons [109] ) and MBPT2 [108] correlation energies for neutral atoms nonrelativistic correlation energy,...

In the next step, the dipole matrix element, Eq. (3), for Cl Z =17) and Mn Z 25) atoms were calculated with the DV-integration method for all possible dipole transitions. The obtained results for the square of the matrix element are shown in Table 1. For comparison, the nonrelativistic atomic HFS calculations were carried out by the use of the computer code of Herman and Skillman (HS) [35] and the dipole matrix elements corresponding to Eq. (3) were evaluated by the conventional numerical integration method. The calculated values are also listed in Table 1 and compared with the DV values. 

The more accurate Hartree-Fock method approximates the wave function as an antisymmetrized product (Slater determinant or determinants) of one-electron spin-orbitals and finds the best possible forms for the spatial orbitals in the spin-orbitals. Hartree-Fock calculations are usually done by expanding each orbital as a linear combination of basis functions and iteratively solving the Hartree-Fock equations (11.12). The Slater-type orbitals (11.14) are often used as the basis functions in atomic calculations. The difference between the exact nonrelativistic energy and the Hartree-Fock energy is the correlation energy of the atom (or molecule). 

Figure 2. Upper panel Log-log plot of total nonrelativistic atomic energies 1

In this section, we report calculations for the He and Be atoms using the exact ground-state Kohn-Sham potentials, the three approximations to the kernel mentioned in the previous section, and including many bound-state poles in Eq. (2), but neglecting the continuum. The technical details are given in Ref. [23]. Table 1 lists the results, which are compared with a highly accurate nonrelativistic variational calculations. In each symmetry class (s, p, and d), up to 38 virtual states were calculated. The errors reported... 

The alternative to patching up a nonrelativistic basis set for four-component calculations is to derive sets of exponents specifically for use in relativistic calculations. This is usually done by optimizing the exponents directly in relativistic atomic calculations, for example by minimizing the energy. An alternative is to use an even-tempered basis, where the exponents form a geometric series such that... 

By contrast, the nonrelativistic equilibrium atomization energy Dg calculated theoretically represents the difference between the nonrelativistic atomic ground-state term energy and... 

The Schrodinger equation is a nonrelativistic description of atoms and molecules. Strictly speaking, relativistic effects must be included in order to obtain completely accurate results for any ab initio calculation. In practice, relativistic effects are negligible for many systems, particularly those with light elements. It is necessary to include relativistic effects to correctly describe the behavior of very heavy elements. With increases in computer capability and algorithm efficiency, it will become easier to perform heavy atom calculations and thus an understanding of relativistic corrections is necessary. 

Relativistic quantum mechanics yields the same type of expressions for the isomer shift as the classical approach described earlier. Relativistic effects have to be considered for the calculation of the electron density. The corresponding contributions to i/ (0)p may amount to about 30% for iron, but much more for heavier atoms. In Appendix D, a few examples of correction factors for nonrelativistically calculated charge densities are collected. Even the nonrelativistically calculated p(0) values accurately follow the chemical variations and provide a reliable tool for the prediction of Mossbauer properties [16]. 

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

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