Dirac-Fock orbital energies

Pseudopotentials (PP) were originally proposed to reduce the computational cost for the heavy atoms with the replacement of the core orbitals by an effective potential. Modern pseudopotentials implicitly include relativistic effects by means of adjustment to quasi-relativistic Har-tree-Fock or Dirac-Hartree-Fock orbital energies and densities [35]. In the present research, we adopted Peterson s correlation-consistent cc-pVnZ-PP (n — D, T, Q, 5) basis sets [23] with the corresponding relativistic pseudopotential for the Br atom. The corresponding cc-pVnZ (n = D, T, Q, 5) basis sets were used for the O and H atoms. The optimized geometries and relative energies for the stationary points are reported in Table 1 and Fig. 3, and the harmonic vibrational frequencies and zero-point vibrational energies are reported in Table 4. 

Fig. 5 Relativistic stabilization of the ns and npi/2 orbitals and the spin-orbit splitting of the np orbitals for the noble gases Xe, Rn and element 118. The Dirac-Fock atomic energies are from [21] and the Hartree-Fock (nonrelativistic) values are from [8]...

K. Rashid, M. Z. Saadi, and M. Yasin, Dirac-Fock total energies, ionization energies, and orbital energies for uranium ions (U I to U XCII), At. Data Nucl. Data Tables 40, 365 (1988). 

The existence of a common momentum profile for the manifold a confirms the weak-coupling binary-encounter approximation. Within these approximations we must make further approximations to calculate differential cross sections. For the probe amplitude of (11.1) we may make, for example, the distorted-wave impulse approximation (11.3). This enables us to identify a normalised experimental orbital for the manifold. If normalised experimental orbitals are used to calculate the differential cross sections for two different manifolds within experimental error this confirms the whole approximation to this stage. An orbital approximation for the target structure (such as Hartree—Fock or Dirac—Fock) is confirmed if the experimental orbital energy agrees with the calculated orbital energy and if it correctly predicts differential cross sections. 

The 5s manifold shows great complexity. For the lowest state S23.4(5s) = 0.37. This value is considerably lower than many structure calculations predict, but the perturbation calculation of Kheifets and Amusia (1992) obtains 0.384. The orbital energy ess (11.18) is 27.6+0.3eV, which is to be compared to the Dirac—Fock value 27.49 eV. The Hartree—Fock value is 25.70 eV. The criterion for the strength of the perturbation, given by the ratio of the standard deviation to the mean of the 5s manifold is 0.18. The ratios S29.i(5s) S23.4(5s) and S23.4(5s) Z/S/(5s) are compared at different momenta in fig. 11.10. The condition for the validity of the weak-coupling binary-encounter approximation is completely satisfied within experimental error. 

In the work of Larsson and Pyykko [51] the energy dependence of the atomic orbitals is defined by a polynomial fit of the charge dependence of the energy of any one AO on the population of the other AOs, as derived from Dirac-Fock calculations [49], Orbital populations are obtained by the Mulliken approximation. The orbital energies are corrected for the Coulomb interaction of the total charge on other atoms. The wavefunctions are of double-zeta variety, the radial parameters being optimised separately for the different spinors. 

Figure 2. Nonrdativistic Hartree-Fock (HF) and relativistic Dirac-Hartree-Fock (DHF) orbital energies e and orbital radius expectation values < r > for the valence shells of the group 4 elements (n = 2,3,4,5,6 for C, Si, Sn, Pb and Eka-Pb, respectively).

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