Dirac-Hartree-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. 

The Dirac-Hartree-Fock iterative process can be interpreted as a method of seeking cancellations of certain one- and two-body diagrams.33,124 The self-consistent field procedure can be regarded as a sequence of rotations of the trial orbital basis into the final Dirac-Hartree-Fock orbital set, each set in this sequence forming a basis for the Furry bound-state interaction picture of quantum electrodynamics. The self-consistent field potential involves contributions from the negative energy states of the unscreened spectrum so that the Dirac-Hartree-Fock method defines a stationary point in the space of possible configurations, rather that a variational minimum, as is the case in non-relativistic theory. 

A. Weigand, X. Cao, V. Vallet, J.-P. Flament, and M. Dolg, Multi-configuration Dirac-Hartree-Fock adjusted energy-consistent pseudopotential for uranium spin-orbit configuration interaction and Fock-space coupled-cluster study of U and J. Phys. Chem. A, 113, 11509-11516(2009). 

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).

Due to the energy-dependence of the Hamiltonian the Wood-Boring approach leads to nonorthogonal orbitals and has been mainly used in atomic finite difference calculations as an alternative to the more involved Dirac-Hartree-Fock calculations. The relation... 

Orbital energies e (a.u.) and radial expectation values (r) (a,u.) for the valence shells of Ce and Lu from multi-conflguration Dirac-Hartree-Fock calculations for the average of the 4f 5d 6s and 4f 5d 6s configurations, respectively. The ratio of relativistic and corresponding nonrelativistic values is given in parentheses, Data taken... 

Fig. 6. Orbital energies of valence s orbitals for group 11 and 12 metals. DHF and HF refer to Dirac-Hartree Fock and Hartree-Fock results, respectively p denotes calculations for the pseudoatoms where the effect of the 4f shell (Au, Hg) and 5f shell (,nE, ujE) has been omitted. Data taken from Bagus et al. (1975) and Seth et...

The proportionality constant k is usually assigned a value of approximately 1.75. The overlap matrix elements Sy are calculated with respect to a set of two component basis functions with lsjm) quantization. The radial parts were chosen to be one or two Slater functions yielding (r ) (k=-l,0,1,2) expectation values as close as possible (Lohr and Jia 1986) to the Dirac-Hartree-Fock or Hartree-Fock results tabulated by Desclaux (1973) for the relativistic and nonrelativistic case, respectively. The diagonal Hamiltonian matrix elements Hu were set equal to the corresponding orbital energies from Desclaux s tables. Due to the use of a two-component lsjm) basis set the matrices H and S are generally complex and of dimension 2nx2n, when is the number of spatial orbitals. 

An efficient approach to improve on the Hartree-Fock Slater determinant is to employ Moller-Plesset perturbation theory, which works satisfactorily well for all molecules in which the Dirac-Hartree-Fock model provides a good approximation (i.e., in typical closed-shell single-determinantal cases). The four-component Moller-Plesset perturbation theory has been implemented by various groups [519,584,595]. A major bottleneck for these calculations is the fact that the molecular spinor optimization in the SCF procedure is carried out in the atomic-orbital basis set, while the perturbation expressions are given in terms of molecular spinors. Hence, all two-electron integrals required for the second-order Moller-Plesset energy expression must be calculated from the integrals over atomic-orbital basis functions like... 

Figure 3 One-particle energies of orbitals and spinors of the (n — 2)f (n - I)d ns configuration of Ce (n = 6) and Th (n = 7) from average-level nonrelativistic (nrel) Hartree-Fock and relativistic (rel) Dirac-Hartree-Fock calculations...

Experimental data for the electronic spectra of lanthanides and actinides are available and may serve to parametrize semiempirical approaches or to calibrate ab initio calculations. Total energies, orbital energies, radial orbital expectation values, and maxima from relativistic Dirac-Hartree-Fock as well as nonrelativistic Hartree-Fock calculations have been summarized by Desclaux and form a useful starting point for (qualitative) discussions of the electronic structure of lanthanide and actinide compounds. 

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