Dirac-Hartree-Fock energy

The best-known and widely-quoted tabulation of atomic Dirac-Hartree-Fock energies was published by Desclaux [11], covered elements in the range Z=1 to Z=120 using finite difference methods. A number of computer packages are available to perform MCDHF calculations [19]. Published DHF and Dirac-Fock-Slater (DFS) calculations for atoms are now too numerous to construct a comprehensive catalogue. It is, however, possible to sort the purposes for which these calculations have been performed into general classes. 

Figure 6. The (Dirac) Hartree-Fock energy — hf and the first order Breit correction are given for helium-like ions. —grows approximately as and grows approximately as for large Z.

Let s look at specific results. For neutral lithium, the lowest-order Dirac-Hartree-Fock energies = e for the 2s and 2p states are given in the first row of Table 4. The Hartree-Fock energies are within 1-2% of the measured removal energies. Including second-order corrections accounts for the major part of the residual difference, while third-order corrections improve the differences with measurement to less than 0.1%, as shown in the table. 

The starting point for our discussion is the Dirac-Hartree-Fock energy expression in Eq. (8.121) which we may write as... 

Introducing, however, a basis set expansion as of Eqs. (10.2) and (10.3) in the relativistic case for the molecular spinors ipi, this leads to a complicated expression for the Dirac-Hartree-Fock energy of Eq. (10.21),... 

The Dirac-Hartree-Fock energy depends on the molecular spinor coefficients Cipi and on a given parameter A that may be, for instance, a nuclear coordinate Ra,k- A first derivative of this energy may be formulated in the most general form by using the chain rule. 

Since we have two different sets of molecular spinor coefficients connected to large- and small-component basis functions, the differentiation of the Dirac-Hartree-Fock energy is more tedious than the differentiation of the nonrelativistic Hartree-Fock energy. For this reason, we proceed with taking the derivative of the Hartree-Fock energy for the sake of simplicity in order to highlight the principles. The Hartree-Fock energy can be written as... 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

Figure 4.13 Excitation energies for the s-d and s-p gaps of the Group 11 elements. Experimental (Cu, Ag and Au) and coupled cluster data (Rg) are from Refs. [4, 91]. For the s-p gap of Rg we used Dirac-Hartree-Fock calculations including Breit and QED corrections.

Figgen, D., Rauhut, G., Dolg, M. and StoD, H. (2005) Energy-consistent pseudopotentials for group 11 and 12 atoms adjustment to multi-configuration Dirac-Hartree-Fock data. Chemical Physics, 311, 227-244. 

Table 6 Matrix Dirac-Hartree-Fock (Edhf) and Hartree-Fock (Ehf) energies calculated using BERTHA. The Gaussian exponential parameters are those of the non-relativistic sets derived by van Duijenveldt and tabulated in Poirier et al [36]. Thejirst-order molecular Breit energy, Eb, v as calculated using methods described in [12] relativistic corrections to Ehf collected in the column labelled E energies are in atomic units.

In Table 6, we present a series of calculations ofthe molecular structures of N2, CO, BF, andNO+ using sets of published Gaussian basis set parameters [36]. The relativistic Dirac-Hartree-Fock electronic energies, and the non-relativistic Hartree-Fock... 

Table 7 Estimates of total relativistic correction, E, and the first-order Breit energy correction,, obtained by combining the atomic or ionic contributions indicated by the second column. They may be compared with the values of the total relativistic correction, Ek. and thefirst-order Breit interaction, Eb, obtained directly from matrix Dirac-Hartree-Fock and Hartree-Fock calculations of the molecular structure using BERTHA [12], Only the results of the Iis7p2d atom-centred basis sets for Ek and Eb are quoted. All energies in atomic units.

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

As long as one is interested only in the total energy of the atomic electron system, the change from the simple but unrealistic PNC to a roughly realistic FNC is much more important than finer details due to variation of the finite nucleus model. This can be seen also from a recently published comparative study on numerical Dirac-Hartree-Fock calculations for... 

Dirac s relativistic theory for the motion of electrons in molecules was introduced in the preceding chapters. The appearance of positron solutions and the four-component form of the wave function looks problematic at first sight but in practice it turns out that the real challenge is, like in non-relativistic electronic structure theory, the description of the correlation between the motion of electrons. The mean-field approximation that is made in the Dirac-Hartree-Fock (DHF) approach provides a good first step, but gives bond energies and structures that are often too inaccurate for chemical purposes. 

Figure 1. Total nonrelativistic multi-configuration Hartree-Fock energy, relativistic corrections (estimated as the difference between the multi-configuration Dirac-Hartree-Fock and Hartree-Fock energies) and correlation contributions (estimated from correlation energy density functional calculations) for the group 4 elements. The multi-configuration treatments were carried out with the atomic structure code GRASP [78] and correspond to complete active space calculations with the open valence p shell as active space. The nonrelativistic results were obtained by multiplying the velocity of light with a factor of 10 .

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).

Figure 3. Relativistic Dirac-Hartree-Fock (DHF) and experimental (Exp.) term energies of LSI levels arising from the ns np configuration of the group 4 elements (J=0 solid lines, J=1 dotted lines, J=2 dashed lines). The experimental result for Eka-Pb actually corresponds to the result of a high level relativistic coupled-cluster calculation [79]. The corresponding results for the nonrelativistic P, and S states (dot-dashed lines) were obtained from Hartree-Fock (HF) calculations.

Figure 5. Relativistic effects on bond lengths and binding energies of group 4 tctrahydrides XH. The bond length contraction (in A) and bond destabilization (in eV) were obtained as the difference between relativistic Dirac-Hartree-Fock calculations based on the Dirac-Coulomb-Gaunt Hamiltonian and corresponding nonrelativistic Hartree-Fock calculations [28,29].

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... 

In the most recent version of the energy-consistent pseudopotential approach the reference data is derived from finite-dilference all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian. As an example the first parametrization of such a potential,... 

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