Results for Total Energies and Radial Functions

Numerical discretization methods pose an interesting consequence for fully numerical Dirac-Hartree-Fock calculations. These grid-based methods are designed to directly calculate only those radial functions on a given set of mesh points that occupy the Slater determinant. It is, however, not possible to directly obtain any excess radial functions that are needed to generate new CSFs as excitations from the Dirac-Hartree-Fock Slater determinant. Hence, one cannot directly start to improve the Dirac-Hartree-Fock results by methods which capture electron correlation effects based on excitations that start from a single Slater determinant as reference function. This is very different from basis-set expansion techniques to be discussed for molecules in the next chapter. The introduction of a one-particle basis set provides so-called virtual spinors automatically in a Dirac-Hartree-Fock-Roothaan calculation, which are not produced by the direct and fully numerical grid-based approaches. 

In short Cl expansions, one may set up explicitly those CSFs which allow us to assign a correlating radial function to a given radial function of the Dirac-Hartree-Fock Slater determinant. This correlating function has got some well-defined properties for instance, the virtual radial function and the one to be correlated should live in the same spatial region. However, this can create additional technical difficulties for the guess of those radial functions which have been introduced to account for the correlation of a particular shell in the Cl expansions with more than one CSF. The above-mentioned model potentials are not the best choice, and additional adjustments to them are necessary so that it can be certain that the shells to be correlated live in the same radial space. 

In this last section, we shall present some examples of results of numerical atomic structure calculations to demonstrate properties of radial functions and the magnitude of specific effects like the choice of the finite nucleus model or the inclusion of the Breit operator. It should be emphasized that the reliability of numerical calculations is solely governed by the affordable length of the Cl expansion of the many-electron wave function since the numerical solution techniques allow us to determine spinors with almost arbitrary accuracy. Expansions with many tens of thousands of CSFs can be routinely handled (with the basis-set techniques of chapter 10, expansions of billions of CSFs are feasible via subspace iteration techniques [372]). 

While the Cl coefficients given in Table 9.3 are equal, the results for the fine-structure splitting (Table 9.4) differ since the total electronic energies in Ref. [184] have been corrected for the Breit interaction within perturbation theory. Moreover, an MCSCF-EAL (extended average level) scheme was employed for the orbital optimization by Das and Grant (cf. Ref. [507]). The Breit 

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Mixed QM/MM Car-Parrinello simulation techniques are among the most powerful computational methods for exact numerical calculations of macroscopic and microscopic properties at finite temperatures of a large variety of condensed matter systems of current theoretical and experimental interest. Physical quantities that may be computed exactly and compared directly to experimental results, where available, include the kinetic, potential, and total energy, the radial distribution function, neutron scattering cross-sections, and so on. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

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