Dirac equation four-component calculations

Relativistic Methods 204 8.1 Connection Between the Dirac and Schrodinger Equations 207 8.2 Many-particle Systems 210 8.3 Four-component Calculations 213 11.4.1 Ab Initio Methods 272 11.4.2 DFT Methods 273 11.5 Bond Dissociation Curve 274 11.5.1 Basis Set Effect at the HF Level 274 11.5.2 Performance of Different Types of Wave Function 276... 

In the preceding chapters, the theory for calculations based on the Dirac equation has been laid out in some detail. The discussion of the methods included a comparison with equivalent nonrelativistic methods, from which it is apparent that four-component calculations will be considerably more expensive than the corresponding nonrelativistic calculations—perhaps two orders of magnitude more expensive. For this reason, there have been many methods developed that make approximations to the Dirac equation, and it is to these that we turn in this part of the book. ... 

Four-component relativistic molecular calculations are based directly on the Dirac equation. They include both scalar relativistic effects and spin-orbit... 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

The relativistic coupled cluster method starts from the four-component solutions of the Drrac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. The Fock-space coupled-cluster method yields atomic transition energies in good agreement (usually better than 0.1 eV) with known experimental values. This is demonstrated here by the electron affinities of group-13 atoms. Properties of superheavy atoms which are not known experimentally can be predicted. Here we show that the rare gas eka-radon (element 118) will have a positive electron affinity. One-, two-, and four-components methods are described and applied to several states of CdH and its ions. Methods for calculating properties other than energy are discussed, and the electric field gradients of Cl, Br, and I, required to extract nuclear quadrupoles from experimental data, are calculated. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

Relativistic effects remarkably influence the electronic structure and the chemical bonding of heavy atoms [15]. In order to calculate the relativistic effects a four-component relativistic formulation by solving the Dirac equation is essential [16]. 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

During the last few decades, general computer programs, based on the Dirac equation, for extensive four-component electronic structure calculations for atoms have been developed by several groups. These programs are being improved and extended continually, incorporating the explicit treatment of the Breit interaction as well as an ever more sophisticated consideration of interelectron correlation and even of QED effects. 

Methods for solid-state calculations have been devised on the basis of the Dirac equation (bei der Kellen and Freeman 1996 Shick et al. 1999 Wang et al. 1992). Very recent progress has been achieved in the framework of four-component density functional theory for solids (Theileis and Bross 2000) (compare also the review on the... 

The Dirac equation with four spinor components demands large computational efforts to solve. Relativistic effects in electronic structure calculations are therefore usually considered by means of approximate one- or two-component equations. The approximate relativistic (also called quasi-relativistic) Hamiltonians consist of the nonrelativistic Hamiltonian augmented with additional... 

Relativistic molecular density functional calculations, as performed today, essentially use results from non-relativistic density functional theory combined with the Dirac (four-component) form of the one-particle equations. In other words, the motion of the electron is treated relativistically but the interaction between particles is still described by the non-relativistic expressions. 

Despite this ubiquitous presence of relativity, the vast majority of quantum chemical calculations involving heavy elements account for these effects only indirectly via effective core potentials (ECP) [8]. Replacing the cores of heavy atoms by a suitable potential, optionally augmented by a core polarization potential [8], allows straight-forward application of standard nonrelativistic quantum chemical methods to heavy element compounds. Restriction of a calculation to electrons of valence and sub-valence shells leads to an efficient procedure which also permits the application of more demanding electron correlation methods. On the other hand, rigorous relativistic methods based on the four-component Dirac equation require a substantial computational effort, limiting their application in conjunction with a reliable treatment of electron correlation to small molecules [9]. 

For most chemical applications, one is not interested in negative energy solutions of a four-component Dirac-type Hamiltonian. In addition, the computational expense of treating four-component complex-valued wave functions often limited such calculations to benchmark studies of atoms and small molecules. Therefore, much effort was put into developing and implementing approximate quantum chemistry methods which explicitly treat only the electron degrees of freedom, namely two- and one-component relativistic formulations [2]. This analysis also holds for a relativistic DFT approach and the solutions of the corresponding DKS equation. 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

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