Dirac-Coulomb-Breit Hamiltonian/method

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

We will start by reviewing some basic relativistic theory to introduce the notation and concepts used. The rest of the chapter is devoted to the three major post-DHF methods that are currently available for the Dirac-Coulomb-Breit Hamiltonian. All formulas will be given in atomic units. 

The DCB CCSD method is based on the Dirac-Coulomb-Breit Hamiltonian... 

An improved basis set with 36s32p24d22fl0g7h6i uncontracted Gaussian-type orbitals was used and all 119 electrons were correlated, leading to a better estimate of the electron affinity within the Dirac-Coulomb-Breit Hamiltonian, 0.064(2) eV [102]. Since the method for calculating the QED corrections [101] is based on the one-electron orbital picture, the 8s orbital of El 18 was extracted from the correlated wave function by... 

Abstract Variational methods can determine a wide range of atomic properties for bound states of simple as well as complex atomic systems. Even for relatively light atoms, relativistic effects may be important. In this chapter we review systematic, large-scale variational procedures that include relativistic effects through either the Breit-Pauli Hamiltonian or the Dirac-Coulomb-Breit Hamiltonian but where correlation is the main source of uncertainty. Correlation is included in a series of calculations of increasing size for which results can be monitored and accuracy estimated. Examples are presented and further developments mentioned. 

The most advanced relativistic approach in relativistic calculations of X-ray spectra, is most likely that based on the Dirac-Coulomb-Breit Hamiltonian and quantum electrodynamic contributions accounted for. In addition, one should also carry out the corresponding correlated-level calculation within these relativistic formalism. To illustrate the role and size of relativistic and QED corrections the core and valence ionisation potentials and excitation energies of noble gases are shown. The relativistic fOTC CASSCE/CASPT2 method together with the restricted active space... 

A reliable prediction of spectroscopic phenomena in heavy-element compounds requires a balanced description of scalar-relativistic, spin-orbit and electron-correlation effects. In some cases one or more of these effects can be dominant, requiring an elaborate method to take this into account, whereas the others may be treated in a more approximate way or can even be completely neglected. The choice of the Hamiltonian is a crucial issue in relativistic calculations of spectroscopic quantities. Four-component methods employing the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian offer the most... 

Spin-Dependent Terms of the Dirac-Coulomb-Breit Hamiltonian, (b) K. G. Dyall,/. Chem. Phys., 109,4201 (1998). Interfacing Relativistic and Nonrelativistic Methods, n. Investigation of a Low-Order Approximation, (c) K. G. Dyall and T. Enevoldsen, J. Chem. Phys., Ill, 10,000 (1999). Interfacing Relativistic and Nonrelativistic Methods. HI. Atomic 4-Spinor Expansions and Integral Approximations, (d) K. G. Dyall,/. Chem. Phys., 115,9136 (2001). Interfacing Relativistic and Nonrelativistic Methods. IV. One- and Two-Electron Scalar Approximations. 

The no-virtual-pair Dirac-Coulomb-Breit Hamiltonian, correct to second order in the fine-structure constant a, provides the framework for four-component methods, the most accurate approximations in electronic structure calculations for heavy atomie and molecular systems, ineluding aetinides. Electron correlation is taken into aeeount by the powerful coupled eluster approaeh. The density of states in actinide systems necessitates simultaneous treatment of large manifolds, best achieved by Fock-space coupled eluster to avoid intruder states, which destroy the convergence of the CC iterations, while still treating a large number of states simultaneously, intermediate Hamiltonian sehemes are employed. 

The most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

Thus we have the situation that the current four-component methods are based on the so-called Dirac-Coulomb-Breit (DCB) operator (or the Dirac-Coulomb operator, if the Breit interaction is omitted). The DCB Hamiltonian is not covariant with respect to formulation in another inertial frame of reference, although the epithet fully relativistic is often used. 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

There is increasing interest in the relativistic treatment of atoms/ molecules/ and solids. A relativistic Hartree-Fock scheme [Hartree-Fock-Dirac (HFD) method] based on the variation in the total energy obtained with a single Slater determinant (in which the one-electron orbitals are four-component Dirac spinors), using a Dirac-type Hamiltonian for each electron and including Coulomb interaction, was developed some time ago.< For the remaining interaction terms the first-order perturbation of the Breit interaction operato reduced to large components (Pauli approximation) is usually taken into account (see, however, the work of Mann and Johnson ). 

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