The Dirac-Coulomb-Breit Hamiltonian

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. 

In order to treat the motion of electrons in a molecule we need a many-electron Hamiltonian that describes both the interaction with the external field (including that of the nuclei which are assumed stationary in the reference frame) and the interactions between the electrons. The first part is done fully relativistic using the Dirac Hamiltonian 

The interaction between the electrons is described by the Coulomb-Breit [5] operator. This operator is usually considered in the zero-frequency limit where it becomes 

To avoid the difficult integration over the third term this operator is often further approximated to the Gaunt [6,7] 

The Dirac-Coulomb-Breit (DCB) Hamiltonian for an N-electron system is 

---

The Dirac-Coulomb-Breit Hamiltonian rewritten in second-quantized... 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

By inserting the equations defining the kinetic energy operators and the pairwise interaction operators into Eq. (8) we obtain the Dirac-Coulomb-Breit Hamiltonian, which is in chemistry usually considered the fully relativistic reference Hamiltonian. 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

Dependent Terms of the Dirac-Coulomb-Breit Hamiltonian. 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

Relativistic PPs to be used in four-component Dirac-Hartree-Fock and subsequent correlated calculations can also be successfully generated and used (Dolg 1996a) however, the advantage of obtaining accurate results at a low computational cost is certainly lost within this scheme. Nevertheless, such potentials might be quite useful for modelling a chemically inactive environment in otherwise fully relativistic allelectron calculations based on the Dirac-Coulomb-(Breit) Hamiltonian. 

Nieuwpoort, W. C., Aerts, P. J. C. and Visscher, L. (1994) Molecular electronic structure calculations based on the Dirac-Coulomb-(Breit) Hamiltonian. In Malli (1994), pp. 59-70. 

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... 

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

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 Dirac-Coulomb-Breit Hamiltonian is derived pertmbationally and thus it is frequently suggested that the Breit correction to the Coulomb interaction should be considered in the perturbation framework and evaluated as the first-order contribution to the energy which follows from the Dirac-Coulomb calculations, and this is the way how it is done, as example, in the atomic GRASP2K package. 

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... 

Relativistic coupled-cluster calculations based on the Dirac-Coulomb-Breit Hamiltonian (CCSD) including dynamical correlations (EHav et al. 1995) reverted to the 6d 7s configuration as the ground state of Rf Here, the 6d7s 7p state is 0.274 eV above the ground state. 

A review of the problems associated with the Dirac Hamiltonian as well as a classification of attempts to avoid them has been given by Kutzelnigg (1984). Recently, the same author developed a direct perturbation theory for relativistic effects (Kutzelnigg 1990, Kutzelnigg et al. 1995, Ottschofski and Kutzelnigg 1995). Earlier work in this direction was published by Rutkowski (1986a-c) and Jankowski and Rutkowski (1987). A modification of the Dirac-Coulomb-Breit Hamiltonian which allows the exact separation of spin-free and spin-dependent terms has been proposed by Dyall (1994b). 

Therefore, we may here start from the Breit interaction to derive the pseudo-relativistic Hamiltonians instead of following the somewhat meandering historical path from 1926 to about 1932, which was mainly based on classical considerations. As usual, a quantum-electrodynamical derivation is also possible and has been presented by Itoh [678], but the sound basis of our semi-classical theory, which we pursue throughout this book, is necessarily the Breit equation. Needless to say, the rigorous transformation approach to the Dirac-Coulomb-Breit Hamiltonian yields results identical to those from the QED-based derivation. 

K. G. Dyall. An exact separation of the spin-free and spin-dependent terms of the Dirac-Coulomb-Breit Hamiltonian. J. Chem. Pfcys., 100(3) (1994) 2118-2127. 

If one neglects the transverse contribution, one arrives at what is termed the Dirac-Coulomb approximation (a standard in quantum chemistry). Inclusion of the transverse term, which describes retardation and magnetic effects, in perturbation theory (weakly relativistic hmit) leads to the Dirac-Coulomb-Breit Hamiltonian. 

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. 

It should be pointed out at the outset that the Dirac-Coulomb(-Breit) Hamiltonian already contains approximations to the full QED Hamiltonian (if one exists ) therefore, we are considering further approximations, with this Hamiltonian as the most rigorous reference point. 

---