Quasi-Relativistic Hamiltonians

With equation (20) as starting point, the first approximation one can make in order to derive quasi-relativistic two-component equations is to assume that the upper (0 ) and the lower (0y) components are identical. Note that the ansatz 

Expression (21) is again a convenient starting point for further approximations. Scalar and spin-orbit contributions can be separated and by omitting the spin-orbit contributions, one-component quasi-relativistic models are obtained. 

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Our work has used the Coulomb (i.e., nonrelativistic) or the Breit-Pauli (i.e., quasi-relativistic) Hamiltonian. Relativistic structures and wavefunc-tions, without or with relativistic radials, follow fhe same concepfs as in the nonrelativistic treatment, only that here, the differences befween the level energies are generally smaller and, hence, the sensitivity of fhe mixing coefficients to the choice of orbitals and of the mixing configurations is expected to be much higher. 

Perturbation Theory Based on Quasi-Relativistic Hamiltonians... 

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... 

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

In this Chapter, we will show how a whole family of one- and two-component quasi-relativistic Hamiltonians can conveniently be derived. The operator difference between the quasi-relativistic Hamiltonians and the Dirac equation can be explicitly identified and used in perturbation expansions. Expressions are derived for a direct perturbation theory scheme based on quasi-relativistic two-component Hamiltonians. The remaining difference between the variational energy obtained using quasi-relativistic Hamiltonians and the energy of the Dirac equation is estimated numerically by applying the direct perturbation theory ap-... 

Transformed Dirac equations are convenient starting points for the derivation of quasi-relativistic Hamiltonians. The transformed Dirac equations can be obtained by using approximate solutions for the small components as ansatze for the wave function. The ansatz can be deduced from the lower half of the Dirac equation by an approximate elimination of the small component. 

The quasi-relativistic Hamiltonians obtained using the general ansatz have usually a metric that include spin-orbit contributions. This can be a undesirable situation since in a perturbation study of spin-orbit effects, the addition of the spin-orbit coupling requires reorthogonalization of the orbitals [69]. However, as seen in equation (18), the relativistic correction term (V) consists of two contributions f and B. B is several orders of magnitude less significant than f. Furthermore, the B operator can also be separated into scalar relativistic and spin-orbit contributions. 

In the last step of the derivation of the quasi-relativistic Hamiltonian (21), it was assumed that the upper (0 ) and the lower components are identical. This was the only approximation made in that derivation. Instead of making this assumption, the difference between the upper and the lower components can be denoted by A. ... 

As seen in equation (26), the quasi-relativistic Hamiltonian and the operators describing the difference between the exact Dirac Hamiltonian and the quasi-relativistic one are now explicitly separated and the direct perturbation theory method can be applied. In the direct perturbation theory approach, the metric is also affected by the perturbation [12]. Note that the interaction matrix is block diagonal at the lORA level of theory, whereas the coupling between the upper and the lower components still appears in the metric. 

The quadrupole coupling term is added to the interaction potential V r) and when one proceeds as outlined in Sections 2 and 3, one obtains a quasi-relativistic Hamiltonian (44) which considers the quadrupole coupling interaction between the electrons and the nuclei. 

As shown in Section 4, quasi-relativistic Hamiltonians such as the lORA, ERA, MIORA and MERA ones can be used as a zeroth-order approximation to the Dirac Hamiltonian. The operator difference between the quasi-relativistic and fully relativistic equations can be used as a perturbation operator and the corresponding energy difference can be considered by using a direct perturbation theory approach. 

The first-order relativistic perturbation energy corrections (in Hartrees) obtained using the lORA quasi-relativistic Hamiltonian as the zeroth-order approximation to the Dirac equation. 

A couple of new quasi-relativistic Hamiltonians have been proposed and the methods have been implemented and tested on some one-electron atoms. The calculations show that the energies obtained with the present quasi-relativistic Hamiltonians are in fairly good agreement with the corresponding Dirac energies. The discrepancy between the quasi-relativistic and the Dirac energies scales with a Z, where Z is the the nuclear charge and a is the fine structure constant. 

General two-component methods have been discussed in various chapters of the first part of this book, for instance in chapter 11 on Two-Component Methods and the Generalised Douglas-Kroll Transformation by Wolf, Reiher and Hess [165], in chapter 12 by Kutzelnigg on Perturbation Theory of Relativistic Effects [166] and in chapter 13 by Sundholm on Perturbation Theory Based on Quasi-Relativistic Hamiltonians [167]. 

D. Sundholm, Perturbation theory based on quasi-relativistic Hamiltonians, in P. Schwerdtfeger (Ed.), Relativistic Electronic Structure Theory, Part 1, Fundamentals, Elsevier, Netherlands, 2002, pp. 763-798. 

All quasi-relativistic Hamiltonians in the following may be thought of as being surrounded by proper projection operators onto electronic bound or... 

Gagliardi and Roos conducted a series of studies on actinide compounds. They follow a combined approach with DKH/AMFI Hamiltonians combined with CASSCF/CASPT2 for the energy calculation and an a posteriori added spin-orbit perturbation expanded in the space of nonrelativistic CSFs. This strategy aims to establish a balance of sufficiently accurate wave function and Hamiltonian approximations. Since the CASSCF wave function provides chemically reasonable but not highly accurate results (as witnessed, for instance, in the preceding section), it is combined with a quasi-relativistic Hamiltonian, namely the sc alar-relativistic DKH one-electron Hamiltonian. Additional effects — dynamic correlation and spin-orbit coupling — are then considered via perturbation theory. 

Current relativistic electronic structure theory is now in a mature and well-developed state. We are in possession of sufficiently detailed knowledge on relativistic approximations and relativistic Hamiltonian operators which will be demonstrated in the course of this book. Once a relativistic Hamiltonian has been chosen, the electronic wave function can be constructed using methods well known from nonrelativistic quantum chemistry, and the calculation of molecular properties can be performed in close analogy to the standard nonrelativistic framework. In addition, the derivation and efficient implementation of quantum chemical methods based on (quasi-)relativistic Hamiltonians have facilitated a very large amount of computational studies in heavy element chemistry over the last two decades. Relativistic effects are now well understood, and many problems in contemporary relativistic quantum chemistry are technical rather than fundamental in nature. 

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