Two-component methods

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]. 

First attempts to calculate molecular parity violating potentials within a two-component framework have been undertaken by Kikuchi and coworkers [168,169]. They have added the Breit-Pauli spin-orbit coupling operator Hso to the usual non-relativistic Hamiltonian Hq 

The disadvantage of this particular two-component realisation is that the full Breit-Pauli spin-orbit coupling operator is not bound from below and therefore critical in a variational procedure. Hence, only relative modest basis sets have been used in the calculations of parity violating effects within this scheme [168,169]. 

Since the first quantitative calculations on parity violating energy differences, which have been reported by Hegstrom, Rein and Sanders [25,107] almost two decades ago, about a dozen groups worldwide have performed calculations on various aspects of molecular parity violation. 

The improved theoretical methods also invited to reinvestigate parity violating effects in biologically relevant systems [125,134,140,141,162,168, 169,178] and it could be demonstrated, that previous claims for a systematic stabilisation of amino acids in water, which based on lower level calculations, were not justified [140]. This finding gave again fresh impetus to the debate on possible relations between parity violating interaction and biochemical homochirality. 

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Manufacture of thiophene on the commercial scale involves reactions of the two component method type wherein a 4-carbon chain molecule reacts with a source of sulfur over a catalyst which also effects cyclization and aromatization. A range of suitable feedstocks has included butane, / -butanol, -butyraldehyde, crotonaldehyde, and furan the source of sulfur has included sulfur itself, hydrogen sulfide, and carbon disulfide (29—32). 

Two-component methods represent the most widely applied principles in sulfone syntheses, including C—S bond formation between carbon and RSOz species of nucleophilic, radical or electrophilic character as well as oxidations of thioethers or sulfoxides, and cheletropic reactions of sulfur dioxide. Three-component methods use sulfur dioxide as a binding link in order to connect two carbons by a radical or polar route, or use sulfur trioxide as an electrophilic condensation agent to combine two hydrocarbon moieties by a sulfonyl bridge with elimination of water. 

III. TWO-COMPONENT METHODS A. S-Substitutlon of Sulfinate Nucleophiles with C-Electrophiles... 

Various approaches can be pursued to compute spin-orbit effects. Four-component ab initio methods automatically include scalar and magnetic relativistic corrections, but they put high demands on computer resources. (For reviews on this subject, see, e.g., Refs. 18,19,81,82.) The following discussion focuses on two-component methods treating SOC either perturbationally or variationally. Most of these procedures start off with orbitals optimized for a spin-free Hamiltonian. Spin-orbit coupling is added then at a later stage. The latter approaches can be divided again into so-called one-step or two-step procedures as explained below. 

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. 

Two-component methods and the generalised Douglas-KroU transformation... 

However, the salient features of this approach can hardly be transfered to two-component methods, and we therefore focus in our presentation in the following only on the common standard representation of the Dirac algebra given above. 

In the following overview, the theory of two-component methods in relativistic quantum chemistry is shortly reviev. ed. The relationship between the large and small component is analysed in section 2, followed by a discussion of possible methods to decouple the Dirac Hamiltonian. The focus is equally distributed... 

In this account, we have presented both the basic theory of two-component methods in quantum chemistry and new developments achieved within the framework of the generalised DK transformation, which have not yet been described elsewhere. Two-component methods have several advantages when compared to the traditional four-component formulation, which is the well-established covariant description of relativistic quantum mechanics. On the one hand, the computational requirements are significantly reduced by transition to a two-component formulation. While on the other hand, the full four-component machinery is primarily necessary to describe both electronic and positronic degrees of freedom, only the electronic degrees of freedom need to be treated explicitly for chemical purposes. It appears therefore justified to claim that two-component formulations are the natural description for relativistic quantum chemistry. 

A. Wolf, M. Reiher, B. Hess, Two-component methods and the generalized Douglas-Kroll transformation, in P. Schwerdtfeger (Ed.), Relativistic Electronic Structure Theory, Part 1, Fundamentals, Elsevier, Netherlands, 2002, pp. 627-668. 

Density functional theory (DFT) provides an accurate and inexpensive access to molecular electronic structure and properties and was already used for scalar relativistic EFG calculations for example on Fe in solids [145] or in molecules [146,147]. We will not report on nonrela-tivistic DFT EFG calculations here but focus on the ZORA [148-151] and especially ZORA-4 methods, the latter including the density from the small component. Since the ZORA method is a two-component method sizable picture change effects will occur when calculating a core property like the EFG. In order to briefly illustrate this method a few of the fundamental equations will be given. A concise treatment of the ZORA formalism and its applications can be found in [152]. 

Calculated SOPP (ARPP) bond lengths (R /pm) and dissociation energies (Dg/meV) for Rn2 using MP2, CCSD and CCSD(T). The RPP results are calculated by two-component methods. 

Our DKH values and the ZORA results for g values are rather close in many cases although for such large Ag shifts as for the heavy-atom radical PdH the absolute difference can become significant. In general, the accuracy of calculated g values for c -metal species, at all levels of approximations presently described in the literature, is notably lower than that of p-element compounds [35] see also Chapter 9 of this volume. Detailed reasons for this misrepresentation still have to be found. Obviously, for g tensors of (heavy) d-metal open-shell systems, one can profit from two-component methods that directly treat spin-orbit interaction which may become too large to be amenable to a perturbation treatment. One may also benefit from specially adapted xc functionals that reflect the relativistic kinematics of the electrons and the Breit contribution to the electron-electron interaction [57]. Despite of the very minor effect of these relativistic corrections on many molecular observables [66], notable changes of the very sensitive g tensor are anticipated. 

Several excellent reviews of the relativistic two-component methods have recently appeared [12-15]. This review is not aimed at the completeness of the... 

The two-component methods, though much simpler than the approaches based on the 4-spinor representation, bring about some new problems in calculations of expectation values of other than energy operators. The unitary transformation U on the Dirac Hamiltonian ho (Eq.4.23 is accompanied by a corresponding reduction of the wave function to the two-component form (Eq.4.26). The expectation value of any physical observable 0 in the Dirac theory is defined as ... 

The equivalence of the lOTC method to the four-component Dirac approach has been documented by calculations of spin orbital energies in several papers [18,20, 63]. The unitary transformation does not affect the energy eigenspectrum, though it reduces the four-component bi-spinors to two-component spinor solutions. Due to this fact the two-component methods are frequently addressed as being quasi-relativistic and it is assumed that some information is lost. It can be demonstrated [22] that the two-component lOTC wave function which is the upper component of the unitarly transformed four-component Dirac spinor I ... 

In the present work we will focus mainly on the infinite order two-component method, lOTC. However, some comparison between the lOTC and DKHn methods will be also presented. So far the discussion has been focus on the block-diagonalization of the one-electron Dirac Hamiltonian. For the N electron system a Hamiltonian may be written as the sum of the one-electron transformed Dirac Hamiltonian plus the Coulomb electron-electron interaction and it is commonly used form of the relativistic Hamiltonian. 

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