Four-component methods

Early four-component numerical calculations of parity-violating effects in diatomic molecules which contain only one heavy nucleus and which possess a Si/2 ground state have been performed by Kozlov in 1985 [149] within a semi-empirical framework. This approach takes advantage of the similarity between the matrix elements of the parity violating spin-dependent term e-nuci,2) equation (114)) and the matrix elements of the hyperfine interaction operator. Kozlov assumed the molecular orbital occupied by the unpaired electron to be essentially determined by the si/2, P1/2 and P3/2 spinor of the heavy nucleus and he employed the matrix elements of e-nuci,2) nSi/2 and n Pi/2 spinors, for which an analytical expres- 

The effective spin-rotational Hamiltonian ifgr reads 

The first four-component calculations on parity violating effects in chiral molecules were performed in 1988 by Barra, Robert and Wiesenfeld [54] within an extended Hiickel framework. Interestingly, this study was on parity violating chemical shift differences in the nuclear magnetic resonance (NMR) spectra of chiral compounds and hence focused as well on the nuclear spin-dependent term of Hpv. Shortly later, however, also the first four-component results on parity violating potentials obtained with an extended Hiickel method were published by Wiesenfeld [150]. 

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which 

A related procedure with individual core potentials for the various atoms was later used by the same authors [154] to estimate parity violating potentials in chiral NFClBr, BiFClBr and NFClAt with intuitively chosen bond lengths and bond angles. 

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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. 

Molecules are more difficult to treat accurately than atoms, because of the reduced symmetry. An additional complication arises in relativistic calculations the Dirac-Fock-(-Breit) orbitals will in general be complex. One way to circumvent this difficulty is by the Douglas-Kroll-Hess transformation [57], which yields a one-component function with computational effort essentially equal to that of a nonrelativistic calculation. Spin-orbit interaction may then be added as a perturbation, implementation to AuH and Au2 has been reported [58]. Progress has also been made in the four-component formulation [59], and the MOLFDIR package [60] has been extended to include the CC method. Application to SnH4 has been described [61] here we present a recent calculation of several states of CdH and its ions [62], with one-, two-, and four-component methods. 

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. 

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

The electronic structure of atoms has been studied for many decades on the basis of four-component methods (Grant 1994 Kim 1993a,b Reiher and Hess 2000 Sapir-stein 1993, 1998). Nevertheless, significant improvements have been achieved in recent years. Even one-electron atoms still give us new insight into four-component electronic structure theory (Andrae 1997 Autschbach and Schwarz 2000 Chen and Goldman 1993 Chen etal. 1994 Pyykko and Seth 1997). In this section we review methodological improvements as well as new implementations and typical applications. 

In general, only small molecules, usually diatomics, have been studied with four-component methods. Often, correlation effects have not yet been taken into account. Those larger molecules, which have also been studied to some extent, exhibit high symmetry like Oh or 7d consisting of only two symmetry-inequivalent atoms. Therefore, hydrides, oxides and halides are by far the most extensively studied molecules. 

Table 2.1 List of molecules studied with four-component methods. The fourth column lists quantities, which have been investigated primary data P = (total electronic energies (E), orbital energies e,-, population analyses PA), ionization energies IE, election affinities EA, atomization energies A, spectroscopic data S = (equilibrium distance re, dissociation energy De, frequencies/wave numbers coe, bond angles 0), electric properties E = (dipole moment fx, quadrupole moment 0, dipole polarizability a, infrared intensities I, excited states ES, electric field gradients EFG, energetics of reaction R.

Apart from studies on single molecules or homologous molecules for the analysis of vertical trends in the periodic table of the elements, some interesting chemical and physical effects—such as lanthanide contraction, phosphorescence phenomena and parity violation—that are perfect areas to be tackled by four-component methods have been investigated. Some of the latest results are discussed in the following subsections. 

A major advantage of four-component methods, in which not only the ground state but also excited states are accessible (Cl, MCSCF or Fock-space CC methods), is that electronic transitions, which are spin forbidden in nonrelativistic theory, can be studied due to the implicit inclusion of spin-orbit coupling. Four-component methods are thus able to describe phosphorescence phenomena adequately. However, only a little work has been done for this type of electronic transitions and almost all studies utilize approximate descriptions of spin-orbit coupling (see, for instance, Christiansen et al 2000). 

Of particular importance in chemistry is the response of a molecular system to an external magnetic field as applied in routinely performed NMR experiments for the identification of compounds, the analysis of reaction mechanisms, and reaction control. Theoretical tools must provide spin-spin coupling constants and shielding tensors in order to calculate quantities, which can be related to experimental data. Needless to say, coupling constants and chemical shifts calculated from shielding tensors can only be obtained from accurate four-component methods for heavy nuclei. The theory of relativistic calculations of magnetic properties has recently been analysed in great detail (Aucar et al. 1999). 

Dirac-Fock calculations were the standard four-component method for electronic structure calculations on molecules during the last decade. However, they are still very demanding or completely infeasible if applied to large unsymmetric molecules with several heavy atoms. In addition, taking properly care of electron correlation increases the computational effort tremendously. Future work will certainly continue the development of relativistic correlation methods, which will be far less expensive. 

Finally, four-component methods will reach a high degree of applicability such that the relativistic approaches will become the standard tool for electronic structure calculations in the next decades. The four-component theories provide the general framework, in which more approximate methods—such as elimination methods for the small components, reduction methods to one-component wave functions, and also the nonrelativistic approaches—elegantly fit. This function of the four-component theories as the theoretical basis will certainly be reflected in algorithms and computer codes to be developed. 

A pilot calculation on CdH using one-, two- and four-component Fock space relativistic coupled-cluster methods has been published by Eliav et al. (1998b). The calculated values obtained were in very good agreement with experiment. While the four-component method gives the best results, one- and two-component calculations include almost all the relativistic effects. 

We have recently developed an efficient computational scheme for the four-component method that employs four-component contraction for molecular basis spinors and the new atomic spinor (AS) integral algorithm [130-132]. In the following sections we will briefly introduce our new relativistic scheme. 

While correlated four-component methods provide excellent results where applicable, they are too expensive to use for large systems, particularly larger molecules. Less computationally demanding methods are available and have proved to be of great value to the study of atomic and molecular systems, as shown in other chapters of this book. Nevertheless, four-component benchmark calculations for carefully selected systems will be necessary to test and calibrate the less expensive and more widely applicable schemes. 

The most rigorous way of incorporating relativity in the calculations is achieved by using four-component methods. However, for many applications a full Dirac-Hartree-Fock treatment is not feasible and reduced schemes are preferred keeping the numerical effort much lower. We... 

We review the Douglas-Kroll-Hess (DKH) approach to relativistic density functional calculations for molecular systems, also in comparison with other two-component approaches and four-component relativistic quantum chemistry methods. The scalar relativistic variant of the DKH method of solving the Dirac-Kohn-Sham problem is an efficient procedure for treating compounds of heavy elements including such complex systems as transition metal clusters, adsorption complexes, and solvated actinide compounds. This method allows routine ad-electron density functional calculations on heavy-element compounds and provides a reliable alternative to the popular approximate strategy based on relativistic effective core potentials. We discuss recent method development aimed at an efficient treatment of spin-orbit interaction in the DKH approach as well as calculations of g tensors. Comparison with results of four-component methods for small molecules reveals that, for many application problems, a two-component treatment of spin-orbit interaction can be competitive with these more precise procedures. 

Diatomic molecules containing heavy elements are often used as benchmark of relativistic methods because they are small enough to be treated by demanding approaches like four-component methods nevertheless, diatomics with a variety of chemical bonds afford a thorough testing. To illustrate the accuracy of the DKH approach to relativistic DF calculations, we chose Auz and Bi2. Au2 shows predominantly scalar relativistic effects, spin-orbit interaction is not important because bonding is predominantly mediated by the interaction of the... 

Neglect of relativistic effects, by using the Schrodinger instead of the Dirac equation. This is reasonably justified in the upper part of the periodic table but not in the lower half. For some phenomena, such as spin-orbit coupling, there is no classical counterpart and only a relativistic treatment can provide an understanding. The relativistic effects may be incorporated by a one-component (mass-velocity and Darwin terms), two-component (spin-orbit) or fuU four-component methods (Figure 8.2). 

Four-component methods are computationally expensive since one has to deal with small-component integrals. Therefore, various two-component methods in which small-component degrees of freedom are removed have flourished in the literature. We focus the present discussion on the X2c theory at one-electron level (X2c-le). The X2c-le scheme consists of a one-step block diagonalization of the Dirac Hamiltonian in its matrix representation via a Foldy-Wouthuysen-type matrix unitary transformation ... 

Although accurate, first-quantized four-component methods are both computationally demanding and plagued by interpretive problems due to the negative-energy states. From a conceptual point of view it is desirable to decouple the upper and lower components of the Dirac Hamiltonian and to obtain a two-component description for electrons only. 

We have already noted in chapter 13 that the Pauli approximation produces a spin-orbit coupling operator that maybe employed in essentially one-component, i.e., nonrelativistic or scalar-relativistic, methods via perturbation theory. Of course, this is an approximation compared with fully fledged four-component methods, but it can be a very efficient one that requires less computational effort without significant loss of accuracy. 

Probably one of the most appealing features of four-component methods is the possibility to test fundamental physical symmetries through accurate electronic structure calculations of molecules. First four[Pg.603]

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... 

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