Four-component Calculations

Terms up to order 1/c are normally sufficient for explaining experimental data. There is one exception, however, namely the interaction of the nuclear quadrupole moment with the electric field gradient, which is of order 1/c. Although nuclei often -a e modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than 1/2 (the dipole and octopole moments vanish by symmetry). As discussed in section 10.1.1, this leads to an interaction term which is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

All of the terms in eqs. (8.29-8.34) may be used as perturbation operators in connection with non-relativistic theory, as discussed in more detail in Chapter 10. It should be noted, however, that some of the operators are inherently divergent, and should not be used beyond a first-order perturbation correction. 

Although relativistic effects can be included in the Schrodinger equation by addition of 

The presence of the momentum operator means that the small component basis set must contain functions which are derivatives of the large basis set. The use of kinetic balance ensures that the relativistic solution smoothly reduces to the non-relativistic wave function as c is increased.-------------------------------------------------------------------- 

Results from fully relativistic calculations are scarce, and there is no clear consensus on which effects are the most important. The Breit (Gaunt) term is believed to he small, and many relativistic calculations neglect this term, or include it as a perturbational term 

The only term surviving the Bom-Oppenheimer approximation is the direct spin-spin coupling, as all the others involve nuclear masses. Furthermore, there is no Fermi-contact term since nuclei cannot occupy the same position. Note that the direct spin-spin coupling is independent of the electronic wave function, it depends only on the molecular geometry. 

The presence of the momentum operator means that the small component basis set must contain functions which are derivatives of the large basis set. The use of kinetic balance ensures that the relativistie solution smoothly reduees to the non-relativistic wave function as c is increased. 

The H e operator is the one-electron part of the spin-orbit interaction, while the H e and He°° operators in eq. (8.36) define the two-electron part. The one-electron term dominates and the two-electron contribution is often neglected or accounted for approximately by introducing an effective nuclear charge in H ° (corresponding to a screening of the nucleus by the electrons). The effect of the spin-orbit operators is to mix states having different total spin, as for example singlet and triplet states. 

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There are also ways to perform relativistic calculations explicitly. Many of these methods are plagued by numerical inconsistencies, which make them applicable only to a select set of chemical systems. At the expense of time-consuming numerical integrations, it is possible to do four component calculations. These calculations take about 100 times as much CPU time as nonrelativistic Hartree-Fock calculations. Such calculations are fairly rare in the literature. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

The GRECP method is described in detail in the above papers and we only add here that it allows one to avoid the complications of the four-component calculations described in the introduction (see also [30]) and to attain very high accuracy, limited in practice by possibilities of the correlation methods, while requiring moderate computational efforts when the IC, OC and V subspaces are appropriately chosen. 

With the restored molecular bispinors, the two-electron integrals on them can be easily calculated. Thus, the four-component transfomation from the atomic basis that is a time-consuming stage of four-component calculations of heavy-atom molecules can be avoided. Besides, the VOCR technique developed in [92] for molecular pseudospinors can be reformulated for the case of molecular pseudospinorbitals to reduce the complexity of the molecular GRECP calculation as is discussed in section 5. 

Relativistic Methods 204 8.1 Connection Between the Dirac and Schrodinger Equations 207 8.2 Many-particle Systems 210 8.3 Four-component Calculations 213 11.4.1 Ab Initio Methods 272 11.4.2 DFT Methods 273 11.5 Bond Dissociation Curve 274 11.5.1 Basis Set Effect at the HF Level 274 11.5.2 Performance of Different Types of Wave Function 276... 

The last decade has seen a vast amount of method and algorithm development to set up computer programs that can be used for efficient four-component calculations of the electronic structure of molecules. These calculations need incredibly large computer resources even for standard noncorrelated methods like Dirac-Hartree-Fock applied to molecules with only one heavy atom. 

One purpose of these calculations is to understand the effect of a four-component treatment for different types of molecules to evaluate the reliability of more approximate treatments like two-component or one-component methods. In other words, those cases must be identified where only four-component calculations yield sufficiently accurate results. In all other cases, more approximate methods, which do... 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

Figure 18. Principal component analysis (PCA) of EXAFS spectra, a) linear combinations of MnC03, y-MnOOH and MnS reference spectra used as unknowns. b) First four components calculated by PCA of the unknowns shown in a), c) Target transformation on y-MnOOH and chalcophanite. The reconstructed and reference spectra for y-MnOOH are identical, which indicates that this species is contained in the series of unknowns. In contrast, the two chalcophanite (ZnMn307-3H20) spectra are very different since this species is absent. 

The potentials (94) and (95) are already quite similar to the leading effective Hamiltonians that have been used so far in one- and four-component calculations of molecular parity violating eflFects. We have assumed above that the fermions are elementary particles. The effective potentials may, however, also be applied for the description of low energy weak neutral scattering events, in which heavy non-elementary fermions like the proton and the neutron or even entire atomic nuclei are involved, provided that properly adjusted vector and axial coupling coefficients py and for non-elementary fermions are used. 

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]. 

An overview of relativistic state-of-the-art calculations on electric field gradients (EFG) in atoms and molecules neccessary for the determination of nuclear quadrupole moments (NQM) is presented. Especially for heavy elements four-component calculations are the method of choice due to the strong weighting of the core region by the EFG operator and the concomitant importance of relativity. Accurate nuclear data are required for testing and verification of the various nuclear models in theoretical nuclear physics and this field represents an illustrative example of how electronic structure theory and theoretical physics can fruitfully interplay. Basic atomic and molecular experimental techniques for the determination of the magnetic and electric hyperfine constants A and B axe briefly discussed in order to provide the reader with some background information in this field. 

Early four-component calculations of hfs constants in atoms... 

A combined approach starting from an unrestricted Hartree-Fock (UHF) scheme in L5-coupling in combination with radial four-component calculations was performed by Desclaux and co-workers [78]. These authors investigated the hyperfine structure of Ga and Br allowing diflFerent radial parts for all the spin orbitals. These functions were then jj recoupled corresponding to the electronic configurations... 

The analysis of the results obtained showed the relevance of the inner shell polarization which was accounted for by the UHF calculations. Since a fully numerical four-component code including exchange was not at hand at that time the authors also had to analyze the impact of Slater s exchange approximation especially its transferability from the nonrelativistic to the relativistic realm. Shortly after these investigations Desclaux and Bessis presented fully numerical four-component calculations now substituting Slater s approximation by an explicit treatment of exchange and reported results for A and B values of Sc, Cu, Ga and Br [79]. 

Pseudopotentials have been the subject of considerable attention in the last two decades, and they have been developed by a number of different groups. They are also the most widely used effective core potentials in chemical applications either for the study of chemical reactions or spectroscopy. A large variety of pseudopotentials are now available, and all the coupling schemes at the SCF step have been implemented four-component, two-component, and scalar relativistic along with spin-orbit pseudopotentials. However, it is well known that four-component calculations can (in the worst cases) be 64 times more expensive than in the non relativistic case. In addition, the small component of the Dirac wave function has little density in the valence region, and pseudopoten-... 

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