Relativistic methods four-component calculations

In preceding sections we have discussed several different relativistic methods four-component Dirac—Fock with and without correlation energy, the second-order Douglas—Kroll method, and perturbation methods including the mass—velocity and Darwin terms. The relativistic effective core potential (RECP) method is another well-established means of accounting for certain relativistic effects in quantum chemical calculations. This method is thoroughly described elsewhere - anJ is basically not different in the relativistic and... 

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. 

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... 

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. 

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. 

Lee and McLean have considered full relativistic all-electron solutions to the Dirac equation for AgH and AuH. In this method, four-component, allelectron spinors are obtained using a LCAS-MS (linear combination of atomic spinor—molecular spinor) method. These authors employ a Slater-type basis for AgH and AuH. However, such relativistic all-electron calculations do not seem to be practicable for molecules other than diatomic hydrides at present. 

Mochizuki and Okamoto applied the Dirac program for the estimation of stabilities of trivalent actinide elements and water or ammine complexes (Mochizuki and Okamoto 2002). Mochizuki and Tatewaki (2002) also carried out the electronic structure calculation on the hexa-hydrated ions of curium and gadolinium. They used the Dirac program and also predicted the fluorescence transition energy using the Complete Open-Shell Configuration Interaction (COSCI) method. Even the hexa-hydrate curium ion needs 2,108 basis functions for the fully relativistic four-component calculation. 

Relativistic calculations of NMR properties of RgH ion (where Rg = Ne, Ar, Kr, Xe), Pt shielding in platinum complexes, and Pb shielding in solid ionic lead(II) halides have been reported in this review period. For the Rg nucleus in the RgH ions, the following methods were used and results compared with each other non-relativistic uncorrelated method (HF), relativistic uncorrelated methods, four component Dirac Hartree-Fock method (DHF) and two-component zeroth order regular approach (ZORA), non-relativistic correlated methods using second order polarization propagator approach SOPPA(CCSD), SOPPA(MP2), respectively coupled cluster singles and doubles or second order Moller-Plesset, and... 

As the fully relativistic (four-component) calculations demand severe computational efforts, several quasirelativistic (two-component) approximations have been proposed in which only large components are treated explicitly. The approaches with perturbative treatment of relativistic effects [507] have also been developed in which a nonrelativistic wavefunction is used as reference. The Breit-Pauli (BP) approximation uses the perturbation theory up to the (p/mc) term and gives reasonable results in the first-order perturbation calculation. Unfortunately, this method cannot be used in... 

While this is by no means the only possible choice for relativistic four-component calculations, it is definitely the most efficient and convenient. It permits programmers to exploit an extensive technology that has been refined and tested through years of development of nonrelativistic methods. The usual complaint about the poor behavior of Gaussian functions close to nuclei is less severe for relativistic calculations, where nuclei of finite size are normally used and not point nuclei. As we discussed in chapter 7, Gaussians actually are particularly suitable for describing the wave function close to a nucleus of finite size. 

Full account for relativistic effects is provided by four-component calculations. A four-component relativistic MP2-F12 method has been recently reported by Ten- no and Yamaki. " In fact, the actual cusp-conditions are still subject to debate, as they depend on the type of two-electron operator used in the relativistic framework. This has been addressed by Kutzelnigg and Li et... 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

In this chapter, we therefore consider whether it is possible to eliminate spin-orbit coupling from four-component relativistic calculations. This is a situation quite different from that of more approximate relativistic methods where a considerable effort is required for the inclusion of spin-orbit coupling. We have previously shown that it is indeed possible to eliminate spin-orbit coupling from the calculation of spectroscopic constants [12,13]. In this chapter, we consider the extension of the previous result to the calculation of second-order electric and magnetic properties, i.e., linear response functions. Although the central question of this article may seem somewhat technical, it will be seen that its consideration throws considerable light on the fundamental interactions in molecular systems. We will even claim that four-component relativistic theory is the optimal framework for the understanding of such interactions since they are inherently relativistic. 

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. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

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