Quasi-relativistic approximations

Given a four-component Dirac-Slater scheme, quasi-relativistic approximations can be derived using the same techniques as in wave 

The Pauli Hamiltonian has already been discussed in volume 1 of this series (see e.g. Ref. [35]) and reads 

The (iterative) use of the Pauli Hamiltonian, the so-called quasi-rela-tivistic (QR) method, must be regarded as obsolete, as the Pauli 

These problems are avoided if one uses regular Hamiltonians which are bounded from below. Many applications are based on the so-called zero order regular approximation (ZORA), which has been extensively investigated by the Amsterdam group [46-50]. It can be viewed as the first term in a clever expansion of the elimination of the small component, an expansion which already covers, at zeroth order, a substantial part of the relativistic effects. In fact ZORA is a rediscovery of the so-called CPD Hamiltonian (named after the authors. Ref. [60]). 

Quasi-relativistic approximations usually start from re-writing the Dirac equation in two-component form. Starting from the Dirac equation (with rest energy subtracted) 

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From here, expanding the operator acting on in terms of powers of one obtains the familiar Pauli approximation as well as numerous two-component quasi-relativistic approximations. 

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]. 

Despite recent implementations of an efficient algorithm for the four-component relativistic approach, the DC(B) equation with the four-component spinors composed of the large (upper) and small (lower) components stiU demands severe computational efforts to solve, and its applications to molecules are currently limited to small- to medium-sized systems. As an alternative approach, several two-component quasi-relativistic approximations have been proposed and applied to chemically interesting systems containing heavy elements, instead of explicitly solving the four-component relativistic equation. 

Within the DFT framework, we apply two different approaches to deal with relativistic effects, the so-called quasi-relativistic (QR) method (73) and the more modem "Zeroth Order Regular Approximation for Relativistic Effects" (ZORA) (14-16). The QR method is also known as the Pauli approach. 

The plus (minus) sign corresponds to E°7+(E°7.) E°9, E°7+ and E°7. correspond to the upper valence band states with T9, T7 and r7 symmetries, respectively. They are called A, B and C bands in that order, conventionally. The three quantities Ai, A2 and A3 can be derived from a theoretical procedure. At is directly estimated from the calculation without the spin-orbit interaction, and A2 and A3 are obtained by fitting the fully relativistically calculated top three energy levels at the T point to the above equations with the help of the obtained value of Ai. If we assume the quasi-cubic approximation (A = Ai, Ao = 3A2 = 3A3) [1,2], the energy splittings E°9 - E°7 can be rewritten as... 

The different approximations for all-electron relativistic calculations using one-component methods have recently been compared with each other and with relativistic ECP calculations of TM carbonyls by several workers (47,55). Table 6 shows the calculated bond lengths and FBDEs for the group 6 hexacarbonyls predicted when different relativistic methods are used. The results, which were obtained at the nonrelativistic DFT level, show the increase in the relativistic effects from 3d to 4d and 5d elements. It becomes obvious that the all-electron DFT calculations using the different relativistic approximations—scalar-relativistic (SR) zero-order regular approximation (ZORA), quasi-relativistic (QR) Pauli... 

Spectroscopic constants and dipole moment curves of the coinage metal diatomic molecules with boron, BCu, BAg, and BAu were investigated using high-level-correlated methods combined with quasi-relativistic Douglas-Kroll (No-pair) spin averaged approximation. 

The non-relativistic PolMe (9) and quasirelativistic NpPolMe (10) basis sets were used in calculations reported in this paper. The size of the [uncontractd/contracted] sets for B, Cu, Ag, and Au is [10.6.4./5.3.2], [16.12.6.4/9.7.3.2], [19.15.9.4/11.9.5.2], and [21.17.11.9/13.11.7.4], respectively. The PolMe basis sets were systematically generated for use in non-relativistic SCF and correlated calculations of electric properties (10, 21). They also proved to be successful in calculations of IP s and EA s (8, 22). Nonrelativistic PolMe basis sets can be used in quasirelativistic calculations in which the Mass-Velocity and Darwin (MVD) terms are considered (23). This follows from the fact that in the MVD approximation one uses the approximate relativistic hamiltonian as an external perturbation with the nonrelativistic wave function as a reference. At the SCF and CASSCF levels one can obtain the MVD quasi-relativistic correction as an expectation value of the MVD operator. In perturbative CASPT2 and CC methods one needs to use the MVD operator as an external perturbation either within the finite field approach or by the analytical derivative schems. The first approach leads to certain numerical accuracy problems. 

The calculations were carried out at B3LYP using a quasi-relativistic small-core ECP with a DZP-quality basis set for Pt and 6-31G(d) for C and H. The bond energies were approximated CCSD(T) values. For details see the original paper J. Uddin, S. Dapprich, G. Frenking and B. F. Yates, Organometallics, 1999,18,457. 

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... 

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. 

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. 

The quasi-relativistic model obtained by using the ZORA ansatz in combination with a fully variational derivation is the infinite-order regular approximation (lORA) previously derived by Sadlej and Snijders [46] and by Dyall and Lenthe [47]. The lORA method has recently been implemented by Klop-per et al. [48]. The ZORA model can be obtained from the lORA equation by omitting the relativistic correction term to the metric. However, the indirect renormalization contribution is as significant as the relativistic interaction operator in the Hamiltonian. This is the reason why ZORA overestimates the... 

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 (39), the block consists of three contributions. Since —f is of the order aP, whereas at the quasi-relativistic level X, B, and A are proportional to an approximate first-order energy correction can be obtained by neglecting the B and X terms in a = c is the fine structure constant. However, this approximation has not been tested numerically. 

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 perturbation theory corrections to the quasi-relativistic energies obtained with the lORA and the ERA Hamiltonian as the zeroth-order approximation to the Dirac equation are given in Table 6 and Table 7, respectively. Direct perturbation theory calculations on top of the MIORA and MERA Hamiltonians has not been studied computationally. 

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. 

One of the shortcomings of the BP approach is that the expansion in (p/mc) is not justified in the case where the electronic momentum is too large, e.g. for a Coulomb-like potential. The zeroth-order regular approximation (ZORA) [142,143] can avoid this disadvantage by expanding in E/ 2mc — P) up to the first order. The ZORA Hamiltonian is variationally stable. However, the Hamiltonian obtained by a higher order expansion has to be treated perturbatively, similarly to the BP Hamiltonian. Other quasi-relativistic methods have been proposed by Kutzelnigg [144,145] and DyaU [146]. 

Three quasi-relativistic approaches that are variationally stable are the Doug-lass-Kroll-Hess transformation of the no-pair Hamiltonian (for example, see Ref. 11, 20, 23-29), the zeroth order regular approximation, ZORA, (for example, see Ref. 30-34), and the approach of Barysz and Sadlej (for example, see Ref. 36). The results of the first two approaches differ considerably even when used by the same authors,which led them to try the third approach. A calibration study suggests that relativistic effects on heavy atom shieldings are significantly underestimated by ZORA in comparison to the four-component relativistic treatment, but that the neighboring proton chemical shifts are closer to experi-... 

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