Zero-Order Regular Approximation

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

In contrast to eq. (8.13), the factor Ej 2mc — V) is always much smaller than 1. K may now be expanded in powers of Ej(2mc — V), analogously to eq. (8.22). Keeping only the zero-order term (i.e. setting K = 1) gives the Zero-Order Regular Approximation (ZORA) method. ... 

It seems natural to suppose that the tetragonal distortion of the tri-anion results from the Jahn-Teller effect. In order to study the problem more thoroughly we undertook recently the DFT calculations of this cluster as well as of several other hexanuclear rhenium chalcohalide clusters. The technical details of these calculations can be found in the original publication [8]. Here we only want to note that the introduction of relativistic corrections for Re atoms is crucial for the correct reproduction of the geometry of clusters. In our calculations, this was done by the zero order regular approximation (ZORA) Hamiltonian [9] within ADF 2000.02 package [10]. 

Quadratic Configuration Interaction with Singles and Doubles Quadratic Configuration Interaction with Singles, Doubles, and Noniterative Approximation of Triples Symmetry Adapted Cluster-Configuration Interaction Split-Valence basis set plus Polarisation functions Zero-Order Regular Approximation Zero-Point Energy... 

Relativistic effects may be also considered by other methods than pseudopotentials. It is possible to carry out relativistic all-electron quantum chemical calculations of molecules. This is achieved by various approximations to the Dirac equation, which is the relativistic analogue to the nonrelativistic Schrodinger equation. We do not want to discuss the mathematical details of this rather complicated topic, which is an area where much progress has been made in recent years and where the development of new methods is a field of active research. Interested readers may consult published reviews . A method which has gained some popularity in recent years is the so-called Zero-Order Regular Approximation (ZORA) which gives rather accurate results ". It is probably fair to say that... 

Table 2 Effect of spin-orbit coupling on the splitting of the T2g and Tig ground states of octahedral [Ee(CN)g] and [Mn(CN)g] complexes, with expressions for the g-tensors and values of the spin-orbit coupling constants from relativistic zero-order regular approximation DPT calculations ...

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

The zero-order regular approximation (ZORA), a two-component form of the fully-relativistic Dirac equation, is currently used for organotin computational calculations using basis sets specifically designed for ZORA. It should be noted that while all-electron calculations, whether non-relativistic or relativistic, can be used for organotin systems, the 6-3IG Pople basis set is not available for tin and therefore, most all-electron calculations involving tin employ the smaller 3-21G basis set. 

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... 

Calculations of the full manifold of the electronic states spanned by these configurations have been done on structures obtained from DFT geometry optimizations using the Perdew-Becke-Ernzerhof (PBE) functional [119, 120], empirical van der Waals corrections [121] for the DFT energy, the scalar relativistic zero-order regular approximation (ZORA) [122], and the scalar relativistically recontracted (SARC) [123] version of the def2-TZVP basis set [124]. 

The Amsterdam Density Functional (ADF) method [118,119] was used for calculations of some transactinide compounds. In a modem version of the method, the Hamiltonian contains relativistic corrections already in the zeroth order and is called the zero-order regular approximation (ZORA) [120]. Recently, the spin-orbit operator was included in the ZORA Fock operator [121]. The ZORA method uses analytical basis fimctions, and gives reliable geometries and bonding descriptions. For elements with a very large SO splitting, like 114, ZORA can deviate from the 4-component DFT results due to an improper description of the pi/2 spinors [117]. Another one-component quasirelativistic scheme [122] applied to the calculations of dimers of elements 111 and 114[116,117]isa modification of the ZORA method. 

SOCI spin-orbit configuration interaction SOFT second-order perturbation theory SOREP spin-orbit relativistic effective potential TD-DFT time dependent density functional theory ZORA zero-order regular approximation... 

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]). 

In BPW91 calculations by the ADF package, the zero-order regular approximation formalism (ZORA) without the spin-orbit effect was used [106, 107]. The ls-2p orbitals for Cu, ls-3d orbitals for Ag, Is—4f orbitals for ft and Au, and ls-2s orbitals for C and N are kept frozen, respectively, in the frozen core approximation, and the valence orbitals based on the Slater-type orbital are expanded within the triple zeta basis set augmented with two polarization functions. Frequency analyses have been used to assess the stability of the optimized structures. Calculated zero-point energy (ZPE) corrections have been incorporated into total energies in estimation of relative energies. 

Baerends. The zero-order regular approximation for relativistic effects The effect of spin-orbit coupling in closed shell molecules. J. Chem. Phys., 105 (19%) 6505-6516. 

E. van Lenthe, P. E. S. Warmer, A. van der Avoird. Density functional calculations of molecular -tensors in the zero order regular approximation for relativistic effects. 

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