Zeroth-order regular approximation Hamiltonian/method

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

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]. 

Another method to avoid the singularities in the vicinity of the nuclei was proposed by van Lenthe et al. [26,27,35]. They suggested an method that also includes the interaction potential, V (r), in the denominator of the ansatz for the small component. This ansatz was used in the derivation of the so called zeroth-order regular approximation (ZORA) Hamiltonian. The ZORA ansatz can thus be the written as... 

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]. 

VMC method [14, 15] by deriving the relativistic local energy of the scalar version of the zeroth-order regular approximation (ZORA) Hamiltonian [16-19] as the first attempt to develop the relativistic QMC method. 

A one-component quasirelativistic DFT method, also a part of the ADF package [92], was extensively used in the calculations for transition element and actinide compounds. (Earlier, the quasirelativistic Hartree-Fock-Slater (QR HFS) method was widely used for such calculations [93]). In this method, the Hamiltonian contains relativistic corrections already in the zeroth-order and is therefore called the zeroth-order regular approximation (ZORA) [94, 95]. The spin operator is also included in the ZORA Fock operator [96]. Other popular quasirelativistic 2c-DFT methods are based on the DKH approximation [97, 98] and implemented in many program packages. The following codes should also be mentioned of Rbsch [99, 100], Ziegler [101], and Case and Young [102]. They were, however, not used for the heaviest elements. A review on relativistic DFT methods for solids can be found in [103]. 

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. 

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