Hamiltonian operators relativistic methods

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

So far we have discussed nonrelativistic ab initio methods they ignore those consequences of Einstein s theory of relativity that are relevant to chemistry (section 4.2.3). These consequences arise in the special (rather than the general) theory, from the dependence of mass on velocity [4. This dependence causes the masses of the inner electrons of heavy atoms to be significantly greater than the electron rest mass since the Hamiltonian operator in the Schrodinger equation contains the electron mass (Eqs (5.36) and (5.37)), this change of mass should be taken into account. Relativistic effects in... 

The alternative to the development of new algorithms to handle relativistic Hamiltonians is to search for a way to extend non-relativistic algorithms such that they can handle the additional couplings. Since most implementations are based on a restricted Hartree-Fock scheme the first step is to mimic the spin-restricted excitation operators used in the non-relativistic methods by Kramers restricted excitation operators. This can be done by employing the so-called X-operator formalism [37]. 

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. 

There are lots of exehange-eorrelation potentials in the literature. There is an impression that their authors worried most about theory/experiment agreement. We ean hardly admire this kind of seienee, but the alternative (i.e., the practiee of ab initio methods with the intact and holy Hamiltonian operator) has its own disadvantages. This is because finally we have to choose a given atomic basis set, and this influences the results. It is true that we have the variational principle at our disposal, and it is possible to tell which result is more accurate. But more and more often in quantum chemistry, we use some non-variational methods (cf. Ch ter 10). Besides, the Hamiltonian holiness disappears when the theory becomes relativistic (cf. Qiapter 3). 

Current relativistic electronic structure theory is now in a mature and well-developed state. We are in possession of sufficiently detailed knowledge on relativistic approximations and relativistic Hamiltonian operators which will be demonstrated in the course of this book. Once a relativistic Hamiltonian has been chosen, the electronic wave function can be constructed using methods well known from nonrelativistic quantum chemistry, and the calculation of molecular properties can be performed in close analogy to the standard nonrelativistic framework. In addition, the derivation and efficient implementation of quantum chemical methods based on (quasi-)relativistic Hamiltonians have facilitated a very large amount of computational studies in heavy element chemistry over the last two decades. Relativistic effects are now well understood, and many problems in contemporary relativistic quantum chemistry are technical rather than fundamental in nature. 

Models related to spin-forbidden reactions are discussed in this chapter. Coupling between two surfaces of different spin and symmetry is given by various levels of approximation for spin-orbit operators from the reduction of relativistic quantum mechanics. Well-established methods such as the Breit-Pauli Hamiltonian exist, but new relativistic methods such as the Douglas-Kroll Hamiltonian and other new transformation schemes are also being investigated and implemented today. 

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]. 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

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. 

The method discussed here for the inclusion of relativistic effects in molecular electronic structure calculations is grounded in the Dirac-Fock approximation for atomic wave functions (29). The premise is that the major relativistic effects of the Dirac Hamiltonian are manifested in the core region, involving the core electrons, and that these effects propagate to the valence electrons. In addition, there are direct relativistic effects on valence electrons penetrating into the core region. Insofar as this is true, the valence electrons can be treated using a nonrelativistic Hamiltonian to which is added an operator, the relativistic effective core potential (REP). The REP formally, incorporates relativistic effects due to core electrons and to interactions of valence electrons with core electrons in an internally consistent way. 

Another method, devised by Cohen et al. to determine oxygen-rate gas collision parameters is to define an effective spin-orbit operator that includes r dependence, Zeff/r3, where the value of Zeff is adjusted to match experimental data (76). Langhoff has compared this technique with all-electron calculations using the full microscopic spin-orbit Hamiltonian for the rare-gas-oxide potential curves and found very good agreement (77). This operator has also been employed in REP calculations on Si (73), UF6 (78), U02+ and Th02 (79), and UF5 (80). The REPs employed in these calculations are based on Cowen-Griffin atomic orbitals, which include the relativistic mass-velocity and Darwin effects but do not include spin-orbit effects. Wadt (73), has made comparisons with calculations on Si by Stevens and Krauss (81), who employed the ab initio REP-based spin-orbit operator of Ermler et al. (35). 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

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 method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

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