Dirac-Fock approximation

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. 

The parameter Ka — 0.24 — 0.33 is determined from nuclear model calculations [49]. These two interactions can be treated together using (104) with Ka K = Ka — 7C2(k — 1/2)1 K. The resulting spin-dependent correction was evaluated in the Dirac-Fock approximation including weak core-polarization corrections. Combining that calculation with the previous spin-independent result, we obtain... 

Maroulis et al.122 have applied their static polarizability, finite field technique to a study of the 22 electron diatomics CP , BC1, CC1+ and PO+. The vibrational contribution to the ground state polarizability has also been calculated. The dipole polarizability and other properties of YbF have been investigate in the unrestricted Dirac-Fock approximation by Parpia123 and the static second hyperpolarizability of the Cu2 dimer has been calculated in a correlation corrected UHF study by Shigemoto et al.124... 

Other interesting calculations of Be ", Ne "", Be, and Ne atom have been carried out by Kenny et al. [85] in which they evaluated perturba-tionally the relativistic corrections to the total energies. In particular, they found that the Breit correction is systematically larger in the Dirac-Fock approximation and calculated the most accurate values of relativistic corrections for the Ne atom to date. These results demonstrate another useful capability of the correlated wavefunction produced by QMC to estimate relativistic effects. Similar study within the VMC method has been done on examples of Li and LiH [86]. We expect that an important future application will be to carry out similar calculations of transition... 

The Pauli approximation may be used in conjunction with this method by neglecting the small component spinors Q) of the Dirac equation, leading to RECPs expressed in terms of two-component spinors. The use of a nonrelativistic kinetic energy operator for the valence region, and two-component spinors leads to Hartree-Fock-like expressions for the pseudoorbitals. Note that the V s (effective potentials) in this expression are not the same for pseudo-orbitals of different symmetry. Thus the RECPs are expressed as products of angular projectors and radial functions. In the Dirac-Fock approximation, the orbitals with different total j quantum numbers, but which have the same / values are not degenerate, and thus the potentials derived from the Dirac-Fock calculations would be y-dependent. Consequently, the RECPs can be expressed in terms of the /y-dependent radial potentials by equa-... 

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

Fig. 7.78 Linear relation of the quadmpole splitting A q = ( jl)eqQ (1 + j /3)l/2 and the isomer shift b for aurous (a) and auric (b) compounds. Also included is a correlation with the relative change in electron density at the gold nucleus, Ali/r(o)P, as derived from Dirac-Fock atomic structure calculations for several electron configurations of gold. An approximate scale of the EFG (in the principal axes system) is given on the right-hand ordinate (from [341])...

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

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. 

There are many problems in e.g. catalysis in which relativity may play a deciding role in the chemical reactivity. These problems generally involve large organic molecules which cannot be handled within the Dirac Fock framework. It is therefore necessary to reduce the work by making additional approximations. Generally used approaches are based on the Pauli expansion or on the Douglas Kroll transformation [3]. 

In this paper we use a regular approximation of the Dirac Fock formalism known as... 

Equation (1) is obtained by using an expansion in E/ 2c - Vc) on the Dirac Fock equation. This expansion is valid even for a singular Coulombic potential near the nucleus, hence the name regular approximation. This is in contrast with the Pauli method, which uses an expansion in (E — V)I2(. Everything is written in terms of the two component ZORA orbitals, instead of using the large and small component Dirac spinors. This is an extra approximation with respect to the original formalism. 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

Fig. 3. A comparison of the eigenvalues of the outermost valence electrons for Pu using relativistic, semi-relativistic and non-relativistic kinematics and the local density approximation (LSD). Dirac-Fock eigenvalues after Desclaux are also shown. The total energies of the atoms (minus sign omitted), calculated with relativistic and non-relativistic kinematics are also shown. HF means Hartree Fock...

We would like to emphasize here that the all-electron Dirac-Fock calculations on TIE and YbF are, in particular, important for checking the quality of the approximations made in the two-step method. The comparison of appropriate results, Dirac-Fock vs. RECP/SCF/NOCR, is, therefore, performed in papers [94, 19] and discussed in the present paper. 

Further we present the results of our calculations of the Li- ike iGplasma satellite lines on the basis of QED PT with ab initio zeroth-order approximation for three-quasiparticle systems, together with the optimized Dirac-Fock results and experimental data for comparison. In Table 4 there are displayed the experimental value (A) for wavelength (in A) of the Ti-like lines dielectron satellites to the ls So-ls3p Pi line of radiation in the K plasma, and the corresponding theoretical results (B) PT on 1/Z (C) QED PT (our data) (D) calculation by the AUTOJOLS method, and (E) MCDF [12, 21],... 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

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