Dirac-Fock equations

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. 

It was previously shown [7] that one can, starting from the Dirac-Fock equation, derive the two component scaled ZORA equation... 

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]. 

Consider the Dirac-Fock equations for a three-electron system Is nlj. Formally they fall into one-electron Dirac equations for the orbitals l5 and nlj with the potential ... 

For a synopsis of the history behind the relativistic equations of Schrodinger and Dirac see Weinberg S (1995) The quantum theory of fields, vol I. Cambridge University Press, Cambridge, UK, Chapter 1, and references therein. This account does not deal specifically with the Dirac-Fock equation... 

To construct the Dirac-Fock equations, it is assumed that the wave function for an atom having N electrons may be expressed as an antisymmetrized product of four-component Dirac spinors of the form shown in Eq. (9). For cases where a single antisymmetrized product is an eigenfunction of the total angular momentum operator J2, the JV-electron atomic wave function may be written... 

If it is assumed that the orbitals representing the core and valence electrons comprise an orthonormal set, the Dirac-Fock equation for a single valence... 

Thus one can solve the Dirac-Fock equation to obtain the relativistic energies and four-component spinor wavefunctions. 

The ideas and concepts concerning the use of basis sets in relativistic calculations which have been described in the previous subsections allow the Dirac-Fock equations for many-electron systems to be formulated within the algebraic approximation. A discussion of these equations lies outside the scope of the present chapter. 

M. J. Esteban, E. Sere. Solutions of the Dirac-Fock Equations for Atoms and Molecules. Comm. Math. Phys., 203(3) (1999) 499-530. 

M. Huber, H. Siedentop. Solutions of the Dirac-Fock equations and the energy of the electron-positron field. Arch. Rut. Mech. Anal, 184(1) (2007) 1-22. 

Other programs that solve the Dirac-Fock equations include BERTFIA and REL4D. ... 

Faas, S., Snijders, J. G., van Lenthe, J. H., van Lenthe, E., 8c Baerends, E. J. (1995). TheZORA formalism applied to the Dirac-Fock equation. Chemical... 

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