Beyond the Dirac equation

There is certainly a need for a more accurate theory. 

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The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

Finally, the Breit equation has been given. The equation goes beyond the Dirac model by taking into account the retardation effects. The Breit-Pauli expression for the Breit Hamiltonian contains several easily interpretable physical effects. 

In quantum chemistry, regardless of which operators we choose for the Hamiltonian, we almost invariably implement our chosen method in a finite basis set. The Douglas-Kroll and Barysz-Sadlej-Snijders methods in the end required a matrix representation of the momentum-dependent operators in the implementation, and the regular methods usually end up with a basis set, even if the potentials are tabulated on a grid. Why not start with a matrix representation of the Dirac equation and perform transformations on the Dirac matrix rather than doing operator transformations, for which the matrix elements are difficult to evaluate analytically It is almost always much easier to do manipulations with matrices of operators than with the operators themselves. Provided proper account is taken in the basis sets of the correct relationships between the range and the domain of the operators (Dyall et al. 1984), matrix manipulations can be performed with little or no approximation beyond the matrix representation itself. 

In this book the interaction between fields and molecules is treated in a semi-classical fashion. Quantum mechanics is used for the description of the molecule, whereas the treatment of the electromagnetic fields is based on classical electromagnetism. A complete quantmn mechanical description using quantmn electrodynamics is beyond the scope of this presentation, although we will make use of the correct value of the electronic g-factor as given by quantum electrodynamics. Furthermore, only ab initio methods derived from the non-relativistic Schrodinger equation are discussed. Nevertheless, the Dirac equation is briefly discussed in order to introduce the electronic spin via the Pauli Hamiltonian. 

The mass dependence of the correction of order a Za) beyond the reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction in (3.4), the result in (3.7) is exact in the small mass ratio m/M, since in the framework of the effective Dirac equation all corrections of order Za) are generated by the kernels with one-photon exchange. In some earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M. Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the IS level in hydrogen) to the leading nonrecoil Lamb shift contribution. 

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]. 

Dirac" achieved the combination of QM and relativity. Relativistic corrections are necessary when particles approach the speed of light. Electrons near heavy nuclei will achieve such velocities, and for these atoms, relativistic qnantnm treatments are necessary for accurate description of the electron density. However, for typical organic molecules, which contain only first- and second-row elements, a relativistic treatment is unnecessary. Solving the Dirac relativistic equation is much more difficult than for nonrelativistic computations. A common approximation is to utilize an effective field for the nuclei associated with heavy atoms, which corrects for the relativistic effect. This approximation is beyond the scope of this book, especially since it is unnecessary for the vast majority of organic chemistry. 

For a heavy element whose atomic number is beyond 50, the relativistic effects (error caused by the nonrelativistic approximation) on the valence state can not be ignored. In such a case, it is necessary to solve Dirac equation instead of nonrelativistic Schrodinger equation usually used for the electronic state calculation. The relativistic effects... 

In projective relativity the field equations contain, in addition to the gravitational and electromagnetic fields, also the relativistic wave equation of Schrodinger and, as shown by Hoffinann (1931), are consistent with Dirac s equation, although the correct projective form of the spin operator had clearly not been found. The problem of spin orientation presumably relates to the appearance of the extra term, beyond the four electromagnetic and ten gravitational potentials, in the field equations. It correlates with the time asymmetry of the magnetic field and spin. 

The purpose of this section is to show how the problem of passing from the four-component Dirac equation to two-component Pauli-like equations can be systematically investigated within the framework of the theory of effective Hamiltonians.Beyond the above-mentioned difficulties, we will be able to derive energy-independent two-component effective Hamiltonians that can be used for variational atomic and molecular calculations. To introduce the subject and the notation, let us first consider the simple case of a free electron. 

There have been several successful applications of the Dirac-Hartree-Fock (DHF) equations to the calculation of numerical electronic wave functions for diatomic molecules (Laaksonen and Grant 1984a, 1984b, Sundholm 1988, 1994, Kullie et al. 1999). However, the use of numerical techniques in relativistic molecular calculations encounters the same difficulties as in the nonrelativistic case, and to proceed to general applications beyond simple diatomic and linear molecules it is necessary to resort to an analytic approximation using a basis set expansion of the wave function. The techniques for such calculations may to a large extent be based on the methods developed for nonrelativistic calculations, but it turns out that the transfer of these methods to the relativistic case requires special considerations. These considerations, as well as the development of the finite basis versions of both the Dirac and DHF equations, form the subject of the present chapter. 

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

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