Relativistic Symmetry

Symmetry is one of the great unifying principles of physics and chemistry. Symmetry governs the geometry of molecules, as well as their spectra and the shapes of their crystals. The symmetry of a system provides fingerprints that show up under various probes of the system. Knowledge of the symmetry of a system provides us with valuable information about its properties and behavior, and symmetry may also be used to simplify quantum chemical calculations on the system. All this— provided we have the theory and tools to extract this information. The purpose of this chapter is to provide some of the tools to deal with the symmetries of relativistic quantum chemistry and examine the symmetries of the Dirac equation. 

In this chapter, we start with a qualitative introduction of double groups and relativistic molecular symmetries, connecting to the more or less phenomenological introduction this subject is frequently accorded in quantum chemistry. The aim is to provide the necessary insight for those who only need an operational familiarity with double groups. We then turn to a more formal discussion of the symmetry invariance of 

The area where chemists normally first meet relativistic effects is in the discussion of the spectra of atoms and molecules. This is an example of how knowledge of the symmetry of a system enables us to make a priori predictions about its properties. The simplest case here is atomic spectra, in particular for one-electron atoms, so let us use that as a starting point. 

We will discuss the Dirac equation for the one-electron atom in more detail in chapter 7. Here we are only interested in the symmetry properties of the Hamiltonian for such systems. We know that for the corresponding nonrelativistic case, angular momentum and spin are normal constants of motion, represented by operators that commute with the Hamiltonian. In particular 

The extension to the case of the four-component Dirac Hamiltonian above follows readily by noting that the spin operator and the orbital angular momentum operator for this case are 

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The construction of relativistic and nonrelativistic E-coefficients is therefore at the centre of an efficient electronic structure program a paper describing how to organize this on modern multi-node computers is nearly finished [15]. Relativistic symmetry relations can be exploited to reduce the number of independent coefficients which need to be calculated. 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

The comparison of the calculated spectra of the free ions and the ones in the crystal is not straightforward. Indeed, in the crystal, the presence of the first coordination shell increases the number of electrons and basis functions in the calculations, resulting in a blow-up of the Cl expansion, mainly due to the generated doubly-excited configurations. One should bare in mind that this increase is about six time as fast in double group symmetries as in the non-relativistic symmetry. In a non effective Hamiltonian method, the only way to keep the size of the DGCI matrix to an affordable size of few million configurations, is to cut down the number of correlated electrons. This may essentially deteriorate the quality of electron correlation as the contributions of the spin-orbit interaction... 

For many applications an approximate consideration of relativistic effects is sufficient. In scalar-relativistic approaches, spin-orbit coupling is neglected, so that wave functions with non-relativistic symmetry are obtained. 

In his detailed analysis of Dirac s theory [6], de Broglie pointed out that, in spite of his equation being Lorentz invariant and its four-component wave function providing tensorial forms for all physical properties in space-time, it does not have space and time playing full symmetrical roles, in part because the condition of hermiticity for quantum operators is defined in the space domain while time appears only as a parameter. In addition, space-time relativistic symmetry requires that Heisenberg s uncertainty relations. 

While we still have no intention of proving the invariance of the Dirac equation under Lorentz transformations, we do want to conclude this discussion of relativistic symmetry by demonstrating the relationship between the Lorentz transformations and the spatial transformations already discussed. To make the syuunetry between space and time more explicit we introduce the variable r, defined as... 

This concludes our introduction to relativistic symmetry. Our aim has been to relate closely to features that should be familiar to the practicing quantum chemist. In particular, we have put some emphasis on the double groups, which represent a rather straightforward extension of the methods and concepts of nonrelativistic symmetry. We have also provided a more general discussion that shows how the double group symmetry arises as the direct product of the underlying symmetries in the two separate physical spaces considered—spin space and the four-space spanned by the Lorentz transformations. In the chapters to follow, we will repeatedly exploit both SU 2) (g) G, G 0(3) symmetry and Kramers symmetry to develop and simplify methods for quantum chemical calculations on relativistic systems. 

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