Non-relativistic theory

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

The Dirac operator incorporates relativistic effects for the kinetic energy. In order to describe atomic and molecular systems, the potential energy operator must also be modified. In non-relativistic theory the potential energy is given by the Coulomb operator. 

All of the terms in eqs. (8.29-8.34) may be used as perturbation operators in connection with non-relativistic theory, as discussed in more detail in Chapter 10. It should be noted, however, that some of the operators are inherently divergent, and should not be used beyond a first-order perturbation correction. 

L. D. Landau and E. M. Lifshitz (1958) Quantum Mechanics Non-Relativistic Theory (Pergamon, London Addison-Wesley, Reading, MA). 

Landau, L. D. and Lifshitz, E. M. (1965). Quantum Mechanics. Non-Relativistic Theory. Pergamon Press, Oxford. 

In order to develop a fully consistent non-relativistic theory for particles and fields we must first determine the non-relativistic limit of Maxwell s equations. This is to some extent hampered by the fact that the equations, and in particular the occurrence of the speed of light c, change according to the unit system chosen, a situation that has been analyzed by Jackson [25] and Kutzelnigg [15,35]. In STbased atomic units, the non-relativistic limit is obtained directly as the limit c 00 and leads to the equations... 

The distinction in standard non-relativistic theory between spin-orbit interaction as relativistic on the one hand and other spin interactions as non-relativistic on the other hand does lead to some inconsistencies. Consider, for instance, a hydrogen-like atom where the coordinate system is shifted from the... 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

As of 1994 the agreement between experiment and theory in the observed transition wavelengths was of the order of 1000 ppm. Then, a new theory of Korobov [7] came in. Fig. 4 shows comparison between experiment and theory. Although Korobov s non-relativistic theory showed a dramatical improvement over the earlier theories, it revealed a systematic discrepancy of the order of 50-100 ppm. This urged Korobov and Bakalov to take into account relativistic corrections [17]. The relativistic corrections are systematically about 50 ppm for the Av = 0 transitions and about 100 ppm for the Av = 2 transitions, accounting for the experimental results very well. 

It is not simple to demonstrate that 7/3 is the correct power of Z for real atoms when Z < 100. In fact Foldy showed that up to this value of Z, Hartree and Hartree-Fock data are slightly better fitted with Z12/6. Nevertheless, for a non-relativistic theory there can be no doubt that for really large values of Z the total binding energy must vary as Z7/3. We shall see below, when we deal with... 

The effect of relativity on the bonding in this molecule is so large that even the qualitative features of the bonding cannot be correctly described by non-relativistic theory which (i) fails to predict any 5d-6s hybridization in the localized bonding orbital (ii) seriously underestimates 5d-6s hybridization in one of the nonbonding orbitals (NBOs) (iii) predicts incorrectly that one of the orbitals with mj 1/2 is entirely tt in character and has pure spin and (iv) erroneously predicts that the bond (in AuH) is formed solely from the interaction between the gold 6s and the hydrogen Is atomic orbitals because the non-relativistic molecular orbital (MO) wavefunction constructed from these two atomic orbitals predicts the AuH molecule to be unbound by 0.19 eV. 

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