Dirac equation nonrelativistic limit

When is a one component scalar function, one can take the square root of Eq. (9-237) and one thus obtains the relativistic equation describing a spin 0 particle discussed in Section 9.4. This procedure, however, does not work for a spin particle since we know that in the present situation the amplitude must be a multicomponent object, because in the nonrelativistic limit the amplitude must go over into the 2-component nonrelativistic wave function describing a spin particle. Dirac, therefore, argued that the square root operator in the present case must involve something operating on these components. 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nonrelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that are normally below the relativistic scale, the Berry phase obtained from the Schrodinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

The Vignale-Rasolt CDFT formalism can be obtained as the weakly relativistic limit of the fully relativistic Kohn-Sham-Dirac equation (5.1). This property has been exploited to set up a computational scheme that works in the framework of nonrelativistic CDFT and accounts for the spin-orbit coupling at the same time (Ebert et al. 1997a). This hybrid scheme deals with the kinematic part of the problem in a fully relativistic way, whereas the exchange-correlation potential terms are treated consistently to first order in 1 /c. In particular, the corresponding modified Dirac equation... 

A physical system is close to the nonrelativistic limit, if all velocities of the system are small compared to the velocity of light. Hence the nonrelativistic limit of a relativistic theory is obtained if we let c, the velocity of light, tend to infinity. In the nonrelativistic theory, there is no limit to the propagation speed of signals. For the Dirac equation, the nonrelativistic limit turns out to be rather singular. If we simply set c = oo, we would just obtain infinity in all matrix elements of the Dirac operator H. We must therefore look for cancellations. 

It took some time until it was realized that the Dirac theory describes the spin correctly because it is a spinor-field theory, and not because it is relativistic [16]. In fact, if one takes the nonrelativistic limit of the Dirac equation, spin survives, and this is consistent with the observation that the Galilei group has spinor representations as well. So, without any doubt, spin is not a relativistic effect. 

The terms etc. in (10) represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (4) in addition to the electron repulsion l/rjj. The radial functions Pn ( ) and Qn/c( ) may be obtained by mmierical integration [20,21] or by expansion in a basis (for more details see recent reviews [22,23]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [24,25], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [26,27]. In the nonrelativistic limit (c oo), the small component is related to the large component by [24]... 

Before leaving the theoretical formalism section, it is important to note that perturbation theory for relativistic effects can also be done at the fo n-con onent level, i.e. before elimination of the small component by a Foldy-Wouthitysen (FW) or Douglas-Kroll transformation. This is best done with direct perturbation theory (DPT) [71]. DPT involves a change of metric in the Dirac equation and an expansion of this modified Dirac eqtiation in powers of c . The four-component Levy-Leblond equation is the appropriate nonrelativistic limit. Kutzelnigg [72] has recently worked out in detail the simultaneous DPT for relativistic effects and magnetic fields (both external and... 

Despite this ubiquitous presence of relativity, the vast majority of quantum chemical calculations involving heavy elements account for these effects only indirectly via effective core potentials (ECP) [8]. Replacing the cores of heavy atoms by a suitable potential, optionally augmented by a core polarization potential [8], allows straight-forward application of standard nonrelativistic quantum chemical methods to heavy element compounds. Restriction of a calculation to electrons of valence and sub-valence shells leads to an efficient procedure which also permits the application of more demanding electron correlation methods. On the other hand, rigorous relativistic methods based on the four-component Dirac equation require a substantial computational effort, limiting their application in conjunction with a reliable treatment of electron correlation to small molecules [9]. 

In order to derive an expression for the energy of interaction between an intrinsic magnetic moment (as for a particle with nonzero spin angular momentum) and the electromagnetic field, one cannot proceed via classical mechanics as for the charged particle. Rather one proceeds via the relativistic Dirac equation and seeks the nonrelativistic limit... 

In summary, the Dirac equation under the influence of an external magnetic field is written at the nonrelativistic limit as... 

In the derivation presented above we subtracted nteC from both radial Dirac equations in order to remove any c-dependence from the upper equation, which then allowed us to easily evaluate the limit c —> oo. A similar effect was also induced in the first discussion on the nonrelativistic limit in section 5.4.3. 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

[Pg.533]

We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrodinger-type equation containing spin. 

Before taking the limit c oo, this equation must be rearranged for two reasons first, because we need to change it to a form where c occurs in some form of denominator—this will provide us with terms that vanish and hopefully other terms that remain finite—and, second, because the nonrelativistic wave function is a scalar function, whereas the Dirac wave function is a four-component vector function. If we use the two-component nonrelativistic Schrodinger equation that we derived in section 4.2, we can write the nonrelativistic wave function in terms of spin-orbitals, which can be transformed to two-component spinors. Then it is only necessary to reduce the Dirac equation from four-component to two-component form. 

We now have a matrix form of the Dirac equation which tends to the right nonrelativistic limits, but we have no guarantee that the solutions of these equations are well-behaved. In particular, we need to ensure that the finite basis expansion does not produce solutions corresponding to energies below the positive energy electronic space, despite all our efforts to cast the equations in a form that should prevent this from occurring (chapter 5 and chapter 8). 

Having recovered the potential surface from the solutions of the Born-Oppenheimer electronic problem, we can now proceed to solve the equation for nuclear motion. The Dirac-type equation for the nuclei can easily be reduced to the corresponding nonrelativistic equation by following the same reduction as we did for taking the nonrelativistic limit of the Dirac equation in section 4.6. Doing this, we abandon all pretense of Lorentz invariance for this part of the system, but we know from experiment that the nuclear relative motion in molecules takes place at rather low energies where relativistic effects may safely be neglected. 

Two critical questions about the removal of the spin dependence from the Dirac equation are, What effect does this have on the eigenvalue spectrum and How is the wave function modified These questions may be answered by a similar process to that used for the atomic Dirac equation in chapter 7. The second question has as an auxiliary question, How does the wave function behave in the nonrelativistic limit ... 

Substituting the second equation into the first gives the Schrodinger equation, with [r the nonrelativistic wave function. The second of these equations implies that ( ) also becomes the nonrelativistic wave function. For a finite nucleus this is indeed the case, but for a point nucleus it is necessary to take care with the nonrelativistic limit, as we did for the Dirac equation. Naively setting c -> oo in the series expansions above would still yield the result near the nucleus, not which is what we would expect from the equation above. To illustrate, consider the ground state of a hydrogenic atom, for which the radial part of goes as with... 

It is apparently this discontinuity that causes the problems in defining the nonrela-tivistic limit. This point has been raised by Kutzelnigg (1989) in his discussion of perturbation theories based on the Dirac equation, where he shows that the limits c -> oo and r -> 0 do not necessarily commute, and for certain kinds of relativistic Hamiltonians there is no well-defined nonrelativistic limit. No such problem occurs for the finite nucleus. 

Solutions of the Dirac equation with negative energies, corresponding to chaige-conjugated degrees of freedom. Nonrelativistic limit... 

In the limit of c oo the lower component vanishes and 4 terms into nonrelativistic solutions of the Schrddinger equation. Simultaneously the Dirac energy for these solutions becomes equal to the non-relativistic energy. For this reason the part of the Dirac spectrum which has the Schrddinger non-relativistic limit for c oo is referred to as the positive electronic energy spectrum and is associated with the dominant contribution of 4 in the 4 D-spinor (4.17). 

---