Small component nonrelativistic limit

For the nitrogen hyperfine tensors, there is no satisfactory empirical scheme for estimating the various contributions, so that Table II compares the total observed tensor to the DSW result. The tensors are given in their principal axis system, with perpendicular to the plane of the heme and along the Cu-N bond. The small values (0.1 - 0.2 MHz) found for A O in the nonrelativistic limit are not a consequence of orbital motion (which must vanish in this limit) but are the result of inaccuracies in the decomposition of the total tensor into its components, as described above. 

In the nonrelativistic limit (c °o), the small component is related to the large component by [20]... 

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... 

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,

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If we set c 00 in (4.81b) after dividing it by c, we find that -> 0, which implies that we must eliminate the small component before allowing c to go to infinity in order to obtain the nonrelativistic limit. Examining (4.81b), we see that it can be rearranged to make rlr the subject, provided that V — E — 2mc is nonzero. Now V is always zero or negative for the nuclear potential, so we must have E > —2mc in order that V — E — 2mc is never zero. (Compared with the free-particle case discussed in... 

In the previous section, we showed that the ratio of the large component to the small component was approximately pl2mc for a free-electron state. Given that the large component becomes the nonrelativistic wave function in two-component form in the limit c oo, we may use the nonrelativistic wave function as an approximation to the large component to gain some idea of how the small component behaves for a... 

For the purposes of comparison with the nonrelativistic limit, we eliminate the small component from these equations, using (7.27b) to express Q in terms of P and substituting into (7.27a), with the result... 

In this form the 2pi/2 does appear better behaved. The large component adds in another power of r in the lowest order, and yields the correct power dependence for a nonrela-tivistic 2p function. The problem of the singularity persists in the lowest order for the small component, but here the normalization factor Mq vanishes in the nonrelativistic limit, and so the small component also tends towards the correct behavior. As discussed in chapter 4, this is not true at r = 0, where the singularity persists. 

In particular, in the early days of relativistic quantum chemistry, attempts to solve the DHF equations in a basis set expansion sometimes led to unexpected results. One of the problems was that some calculations did not tend to the correct nonrelativistic limit. Subsequent investigations revealed that this was caused by inconsistencies in the choice of basis set for the small-component space, and some basic principles of basis-set selection for relativistic calculations were established. The variational stability of the DHF equations in a finite basis has also been a subject of debate. As we show in this chapter, it is possible to establish lower variational bounds, thus ensuring that the iterative solution of the DHF equations does not collapse. 

This condition, which ensures that the kinetic energy is properly represented in the nonrelativistic limit, is known as kinetic balance (Stanton and Havriliak 1984). It is an important condition on the basis sets because it defines a relationship between the large- and small-component basis functions that must be satisfied in the nonrelativistic limit. ... 

We see that the magnetic dipole operator connects large and small components, but the prefactor c ensures that the results are of the same magnitude as expectation values between two large components. We can demonstrate this point by substituting for the small component using the approximate expression from the nonrelativistic limit... 

This is a valid definition, as it only requires that the small component be integrable, which is certainly the case sinee must be square integrable in order to normalize the Dirac wave function. The pseudo-large component now has the same symmetry properties as the large component. The nonrelativistic limit of the pseudo-large component is the large component, since... 

The consequence is that we must treat the spin-orbit and the spin-other-orbit interactions separately we cannot combine them as in the Breit-Pauli Hamiltonian. The reason is that the functions on which the momentum operators operate are derived from the small component, and only in the nonrelativistic limit where the large and small components are related by kinetic balance can we rewrite the spatial part of the spin-other-orbit interaction in the same form as the spin-orbit interaction. The reader... 

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]. 

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