Pauli expansion

Flere, we only keep terms having order of magnitude mc o. One can compare this expansion with the one-fermion Pauli expansion presented in equation (53), given the fact that in the one-fermion case 8 — qcpl and = vector algebra. The resulting two-fermion Flamiltonian can be extended to many-fermions quite trivially and the NRF of the many-fermion Flamiltonian can... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

We have focused on the prohlems associated with extending Dirac s one-fermion theory smoothly to many-fermion systems. A brief discussion of QED many-fermion Hamiltonians also was given. A comprehensive account of the problem of decoupling Dirac s four-component equation into two-component form and the serious drawbacks of the Pauli expansion were presented. The origins of the DSO and FC operators have been addressed. The working Hamiltonian which describe NMR spectra is derived. 

There are many problems in e.g. catalysis in which relativity may play a deciding role in the chemical reactivity. These problems generally involve large organic molecules which cannot be handled within the Dirac Fock framework. It is therefore necessary to reduce the work by making additional approximations. Generally used approaches are based on the Pauli expansion or on the Douglas Kroll transformation [3]. 

Both spin-orbit coupling and Zeeman interactions are obtained as terms in the Pauli expansion of the Dirac equation. 

The Pauli expansion results from taking 2mc out of the denominator of the equation for the elimination of the small component (ESC). The problem with this is that both E and V can potentially be larger in magnitude than 2mc and so the expansion is not valid in some region of space. In particular, there is always a region close to the nucleus where V - E /2mc > 1. An alternative operator to extract from the denominator... 

We now define the tliree-dimensional vectors, Kand fl, consisting of the coefficients of the Pauli matrices in the expansion of p and//, respectively ... 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

The 2p orbital radius may be considered anomalously small (of the same order as the 2s orbital radius) because there is no inner shell of the same angular symmetry that exerts outward steric pressure due to the Pauli exclusion principle. (A similar exception causes the first transition series to appear anomalous compared with later lanthanides, since 3d orbitals form the innermost d shell.) The 2p -> 3p expansion therefore appears to be relatively more pronounced than 2s —> 3s expansion. 

Equation (1) is obtained by using an expansion in E/ 2c - Vc) on the Dirac Fock equation. This expansion is valid even for a singular Coulombic potential near the nucleus, hence the name regular approximation. This is in contrast with the Pauli method, which uses an expansion in (E — V)I2(. Everything is written in terms of the two component ZORA orbitals, instead of using the large and small component Dirac spinors. This is an extra approximation with respect to the original formalism. 

Here we have used the natural expansion (33), with spin-orbitals written in the form (29). The second term in (41), absent in a Pauli-type approximation, contains the correction arising from the use of a 4roomponent formulation it is of order (2tmoc) and is usually negligible except at singularities in the potential. As expected, for AT = 1, (41) reproduces the density obtained from a standard treatment of the Dirac equation but now there is no restriction on the particle number. 

Though the ESR Hamiltonian is typically expressed in terms of effective electronic and nuclear spins, it can, of course, also be derived from the more fundamental Breit-Pauli Hamiltonian, when the magnetic fields produced by the moving nuclei are explicitly taken into account. In order to see this, we shall recall that in classical electrodynamics the magnetic dipole equation can be derived in a multipole expansion of the current density. For the lowest order term the expansion yields (59)... 

A complete derivation involves a complete expansion of the Pauli Hamiltonian and the recognition that for the two complexified vector potentials A11 and A 2 that one has the term... 

It can be proven [31] that all possible Slater determinants of N particles constructed from a complete system of orthonormalized spin-orbitals 4>k form a complete basis in the space of normalized antisymmetric (satisfying the Pauli principle) functions, of N electrons i.e. for any antisymmetric and normalizable (K one can find expansion amplitudes so that ... 

Table 1-1. Convergence of the polarization expansion for the interaction of two ground-state helium atoms at R = 1 and 5.6 bohr, and of the lithium and hydrogen atoms in their ground states at R = 10 and 12 bohr. The Coulomb energies represent 53.50% (He2) R = 5.6bohr), 73.4% (LiH, R = 10 bohr), and 85.53% (LiH, R= 12 bohr) of the energies of the fully symmetric (Pauli forbidden) states. The quantity S(n) represents the percent error of the perturbation series through the nth-order with respect to the variational interaction energy of the Pauli forbidden state...

---