Pauli elimination

We recover the rest energy -2 and the nonrelativistic contributions the leading terms of the Foldy-Wouthuysen series. The term fp] offers low-order relativistic corrections to them. The form of the operator in the first term of Eq. (11.84) resembles the classical expression derived in Eq. (3.122). We shall investigate the physical meaning of the second term on the right-hand side of Eq. (11.84) in section 11.5.2. Moreover, we will encounter these operators again in section 13.1 where we will discuss how the Pauli elimination produces relativistic corrections to the nonrelativistic kinetic energy operator of the Pauli Eq. (5.140). 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

If p 7 0 Eqs. (21) and (22) are coupled, but the relations between components of the wavefunction are much simpler than in the standard Dirac-Pauli representation. By the elimination of and respectively from Eq. (21) and from Eq. (22), we get two decoupled second-order equations for and ... 

An alternative treatment of the correction of order Z a) Za) m/M)m was given in [4]. The idea of this work was to modify the standard definition of the proton charge radius, and include the first order quantum electrodynamic radiative correction into the proton radius determined by the strong interactions. Prom the practical point of view for the nS levels in hydrogen the recipe of [4] reduces to elimination of the constant 11/72 in (5.6) and omission of the Pauli correction in (5.7). Numerically such a modification reduces the contribution to the lA energy level in hydrogen by 0.14 kHz in comparison with the naive result in (5.6), and increases it by 0.03 kHz in comparison with the result in (5.8). Hence, for all practical needs at the current level of experimental precision there are no contradictions between our result above in (5.8), and the result in [4]. 

Because often only the field-free Pauli Hamiltonian is presented in literature, we shall briefly sketch the derivation of the Hamiltonian hPauh(i) within an external field. For this, we start with the elimination of the small component in the one-electron Dirac equation by substitution of the small component of Eq. (15) to obtain an expression of the large component only... 

The sum over internal lines is unrestricted so (linked) exclusion principle violating (e-v38 or EPV27) terms are included in the sums. Note that either restricting sums to eliminate both unlinked and linked EPV diagrams or unrestricting summations and doing a simple cancellation of unlinked terms satisfies the Pauli exclusion principle.27... 

For ordinary matter, terms of higher than second order in (v/c) are often very small. Hence it is desirable to have a set of weakly relativistic (Pauli-type rather than Dirac-type) BdG equations. By systematic elimination of the lower components of the DBdG equations (i.e. those which are suppressed by factors of (v/c) in the weakly relativistic limit) we obtain from Equations (5.18) and (5.19) the two equations (Capelle and Gross 1995,1999b) ... 

In order to eliminate problems assoeiated with the differenees in surface properties and shapes of coimnercially available dehydrated vermicelli, Andre and Pauli (1978) ground the samples down to powders, then formed tablets from the powders using a hydraulic press. The 40 mm diameter by 10 mm thick tablets could be measured directly on the head of the colorimeter. Good correlations were found between colour co-ordinate values and the / -carotene content derived from the egg component of the pastas. Pastas made from different flours could also be differentiated in terms of colour values. 

The historically first reduction of the Dirac equation to two-component form is the Pauli approximation, which can be obtained from Eq. (26) by trancating the series expansion for cu after the first two terms, and eliminating the energy dependence by means of a systematic expansion in c. The result is the familiar Pauli Hamiltonian... 

This procedure is usually known as the elimination of the small component (ESC), and Eq. (34) is still equivalent to the original Dirac equation. Although the equation has been reduced to a two-component form, nothing is gained since we now have an energy-dependent Hamiltonian, and one must introduce further approximations to transform Eq. (34) into a form useful for actual calculations. The principal difference between the Pauli and the ZORA Hamiltonian is that to obtain the Pauli Hamiltonian, one uses an expansion in c ... 

Because of the Pauli principle antisymmetry requirement, the ground-state wave function has nodal surfaces in 3n-dimensional space, and to ensure that the walkers converge to the ground-state wave function, one must know the locations of these nodes and must eliminate any walker that crosses a nodal surface in the simulation. In the fixed-node (FN) DQMC method, the nodes are fixed at the locations of the nodes in a known approximate wave function for the system, such as found firom a large basis-set Hartree-Fock calculation. This approximation introduces some error, but FN-DQMC calculations are variational. (In practice, the accuracy of FN-DQMC calculations is improved by a procedure called importance sampling. Here, instead of simulating the evolution of with t, one simulates the evolution off, where / = where is a known accurate trial variation function for the ground state.)... 

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