Pseudo-large component

In order to analyze this question, the best point to start with is the so-called modified Dirac equation [547,718]. The modified Dirac equation is the basis of the so-called normalized elimination of the small component (NESC) worked out by Dyall [608,719-721]. Here, the small component ip of the 4-spinor tp is replaced by a pseudo-large component

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In chapter 10, we have already discussed how the size of the small-component basis set can be made equal to that of the large-component basis set by absorbing the kinetic-balance operator into the one-electron Hamiltonian. In this chapter, we have elaborated on this by introducing a pseudo-large component that has led to the modified Dirac equation. 

It is this operator that determines the symmetry characteristics of the small component relative to the large component the premultiplying factor belongs to the totally symmetric irrep. We will therefore introduce a pseudo-large component, ( ), defined by... 

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... 

At any finite value of c, however, the pseudo-large component differs from the large component. 

It should be noted that the choice of the pseudo-large component is not the only choice that could be made to effect the spin separation (Sadlej and Snijders 1994, Visscher and Saue 2000). We could have multiplied (or p) by any function of the coordinates to obtain a separation. What makes this ehoice unique (up to a scaling factor) is that the metric is spin free. Any other ehoiee results in a metric that has a spin-free part and a spin-dependent part. 

Note that it is the same angular and spin function for both large and pseudo-large components. 

In terms of the large and pseudo-large components, the integrals can be classified into three basic integral types. 

One important advantage of the modified Dirac equation is that, since the large component and the pseudo-large component have the same symmetry, we can use the same primitive basis set for both. However, if we want to use a contracted basis set, the contraction coefficients for these functions will differ. We will therefore distinguish the basis sets for the components when we expand them, which we do now ... 

We use the superscript P for the pseudo-large component, which represents both the initial letter, and also the fact that it is related to the small component through the momentum operator p. The one-electron modified Dirac equation in this basis set is then... 

The kinetic energy matrix is a full matrix of dimension rp- in both cases. As for the other two, the unmodified Dirac kinetic energy matrix is a quaternion matrix, with ArP unique quantities, whereas the spin-fl-ee modified Dirac kinetic energy matrix is a real matrix with rp unique quantities, and the resultant reduction is a factor of 4. However, if an uncontracted basis is used, the spin-free modified Dirac kinetic energy matrix is symmetric, and is the same as the pseudo-large-component overlap matrix. 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

There is a clear connection of these operators with those of the modified Dirac equation, except that they are operating on the large component rather than on the pseudo-large component... 

This equation has the advantages that the large and pseudo-large components have the same symmetry and can be expanded in the same basis set, and the way in which the speed of light appears makes it easy to identify relativistic terms of different orders. We expand the large and pseudo-large components in a basis set... 

In the direct elimination, we invert the second line of (19.3) to obtain a relation between the large- and pseudo-large-component coefficient vectors,... 

The issue of different energies in the denominator of (19.6) is a soluble problem. Consider the whole set of exact, positive-energy solutions, whose large- and pseudo-large-component vectors are and and whose eigenvalues are collected into a diagonal matrix E. The matrix equations are... 

Flooding and Pseudo-First-Order Conditions For an example, consider a reaction that is independent of product concentrations and has three reagents. If a large excess of [BJ and [CJ are used, and the disappearance of a lesser amount of A is measured, such flooding of the system with all components butM permits the rate law to be integrated with the assumption that all concentrations are constant except A. Consequentiy, simple expressions are derived for the time variation of A. Under flooding conditions and using equation 8, if x happens to be 1, the time-dependent concentration... 

Most distillation systems ia commercial columns have Murphree plate efficiencies of 70% or higher. Lower efficiencies are found under system conditions of a high slope of the equiHbrium curve (Fig. lb), of high Hquid viscosity, and of large molecules having characteristically low diffusion coefficients. FiaaHy, most experimental efficiencies have been for biaary systems where by definition the efficiency of one component is equal to that of the other component. For multicomponent systems it is possible for each component to have a different efficiency. Practice has been to use a pseudo-biaary approach involving the two key components. However, a theory for multicomponent efficiency prediction has been developed (66,67) and is amenable to computational analysis. 

Figures 18.13, through 18.17 show the experimental data and the calculations based on model I for the low temperature oxidation at 50, 75, 100, 125 and 150TZ of a North Bodo oil sands bitumen with a 5% oxygen gas. As seen, there is generally good agreement between the experimental data and the results obtained by the simple three pseudo-component model at all temperatures except the run at 125 TT. The only drawback of the model is that it cannot calculate the HO/LO split. The estimated parameter values for model I and N are shown in Table 18.2. The observed large standard deviations in the parameter estimates is rather typical for Arrhenius type expressions.

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