Dirac equation modified

In the quaternion modified Dirac equation the spin-free equation is thereby obtained simply by deleting the quaternion imaginary parts. For further details, the reader is referred to Ref. [13]. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

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... 

The latest major achievement in the field of elinnination techniques for the small component is due to Dyall and has been worked out to an efficient computational tool for quantum chemistry within the last few years [36-39]. This method is commonly dubbed normalised elimination of the small component (NESC) and is based on the modified Dirac equation [40,41], where the small component (f> of the 4-spinor 4> is replaced by a pseudolarge component defined by the relation... 

An insertion of this relation into the split-form of the Dirac equation, Eq. (5) and (6), yields the modified Dirac equation... 

In order to derive a useful perturbation theory expression with equation (21) as the zeroth-order equation, the modified Dirac equation (20) has to be reformulated in such a way that the operator difference between equations (20) and (21) can be identified and used as a perturbation operator. 

By inserting equation (25) into the modified Dirac equation (20) a new exact expression for the Dirac equation (26) is obtained. 

The procedure not to apply the (a p)-operator onto the basis functions but to consider it explicitly and rewrite the one-electron equation is a major trick which has been dicsussed by many authors and which is intimately connected with the modified Dirac equation and the so-called exact-decoupling methods discussed in detail in section 14.1. 

From the Modified Dirac Equation to Exact-Two-Component Methods... 

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

[Pg.531]

The modified Dirac equation can now be viewed from two different perspectives. The first perspective is the fact that the approximate kinetic balance condition of Eq. (5.137) has been exploited. The normalized elimination procedure then results in energy eigenvalues which deviate only in the order c from the correct Dirac eigenvalues, whereas the standard un-normalized elimination techniques are only correct up to the order c. In addition, the NESC method is free from the singularities which plague the un-normalized methods and can be simplified systematically by a sequence of approximations to reduce the computational cost [562,720,721]. From a second perspective, Eq. (14.1) defines an ansatz for the small component, which, as such, is not approximate. Hence, Eq. (14.4) can be considered an exact starting point for numerical approaches that aim at an efficient and accurate solution of the four-component SCF equations (without carrying out the elimination steps). We will discuss this second option in more detail in the next section. 

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]

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. 

L. Visscher, T. Saue. Approximate relativistic electronic structure methods based on the quaternion modified Dirac equation. /. Chem. Pkys., 113(10) (2000)3996-4002. 

K. G. Dyall. Interfacing relativistic and nonrelativistic methods. 1. Normalized elimination of the small component in the modified Dirac equation. J. Chem. Phys., 106(23) (1997) 9618-9626. 

In most cases, spin-free relativistic effects dominate the relativistic corrections to electronic structure. We will show later that in a perturbation expansion based on the nonrelativistic wave function, the spin-free effects for a closed-shell system enter in first order, whereas the spin-dependent effects make their first contribution in second order. Thus a reasonable approach to the treatment of relativistic effects is to include the spin-free effects fully and treat the spin-dependent effects as a perturbation. We discuss the latter task in chapter 21. In this chapter, we will examine a modification to the Dirac equation that permits the spin-fi ee and spin-dependent terms to be separated (Kutzelnigg 1984, Dyall 1994). This separation is exact, in that no approximations have been made to obtain the separation, and therefore results obtained with the modified Dirac equation are identical to those obtained with the unmodified Dirac equation. The advantage of the separation is the identification of the genuine spin-dependent terms and the possibility of their omission in approximate calculations. This development also provides a basis for discussion and analysis of spin-free and spin-dependent operators in other approximations. 

In the next section, we will examine some properties of the solutions of the spin-free modified Dirac equation, and we then proceed to a closer inspection of the one-electron operators in the modified formalism before treating the two-electron terms. 

Solutions of the Spin-Free Modified Dirac Equation... 

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