The Modified Dirac Equation

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. 

We will work with the Dirac equation in 2-spinor form. 

The second term on the right-hand side vanishes if u = v, and hence the spin dependence of the product is eliminated. To make use of this relation, we need to introduce another scalar product involving r into (15.1). This is achieved as follows. 

The operator (a-p) appears in the relation between the small and large components. 

For open shell systems, the z component of the spin-orbit interaction can contribute at first order since it preserves rris- 

---

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. 

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. 

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 ... 

The transformed two-electron operator bears a striking resemblance to the operator from the modified Dirac equation given in (15.43). We need only define [f = and = [c /( p- -mc )],4 [f andtheidentityiscomplete. Theanalysisoftheterms of the modified Dirac equation into scalar and spin-orbit terms in section 15.4 can then be transferred directly to the above equation. The kinematic factors are reintroduced at the end to obtain the final expressions. 

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... 

The mass-velocity term is therefore the lowest-order term from the relativistic Hamiltonian that comes from the variation of the mass with the velocity. The second relativistic term in the Pauli Hamiltonian is called the Darwin operator, and has no classical analogue. Due to the presence of the Dirac delta function, the only contributions for an atom come from s functions. The third term is the spin-orbit term, resulting from the interaction of the spin of the electron with its orbital angular momentum around the nucleus. This operator is identical to the spin-orbit operator of the modified Dirac equation. 

---