Douglas-Kroll operator

What we must not do is to add an all-electron spin-orbit operator, such as the Breit-Pauli or Douglas-Kroll operator, because the effect of the core is not included in these operators, nor is the removal of the core tail. The all-electron spin-orbit operators behave as 1/r, and since the pseudospinors have minimal core amplitude, the spin-orbit effect will be grossly underestimated. 

It is clear from Ho that the Douglas-Kroll transformation makes use of a model space of relativistic free-particle spinors, and that it is defined by a perturbative expansion with the external potential as perturbation. Indeed, using the formulas given above, we get the familiar expressions for the second-order Douglas-Kroll-transformed Dirac operator, which is often dubbed Douglas-Kroll-Hess (DKH) operator... 

If a multiparticle system is considered and the election interaction is introduced, we may use the Dirac-Coulomb-Breit (DCB) Hamiltonian which is given by a sum of one-particle Dirac operators coupled by the Coulomb interaction 1 /r,7 and the Breit interaction Bij. Applying the Douglas-Kroll transformation to the DCB Hamiltonian, we arrive at the following operator (Hess 1997 Samzow and Hess 1991 Samzow et al. 1992), where an obvious shorthand notation for the indices pi has been used ... 

DK approximation and it will be shown that the result is independent of the chosen parametrisation. This approach has not been investigated in the literature so far. We will denote the resulting operator equations as the generalised Douglas-Kroll transformation. We conclude this section by a presentation of some technical aspects of the implementation of the DK Hamiltonian into existing quantum chemical computer codes. 

In the Douglas-Kroll-Hess spin-free relativistic Hamiltonians (second-order and third-order) [11,13], the T andF operators in Eq. (4) are... 

Operators that result from a DK transformation are directly given in the momentum representation. Hess et al. [29,31] developed a very efficient strategy to evaluate the corresponding matrix elements in a basis set representation it employs the eigenvectors of the operator as approximate momentum representation [29,31]. In practice, the two-component DK Hamiltonian is built of matrix representations of the three operators p, V, pVp + id(pV x p). This Douglas-Kroll-Hess (DKH) approach became one of the most successful two-component tools of relativistic computational chemistry [16,74]. In particular, many applications showed that the second-order operator 2 Is variationally stable [10,13,14,31,75,76,87]. 

The reduction of the relativistic many-electron hamiltonian by expansion in powers of the external field is the second-order Douglas-Kroll transformation [29], and has been used with success by Hess and co-workers [30]. The operators which result from this transformation are non-singular, but the integrals over the resulting operators are complicated and have to be approximated, even for finite basis set expansions. The reduction of the Dirac-Coulomb-Breit equation to two-component form using direct perturbation theory has been described by Kutzelnigg and coworkers [26, 27, 31], Rutkowski [32], and van Lenthe et al. [33]. 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

Breit-Pauli and Douglas-Kroll Spin-Orbit-Coupling Operators... 

Two detailed discussions about the Breit-Pauli (BP-) and Douglas-Kroll (DK-) SOC operators are given in our recent publications [175,180] and here we only give a short overview on this subject, as the background knowledge of our development of DK-SOC-adapted MCP. The one-electron BP-SOC and DK-SOC operators are related via... 

The above idea was the basis of the BSS method formulated by Barysz and Sadlej [8]. The BSS method has its roots in the historically earlier Douglas-Kroll-Hess (DKH2 and DKH3) [9, 10] approximation. In the BSS approximation the fine struc-tme constant a is the pertmbative parameters and it differs from the DKH method where the potential V is nsed as the pertmbation. Formally the BSS and DKH methods are of the infinite order in a or V. However, the necessity to define the analytical form of the R operator and the Hamiltonian in each step of the iteration, makes the accmacy of both methods limited to the lowest order in a or V. 

Of course, what has just been stated for the one-electron Dirac Hamiltonian is also valid for the general one-electron operator in Eq. (11.1). However, the coupling of upper and lower components of the spinor is solely brought about by the off-diagonal ctr p operators of the free-partide Dirac one-electron Hamiltonian and kinetic energy operator, respectively. We shall later see that the occurrence of any sort of potential V will pose some difficulties when it comes to the determination of an explicit form of the unitary transformation U. A universal solution to this problem will be provided in chapter 12 in form of Douglas-Kroll-Hess theory. 

Chapter 11 introduced the basic principles for elimination-of-the-small-component protocols and noted that the Foldy Wouthuysen scheme applied to one-electron operators including scalar potentials yield ill-defined 1 /c-expansions of the desired block-diagonal Hamiltonian. In contrast, the Douglas Kroll-Hess transformation represents a unique and valid decoupling protocol for such Hamiltonians and is therefore investigated in detail in this chapter. 

Historically, the first derivations of approximate relativistic operators of value in molecular science have become known as the Pauli approximation. Still, the best-known operators to capture relativistic corrections originate from those developments which provided well-known operators such as the spin-orbit or the mass-velocity or the Darwin operators. Not all of these operators are variationally stable, and therefore they can only be employed within the framework of perturbation theory. Nowadays, these difficulties have been overcome by, for instance, the Douglas-Kroll-Hess hierarchy of approximate Hamiltonians and the regular approximations to be introduced in a later section, so that operators such as the mass-velocity and Darwin terms are no... 

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