Douglas-Kroll second-order

DKH2 Second order Douglas-Kroll-Hess transformation... 

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... 

The Douglas-Kroll (DK) approach [153] can decouple the large and small components of the Dirac spinors in the presence of an external potential by repeating several unitary transformations. The DK transformation is a variant of the FW transformation [141] and adopts the external potential Vg t an expansion parameter instead of the speed of light, c, in the FW transformation. The DK transformation correct to second order in the external potential (DK2) has been extensively studied by Hess and co-workers [154], and has become one of the most familiar quasi-relativistic approaches. Recently, we have proposed the higher order DK method and applied the third-order DK (DK3) method to several systems containing heavy elements. 

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]. 

A self-consistent scalar-relativistic (SR) version of the LCGTO-DF method has also been developed recently." "The SR variant employs a unitary second-order Douglas-Kroll-Hess (DKH) "" transformation for decoupling large and small components of the full four-component spinor solutions to the Dirac-Kohn-Sham equation. The approximate DKH transformation, very appropriate and efficient for molecular calculations, has been implemented this variant utilizes nuclear potential-based projectors and leaves the electron-electron interaction untransformed. 

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 scalar relativistic (SR) corrections were calculated by the second-order Douglas-Kroll-Hess (DKH2) method [53-57] at the (U/R)CCSD(T) or MRCI level of theory in conjunction with the all-electron aug-cc-pVQZ-DK2 basis sets that had been recently developed for iodine [58]. The SR contributions, as computed here, account for the scalar relativistic effects on carbon as well as corrections for the PP approximation for iodine. Note, however, that the Stuttgart-Koln PPs that are used in this work include Breit corrections that are absent in the Douglas-Kroll-Hess approach [58]. 

The chapter consists of two parts. In the first part I discuss some aspects of relativistic theory, the accuracy of the infinite order two-component relativistic lOTC method and its advantage over the infinite order Douglas-Kroll-Hess (DKHn) theory, in the proper description of the molecular spectroscopic parameters. Spin-free and spin dependent atomic mean filed (AMFI) theory is presented. Additionally, the accuracy of the relevant potential energy curves is discussed as well. In the second part I show the role of the QED corrections and that they are necessary for the correct description of the spectroscopic properties of atoms for the X-ray spectra. Some examples of the molecular QED calculations will be discussed here as well. 

Douglas and Kroll (1974) modified the elimination of the positronic one-particle states in a way that leads to relativistic operators suitable for the variational approaches used in quantum chemistry. Instead of expansions in powers of d the transformation is constructed to lead to expansions in powers of the external potential. The ideas of Douglas and BCroll were followed and implemented by Hess (1986), Jansen and Hess (1989a) and Samzow et al. (1992). Correct to second order in the potential the Douglas-Kj-oll-Hess (DKH) one-electron Hamiltonian is... 

In preceding sections we have discussed several different relativistic methods four-component Dirac—Fock with and without correlation energy, the second-order Douglas—Kroll method, and perturbation methods including the mass—velocity and Darwin terms. The relativistic effective core potential (RECP) method is another well-established means of accounting for certain relativistic effects in quantum chemical calculations. This method is thoroughly described elsewhere - anJ is basically not different in the relativistic and... 

In most current applications of the Douglas-Kroll transformation, the Hamiltonian is truncated at second order in the successive unitary transformation, that is, Ui, and the resulting Hamiltonian can be written in... 

The second-order one-electron Douglas-Kroll Hamiltonian has found wide application in quantum chemistry programs through approximations that are discussed in the next two sections. Although it is a considerable improvement on the first-order Hamiltonian, for some heavy elements the error is significant. Hamiltonians through fifth order have been derived by Nakajima and Hirao (2000). The third-order Hamiltonian is given by... 

Equation (16.69) can now solved by an iterative procedure, making use of the orders of the various terms in powers of 1/c to define a procedure that should be convergent. Since (Hi)22 is of order and ( ii)2i is of order the maximum order of Xi is c (as was found for >Vi in the Douglas-Kroll transformation). The three terms on the right-hand side of (16.69) are then of order c, c, and c, respectively. The iteration is started by neglecting the second and third terms of (16.69). As H )22 is of order c, this gives a transformation that is correct to order c ... 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

To illustrate, we use the second-order Barysz-Sadlej-Snijders transformation, which is more transparent than the Douglas-Kroll transformation. Introducing a perturbation parameter X, the Hamiltonian including the electric perturbation is... 

This is a scalar term the spin-dependent terms that are second order in the potential have a higher leading power of l(mc). The Douglas-Kroll correction including the nuclear potential term can be derived from the lowest-order part of the term,... 

The first parenthesis contains the first-order term from the free-particle Foldy-Wouthuysen transformation, and the second parenthesis contains the second-order Douglas-Kroll term. Both A and Ep are spin-free, so the spin-dependence comes from the transformations involving 1 2, which was defined as... 

The Second-Order Term of the Douglas-Kroll Expansion... 

In this appendix, we examine the content of the second-order term of the Douglas-Kroll expansion given by (16.61). For this purpose, we expand the products of the operators involving Wi. It is only necessary for this purpose to consider which expands to... 

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