Douglas-Kroll-Hess operator

In the case of the Douglas-Kroll-Hess method, no special treatment of relativistic effects is required apart from the use of the one-electron Douglas-Kroll-Hess operator for the valence electrons. The direct and exchange potentials for the core electrons are treated in exactly the same way as in the nonrelativistic case but using the atomic Douglas-Kroll-Hess orbitals. However, only the unmodified part of the nuclear attraction is partitioned into a core and a valence part the core part is included with the core direct potential just as in the nonrelativistic case. 

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... 

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 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]. 

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... 

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. 

The transformed Hamiltonians that we have derived allow us to calculate intrinsic molecular properties, such as geometries and harmonic frequencies. We would like to be able to calculate response properties as well, with wave functions derived from the transformed Hamiltonian. If we used a method such as the Douglas-Kroll-Hess method, it would be tempting to simply evaluate the property using the nonrelativistic property operators and the transformed wave function. As we saw in section 15.3, the property operators can have a relativistic correction, and for properties sensitive to the environment close to the nuclei where the relativistic effects are strong, these corrections are likely to be significant. To ensure that we do not omit important effects, we must derive a transformed property operator, starting from the Dirac form of the property operator. 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

The lowest-order effect of relativity on energetics of atoms and molecules—and hence usually the largest—is the spin-free relativistic effect (also called scalar relativity), which is dominated by the one-electron relativistic effect. For light atoms, this effect is relatively easily evaluated with the mass-velocity and Darwin operators of the Pauli Hamiltonian, or by direct perturbation theory. For heavier atoms, the Douglas-Kroll-Hess method or the NESC le method provide descriptions of the spin-independent relativistic effect that are satisfactory for all but the highest accuracy. 

NR - nonrelativistic, PT-MVD - pCTturbative treatment of mass-velocity and Darwin operators (only SCF), DKH - Douglas-Kroll-Hess, RECP — relativistic effective core potential, DC - four-component Dirac-Coulomb, Exp - experiment. 

Liu W, Peng D. Exact two-component Hamiltonians revisited. J Chem Phys. 2009 131 031104. Nakajima T, Hirao K. The Douglas-Kroll-Hess Approach. Chem Rev. 2011 112 385-402. Belpassi L, Storchi L, Quiney HM, Tarantelli F. Recent advances and perspectives in four-component Dirac-Kohn-Sham calculations. Phys Chem Chem Phys. 2011 13 12368-12394. Peng D, Reiher M. Exact decoupling of the relativistic Fock operator. Theor Chem Acc. 2012 131 1081. 

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 ... 

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]. 

There have been many calculations performed on diatomic molecules that include spin-orbit coupling. Hess, Marian, and Peyerimhoff have reviewed calculations of transition metal containing diatomic molecules using their Douglas-Kroll spin-orbit operators. A compilation of relativistic ECP-based spin-orbit Cl calculations for transition metal containing diatomics is given by Balasubramanian. More recent diatomic molecules that have been studied include N2 HI and DI, TeH and TeLi, HCl ... 

A regular alternative to the Foldy-Wouthuysen transformation was given by Douglas and Kroll and later developed for its use in electronic structure calculations by Hess et al. The Douglas-Kroll (DK) transformation defines a transformation of the external-field Dirac Hamiltonian Hq of equation (11) to two-component form which leads, in contrast to the Foldy-Wouthuysen transformation, to operators which are bounded from below and can be used variationally, similarly to those of the regular approximations discussed above. As in the FW transformation, it is not possible in the DK formalism to give the transformation in closed form. Rather, it is... 

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... 

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