Douglas-Kroll-Hess Hamiltonian/method

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

This Hamiltonian can then be used variationally in quantum chemical calculations, since because of its derivation no negative energy states can occur. It should be anticipated that this Hamiltonian is conceptually equivalent to the infinite-order Douglas-Kroll-Hess Hamiltonian to be discussed in section 12.3, because both schemes do not apply any expansion in 1/c. Also the expressions for Ep and Ap are strictly evaluated in closed form within both approaches. However, whereas Douglas-Kroll-Hess theory yields analytic exressions for each order in V, the infinite-order two-component method summarizes all powers of V in the final matrix representation of/+. 

We extend the method over all three rows of TMs. No systematic study is available for the heavier atoms, where relativistic effects are more prominent. Here, we use the Douglas-Kroll-Hess (DKH) Hamiltonian [14,15] to account for scalar relativistic effects. No systematic study of spin-orbit coupling has been performed but we show in a few examples how it will affect the results. A new basis set is used in these studies, which has been devised to be used with the DKH Hamiltonian. 

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

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. 

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 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. 

The second major method leading to two-component regular Hamiltonians is based on the Douglas-Kroll transformation (Douglas and Kroll 1974 Hess 1986 Jansen and Hess 1989). The classical derivation makes use of two successive unitary transformations... 

General two-component methods have been discussed in various chapters of the first part of this book, for instance in chapter 11 on Two-Component Methods and the Generalised Douglas-Kroll Transformation by Wolf, Reiher and Hess [165], in chapter 12 by Kutzelnigg on Perturbation Theory of Relativistic Effects [166] and in chapter 13 by Sundholm on Perturbation Theory Based on Quasi-Relativistic Hamiltonians [167]. 

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