Douglas-Kroll Hess Transformation

Reiher, M. and Wolf A. (2004) Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas—Kroll—Hess transformation up to arbitrary order. Journal of Chemical Physics, 121, 10945-10956. 

Molecules are more difficult to treat accurately than atoms, because of the reduced symmetry. An additional complication arises in relativistic calculations the Dirac-Fock-(-Breit) orbitals will in general be complex. One way to circumvent this difficulty is by the Douglas-Kroll-Hess transformation [57], which yields a one-component function with computational effort essentially equal to that of a nonrelativistic calculation. Spin-orbit interaction may then be added as a perturbation, implementation to AuH and Au2 has been reported [58]. Progress has also been made in the four-component formulation [59], and the MOLFDIR package [60] has been extended to include the CC method. Application to SnH4 has been described [61] here we present a recent calculation of several states of CdH and its ions [62], with one-, two-, and four-component methods. 

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

Since it is actually impossible to sum the infinite series, this summation is terminated at particular numbers of unitary transformations 2 in DK2, 3 in DK3 and so forth. This method is also called the Douglas-Kroll-Hess transformation, because it was revised by Hess and coworkers (Jansen and Hess 1989). 

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. 

For the discussion of the Douglas-Kroll-Hess transformation in chapter 12 the Fourier transformation of a product of two functions, h x) = f(x)gix), has been employed. In one dimension it is given by the convolution integral of the Fourier transformations of / and g. 

For the discussion of the Douglas-Kroll-Hess transformation in chapter 12 it has been advantageous to consider the momentum-space representation of the Coulomb potential, which may be obtained via a Fourier transformation of V (r). It is given by... 

J. C. Boettger. Approximate two-electron spin-orbit coupling term for density-functional-theory DFT calculations using the Douglas-Kroll-Hess transformation. Phys. Rev. B, 62(12) (2000) 7809-7815. 

R. Fukuda, M. Hada, H. Nakatsuji. Quasirelativistic theory for the magnetic shielding constant. I. Farmulation of Douglas-Kroll-Hess transformation for the magnetic field and its application to atomic systems. J. Chem. Phys., 118(3) (2003) 1015-1026. 

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