Hamiltonian Douglas-Kroll

All calculations are scalar relativistic calculations using the Douglas-Kroll Hamiltonian except for the CC calculations for the neutral atoms Ag and Au, where QCISD(T) within the pseudopotential approach was used [99], CCSD(T) results for Ag and Au are from Sadlej and co-workers, and Cu and Cu from our own work, using an uncontracted (21sl9plld6f4g) basis set for Cu [6,102] and a full active orbital space. 

We have presented results from a theoretical study of the electronic and molecular properties of the PbO molecule. A multiconfigurational approach has been used, which essentially works with non-relativistic wave functions. Relativity is introduced in two steps. First scalar relativistic elfects are included through the use of the Douglas-Kroll Hamiltonian and a correspondingly constructed AO basis set. Spin-orbit effects are included by allowing different CASSCF wave functions to mix under the influence of a spin-orbit Hamiltonian. 

The spin-orbit operators in the Douglas-Kroll Hamiltonian still contain the 1/r term however, that term is offset by the l/( , -b term, where as... 

Models related to spin-forbidden reactions are discussed in this chapter. Coupling between two surfaces of different spin and symmetry is given by various levels of approximation for spin-orbit operators from the reduction of relativistic quantum mechanics. Well-established methods such as the Breit-Pauli Hamiltonian exist, but new relativistic methods such as the Douglas-Kroll Hamiltonian and other new transformation schemes are also being investigated and implemented today. 

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... 

Notice that the additional term only involves VVi. This is an instance of the familiar 2n- -1 rule of perturbation theory. Here, the operators up to VV are all that are needed to determine the Hamiltonian of order 2n - -1. Higher-order transformations have also been derived and examined by Wolf et al. (2002), to which the reader is referred for details. The Douglas-Kroll Hamiltonian of order n is often written as Hdkii or... 

The reader can easily verify that in the limit p -> 0 in the kinematic factors this yields the expression obtained in the Breit-Pauli Hamiltonian. Thus, for the Douglas-Kroll Hamiltonian, calculating the primitive integrals over basis funetions for these operators will involve the same work as for the Breit-Pauli Hamiltonian, but at the same time the kinematic factors will have to be accounted for. 

Hobza and coworkers performed a comparative study of Ag, Au, and Pd atoms binding to graphene [114] with electron correlation (CCSD(T) and MP2 with Douglas-Kroll Hamiltonians), conventional (i.e., LDA and GGA) and dispersionaccounting DFT methods (PBE-D3, M06-2X, vdW-DF, and EE -I- vdW). Binding of these metals to graphene is of varied nature, but it is due to electron correlation, as ROHF/ANO-RCC-VTZP benzene-metal potential energy curves have no minima. 

Symmetry-adapted perturbation theory Computation with the Douglas-Kroll Hamiltonian... 

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. 

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. 

Thakkar and Lupinetti5 have used the coupled-cluster method in conjunction with the Douglas-Kroll relativistic Hamiltonian to obtain a very accurate value for the static dipole polarizability of the sodium atom. Their revised value for a(Na) = 162.88 0.6 au resolves a previous discrepancy between theory and experiment and when combined with an essentially exact value for lithium, establishes the ratio a(Li)/a(Na) = 1.0071 0.0037, so that, because of the... 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

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. 

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... 

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