Four component regular relativistic Hamiltonians and the perturbational treatment of Dirac’s equation (original) (raw)
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Perturbation energy expansions based on two-component relativistic Hamiltonians
Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), 2003
The approximate elimination of the smallcomponent approach provides ansa¨tze for the relativistic wave function. The assumed form of the small component of the wave function in combination with the Dirac equation define transformed but exact Dirac equations. The present derivation yields a family of two-component relativistic Hamiltonians which can be used as zerothorder approximation to the Dirac equation. The operator difference between the Dirac and the two-component relativistic Hamiltonians can be used as a perturbation operator. The first-order perturbation energy corrections have been obtained from a direct perturbation theory scheme based on these two-component relativistic Hamiltonians. At the two-component relativistic level, the errors of the relativistic correction to the energies are proportional to a 4 Z 4 , whereas for the relativistic energy corrections including the first-order perturbation theory contributions, the errors are of the order of a 6 Z 6 -a 8 Z 8 depending on the zeroth-order Hamiltonian.
Relativistic regular two-component Hamiltonians
International Journal of Quantum Chemistry, 1996
In this paper, potential-dependent transformations are used to transform the four-component Dirac Hamiltonian to effective two-component regular Hamiltonians. To zeroth order, the expansions give second order differential equations (just like the Schrodinger equation), which already contain the most important relativistic effects, including spin-orbit coupling. One of the zero order Hamiltonians is identical to the one obtained earlier by Chang, Pelissier, and Durand [Phys. Ser. 34, 394 (1986)]. Self-consistent all-electron and frozen-core calculations are performed as well as first order perturbation calculations for the case of the uranium atom using these Hamiltonians. They give very accurate results, especially for the one-electron energies and densities of the valence orbitals.
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International Journal of Quantum Chemistry, 1986
A transformation decoupling the Dirac second-order equations for the large and small components of the radial atomic wavefunction has been derived. The decoupling procedure is exact for one-electron atoms and appmximate, though very accurate, for a general case of an atomic spherical potential. Its connection with the quasirelativistic theory of atoms has been discussed. The transformation has been applied to deriving an optimized quasirelativistic equation and to establishing relations between the quasirelativistic and the Dirac wavefunctions. Formulas resulting from these considerations allow improvement of quasirelativistic expectation values of operators so that the error, relative to the Dirac values, is reduced by a factor of 10.
Chemical Physics, 2008
Direct perturbation theory (DPT) and its quasi-degenerate version (QD-DPT) in a matrix formulation, i.e. DPT-mat and QD-DPT-mat are derived from the matrix representation of the Dirac operator in a kinetically balanced basis, both in the intermediate and the unitary normalization. The results are compared with those of an earlier formulation in terms of operators and wave functions. In the wave function formulation it is imperative to describe the weak singularities of the wave function at the position of a point nucleus correctly and to satisfy the key relation between large and small components locally. This formulation is incompatible with an expansion in a regular basis. In a matrix formulation in a kinetically balanced basis both the large and the small component are expanded in regular basis sets and the key relation is only satisfied in the mean. DPT is essentially a theory at bispinor level. Although it is possible to eliminate the small component to arrive at a quasi-relativistic theory, this requires some care. A both compact and numerically stable formulation is in terms of the large and the small component. The generalization from a theory for one state to a quasi-degenerate formulation for a set of states, is very simple in the matrix formulation, but rather complicated and somewhat indirect at wave function level, where an intermediate quasi-relativistic step is needed. The advantage of the matrix formulation is particularly pronounced in the unitary normalization.
Dirac equation in very special relativity for hydrogen atom
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The improved Dirac equation is completely solved in the case of the hydrogen atom. A method of separation of variables in spherical coordinates is used. The angular functions are the same as with the linear Dirac equation: they account for the spin 1/2 of the electron. The existence of a probability density governs the radial equations. This gives all the quantum numbers required by spectroscopy, the true number of energy levels and the true levels obtained by Sommerfeld’s formula.
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