Hamiltonian 4-component

It is instructive to compare the 2-component Hamiltonian T-Cl with the Pauli Hamiltonian, which was derived in the previous section. The term Ep contains the relativistic free-particle energy, which is well-behaved for all values of the momentum. Ep, which is the kinetic energy operator for the positive-energy states, is a positive definite operator. In the Pauli Hamiltonian we see that this operator is expanded in powers of p/mc, which does not converge if p/mc > 1—a situation that will occur in any potential if the electron is sufficiently close to the nucleus. As mentioned above, the mass-velocity term is not bounded from below and so cannot be used variationally. 

All of the above procedures provide an approximate block diagonalization of the starting 4-component one-electron Hamiltonian from which a 2-component Hamiltonian may be extracted... 

Kedziera, D. (2005) Convergence of approximate two-component Hamiltonians How far is the Dirac limit. Journal of Chemical Physics, 123, 074109-1-074109-5. 

A second approach to achieve a reduction of the 4-component Hamiltonian to an electrons-only Hamiltonian is to introduce approximations by eliminating the small components of the wave function (41-53). Also here, different protocols have been successfully exploited in quantum chemistry. 

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

Sadlej AJ, Snijders JG, van Lenthe E, Baerends EJ (1995) Relativistic regular two-component Hamiltonians, four component regular relativistic Hamiltonians and the perturbational treatment of Dirac s equation. J. Chem. Phys. 102 1758-1766... 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

Once the matrix representation of the i -operator is known, it can immediately be used to determine the matrix representation of the two-component Hamiltonian h+, which is, of course, given by Eq. (44) and reads... 

The restriction to scalar terms and hence to a one-component Hamiltonian is sometimes referred to as a spin averaged formulation. However, the names spin-free or scalar relativistic formulation seem to be more appropriate. The matrix representations of all terms occurring after these manipulations are already available within every quantum chemical program package except for the PVP expressions. Their evaluation can, however, be reduced to the representation of the external potential via the relation... 

In this Chapter, we will show how a whole family of one- and two-component quasi-relativistic Hamiltonians can conveniently be derived. The operator difference between the quasi-relativistic Hamiltonians and the Dirac equation can be explicitly identified and used in perturbation expansions. Expressions are derived for a direct perturbation theory scheme based on quasi-relativistic two-component Hamiltonians. The remaining difference between the variational energy obtained using quasi-relativistic Hamiltonians and the energy of the Dirac equation is estimated numerically by applying the direct perturbation theory ap-... 

By inserting equation (10) into the energy expression for the Dirac equation (1) one obtains the effective four-component Hamiltonian HDirac)... 

Since we treat the hyperfine energy contribution by first-order perturbation theory we have a four-component Hamiltonian describing the hyperfine interaction according to... 

The two-component Hamiltonian of Eq. (1) is invariant under the time reversal operation. In the one-electron case and for a special choice of phases, the time reversal operator T is given by... 

More accurate two-component Hamiltonians can be obtained by successive application of higher-order DK transformations [12,49],... 

Recently, the accuracy of two-component Hamiltonians was analyzed in terms of [90,92],... 

In recent years, higher orders of the DK transformation were formulated and explored in benchmark calculations on small molecules. Furthermore, it was shown that highly accurate transformed two-component Hamiltonians can be generated via the DK transformations of higher orders. These Hamiltonians converge quite well for the known elements of the periodic table limits of accuracy become noticeable only for elements with Z > 120. Higher orders of DK transformed Hamiltonians yield only small corrections for molecular observables thus, for most applications with normal demands of accuracy, DK2 is a reasonable, efficient, and well established choice. A valuable alternative is provided by the ZORA scheme, as comparison of available results shows. On the other hand, in the near future, accurate four-component approaches are expected to be essentially restricted to benchmark calculations due to their computational requirements. 

Van Lenthe, E., R. van Leeuwen, E.J. Baerends and J.G. Snijders, 1994, Relativistic regular two-component Hamiltonians, in New Challenges in Computational Quantum Chemistry, eds R. Broer, P.J.C. Aerts and PS. Bagus (Department of Chemical Sciences and Material Science Centre, University of Groningen, Netherlands). 

Because of the li (i) operators in this expression, this Hamiltonian is (less rigorously) called a four-component Hamiltonian in order to distinguish it from more approximate Hamiltonians that contain one-electron operators refering to 2-spinor representations, which are therefore called two-component Hamiltonians. In Eq. (8.66) the nucleus-nucleus repulsion operators are incorporated in the last term on the right-hand side of that equation abbreviated as... 

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