Relativistic Hamiltonian

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

In this section, we review the relativistic Hamiltonian with special emphasis on how interactions are introduced. Within the Bom-Oppenheimer approximation the electronic Hamiltonian, relativistic or not, has the generic form... 

Hess, B. A. Dolg, M. Relativistic Quantum Chemistry with Pseudopotentials and Transformed Hamiltonians, Relativistic Effects in Heavy-Element Chemistry and Physics -, Ed. Hess, B. A. Wiley Chichester, 2002, pp. 89-122. 

Our work has used the Coulomb (i.e., nonrelativistic) or the Breit-Pauli (i.e., quasi-relativistic) Hamiltonian. Relativistic structures and wavefunc-tions, without or with relativistic radials, follow fhe same concepfs as in the nonrelativistic treatment, only that here, the differences befween the level energies are generally smaller and, hence, the sensitivity of fhe mixing coefficients to the choice of orbitals and of the mixing configurations is expected to be much higher. 

Y. Ishikawa, K. Koc. Relativistic many-body perturbation theory based on the no-pair Dirac-Coulomb-Breit Hamiltonian Relativistic correlation energies for the noble-gas sequence through Rn (Z=86), the group-llB atoms through Hg, and the ions of Ne isoelectronic sequence. Phys. Rev. A, 50(6) (1994) 4733-4742. 

E. Eliav, U. Kaldor, Y. Ishikawa. Relativistic Coupled Cluster Theory Based on the No-Pair Dirac-Coulomb-Breit Hamiltonian Relativistic Pair Corrdation Energies of the Xe Atom. Int. J. Quantum Chem. Quantum Chem. Symp., 28 (1994) 205-214. 

In this general definition of the Hamiltonian, relativistic effects are neglected. These effects are normally negligible for the first three rows in the periodic table (i.e. Z < 36), but become important for the fourth and fifth rows, and for transition metals. Other operators such as the ones describing spin-orbit, orbit-orbit, or spin-spin couplings are also neglected because their contributions are, in most cases, rather small. 

Expansion techniques (variational Hamiltonians - relativistic perturbation theory)... 

An Extended (Sufficiency) Criterion for the Vanishing of the Tensorial Field Observability of Molecular States in a Hamiltonian Formalism An Interpretation Lagrangeans in Phase-Modulus Formalism A. Background to the Nonrelativistic and Relativistic Cases Nonreladvistic Electron... 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

The most common way of including relativistic effects in a calculation is by using relativisticly parameterized effective core potentials (RECP). These core potentials are included in the calculation as an additional term in the Hamiltonian. Core potentials must be used with the valence basis set that was created for use with that particular core potential. Core potentials are created by htting a potential function to the electron density distribution from an accurate relativistic calculation for the atom. A calculation using core potentials does not have any relativistic terms, but the effect of relativity on the core electrons is included. 

A number of types of calculations can be performed. These include optimization of geometry, transition structure optimization, frequency calculation, and IRC calculation. It is also possible to compute electronic excited states using the TDDFT method. Solvation effects can be included using the COSMO method. Electric fields and point charges may be included in the calculation. Relativistic density functional calculations can be run using the ZORA method or the Pauli Hamiltonian. The program authors recommend using the ZORA method. 

There are at least two ways forward, and the first was proposed by Schrddinger. Instead of the non-relativistic Hamiltonian for a free electron, he started from the correct relativistic expression... 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

In some cases, one is not interested in the Green function but in the Hamiltonian. Grigore, Nenciu and Purice (1989) and Thaller (1992, p. 184) gave " ormula for the relativistic corrections to the non relativistic eigenstates of energy Eq. The following discussion is a bit abstract, but it will be illustrated by examples in the next two sections. Equation (2) is rewritten as... 

In this review, we have mainly studied the correlation energy connected with the standard unrelativistic Hamiltonian (Eq. II.4). This Hamiltonian may, of course, be refined to include relativistic effects, nuclear motion, etc., which leads not only to improvements in the Hartree-Fock scheme, but also to new correlation effects. The relativistic correlation and the correlation connected with the nuclear motion are probably rather small but may one day become significant. 

Let us, therefore, assume that the amplitude >ft x) describing a relativistic spin particle is an -component object. We are then looking for a hermitian operator H, the hamiltonian or energy operator, which is. linear in p and has the property that H2 = c2p2 + m2c4 = — 2c2V2 + m2c4. We also require H to be the infinitesimal operator for time translations, i.e., that... 

In the present section we shall make this difficulfy apparent in a somewhat different way by showing that it is not possible to satisfy the asymptotic condition when the theory is formulated in terms of an unsubtracted hamiltonian of the form jltAll(x) — JS0JV. We shall work in the Lorentz gauge, where the relativistic invariance of the theory is more obvious. 

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. 

Ilias, M. and Saue, T. (2007) An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation. Journal of Chemical Physics, 126, 064102-1-064102-9. 

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