Relativistic terms

The most important relativistic corrections are the one-electron spin-orbit operator, and the relativistic correction to the spin-Zeeman operator. 

Other relativistic corrections, such as the mass-velocity and Darwin terms, affect the wave function but do not lead to operators associated with molecular 

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The terms before the square brackets give the nonrelativistic part of the Hamilton-Jacobi equation and the continuity equation shown in Eqs. (142) and (141), while the term with the squaie brackets contribute relativistic corrections. All terms from are of the nonmixing type between components. There are further relativistic terms, to which we now turn. 

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. 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

In this way it was shown that the opt values derived from data for MCl, MFi, and MFg species gave excellent correlations with the occupation number, q, and that the n -<-75 (ys) peak positions could be well reproduced using Eqs. 5 (7) and 5 (2), with spin-orbit corrections. In all cases the correlations were significantly better when the relativistic terms were included than when they were omitted, and in Table 30 we list the xopt and oPt values derived from the 5 d data for MFg, MFg, and MF6 complexes. In the Table we also show the observed and the calculated band positions using the corrected forms of Eqs. 5 (7) and 5 (2). Once again the xopt vs. q plots yield slopes in excellent agreement with the ( —ri) values deduced from these equations. Finally, in Figs. 16... 

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

Conditions in Weak Fields,—If the relativistic term is more important than the electric, we have to take it into account first. The motion without electric field is then the relativistic motion, and, instead of the first equation (7), we should take the equation corresponding to this case. The main feature of the relativistic motion is that in it the characteristic value Eo is a ftmction, not only of the quantic number /, but also of n. This is, in fact, the only point of any importance for our piupose if we bear it in mind, we may neglect relativity in every other respect and use the same analysis as in the previous case. Instead of (15), we shall have for the total quantic number 1 = 2 two possible expressions of 0, corresponding to w = 1 and w = 0,... 

The first term on the right is the operator for the electrons kinetic energy the second term is the operator for the potential energy of attraction between the electrons and the nucleus (r, being the distance between electron i and the nucleus) the third term is potential energy of repulsion between all pairs of electrons ru being the distance between electrons / and j) the last term is the spin-orbit interaction (discussed below). In addition, there are other relativistic terms besides spin-orbit interaction, which we neglect. 

Thus, a more energetic ion will tend to lose energy at a lower rate than a less energetic ion. Be careful to note that we have ignored the relativistic terms, y2 and (32, in the parentheses, which produce a minimum in the complete function near (3 0.96 and a small rise at higher velocities. (Particles with (3 0.96 are called minimum ionizing particles.) The proportionality of the stopping power... 

The first two terms, the mass-velocity and the Darwin operators, are called scalar relativistic terms since they do not involve the electron spin. They are given by... 

For Kramers (e.g. one-electron) states where the eigenvalue of T2, A = — 1, the metric gAA is antisymmetric, and so relating to symplectic algebras (in relativistic terms to pure torsion rather than to curvature), rather than symmetric as for non-Kramers systems. The joint action of Hermitian conjugation and time reversal (which is not commutative) is summarized with the above results for these individual operations in Table 1. 

Unimolecular reactions can, of course, also be induced by UV-laser pulses. As pointed out above, in order to reach a specific reaction channel, the electric field of the laser pulse must be specifically designed to the molecular system. All features of the system, i.e., the Hamiltonian (including relativistic terms), must be completely known in order to solve this problem. In addition, the full Schrodinger equation for a large molecular system with many electrons and nuclei can at present only be solved in an approximate way. Thus, in practice, the precise form of the laser field cannot always be calculated in advance. 

The closest theoretical result, the Unified theory [68], differs by more than 300 times the experimental uncertainty. This discrepancy should be partially removed by analysis including an estimate of the order (Za)4a2mec2 relativistic term and a complete calculation of the two-electron Bethe-logarithm [92]. The 14,i5N5+ 21S o — 23Pi isotope shift was measured to be —1.6623(10) cm-1, in fair agreement with an estimate based on [68]. The hyperfine corrected 3Pq —3Pi... 

The IS — 2S transition obeys the selection rule AF = 0, Am = 0 and is almost field-independent However, the g-factor for the bound electron is slightly less than for free space due to relativistic effects, and this gives the transition a small first-order field dependence. In the IS state the g-factor is g(lS) = g0 (1 - a2/3) [10]. The relativistic term is proportional to the binding energy so that g(2S) = ge (1 - a2/12). Thus, the field-dependence of the transition IS- 2S, (F=l,m=l AF=0,Am=0) leads to a frequency shift... 

Next we can dispose of the matrix elements of the one-electron operator Hy,Eq. (3-3 b) the kinetic energy operator, the electron-nuclear attraction potential arising from the metal nucleus, and the spin-independent relativistic terms have spherical symmetry and can be treated through the definition of the basis orbitals ip, Eq. (3-11). The spin-... 

HgH.—Das and Wahl have carried out a calculation on the HgH molecule which has many points of interest for the practical implementation of pseudopotentials on heavy-atomic molecular systems. As the nuclear charge increases so does the importance of the relativistic terms in the hamiltonian, and their influence is not only confined to the core orbitals (e.g. the Hg Is) where the kinetic energy of the electron is comparable with its rest mass, but even afiects the valence (Hg 6s) orbitals (Grant °) and the binding energy of Hgj (Grant and Pyper ),... 

Our new coordinates q[, qz, q t though dependent on the direction of the electric field, will not yet quite correspond to the parabolic coordinates used in the usual theory of the Stark effect. In order to obtain these, it would be necessary to go further in the approximation, by supposing that even the second order terms of the electric pertiurbation dominate over the relativistic terms. 

Conditions in Weak Fields,— When the field is so weak that the relativistic pertiurbation term dominates over the electric, the effect of relativity must be taken into account first. It is not necessary to discuss the mathematical side of this step, since the effect of relativity is fully known from the work of Sommerfeld. We use the same variables as in the unpertiu bed motion, but the choice of the momentum is now restricted by an additional quantum condition introduced by the relativistic term p2 = n2h/2T, while p remains arbitrary. Together with (10), this condition means that both the major axis and the eccentricity of the orbit are restricted by quantum conditions, while its orientation is arbitrary. The Hamiltonian fimction of the system is according to Sommer-feld... 

Except for the efforts mentioned above, relativistic calculations of shielding evaluate the main relativistic effects using one or two component limits of the four-component formalism, quasi-relativistic approaches. These avoid the variational collapse in the calculation of the scalar relativistic terms by employing frozen cores, or effective core potentials. Some include the one-electron spin-orbit terms, and sometimes the higher order spin-orbit terms too. Others include both scalar and spin-orbit terms. Ziegler... 

The best-known example is for the first term in the expansion when the Bethe formulation is employed. In this case, the first, or Bethe-Bom term in Lq can be written, including the relativistic terms [3], as... 

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