Hamiltonian with relativistic terms

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 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. 

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. 

Thus, we have expressed the non-relativistic Hamiltonian of a many-electron atom with relativistic corrections of order a2 in the framework of the Breit operator (formulas (1.15), (1.18)—(1.22)) in terms of the irreducible tensorial operators (second term in (1.15), formulas (19.5)—(19.8), (19.10)— (19.14), (19.20), respectively). 

The AREP has the advantage that it may be used in standard molecular calculations that are based on A-S coupling. The AREP may be interpreted as containing the relativistic effects included in the Dirac Hamiltonian, with the exception of spin-orbit coupling. This form is the same as that presented by Kahn et al. (33) which is based on the relativistic treatment of Cowan and Griffin (34). The Hamiltonian employed by Cowan and Griffin is based on the Pauli approximation to the Dirac Hamiltonian with the omission of the spin-orbit term. 

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 ),... 

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... 

While using (4.14) and co = E — 2, the expression in curly brackets is zero for the case of the exact wave functions. This supports the conclusion of Drake [83] that for electric dipole transitions, by considering the commutator of QW with the atomic Hamiltonian in the Pauli approximation, we obtain Q q with relativistic corrections of order v2/c2 (see (4.18)-(4.20)). However, for many-electron atoms and ions, one has to use approximate (e.g., Hartree-Fock) wave functions, and then this term gives non-zero contribution, conditioned by the inaccuracy of the model adopted. 

We will deal only with computational procedures that are normal for molecnles encoimtered in organic photochemistry. These methods depend heavily on the assumption that the spin-orbit conpling term is only a minor perturbation. In snch an instance, it is common to not include small relativistic terms such as spin-orbit coupling in the Hamiltonian from the beginning bnt to include them as an afterthought after an ordinary nonrelativistic calculation. This is usually done nsing pertnrbation theory or response theory. 

For heavy atoms and molecules, many-electron theory can be made to start with relativistic equations. Though the exact relativistic Hamiltonian is not known it seems a good approximation to base the theory on the relativistic Hartree-Fock Hamiltonian corrected by the non-relativistic 1/r,-, terms. 

Inclusion of the spin-orbit and other relativistic terms in Eq. (5), as we have done, is, strictly speaking, the most correct approach. This yields, as we have seen, a set of nuclear wave functions Xa(R) whose uncoupled motion is governed by the potentials (7a (R) and which are coupled only by the nuclear-derivative terms Fa (R) and Ga (R). In practice, though, Hrei(R) is difficult to treat on an equal footing with the coulombic terms in the Hamiltonian. Therefore one sometimes works with... 

In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part is added to the Hamiltonian. The effects of the spin-orbit term on the energies and wave functions are commonly estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry [2, 3]. 

In this spirit, the simplest approach would be to consider the relativistic terms as a perturbation to the nonrelativistic Hamiltonian in the Schrodinger equation. From the expansion of Eq. [89], it is quite natural to apply standard Rayleigh—Schrodinger perturbation theory with as the unperturbed Ham-... 

We can also derive a bound that is related to the expectation of the nonrela-tivistic Hamiltonian with a relativistic correction. To proceed, we use the matrix relation (11.13) twice to partition the second term of the Rayleigh quotient, R2, with A = 2mc S and B = — ES. The first application yields... 

The mass-velocity term is therefore the lowest-order term from the relativistic Hamiltonian that comes from the variation of the mass with the velocity. The second relativistic term in the Pauli Hamiltonian is called the Darwin operator, and has no classical analogue. Due to the presence of the Dirac delta function, the only contributions for an atom come from s functions. The third term is the spin-orbit term, resulting from the interaction of the spin of the electron with its orbital angular momentum around the nucleus. This operator is identical to the spin-orbit operator of the modified Dirac equation. 

To formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

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