Relativistic terms in the Hamiltonian

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. 

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 ),... 

The discussion of electric and magnetic properties of molecules retains its importance, especially with the growth of new and sensitive experimental techniques in spectroscopy and magnetic resonance. Chapter 11, on static properties, is a rewritten and expanded version of the corresponding chapter in the first edition and covers effects due to the presence of small (relativistic) terms in the Hamiltonian and to the application of external fields. Nowadays, however, dynamic effects are also of growing importance, and a new Chapter 12 is devoted to time-dependent perturbations and the general theory of linear response. 

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. 

So far in this book we have only discussed non-relativistic Hamiltonian operators but when atomic or moleoular spectra are considered it is necessary to account for relativistic effects. These lead to additional terms in the Hamiltonian operator which can be related to the following phenomena ... 

It is important to notice, however, that consistent neglect of all terms of the order 1/c (which has not been treated consistently in the weakly relativistic expansion) in the Hamiltonian allows a proof of a HK-theorem on the basis of the variables n and/. In other words Only a fully relativistic approach combines consistency in 1/c with gauge invariance. It remains to be investigated explicitly, whether inclusion of all relevant terms to order 1/c allows to reinstate the physical current j x) as basic variable also in this order as one would expect from the fully relativistic theory. 

The potential-energy terms in the Hamiltonian are only those due to Coulomb s law. We exclude magnetic and relativistic effects completely. 

The last three terms in the Hamiltonian in Eq. (5.28) define an effective one-body potential, Veff(r), which transforms the density of the non-interacting system into the real density. Then, by choosing Vj(r) = Veffir) in Eq. (5.25) the effective potential is found. As was the case with HE, the one-electron Hamiltonian in the KS equations is solved iteratively and self-consistently. If the exact expression for Exc [p(r)] was known, the KS equations would provide the exact non-relativistic ground state solution within the space spanned by a given basis set, including all electron correlation effects. The latter are missing in HE. This is an important difference between HE and KS. 

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... 

Interpretation 2 (Dutch explanation) the change of the contributions of relativistic, i.e. c-dependent, terms in the Hamiltonian H/dr) due to changes in the wavefunction upon molecular formation d W/dR, causes relativistic bond length changes ... 

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. 

Given an approximation to the ground state energy of the BO-Hamiltonian by some method, one needs to introduce the smaller field- and spin-dependent terms in the Hamiltonian fliat give rise to the interactions one actually probes by EPR spectroscopy. These terms can be derived through relativistic quantum chemistry, which is outside the scope of this chapter. Among the many terms that arise, we will mainly need the following interactions ... 

To illustrate the magnimde of this term, recall that for a hydrogen-like model this correetion is of order (Za). Hence, in the heavy elements, where Z is comparable to the inverse of the fine structure constant, l/oc = 137, this term cannot be ignored. Because this term in the Hamiltonian is an operator on the spatial wavefunction of the eleetron, it is referred as a scalar relativistic effect, to distinguish it from the vector terms that depend on more than one component of the spin states, as we discuss below. 

The second relativistic contribution of scalar nature is the one-electron Darwin term, Wd. This term derives from the non-relativistic expansion of Dirac s equation, in powers of (v/c), and results in a non-local interaction between the electron and the nucleus. The interaction extends over a region in space of size roughly that of the Compton wavelength of the electron. The order of magnimde of this term in the Hamiltonian is also (Za) making it non-negligible for heavy elements. [12] These scalar relativistic terms have significant effects on the radial extent of the inner core orbitals. 

Molecules are described in terms of a Hamiltonian operator that accounts for the movement of the electrons and the nuclei in a molecule, and the electrostatic interactions among the electrons and the electrons and the nuclei. Unlike the theory of the nucleus, there are no unknown potentials in the Hamiltonian for molecules. Although there are some subtleties, for all practical purposes, this includes relativistic corrections, [2] although for much of light-element chemistry those effects are... 

Generally, it is not required to retain all the terms in the resulting approximate Hamiltonian, except those operators which describe the actual physical processes involved in the problem. For example, in the absence of an external electromagnetic field, the non-relativistic energy calculations only requires... 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

Atomic units will be used throughout. The explicit density functionals representing the different contributions to the energy from the different terms of the hamiltonian are found performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. Those representing the first relativistic corrections are calculated in the Appendix. 

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