Single-electron Dirac Hamiltonian

The spectrum of the single-electron Dirac operator Hd and its eigenspinors (/> for Coulombic potentials are known in analytical form since the early days of relativistic quantum mechanics. However, this is no longer true for a many-electron system like an atom or a molecule being described by a many-particle Hamiltonian H, which is the sum of one-electron Dirac Hamiltonians of the above kind and suitably chosen interaction terms. One of the simplest choices for the electron interaction yields the Dirac-Coulomb-Breit (DCB) Hamiltonian, where only the frequency-independent first-order correction to the instantaneous Coulomb interaction is included. 

Once we know the solution of the Dirac equation for a single electron moving in the external potential it is tempting to build the relativistic theory of many-electron systems in a similar way as the non-relativistic theory is built, i.e., by combining the one-electron Dirac Hamiltonian for each electron with the interaction between electrons. 

Hence, the Dirac statement finds its Umitation caused by technical problems as soon as the number of electrons is large. Traditional treatment of a many-hody problem of N electrons is in practical very difficult, requiting severe approximations. DFT is an efficient way to include correlation, preserving the simplicity of the treatment of a single electron. This is the main reason for its success. Its advantages are that it requires much less computation than the IC or VB methods and that it is adapted to solids and metal-metal bonds and easy to use. The disadvantages are that it is less reliable than IC or VB methods and is not ah initio in a strict sense since an approximate (fitted) term is introduced in the Hamiltonian. It also does not allow comparison of results obtained using different functionals. 

The Dirac equation represents an approximation- and refers to a single particle. What happens with larger systems Nobody knows, but the first idea is to construct the total Hamiltonian as a sum of the Dirac Hamiltonians for individual particles plus their Coulombic interaction (the Dirac-Coulomb apjmmmation). This is practised routinely nowadays for atoms and molecules. Most often we use the mean-field approximation (see Chapter 8) with the modification that each of the one-electron functions represents a four-component bispinor. Another approach is extreme pragmatic, maybe too pragmatic we perform the non-relativistic calculations with a pseudopotential that mimics what is supposed to happen in a relativistic case. 

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

We have also used both fi and ii for the one-electron Hamiltonian operator. The latter is used for the free-particle Dirac Hamiltonian where a distinction between it and the full one-electron Hamiltonian is necessary, and is also used in a sum over one-electron Hamiltonians for a single electron. The former is usually used in formal developments, and to represent the total Hamiltonian. In many of the formal developments, the total Hamiltonian is simply the one-electron Hamiltonian, so FL is used. However, for the one-electron Hamiltonian matrix elements, lower case is always used, and for the A -electron Hamiltonian matrix elements, upper case is always used. 

In the last section, a non-relativistic Hamiltonian for a spin-less particle was derived. However, electrons have spin and in general it would be desirable to use a Hamiltonian operator that fulfills the requirements of special relativity. The so-called Dirac Hamiltonian operator is such a relativistic operator for a single particle in the presence of an electromagnetic field. It can be derived in the same ways as the non-relativistic analogue was obtained in the previous section. 

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. 

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

It has been known for a long time, especially from the work of Dirac [15] and of Lowdin [78], that the (now idempotent) 1 DM is sufficient to determine the N-electron wave function for the case of a single Slater determinant. It has been equally clear to many workers in the field that such knowledge of the 1DM cannot be adequate to reconstruct the AT-body wave function for the fully interacting electron system, without appeal to the total Hamiltonian. 

The formalism described here to derive energy-consistent pseudopotentials can be used for one-, two- and also four-component pseudopotentials at any desired level of relativity (nonrelativistic Schrbdinger, or relativistic Wood-Boring, Douglas-Kroll-Hess, Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian implicit or explicit treatment of relativity in the valence shell) and electron correlation (single- or multi-configurational wavefunctions. The freedom... 

Starting from the Dirac-Coulomb Hamiltonian and a single Slater determinant as many-electron wavefimction Swirles (1935, 1936) derived the relativistic DHF equations... 

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