Hamiltonian Dirac, derivation

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... 

Apparently, a large number of successful relativistic configuration-interaction (RCI) and multi-reference Dirac-Hartree-Fock (MRDHF) calculations [27] reported over the last two decades are supposedly based on the DBC Hamiltonian. This apparent success seems to contradict the earlier claims of the CD. As shown by Sucher [18,28], in fact the RCI and MRDHF calculations are not based on the DBC Hamiltonian, but on an approximation to a more fundamental Hamiltonian based on QED which does not suffer from the CD. At this point, let us defer further discussion until we review the many-fermion Hamiltonians derived from QED. 

Figure 2. The same as in Fig. 1 but the variational energies are derived from the algebraic Dirac equation in the Biedenharn representation (41). The bottoms of the valleys at /3 = 90 (left figures) and at 6 = 0.754 (right figures) correspond to the exact eigenvalues of the Dirac Hamiltonian. The lower figures show the same function as the upper ones, but over a wider range of parameters.

In section 3.2 we pointed out that the Dirac Hamiltonian contains operators which connect states of positive and negative energy. What we now seek is a Hamiltonian which is relativistically correct but which operates on the two-component electron functions of positive energy only. We require that this Hamiltonian contain terms representing electromagnetic fields, and Foldy and Wouthuysen [12] showed, by a series of unitary transformations, that such a Hamiltonian can be derived. The Dirac Hamiltonian... 

The first stage in deriving a molecular Hamiltonian is to reduce the Breit equation to non-relativistic form and Chraplyvy [17] has shown how this reduction can be performed by using an extension of the Foldy-Wouthuysen transformation. First let us remind ourselves of the most important features in the transformation of the Dirac Hamiltonian. The latter was written (see (3.57) and (3.58)) as... 

This result cannot be reconciled with that of Dirac [13] and with the structure of the second-quantized Hamiltonian [12,14]. The implication is that the locality hypothesis fails for Ex and probably for Ec, which can be incorporated in a formally exact extension of Dirac s derivation. [17] Thus restriction to local exchange and correlation potentials is inconsistent with exact linear-response theory. 

Dealing with electrons we know that the dominant interaction between them is the Coulomb repulsion corrected, because electrons are fermions, by interactions induced by their spin. The spin-orbit interaction is already included in the one-electron Dirac Hamiltonian but the two-electron interaction should also include interactions classically known as spin-other-orbit, spin-spin etc... Furthermore a relativistic theory should incorporate the fact that the speed of light being finite there is no instantaneous interaction between particles. The most common way of deriving an effective Hamiltonian for a many electron system is to start from the Furry [11] bound interaction picture. A more detailed discussion is given in chapter 8 emd we just concentrate on some practical considerations. 

There are other reasons why methods based on Dirac Hamiltonians have been unpopular with quantum chemists. Dirac theory is relatively unfamiliar, and the field is not well served with textbooks that treat the topic with the needs of quantum chemists in mind. Matrix self-consistent-field equations are usually derived from variational arguments, and as a result of the debates on variational collapse and continuum dissolution , many people believe that such derivations are invalid for relativistic problems. Most implementations of the Dirac formalism have made no attempt to exploit the rich internal structure of Dirac... 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order... 

In this Chapter, a method to derive general quasi-relativistic Hamiltonians has been presented. The Dirac equation has been reformulated in such a way that quasi-relativistic Hamiltonians can be identified as the zeroth-order approximation to the Dirac Hamiltonian. The operator difference between the quasi-relativistic and the Dirac Hamiltonians can be treated as the perturba-... 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

All the derivations made up to now have been rigorous. We are interested in comparing the relativistic and the non-relativistic Hamiltonian terms, which can be done approximately in the following way. The Dirac Hamiltonian is assumed in the form of the dominating term for the positive energy solutions E0 and the small perturbation HNR due to a slow-moving electron... 

From Dirac to Schrodinger How Is the Non-Relativistic Hamiltonian Derived ... 

Note that the derivation of the Dirac equation in chapter 5 holds for any freely moving spin-1 /2 fermion and hence also for the proton we come back to such general two-particle Hamiltonians at the beginning of chapter 8 (for an explicit solution of the corresponding two-fermion eigenvalue equation we refer the reader to the work by Marsch [112-115]). In the field-free Dirac Hamiltonian only the rest mass determines which fermion is considered. Accordingly, the total wave function of the hydrogen atom reads... 

So far only the position-space formulation of the (stationary) Dirac Eq. (6.7) has been discussed, where the momentum operator p acts as a derivative operator on the 4-spinor Y. However, for later convenience in the context of elimination and transformation techniques (chapters 11-12), the Dirac equation is now given in momentum-space representation. Of course, a momentum-space representation is the most suitable choice for the description of extended systems under periodic boundary conditions, but we will later see that it gains importance for unitarily transformed Dirac Hamiltonians in chapters 11 and 12. We have already encountered such a situation, namely when we discussed the square-root energy operator in Eq. (5.4), which cannot be evaluated if p takes the form of a differential operator. 

A key element for the reduction to two-component form is the analysis of the relationship between the large and small components of exact eigenfunctions of the Dirac equation, which we have already encountered in section 5.4.3. This relationship emerges because of the (2 x 2)-superstructure of the Dirac Hamiltonian, see, e.g., Eq. (5.135), which turned out to be conserved upon derivation of the one-electron Fock-type equations as presented in chapter 8. Hence, because of the (2 x 2)-superstructure of Fock-type one-electron operators, we may assume that a general relation. 

The historically first attempt to achieve the block-diagonalization of the Dirac Hamiltonian is due to Foldy and Wouthuysen and dates back to 1950 [609]. They derived the very important closed-form expressions for both the unitary transformation and the decoupled Hamiltonian for the case of a free particle without invoking something like the X-operator. Because of the discussion in the previous two sections, we can directly write down the final result since the free-particle X-operator of Eq. (11.10) and hence Uv=o = Uq are known. With the arbitrary phase of Eq. (11.23) being fixed to zero it is given by... 

An expansion in terms of V, i.e., the Douglas-Kroll-Hess expansion, is the only valid analytic expansion technique for the Dirac Hamiltonian, where the final block-diagonal Hamiltonian is represented as a series of regular even terms of well-defined order in V, which are all given in closed form. For the derivation, the initial transformation step has necessarily to be chosen as the closed-form, analytical free-particle Foldy-Wouthuysen transformation defined by Eq. (11.35) in order to provide an odd term depending on the external potential that can then be diminished. We now address these issues in the next chapter. 

If applying a transformation derived from the decoupling of the unperturbed Dirac Hamiltonian was potentially unreliable for electric properties, for magnetic properties it simply will not work. The reason is that magnetic perturbations enter through the vector potential A and thus are odd operators. 

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