Dirac Hamiltonian/operator

Since the structure of the many-electron Dirac Hamiltonian operator for atoms allows us to separate radial and angular degrees of freedom (as in the case of the one-electron atoms solved analytically in chapter 6), the coordinates in the 4-spinor s components may be separated into radial and spherical spinor parts. 

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. 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

From (27) and (29) it follows that every component of the total angular momentum operator J = L + S and J2 commute with the Dirac Hamiltonian. The eigenvalues of J2 and Jz are j(j + 1 )h2 and rrijh respectively and they can be defined simultaneously with the energy eigenvalues E. 

The classical motion corresponding to the quantum dynamics generated by the Dirac-Hamiltonian (2) can most conveniently be obtained by considering the limit h — 0 in the Heisenberg picture Consider an operator B that is a Weyl quantisation of some symbol (see (Dimassi and Sjostrand, 1999))... 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

The DC or DCB Hamiltonians may lead to the admixture of negative-energy eigenstates of the Dirac Hamiltonian in an erroneous way [3,4]. The no-virtual-pair approximation [5,6] is invoked to correct this problem the negative-energy states are eliminated by the projection operator A+, leading to the projected Hamiltonians... 

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

Symmetrically opposite recipes are valid for a Hamiltonian operator in momentum space.) When magnetic fields are present, then the momentum vector receives an additional term, the vector potential A. At relativistic speeds the Dirac equation shall be used. 

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... 

Now Foldy and Wouthuysen suggested that if Xis the Dirac Hamiltonian (3.58), the operator S should be given by... 

We saw in section 3.3 that neither the orbital (TiL = R a P) nor the spin (1/2)7/a angular momenta commute with the Dirac Hamiltonian and are not therefore separate constants of motion, although their sum is. However, we can construct the mean orbital angular momentum and mean spin angular momentum operators in the same way as in equation (3.129). These operators are, respectively,... 

The method discussed here for the inclusion of relativistic effects in molecular electronic structure calculations is grounded in the Dirac-Fock approximation for atomic wave functions (29). The premise is that the major relativistic effects of the Dirac Hamiltonian are manifested in the core region, involving the core electrons, and that these effects propagate to the valence electrons. In addition, there are direct relativistic effects on valence electrons penetrating into the core region. Insofar as this is true, the valence electrons can be treated using a nonrelativistic Hamiltonian to which is added an operator, the relativistic effective core potential (REP). The REP formally, incorporates relativistic effects due to core electrons and to interactions of valence electrons with core electrons in an internally consistent way. 

We are now in a position to express the Dirac Hamiltonian in terms of radial operators and a four-component spin—orbital operator K defined by... 

The Dirac operator Ho is interpreted as the operator corresponding to the energy of a free particle, which fits well to its role as the generator of the time evolution. As a consequence of the anti-commutation relations (5), the square of the Dirac Hamiltonian is a diagonal matrix. It is simply given by... 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

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