Hamiltonian Dirac

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... 

Reiher, M. and Wolf, A. (2004) Exact decoupling of the Dirac Hamiltonian. I. General theory. Journal of Chemical Physics, 121, 2037-2047. 

The Dirac Hamiltonian in the presence of an external field is therefore given by... 

For a Dirac particle in a central field 4> is spherically symmetrical and A = 0. Setting the potential energy V(r) — e(r), the Dirac Hamiltonian becomes... 

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... 

From the classical electrodynamics, the Dirac Hamiltonian of a hydrogen molecule moving in a constant magnetic field B is [102]... 

In relativistic theory, we apply the minimal coupling recipe to the Dirac Hamiltonian... 

All solutions of this Hamiltonian are thereby electronic, whether they are of positive or negative energy and contrary to what is often stated in the literature. Positronic solutions are obtained by charge conjugation. From the expectation value of the Dirac Hamiltonian (23) and from consideration of the interaction Lagrangian (16) relativistic charge and current density are readily identified as... 

Here, X is to be determined by imposing that the resulting transformed Dirac Hamiltonian is block diagonal. It is fairly easy to see that this leads to an equation... 

In this case, X is to be determined by requiring that the off-diagonal blocks of the resulting transformed Dirac Hamiltonian vanishes. It can be shown that the equation for X is identical to the one we obtained in the case of a unitary transformation as given in equation (38). In this case, the effective Hamiltonian hn and wave function xp, can be written as... 

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 radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

It is highly desirable to formulate a variational principle valid for the Dirac Hamiltonian. The first attempts are due to Drake and Goldman [9], Wood etal. [10], Talman [4] and Datta and Deviah [5], Very recently the subject has been discussed in detail by Griesemer and Siedentop [6] and also by Kutzelnigg [7] andby Quiney efa/. [11],... 

We assume for simplicity that the two-electron atom is described by a Hamiltonian (29) in which //(I) and //(2) are the hydrogen-like Dirac Hamiltonians and h, 2) = 0. Apparently, after this simplification the problem is trivial since it becomes separable to two one-electron problems. Nevertheless we present this example because it sheds some light upon formulations of the minimax principle in a many-electron case. 

Fig. 6. The application of the weak minimaxprinciple to the ground state of a Z = 90 two-electron atom described by the simplified two-electron Dirac Hamiltonian using the hydrogen-tike basis with L = S = 1. The thin solid lines represent the energy as a function of a when (3 = broad solid lines give the energy...

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 ... 

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