The Dirac equation

The symmetric two-electron operator V is given in terms of the electron-electron potentials by 

The convention for writing two-electron matrix elements is that ju and a belong to one electron, v and p to the other. 

Note that the commutation rules (3.137—3.139) and the symmetric operators (3.142,3.149) have been derived from properties of determinants. We have not assumed that the orbitals p), v) are orthogonal. In evaluating matrix elements care must be taken to keep track of the scalar products of orbitals that are not orthogonal, such as bound orbitals and plane waves. The iV-electron target configurations are conveniently normalised by (3.123). The normalisation of the continuum orbitals is discussed in chapter 6. 

The relativistically-covariant description of the motion of an electron in quantum mechanics was first given by Dirac (1928). We consider the relativistic motion of an electron in the potential of an atom. 

For a free electron the Hamiltonian H is the total energy, given in atomic units by 

The analysis of molecular spectra requires the choice of an effective Hamiltonian, an appropriate basis set, and calculation of the eigenvalues and eigenvectors. The effective Hamiltonian will contain molecular parameters whose values are to be determined from the spectral analysis. The theory underlying these parameters requires detailed consideration of the ftmdamental electronic Hamiltonian, and the effects of applied magnetic or electrostatic fields. The additional complications arising from the presence of nuclear spins are often extremely important in high-resolution spectra, and we shall describe the theory underlying nuclear spin hyperfine interactions in chapter 4. The construction of effective Hamiltonians will then be described in chapter 7. 

The problem of units is discussed in the Preface to this book. In this and all of the following chapters, we use SI units It is a fact, however, that almost all of the relevant original literature gives expressions in terms of cgs units. The relationships between cgs and SI units are straightforward and are summarised in General Appendix E. We 

We start by considering the relativistic classical mechanics of a particle in free space. As we have already seen in chapter 2, the momentum P of the particle is given by 

Now the energy of the particle is given by the sum of the kinetic and potential energies, and is equal to 

Note from this equation that when the particle is at rest (V = 0) the energy E is equal to me1, which is therefore known as the rest energy. Note also that for small velocities, 

The term in the first square bracket is the same as in the equation without spin. The effect of the first term in the second square bracket can be demonstrated by letting it operate on a function For the case of an arbitrary vector a, we have the general vector relation 

If we apply this and insert the definition of the vector potential, (V x A) = B, we get 

Thus we end up with an extra term in the nonrelativistic Schrodinger equation that accounts for the interaction of the spin of the electron with a magnetic field  

The fourth term is the spin Zeeman term and represents the interaction of the electron spin with the external magnetic field, and under the substitution fur = 2s it takes the form e/m)s B. 

We have demonstrated that by replacing it with a-it we can indeed introduce spin into the nonrelativistic Schrodinger equation. In this form, spin appears explicitly in the wave function through the Pauli spinors (or products of these for many-electron systems), and its interaction with magnetic fields appears naturally in the Hamiltonian and need not be grafted on ad hoc when required. However, apart from the fact that it yields a convenient form of the Schrodinger equation, it is not immediately evident why the operator a it should be used. And we still have the problem that the free-electron part of the Hamiltonian is not Lorentz invariant. So we must look for an alternative route to a relativistic quantum theory for the electron, one which preferably also accounts for spin. Our experiences from the derivations in this section show us that this route may lead to multicomponent wave functions. 

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By following [323], we substitute in the Lagrangean density, Eq. (149), from the Dirac equations [322], namely, from... 

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

Electronic properties of CNTs, in particular, electronic states, optical spectra, lattice instabilities, and magnetic properties, have been discussed theoretically based on a k p scheme. The motion of electrons in CNTs is described by Weyl s equation for a massless neutrino, which turns into the Dirac equation for a massive electron in the presence of lattice distortions. This leads to interesting properties of CNTs in the presence of a magnetic field including various kinds of Aharonov-Bohm effects and field-induced lattice distortions. 

The Schrodinger equation and the Klein-Gordon equation both involve second order partial derivatives, and to recover such an equation from the Dirac equation we can operate on equation 18.12 with the operator... 

In the presence of electric and magnetic fields the Dirac equation is modified to... 

In the time-independent case the Dirac equation may be written as... 

The Dirac equation automatically includes effects due to electron spin, while this must be introduced in a more or less ad hoc fashion in the Schrodinger equation (the Pauli principle). Furthermore, once the spin-orbit interaction is included, the total electron spin is no longer a good quantum number, an orbital no longer contains an integer number of a and /) spin functions. The proper quantum number is now the total angular momentum obtained by vector addition of the orbital and spin moments. 

Neglect of relativistic effects, by using the Schrodinger instead of the Dirac equation. This is reasonably justified in tlie upper part of the periodic table, but not in the lower half. For some phenomena, like spin-orbit coupling, there is no classical counterpart, and only a relativistic treatment can provide an understanding. 

In this paper, for functions (pi r) we shall use the four-component spinors r) being solutions of the Dirac equation... 

The solutions (q, r) of the Dirac equation (1) for the SDW are represented as a linear combination of r) with variational parameters... 

In the relativistic KKR method the trial function inside the MT-sphere is chosen as a linear combination of solutions of the Dirac equation in the center-symmetrical field with variational coefficients C7 (k)... 

Morrison, J., and Moss, R., 1980, Approximate solution of the Dirac equation using the Foldy-Wouthuysen Hamiltonian , A/o/. Phys. 41 491. 

Thaller, B., 1992, The Dirac Equation , Springer-Verlag, Berlin. 

In summary, the OP-term introduced by Brooks and coworkers has been transferred to a corresponding potential term in the Dirac equation. As it is demonstrated this approach allows to account for the enhancement of the spin-orbit induced orbital magnetic moments and related phenomena for ordered alloys as well as disordered. systems by a corresponding extension of the SPR-KKR-CPA method. 

The isomorphism between the tilde operation and hermitian conjugation, implies that upon performing this -operation on the Dirac equation we find that [Pg.524]

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

The condition that serves to determine 8(A) is the requirement of relativistic invariance f must satisfy the Dirac equation in the new coordinate system i.e., f(x ) must satisfy the equation... 

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

The Dirac Equation in a Central Field.—The previous sections have indicated that at times it is useful to have an explicit representation of the matrix element <0 (a ) n> where tfi(x) is the Heisenberg operator satisfying Eq. (10-1). Of particular interest is the case when the external field A (x) is time-independent, Ae = Ae(x), so that the states > can be assumed to be eigenstates of the then... 

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