Equation Dirac, extended

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

Another subtlety is that the assumption nuclei behave as Dirac particles, amounts to assuming that all nuclei have spin 1/2. However, it is not uncommon to have nuclei with spin as high as 9/2 worse nuclei with integer spins are bosons and do not obey Fermi-Dirac statistics. The only justification to use equation (75) for such a case is that the resulting theory agrees with experiment. Under the assumption, we are in a position to extend our many-fermion Hamiltonian to molecules assuming that the nuclei are Dirac particles with anomalous spin. The molecular Hamiltonian may then be written as... 

We have focused on the prohlems associated with extending Dirac s one-fermion theory smoothly to many-fermion systems. A brief discussion of QED many-fermion Hamiltonians also was given. A comprehensive account of the problem of decoupling Dirac s four-component equation into two-component form and the serious drawbacks of the Pauli expansion were presented. The origins of the DSO and FC operators have been addressed. The working Hamiltonian which describe NMR spectra is derived. 

From the symmetric set, an extended set of Maxwell equations was exhibited in Section V.E. This set contains currents and sources for both fields E, B. The old conjecture of Dirac s is vindicated, but the origin of charge density is always electric (i.e., no magnetic monopole). Standard Maxwell s equations are a limiting case in far field. 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

Dirac equation. Leading-order Lorentz violating energy shifts 61/12 and di/34 can be obtained from a Hamiltonian using perturbation theory and relativistic two-fermion techniques. For our observed transitions at the strong magnetic field of 1.7 T, dominantly only muon spin flip occurs so the energy shifts are characterized by the muon parameters alone of the extended theory. The results of this approach are [4] ... 

In the preceeding sections we have discussed the DV scheme as it has been applied in nonrelativistic DF theory here we extend the discussion to solutions of the relativistic Dirac equation,... 

The first statistical models of these interactions are the well-known Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) theories based on the idea of approximating the behavior of electrons by that of the uniform negatively charged gas. Some authors (Sheldon, 1955 Teller, 1962 Balazs, 1967 Firsov, 1953,1957 Townsend and Handler, 1962 Townsend and Keller, 1963 Goodisman, 1971) proved that these theories provide an adequate description of purely repulsive diatomic interactions. Abraham-son (1963, 1964) and Konowalow et al. (Konowalow, 1969 Konowalow and Zakheim, 1972) extended this region to intermediate internuclear distances, but Gombas (1949) and March (1957) showed that the Abraham-son approach is incorrect, and so the question of how adequately the TFD theory provides diatomic interactions for closed-shell atoms is still open. Here we need to note that until recently, there has existed only work by Sheldon (1955), as far as we know, in which the TFD interaction potential is actually calculated by solving the TFD equation for a series of internuclear distances (see also, Kaplan, 1982). 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

During the last few decades, general computer programs, based on the Dirac equation, for extensive four-component electronic structure calculations for atoms have been developed by several groups. These programs are being improved and extended continually, incorporating the explicit treatment of the Breit interaction as well as an ever more sophisticated consideration of interelectron correlation and even of QED effects. 

The MS-Xa method was extended by Yang et al. to include relativistic effects starting from the Dirac equations [61] but this calculation could not be iterated to self-consistency for technical reasons. The results, in column 2 of Table 3 are rather poor, in that the HOMO-LUMO gap is far too small, and the highest filled levels are again derived from ng. 

The derivation by Dirac [13] of TDHF can be expressed in terms of orbital functional derivatives [12]. This derivation can be extended directly to a formally exact time-dependent orbital-functional theory (TDOFT). The Hartree-Fock operator H is replaced by the OFT operator Q = TL + vc throughout Dirac s derivation. In the linear-response limit, this generalizes the RPA equations [14] to include correlation response [17]. 

DFT is the ground-state limit of a more general OFT. The OEL equations of OFT do not necessarily contain local potential functions. Tests of locality fail for the effective exchange potential in the UHF exchange-only model. Dirac s derivation of TDHF theory can readily be extended to a TDOFT that includes electronic correlation. The exchange-only limit of this theory is consistent with TDHF and with the RPA in many-body theory. 

In Section 2.3 below we are going to show to what extend the anticommutation relations determine the properties of the Dirac matrices. Here we just note that these relations do not define the Dirac matrices uniquely. If (/ , 0, ) is a set of Hermitian matrices satisfying (5), then 0 = S0S and a). = SakS with some unitary matrix S is another set of Hermitian matrices obeying the same relations. Any specific set is said to define a representation of Dirac matrices. With respect to a given representation, the Dirac equation is a system of coupled linear partial differential equations. It is of first order in space and time derivatives. 

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)... 

For a consistent treatment an extended Dirac equation is considered... 

We could not show here the results of solving the relativistic Dirac-Coulomb equation. The FC method can be extended to the case of the Dirac-Coulomb equation with only a small modification [36]. It is important to use the inverse Dirac-Coulomb equation to circumvent the variational collapse problem which often appears in the relativistic calculations [37]. 

To have the electron magnetic moment show up, it is necessary to make it interact with an external magnetic field and to have its spin momentum appear, it has to be combined with an orbital momentum. Equation (2.11) was thus extended to include interactions with an electromagnetic field. Let us call A4 and A the scalar and vector potentials in MKSA units (in earlier formulations of the Dirac equation [5, 6], A was divided by c due to the use of cgs units). We can write... 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

---