Dirac theory structure

The Dirac-Pauli representation is most commonly used in all applications of the Dirac theory to studies on electronic structure of atoms and molecules. Apart of historical reasons, there are several features of this representation which make its choice quite natural. Probably the most important is a well defined symmetry of and in the case of spherically-symmetric potentials V. The Dirac Hamiltonian... 

Later, after experiments performed by Rabi, Lamb and Kusch and their colleagues, it was discovered that the actual hydrogen spectrum was in part in contradiction to Dirac theory (see Fig. 1). In particular, the theory predicted a value of hyperfine structure interval in the ground state of the hydrogen atom, different from the actual one by one part in 103, and no splitting between 2si/2 and... 

Figure 1 Fine structure of Balmer-a according to the Dirac theory.

Herzberg s paper [61] also reports measurements on the He+ line 1640 A (n = 3—2), excited and analysed in the same way as 4686 A. The fine structure is the same as that of Ha enlarged by the factor 16, except that the Lamb shifts are increased by a slightly smaller factor. All seven predicted components can be seen (one with difficulty), in disagreement with the exact coincidences of some of them which one would expect on the Dirac theory. The measured intervals agree with the radiation theory to within the accuracy of measurement ( 0 02 cm 1),... 

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... 

The most serious reservation expressed about relativistic formulations of quantum chemistry in [2] is to know to what question they are supposed to be the answer, in the circumstances in which we And ourselves . This article is intended to demonstrate that one may formulate a valid theory of quantum chemistry without invoking the Schrodinger equation, and that the conventional quantum chemistry that we would And in, say, [3, 4] may be obtained as a limiting case of our formulation. We prefer to keep the relativistic structure of the Dirac theory intact at all times, which means that our formulation does not contain relativistic corrections . This is quite deliberate, and a feature which we will turn into a computational advantage. 

Since one is often more familiar with nonrelativistic Schrodinger quantum mechanics for atoms because of its simpler structure, we recall its basic ingredients, which serve two purposes. For one, the similarities to the Dirac theory... 

We have already seen above that the choice of a point-like atomic nucleus limits the Dirac theory to atoms with a nuclear charge number Z < c, i.e., Zmax 137. A nonsingular electron-nucleus potential energy operator allows us to overcome this limit if an atomic nucleus of finite size is used. In relativistic electronic structure calculations on atoms — and thus also for calculations on molecules — it turned out that the effect of different finite-nucleus models on the total energy is comparable but distinct from the energy of a point-like nucleus (compare also section 9.8.4). 

Two-photon excitation was predicted in her doctoral thesis by Maria Goppert-Mayer [88], who recognized that it was a corollary of the Kramers-Heisenberg-Dirac theory of light scattering. It was not observed experimentally until 30 years later, when pulsed ruby lasers finally provided the high photon flux that was required [89]. Goppert-Mayer received the physics Nobel prize in 1963 for imre-lated work on nuclear structure. 

This appeared to be the final solution to the problem of the hydrogen atom. However, it should be remembered that a few experimentalists were not entirely happy and questioned whether the Dirac theory of the fine structure was completely correct. There was therefore a need to develop new experimental techniques which would increase the accuracy of fine and hyperfine-structure measurements. 

This simplified treatment does not account for the fine-structure of the hydrogen spectrum. It has been shown by Dirac (22) that the assumption that the system conform to the principles of the quantum mechanics and of the theory of relativity leads to results which are to a first approximation equivalent to attributing to each electron a spin that is, a mechanical moment and a magnetic moment, and to assuming that the spin vector can take either one of two possible orientations in space. The existence of this spin of the electron had been previously deduced by Uhlenbeck and Goudsmit (23) from the empirical study of line spectra. This result is of particular importance for the problems of chemistry. 

No theoretical proof of the Pauli principle was given originally. It was injected into electronic structure theory as an empirical working tool. The theoretical foundation of spin was subsequently discovered by Dirac. Spin arises naturally in the solution of Dirac s equation, the relativistic version of Schrodinger s equation. 

The development of the method started in the mid 1920 s with the work of Thomas and Fermi [8, 9]. The aim was to formulate an electronic structure theory for the solid state, based on the properties of a homogeneous electron gas, to which we introduce a set of external potentials (i.e. the atomic nuclei). The original formulation, with later additions by Dirac [10] and Slater [11], was, however, inadequate for accurate description of atomic and molecular properties, and it was not until the ground-breaking work of Kohn and coworkers in the mid 1960 s that the theory was put in a form more suited to computational chemistry [12,... 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

A full account of the theory of relativistic molecular structure based on standard QED in the Furry picture will be found in a number of publications such as [7, Chapter 22], [8, Chapter 3]. These accounts use a relativistic second quantized formalism. For present purposes, it is sufficient to present the structure of BERTHA in terms of the unquantized effective Dirac-Coulomb-Breit (DCB) A-electron Hamiltonian ... 

Following the development of quantum theory by Heisenberg [1] and Schrodinger [2] and a few further discoveries, the basic principles of the structure of atoms and molecules were described around 1930. Unfortunately, the complexity of the Schrodinger equation increases dramatically with the number of electrons involved in a system, and thus for a long time the hydrogen and helium atoms and simple molecules as H2 were the only species whose properties could really be calculated from these first principles. In 1929, Dirac [3] wrote ... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

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