Relativistic methods equations

The relativistic Schrodinger equation is very difficult to solve because it requires that electrons be described by four component vectors, called spinnors. When this equation is used, numerical solution methods must be chosen. 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

In table 2 our result is compared with the UV spectroscopic result of Klein et al. [26], Also shown are the theoretical results of Zhang et al. [2], Plante et al. [27], and Chen et al. [28], The first of these uses perturbation theory, with matrix elements of effective operators derived from the Bethe-Salpeter equation, evaluated with high precision solutions of the non-relativistic Schrodinger equation. This yields a power series in a and In a. The calculations of Zhang et al. include terms up to O(o5 hi a) but omit terms of 0(ary) a.u. The calculations of Plante et al. use an all orders relativistic perturbation theory method, while those of Chen et al. use relativistic configuration interaction theory. These both obtain all structure terms, up to (Za)4 a.u., and use explicit QED corrections from Drake [29],... 

The Born Oppcnheimer electronic Hamiltonian was given previously in (6.77) and a method of obtaining exact solutions is described in appendix 6.1. If the Bom expansion (6.131) is substituted into the complete non-relativistic Schrodinger equation, using the Hamiltonian (6. 130), we obtain a set of coupled differential equations for the functions... 

Relativistic Methods 204 8.1 Connection Between the Dirac and Schrodinger Equations 207 8.2 Many-particle Systems 210 8.3 Four-component Calculations 213 11.4.1 Ab Initio Methods 272 11.4.2 DFT Methods 273 11.5 Bond Dissociation Curve 274 11.5.1 Basis Set Effect at the HF Level 274 11.5.2 Performance of Different Types of Wave Function 276... 

The method outlined on p. 23 for constructing a relativistic wave equation is not the only possible approach. It suffers from the disadvantage that the Hamiltonian function obtained... 

Let us summarise in the effective Hamiltonian language [20] the general fea-tmes of a CP /SO method, in its simple Bloch-type version. In a first step, the scalar relativistic secular equations for states under interest are solved, and extensive Cl calculations define a determinant target space dim providing accurate energies and the corresponding multiconfigurational states of interest Om) m E In a second step, a determinant inter-... 

Despite this ubiquitous presence of relativity, the vast majority of quantum chemical calculations involving heavy elements account for these effects only indirectly via effective core potentials (ECP) [8]. Replacing the cores of heavy atoms by a suitable potential, optionally augmented by a core polarization potential [8], allows straight-forward application of standard nonrelativistic quantum chemical methods to heavy element compounds. Restriction of a calculation to electrons of valence and sub-valence shells leads to an efficient procedure which also permits the application of more demanding electron correlation methods. On the other hand, rigorous relativistic methods based on the four-component Dirac equation require a substantial computational effort, limiting their application in conjunction with a reliable treatment of electron correlation to small molecules [9]. 

The condition for nontrivial solution determines the energy eigenvalues E jk) for each wave vector k. The normalized sets of solutions for A determine the wave functions. The relativistic method (Loucks, 1965) is very similar in principle except that the variational equations are deduced from the Dirac equation and the trial wave functions are spinors. The spin-orbit coupling is automatically included in the calculation. 

In this section a recapitulatory description of the most common ah initio and density functional approaches will be presented. Ah initio methods calculate the electron properties of atoms and molecules at the absolute temperature (T = 0). The starting point is in most cases the non-relativistic Schrodinger equation... 

S. Komorovsy, M. Repisky, O. L. Malkina, V. G. Malkin, 1. Malkin, M. Kaupp. A fully relativistic method for calculation of nudear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the matrix Dirac-Kohn-Sham equation. /. Chem. Phys., 128 (2008) 104101. 

In most cases, the Hartree-Fock method provides a qualitatively correct description of the electronic structure of a molecular system. Usually, the Hartree-Fock method gives 99 % of the total energy of the molecule described by the non-relativistic Schrddinger equation and the clamped nucleus Hamiltonian. The difference between the best Hartree-Fock energy, i.e the Hartree-Fock energy in the limit of an infinite basis, and the exact energy is called the electronic correlation energy. 

The most significant difference of Dirac s results from those of the non-relativistic Pauli equation is that the orbital angular momentum and spin of an electron in a central field are no longer separate constants of the motion. Only the components of J = L - - S and J, which commute with the Hamiltonian, emerge as conserved quantities [1]. Dirac s equation, extended to general relativity by the method of projective relativity [2], automatically ensures invariance with respect to gauge, coordinate and spinor transformations, but has never been solved in this form. 

In this book the interaction between fields and molecules is treated in a semi-classical fashion. Quantum mechanics is used for the description of the molecule, whereas the treatment of the electromagnetic fields is based on classical electromagnetism. A complete quantmn mechanical description using quantmn electrodynamics is beyond the scope of this presentation, although we will make use of the correct value of the electronic g-factor as given by quantum electrodynamics. Furthermore, only ab initio methods derived from the non-relativistic Schrodinger equation are discussed. Nevertheless, the Dirac equation is briefly discussed in order to introduce the electronic spin via the Pauli Hamiltonian. 

At this point, we are flirting with the more difficult relativistic Dirac equation for the F atom. That method involves representing the spin function, as 2 x 2 Pauh matrices. Not only is that method... 

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