Schrodinger equation relativistic methods

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. 

Having stated the limitations (non-relativistic Hamilton operator and the Bom-Oppenheimer approximation), we are ready to consider the electronic Schrodinger equation. It can only be solved exactly for the Hj molecule, and similar one-electron systems. In the general case we have to rely on approximate (numerical) methods. By neglecting relativistic effects, we also have to introduce electron spin as an ad hoc quantum effect. Each electron has a spin quantum number of 1 /2. In the presence of an... 

We would like to point out some steps of derivation of the nonrelativistic limit Hamiltonians by means of the Foldy-Wouthuyisen transformation (Bjorken and Drell, 1964). The method is based on the transformation of a relativistic equation of motion to the Schrodinger equation form. 

The scalar ZORA method has been implemented in the standard non relativistic Ab Initio electronic structure program GAMESS-UK [8]. The technical details of this implementation will be given in the following section. Comparing the Schrodinger equation with the ZORA equation (7) one sees that application of the ZORA method has resulted in a potential dependent correction on the kinetic energy term. 

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... 

The Klein-Gordon equation (Schrodinger s relativistic equation) has been used in the description of a relativistic particle with spin zero (see, e.g., Schiff, 1968) and can be treated using the so(2,1) algebraic methods (Barut, 1971 Cizek and Paldus, 1977, and references therein). It is obtained from the energy-momentum relationship... 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

Relativistic effects may be also considered by other methods than pseudopotentials. It is possible to carry out relativistic all-electron quantum chemical calculations of molecules. This is achieved by various approximations to the Dirac equation, which is the relativistic analogue to the nonrelativistic Schrodinger equation. We do not want to discuss the mathematical details of this rather complicated topic, which is an area where much progress has been made in recent years and where the development of new methods is a field of active research. Interested readers may consult published reviews . A method which has gained some popularity in recent years is the so-called Zero-Order Regular Approximation (ZORA) which gives rather accurate results ". It is probably fair to say that... 

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... 

Schrodinger equation is then solved exactly. Note that exact is this context is not the same as the experimental value, as the nuclei are assumed to have infinite masses (Bom-Oppenheimer approximation) and relativistic effects are neglected. Methods which include eleclroti correlation are thus two-dimensional, the larger the one electron-expansion (basis set size) and the larger the many-electron expansion (number of determinants), the better are the results. This is illustrated in Figure 4.2. 

The development of approximate methods of applying quantum mechanics to molecules and molecular systems leads immediately to the introduction of models. Seldom is discussion of the properties of a given molecule or molecular complex based on the direct solution of the molecular Schrodinger equation (or its relativistic generalization). Instead, approximations are invoked which not only make the computation tractable but also introduce concepts and models upon which our interpretation of the properties of the molecule are based. 

In the time concept of the pre-relativistic mechanics, the observable quantities, time t and energy E, have to be considered as another canonically conjugate pair, as in classical mechanics. The dynamic law (time-dependent energy term) of the Schrodinger equation will then completely disappear [19]. A good occasion for Weyl to introduce the relativistic view would have been his contributions to Dirac s electron theory. His other colleagues developed the method of the so-called second quantization that seemed easier for the entire community of physicists and chemists to accept. 

The first is the Bom-Oppenheimer or fixed-nucleus approximation [32] wherein the more massive nuclei are assumed to be stationary with the electrons moving rapidly about them. The solution of Eq (3) reduces to finding the energies and trajectories of the electrons only, i.e. solution of the so-called many-electron Schrodinger equation. A further simplification which is often assumed, at least initially, is that relativistic effects are negligible. The starting point for so-called ab initio methods is therefore the non-relativistic many-electron Schrodinger equation within the Born-Oppenheimer approximation. 

So far we have discussed nonrelativistic ab initio methods they ignore those consequences of Einstein s theory of relativity that are relevant to chemistry (section 4.2.3). These consequences arise in the special (rather than the general) theory, from the dependence of mass on velocity [4. This dependence causes the masses of the inner electrons of heavy atoms to be significantly greater than the electron rest mass since the Hamiltonian operator in the Schrodinger equation contains the electron mass (Eqs (5.36) and (5.37)), this change of mass should be taken into account. Relativistic effects in... 

A straightforward relativistic ab initio AE calculation might appear to be the most rigorous approach to a problem in electronic stmcture theory, however, one has to keep in mind that the methods to solve the Schrodinger equation used in connection with both the AE Hamiltonian and the ECP VO model Hamiltonian usually also rely on approximations, e.g., the choice of the one- and many-particle basis sets, and therefore lead to more or less significant errors in the results. In some cases the introduction of ECPs even helps to avoid or reduce errors, e.g., the basis set superposition error (BSSE), or allows a higher quality treatment of the chemically relevant valence electron subsystem compared to the AE case. 

The indices i and j denote electrons, X and ju nuclei. is the charge of the nucleus X. For the one- and two-particle operators h and g various expressions can be inserted (e.g., relativistic, quasirelativistic or nonrelativistic all-electron or valence-only). The basic goal of quantum chemical methods is usually the approximate solution of the time-independent Schrodinger equation for a specific Hamiltonian, the system being in the state 7, i.e.. 

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