Wave function small component

Here T l and T s arc (large and small) two-component wave functions which include the a and /3 spin functions. The latter equation can be solved for T s. 

In the non-relativistic limit the small component of the wave function (eq. (8.14)) is... 

For a hydrogen atom the small component accounts for only 0.4% of the total wave function and only 10 % of the electron density, but for a uranium Is-electron it is a third of the wave function and 10% of the density. 

The presence of the momentum operator means that the small component basis set must contain functions which are derivatives of the large basis set. The use of kinetic balance ensures that the relativistie solution smoothly reduees to the non-relativistic wave function as c is increased. 

Since working with the full four-component wave function is so demanding, different approximate methods have been developed where the small component of the wave function is eliminated to a certain order in 1/c or approximated (like the Foldy-Wouthuyserd or Douglas-Kroll transformations thereby reducing the four-component wave function to only two components. A description of such methods is outside the scope of this book. 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

Numerous earlier studies of the ec GG methods clearly indicate that the modest size MR GISD wave functions represent the most suitable and easily available source of higher-than-pair clusters for this purpose (see Ref. [21] for an overview). Indeed, these wave functions can be easily transformed to a SR form, whose cluster analysis is straightforward. Moreover, the resulting three- and four-body amplitudes represent only a very small subset of all such amplitudes, namely those which are most important, and which at the same time implicitly account for all higher-order cluster components that are present in the MR GISD wave functions. 

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a li orbital on each first row atom in the 6-3IG basis set) are contracted into sums of Gaussians with fixed coefficients and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules." ... 

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation... 

A second approach to achieve a reduction of the 4-component Hamiltonian to an electrons-only Hamiltonian is to introduce approximations by eliminating the small components of the wave function (41-53). Also here, different protocols have been successfully exploited in quantum chemistry. 

Thus, the non-relativistic wave function (1.14) of an electron is a two-component spinor (tensor having half-integer rank) whereas its relativistic counterpart is already, due to the presence of large (/) and small (g) components, a four-component spinor. The choice of / in the form (1 + l — l ) is conditioned by the requirement of a standard phase system for the wave functions (see Introduction, Eq. (4)). 

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation ... 

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