Electronic structure calculations - the numerical approach

Electronic structure calculations - The numerical approach We will focus in this subsection on electronic structure calculations (non-relativistic and relativistic) using numerical techniques, i.e., we do not use an expansion of single-electron functions (orbitals or spinors) in terms of analytic basis functions. Nowadays such numerical calculations are routinely feasible only for atoms and diatomics. Within the non-relativistic approach we have to determine rad.ial functions Pj(r) (z = nl) as parts of single-centre functions 

In the case of atoms Eq. (Ill) represents the complete spinor, whereas linear combinations of these single-centre spinors can be used to build up molecular spinors (or crystal spinors) in multi-centre problems. We briefly give an overview on the equations which finally have to be solved for these radial functions in non-relativistic and relativistic calculations. The form 

The eigenvalue problem in the non-relativistic case may be cast into the general form of a two-point boundary value problem [69-71], 

The latter always contains a centrifugal part and the modified electron-nucleus potential The additional term U r) is optional, 

The eigenvalue problem to be solved in the relativistic case leads finally to sets of pairs of coupled first-order differential equations of the following general form [72-74]  

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