Dirac-Fock one-centre method

The Dirac-Fock one-centre method was the first approximation used for relativistic molecular structure calculations and is now only of historical importance. In this method the electron-electron interaction is handled exactly and the one-electron wave functions are four component Dirac spinors. On the other hand both the nuclear potentials and all the one-electron orbitals are expanded about a single common centre taken to be the position of the nucleus of the heaviest atom of the system under consideration. Because of this expansion, the method is restricted to hydrides XH and even for them the expansion is only slowly convergent. Nevertheless, experience gained with non-relativistic calculations has shown surprisingly good results for equilibrium distances of X-H bonds and for force constants. 

To construct the potential due to the off centre nuclei, for each of them we choose a coordinate system with its origin at the expansion centre and the z axis passing through the nucleus for which we expand the potential. In doing so we obtain an expression for each of the off centre nuclei. The next step is to rotate all these contributions to a common coordinate system which requires the Wigner rotation matrices. The explicit results for the many symmetries applied are published elsewhere [45] and we write the potential created by the protons located at a distance R from the centre of expansion in the generic form  

This simple example is sufficient to point out that, to be able to solve the self-consistent equations of the one-centre method, an atomic program needs to be modified only in the following form  

To conclude this short description of the Dirac-Fock one-centre expansion method (a more extensive presentation can be found in Ref 45) we list in the table below most of the model molecular systems computed with that method and the main conclusions drawn from these calculations (see Table 7.3 in Ref 2 for a full list of references to the results summarized here). 

CH4 to PbR, Bond length contraction and increase of force constants 

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