Consequences for the Electron Density Distribution

A hideous feature of transformation schemes is the fact that the density is no longer calculated from the squared transformed orbitals. Especially close to atomic nuclei this leads to significant deviations of squared transformed orbitals from squared spinors, while the effect becomes negligible at larger distances [639,880]. Already in the case of a one-electron system, the absolute square of a four-component spinor (r) is related to the square of two-(or one)-component wave functions fiif) only by introducing a distance-dependent error Ap(r) 

the DKH density cannot simply be obtained from a sum of squared DKH orbitals, though this can be a very good approximation for spatial regions that do not penetrate the vicinity of the nucleus. We must DKH-transform the corresponding operator for the density at the position r 

The orbital contribution to the expectation value for the density then reads 

If the unitary matrix U is represented by the sequence of infinitely many unitary matrices Um, this sequence may be truncated with respect to a predefined DKH order yielding an approximate decoupling scheme 

Finally, we may note that the continuity equation, which we used to define the electron density in chapter 8 (compare also chapter 4), reads 

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