Accurate effective potential for density amplitude and the corresponding Kohn-Sham exchange-correlation potential calculated from approximate wavefunctions

It is well known that direct inversion of an accurate but approximate density leads to KS exchange-correlation potential which has spuriously large deviations from the exact one. On the other hand, a different approach which employs wavefunction directly to obtain KS exchange-correlation potential is found to lead to potentials close to the exact one. This approach utilises the Levy-Perdew-Sahni expression for the effective potential used in the equation for the square root of the density.
We present why the Kohn-Sham exchange-correlation potential obtained by using an approximate wavefunction in the Levy-Perdew-Sahni expression for effective potential is accurate while the direct inversion of density associated with the same wavefunction may lead to
pathological features in it. By analysing the Levy-Perdew-Sahni equation, we show that quantities like effective and exchange-correlation potentials, calculated from approximate wavefunctions are free from spurious features and close to the corresponding exact ones. The
study also suggests possibility of a new approach for the calculation of accurate densities from approximate wavefunctions. These densities are closer to the exact ones than those obtained from the wavefunction directly.