Going beyond Kohn and Sham (KS): determining accurate ground and first excited states

The Total energy in KS is written as \[E=\frac{1}{2}\sum\int\nabla\psi^*\cdot\nabla\psi+ \frac{1}{2}\int\frac{\sum\psi^*\psi(r)\sum\psi^*\psi(r^{\prime})} {(r-r^{\prime})}+\int\sum\psi^{*}\psi V_{nuclei}+Exc\] The KS procedure continues by minimizing the energy with respect the wavefunctions $\psi$. The equation for the wave functions is similar to the one-particle Schroedinger equation. In our talk we will present results obtained in the following way: we add an external potential $V_{add}$ to the nuclei potential $V_{nuclei}$ and, after the calculation is completed, we subtract what we added, namely. $-\int\sum\psi^{*}\psi V_{add}$. The result is a calculation according to the Eq. above but with wavefunctions not satisfying the KS equations. If the exchange-correlation term were reliable one would expect that the calculated energy would be larger than the KS energy. The added potential $V_{add}$ is what is being used in the {\it LDA-1/2} method and is dependent on a cut-off parameter $C$. Making the extremization of the total energy with respect to $C$ we obtain (1) a point of maximum, which frequently will be shown to be the first excited state, (2) a minimum, with an energy lower than the KS ($C=0$) ground state and with improved lattice parameter.