### Orbital Free Density Functional Theory

In the case of WDM there is an issue with the technical problem of solving for the Kohn-Sham orbitals and eigenvalues. Since the computational load from the eigenvalue problem scales, in general, as order \(N^3\) where \(N\) is the number of orbitals, the growth in the number of non-negligibly occupied KS orbitals with increasing temperature is a clear computational bottleneck. For complicated systems, the same bottleneck is encountered in ground-state simulations which use the DFT Born-Oppenheimer energy surface to drive the ionic dynamics.

One result has been the emergence of active research on orbital-free DFT (OFDFT), that is, approximate functionals for the ingredients of the KS free energy, namely the KS KE \({\mathcal T}_s\), entropy \({\mathcal S}_s\), and XC free energy \(\mathcal{F}_{xc}\) or their ground-state counterparts. Almost all of this effort has been for ground-state OFKE functionals. With OFDFT functionals in hand, the DFT extremum condition yields a single non-linear Euler equation for the density which supplants the KS equation for the orbitals. (Note that most of the OFKE literature invokes the KS separation of the KE in order to use existing \(E_{xc}\) approximations consistently.)

The finite-temperature OFDFT work is dominated by variants on Thomas-Fermi-von Weizsäcker theory. That type of theory, however, is known (on both fundamental and computational grounds) to be no more than qualitatively accurate in many circumstances. For more refined approximate OFDFT functionals at finite temperature, little or nothing is known about their accuracy for realistic systems and there is little fiduciary data for comparison.