Kohn-Sham kinetic energy

Zhao, Q., R. C. Morrison, and R. G. Parr. 1994. From electron density to Kohn-Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange-correlation energies. Phys. Rev. A 50, 2138. 

Finally we mention some basic relations which are essential in the discussion of explicitly orbital dependent functionals. Examples of such functionals are the Kohn-Sham kinetic energy and the exchange energy which are dependent on the density due to the fact that the Kohn-Sham orbitals are uniquely determined by the density. The functional dependence of the Kohn-Sham orbitals on the density is not explicitly known. However one can still obtain the functional derivative of orbital dependent functionals as a solution to an integral equation. Suppose we have an explicit orbital dependent approximation for in terms of the Kohn-Sham orbitals then... 

A very different approach has been followed by Zhao et al. [54-57] who based their method on the constrained search definition of the Kohn-Sham kinetic energy. It follows from this definition that, from all Slater determinants which yield a given density, the Kohn-Sham determinant will minimize the kinetic energy. Suppose we have an exact density po- If one minimizes the Kohn-Sham kinetic energy under the constraint... 

This follows directly from the definition of the kinetic energy and the formula at Eq. (99). Note that the first term in the above formula is the von Weiszacker kinetic energy Tyf[jp which for a two-electron system is equal to the Kohn-Sham kinetic energy The term with Vtm therefore contributes to the... 

Fig. 11. The Cl, HF, LDA and Kohn-Sham kinetic energies (in a.u.) for H2 as functions of the bond distance...

The limits on the summation indicate that the sum only runs over those orbitals which are occupied (according to the aufbau rule) moreover, in this section, we restrict ourselves to closed shell systems and so each orbital is doubly occupied. Fortunately, the correction to the Kohn-Sham kinetic energy, [p], is quite small. For notational simplicity and compactness, we shall henceforth employ atomic units. 

Another recent beyond-GGA development is the emergence of so-called Meta-GGAs, which depend, in addition to the density and its derivatives, also on the Kohn-Sham kinetic-energy density r(r) [2, 112, 113]... 

The other reason that researchers choose to approximate the Kohn-Sham kinetic energy is that this functional has simpler mathematical properties. For example, the Kohn-Sham kinetic energy is homogeneous of degree two with respect to coordinate scaling of the electron density ... 

Comparison of Exact Kohn-Sham Kinetic Energies with Kinetic Energies Obtained Using Ground-State Functionals of Equations 5.4 and 5.21 and Functionals of Equations 5.23 and 5.24... 

Here P is a Slater determinant composed of the Kohn-Sham orbitals and Tks [p] is the corresponding Kohn-Sham kinetic energy. This energy can be calculated from the occupied orbitals of the non-interacting system. 

In order to find a more explicit representation of vks(t) we now rewrite the energy functional O Eq. 16.1 using the Kohn-Sham kinetic energy term O Eq. 16.3 and the explicit expression for the classic electronic interaction (Coulomb) energy,... 

Here, is the Kohn-Sham kinetic energy, and J[p is the classical coulomb... 

---